identifier users guide - industrial automation and control ... · apc identifier user’s guide an...

Advanced Process Control

APC IdentifierUser’s Guide

an Integrated Part of

Profit Design Studio (APCDE)

TP-SWIDNTRevision 1.5

5/01

AP09-200

Advanced Process Control

APC IdentifierUser’s Guide

an Integrated Part of

Profit Design Studio (APCDE)

TP-SWIDNT

Revision 1.5

5/01

AP09-200

Copyright, Notices, and Trademarks

Copyright 2001 by Honeywell International Inc.

While this information is presented in good faith and believed to be accurate, Honeywelldisclaims the implied warranties of merchantability and fitness for a particular purpose andmakes no express warranties except as may be stated in its written agreement with and for

its customer.In no event is Honeywell liable to anyone for any indirect, special or consequential

damages. The information and specifications in this document are subject to change withoutnotice.

Profit, TDC 3000, and TotalPlant are registered trademarks of Honeywell International Inc.

Other product names are trademarks of their respective owners.

HoneywellIndustrial Automation and Control

16404 N. Black Canyon HwyPhoenix, AZ 85053

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Table of Contents

COPYRIGHT, NOTICES, AND TRADEMARKS .......................................................................II

TABLE OF CONTENTS ...........................................................................................................III

ABOUT THIS PUBLICATION.................................................................................................. XIStatement of Work ............................................................................................................... xiWho Should Use This Book .................................................................................................. xiProfit Course Information ...................................................................................................... xiWriting Conventions Used in This Book ..................................................................................xii

REFERENCES....................................................................................................................... XIIIDocumentation ...................................................................................................................xiiiTitle...................................................................................................................................xiiiNumber .............................................................................................................................xiiiGeneral .............................................................................................................................xiiiOpen.................................................................................................................................xiiiTPS System.......................................................................................................................xiiiEmbedded Uniformance ......................................................................................................xiii

FOR TECHNICAL ASSISTANCE..........................................................................................XIVIf You Need Assistance .......................................................................................................xivInternational Customers.......................................................................................................xivCustomers Inside the United States ......................................................................................xivArizona Customers .............................................................................................................xivServices Provided ...............................................................................................................xivTime Saving Tip..................................................................................................................xiv

SECTION 1 — APC IDENTIFIER / PROFIT DESIGN STUDIO (APCDE)OVERVIEW................................................................................................................................1

1.1 APC Identifier Overview.....................................................................................................1APC Identifier .......................................................................................................................1Variables .............................................................................................................................1Models ................................................................................................................................1Problem Size........................................................................................................................2Collecting Data .....................................................................................................................2Saving Data .........................................................................................................................2

1.2 Profit Design Studio Overview ................................................................................................3Profit Design Studio ..............................................................................................................3APC Identifier .......................................................................................................................3Profit Controller (RMPCT) ......................................................................................................4Profit Optimizer (DQP)...........................................................................................................4Profit PID (RPID) ..................................................................................................................4Profit Loop ...........................................................................................................................4Step Test Builder ..................................................................................................................4Point Builder.........................................................................................................................4Data Converter .....................................................................................................................5Model Converter ...................................................................................................................5

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Data Operation Tools ............................................................................................................5Profit Toolkit .........................................................................................................................5Profit Sensor ........................................................................................................................5

SECTION 2 — INSTALLING PROFIT DESIGN STUDIO (APCDE) AND THE APCIDENTIFIER............................................................................................................................... 7

2.1 Overview............................................................................................................................7In This Section......................................................................................................................7

2.2 System and Software Requirements .................................................................................8Software Requirements .........................................................................................................8Do I have to Install the Identifier Separately? ............................................................................8PC Requirements..................................................................................................................8

2.3 Quick Reference to Installation..........................................................................................9How to Use the Quick Reference ............................................................................................9Quick Reference Table ..........................................................................................................9

2.4 Installing the Profit Design Studio and the APC Identifier................................................10PC Installation ....................................................................................................................10Installing the Dongle............................................................................................................11Check the Log File ..............................................................................................................13Check the ini File ................................................................................................................14Other Options .....................................................................................................................15Caution..............................................................................................................................15Starting the PC Application...................................................................................................15

SECTION 3 — APC IDENTIFIER CONCEPTS ...................................................................... 173.1 Conceptual Overview of the APC Identifier......................................................................17

Identification—A Science and an Art......................................................................................17The Identification Process ....................................................................................................17Identification Environment ....................................................................................................18Identification Approach ........................................................................................................18Fitting FIR Models...............................................................................................................18Fitting PEM Models .............................................................................................................19Fitting Parametric Models.....................................................................................................19Fitting Final System Models .................................................................................................19

3.2 Theory Overview by Topic ...............................................................................................20Key Topics .........................................................................................................................20

3.3 General Problem Statement ............................................................................................21Identification Structure .........................................................................................................21Quadratic Norm Formulation ................................................................................................21Robust Norm Formulation ....................................................................................................22

3.4 Model Structures..............................................................................................................24Overview............................................................................................................................24FIR Models ........................................................................................................................24FIR Structure......................................................................................................................24PEM Models.......................................................................................................................26PEM Structure ....................................................................................................................26Model for Order and Variance Reduction................................................................................28ARX Parametric Models (Discrete Time) ................................................................................28Output Error Models (Discrete Time) .....................................................................................28Laplace Domain Parametric Models ......................................................................................29Final Model Form ................................................................................................................29

3.5 Solutions ..........................................................................................................................30Overview............................................................................................................................30Linear Solutions FIR Models.................................................................................................30

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Linear Solutions PFX Models (Pre-Filtered ARX) ....................................................................32Nonlinear Solutions .............................................................................................................33Solution Procedure..............................................................................................................33PEM Formulation ................................................................................................................35OE Formulation ..................................................................................................................36Laplace Formulations ..........................................................................................................37Starting Conditions..............................................................................................................37Delay Estimation.................................................................................................................37

3.6 Model Properties..............................................................................................................40Overview ...........................................................................................................................40FIR Bias ............................................................................................................................40FIR Consistency .................................................................................................................41PEM..................................................................................................................................41Summary...........................................................................................................................42

3.7 FIR Statistics ...................................................................................................................45Statistical Properties ...........................................................................................................45Confidence limits and Noise bounds......................................................................................46

3.8 Factorizations ..................................................................................................................49Background........................................................................................................................49Normal vs. Orthonormal.......................................................................................................49Sensitivity and Accuracy ......................................................................................................50An Ill-conditioned example ...................................................................................................52QR Solution .......................................................................................................................53Cholesky Solution ...............................................................................................................54SVD Solution......................................................................................................................56Sensitivity of Ill-conditioned Problem .....................................................................................57Pseudorank........................................................................................................................58A Rank Deficient Example....................................................................................................59Zero Value Solution.............................................................................................................61Minimum Norm Minimum length QR Solution .........................................................................62MATLAB Solutions..............................................................................................................63Perturbed Solution and Pseudorank ......................................................................................64Timing ...............................................................................................................................65

3.9 Summary .........................................................................................................................67Future Perspective..............................................................................................................68

SECTION 4 — GETTING STARTED - THE IDENTIFICATION ENVIRONMENT ..................694.1 Overview..........................................................................................................................69

In This Section ...................................................................................................................69Profit Design Studio (APCDE) ..............................................................................................69

4.2 Starting an Identification Session ....................................................................................70File Types and File Extensions .............................................................................................70

4.3 Creating a Profit Controller (RMPCT) Model File.............................................................72Creating an RMPCT Model File ............................................................................................72Data Source - Data Files......................................................................................................73Data Source - Manually Entered ..........................................................................................74Entering or Changing Variable Information .............................................................................74

4.4 Creating a Robust PID Model File ...................................................................................77Creating an RPID Model File ................................................................................................77Data Source - Data Files......................................................................................................78Data Source - Manually Entered ..........................................................................................79

4.5 Reading in Data...............................................................................................................80Getting Test Data................................................................................................................80Single Point Data Files ........................................................................................................80

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Single Point Data—An Example File......................................................................................80Multiple Point Data File ........................................................................................................81Multiple Point Data—An Example File....................................................................................82Saving an .mdl or .pid File....................................................................................................82

4.6 Reading Model Files Created by Other Applications .......................................................83What the Identifier Expects ..................................................................................................83Non Native FIR Files ...........................................................................................................83Sample File........................................................................................................................83Non Native XFR Files ..........................................................................................................84Sample..............................................................................................................................84

4.7 Hierarchical Overview ..........................................................................................................85Identifier Main Menu............................................................................................................85Keyboard Selection .............................................................................................................91APCDE32.INI .....................................................................................................................92

SECTION 5 — MULTIPLE VIEWS AND THE PRESENTATION OF DATA .......................... 975.1 Overview..........................................................................................................................97

In This Section....................................................................................................................975.3 Basic Views .....................................................................................................................98

Primary Functions ...............................................................................................................985.3 Viewing, Selecting and Marking Data ............................................................................103

Working with different Windows ..........................................................................................103Plotting Raw Data .............................................................................................................103Single-Graph Plots ............................................................................................................104Changing the Plot Size .....................................................................................................105Viewing Single Graph Plots ................................................................................................105Reconfigure Single-Graph Plots ..........................................................................................107Plot Modes.......................................................................................................................108Selecting Time Ranges......................................................................................................109Selecting Ranges..............................................................................................................109Reading the Plots..............................................................................................................110Marking Data Bad at the Global Level..................................................................................111Marking Data Bad at the Regression Level ...........................................................................113Marking Data Bad at the Prediction Level.............................................................................114Scatter Matrix ...................................................................................................................116Multi-Graph/Scatter Plots ...................................................................................................116Multi-Graph Mode .............................................................................................................116Scatter Plot Mode .............................................................................................................117

SECTION 6 — EDIT, MERGE AND RECONFIGURE FUNCTIONS .................................... 1196.1 Overview........................................................................................................................119

In This Section..................................................................................................................119Data and File Manipulation.................................................................................................120

6.2 Edit Functions ................................................................................................................121Basic Edit Characteristics ..................................................................................................122Special Edit Functions .......................................................................................................126Copy Trial Information .......................................................................................................127Edit Variable Attributes ......................................................................................................129Entering or Changing Information........................................................................................130Document without raw data ................................................................................................131Empty Document ..............................................................................................................131

6.3 Combining Files and Rearranging Variables/Data /Models ...........................................133Copying Models/Data From One File to Another Using Copy/Paste .........................................133Copying Models/Data From One File to Another Using Drag-Drop...........................................133

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Rearranging Models and Variables Within a Given File Using Drag-Drop .................................135Copy Data From One File To Another Using Drag-Drop.........................................................135Merging Data....................................................................................................................136Merging Data Marks/Selection Ranges ................................................................................138

SECTION 7 — DATA OPERATIONS....................................................................................1417.1 Overview........................................................................................................................141

In This Section .................................................................................................................141Basic Functions ................................................................................................................141Supported Operations .......................................................................................................141

7.2 Block Manipulations.......................................................................................................142Invoking Block Manipulations .............................................................................................142Options and Their Use.......................................................................................................143

7.3 Vector Calculations........................................................................................................144Source and Destination Variables- Remembering Past events................................................145Invoking Vector Calculations ..............................................................................................145Vector Functions...............................................................................................................147Transformations................................................................................................................148Special Transformations ....................................................................................................151Polynomial .......................................................................................................................151Piecewise Linear...............................................................................................................161Installed Valve Characteristics............................................................................................171Transformations without Data .............................................................................................172Filter................................................................................................................................174Statistics..........................................................................................................................180Edit Data .........................................................................................................................181Combine Variables............................................................................................................185User Notes.......................................................................................................................186Saving and Recovering Vector Calculations .........................................................................187Merging Vector Calculations...............................................................................................193Saving Transformation For Use with Profit Controller.............................................................194

SECTION 8 - OVERALL IDENTIFICATION FUNCTIONS....................................................1978.1 Overview........................................................................................................................197

In This Section .................................................................................................................197Main Functions .................................................................................................................197Overall Options.................................................................................................................197Load & Go .......................................................................................................................198

8.2 Overall Model Setup ......................................................................................................199Setting Overall Options......................................................................................................199Data Rate / Trial Specification ............................................................................................199MIMO Discrete Model Specification.....................................................................................201Initial Conditions and Model Forms .....................................................................................201

8.3 FIR Setup ......................................................................................................................202Configuring FIR Models .....................................................................................................202Max Settle T (Settling Time) ...............................................................................................202# of Coefficients................................................................................................................203FIR Model Form................................................................................................................203FIR Initial Conditions .........................................................................................................204

8.4 PEM Setup.....................................................................................................................205General Guidelines ...........................................................................................................205Auto Setup.......................................................................................................................207Detailed Setup..................................................................................................................207PEM Initial Conditions and Model Form ...............................................................................209

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8.5 Overall Model Setup Options .........................................................................................210Calculation Options ...........................................................................................................210Data Options ....................................................................................................................212FIR Only Options ..............................................................................................................212Data Scaling.....................................................................................................................212Null Model Treatment ........................................................................................................212Regression Selection Options.............................................................................................213PEM Only Options.............................................................................................................213Factorization Options.........................................................................................................214Search Options.................................................................................................................215

8.6 Running Load & Go .......................................................................................................216Load & Go........................................................................................................................216Default Model Settings.......................................................................................................216

SECTION 9 - CREATING FINITE IMPULSE RESPONSE OR PREDICTIONERROR MODELS ................................................................................................................. 217

9.1 Overview........................................................................................................................217In This Section..................................................................................................................217About the FIR Model..........................................................................................................217About the PEM Models ......................................................................................................217

9.2 Procedure ......................................................................................................................219Fitting the FIR/PEM Mode ..................................................................................................219Fit Fir/PEM Models Dialog Box and Associated View.............................................................219Show & Select Vars...........................................................................................................219Set Overall Options ...........................................................................................................220Set Options per Sub Model ................................................................................................221Options per MV/DV ...........................................................................................................222Excluding Data From the Regression...................................................................................223Fit FIR/PEM Models ..........................................................................................................225Model Example.................................................................................................................225Model Descriptors .............................................................................................................226Checking Trial Dependent information .................................................................................227FIR/PEM Step Responses .................................................................................................229Interpreting Results ...........................................................................................................230

9.3 Statistics ........................................................................................................................232Background......................................................................................................................232Guidelines........................................................................................................................233Special Consideration........................................................................................................234Interpretation of Model Rank...............................................................................................235Overview..........................................................................................................................236Correlation View MV/DV to MV/DV......................................................................................237Correlation View CV to MV/DV...........................................................................................238Confidence/Null Hypothesis View........................................................................................239Statistical Summary View...................................................................................................240Descriptors.......................................................................................................................241Positional Form / 1 Trial .....................................................................................................242Impact of Exclude data Options ..........................................................................................244

SECTION 10 - CREATING PARAMETRIC MODELS .......................................................... 24910.1 Overview.......................................................................................................................249

In This Section..................................................................................................................249What Are Parametric Models Used For? ..............................................................................249

10.2 Procedure .....................................................................................................................250Fitting the Parametric Models .............................................................................................250

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Fit Parametric Models Dialog Box and Associated View.........................................................250Show & Select Sub-models ................................................................................................252Overall Options.................................................................................................................252Discrete Model Information ................................................................................................253Individual Options .............................................................................................................256Dialog Box Information ......................................................................................................258Parametric Options Per Trial ..............................................................................................259Viewing the Transfer Function ............................................................................................260Example of Legal Polynomials ............................................................................................262Step Response Overview...................................................................................................262All Responses ..................................................................................................................263

SECTION 11 - SELECTING FINAL MODELS ......................................................................26511.1 Overview.......................................................................................................................265

In This Section .................................................................................................................265Final Models Defined.........................................................................................................265Searching for the Best Final Models ....................................................................................266Two Procedures ...............................................................................................................266

11.2 Procedure .....................................................................................................................267Selecting Final Trials/Finding Final Models...........................................................................267Trial Source .....................................................................................................................268Show & Select Sub-models ................................................................................................271Excluding Data From the Prediction Calculations ..................................................................272Select Trial Manually .........................................................................................................273Dialog Box Information ......................................................................................................273Update Trial .....................................................................................................................274Stop ................................................................................................................................274Plot Predictions ................................................................................................................274Load Source to Final .........................................................................................................278Null Final Model................................................................................................................281

11.3 Final and Model Summary Views .................................................................................283Final Model View ..............................................................................................................283Model Summary View........................................................................................................284Copy Trials from One Source to Another..............................................................................285

SECTION 12 - ANNOTATION ...............................................................................................28712.1 Overview.......................................................................................................................28712.2 Annotation Access and Update ....................................................................................288

Access Overview ..............................................................................................................288Detailed Access and Update ..............................................................................................289Annotation Example ..........................................................................................................291

SECTION 13 - TUTORIAL.....................................................................................................29913.1 Overview.......................................................................................................................29913.2 Rich Input Signals.........................................................................................................300

RichDoc1.........................................................................................................................300WafrDoc1 ........................................................................................................................305

13.3 Typical Input Signals.....................................................................................................309TowrDoc1 ........................................................................................................................309ColDoc1 ..........................................................................................................................316

13.4 Limited Input Signals ....................................................................................................320LevDoc1 ..........................................................................................................................320BlecDoc2.........................................................................................................................322

13.5 Creating PEM models...................................................................................................325Synthetic Data..................................................................................................................325

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Pressure Data ..................................................................................................................328Furnace Data ...................................................................................................................329Large disturbance .............................................................................................................330Demo Data.......................................................................................................................332ColDoc1...........................................................................................................................333WafrDoc1 ........................................................................................................................334BlecDoc2 .........................................................................................................................335

APPENDIX A - SAMPLE OF A FIR MODEL FILE................................................................ 337

APPENDIX B - EXAMPLE OF AN XFR FILE ....................................................................... 357

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About This Publication

Statement of

Work

The following table describes the audience, purpose, and scope of this book:

Purpose This book explains how to work with process data to identifya model of a process.

Audience Process and control engineers

For ProductRelease

• All Profit Controller (RMPCT) releases 200.00 andabove,

• Profit Design Studio (APCDE) release 220.00 andabove,

• Profit Optimizer release 200.00 and above,

• RPID release 115 and above.

Who Should Use

This Book

Anyone responsible for creating process models based on either plant data orexisting models. All models identified are structured for seamless integration intoProfit® Controller (RMPCT), Profit Optimizer (DQP), Profit PID (RPID).

Profit Course

Information

Honeywell offers several courses that explain the math and conceptual underpinningsas well as application implementation of the Advanced Process Control suite ofproducts.

Engineers wanting a more technical exposure to Profit products can contact:

Honeywell Automation College2500 W. Union Hills Drive

Phoenix, AZ 85027

About this Publication

xii APC Identifier User’s Guide 5/01

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Writing

Conventions Used

in This Book

The following writing conventions have been used throughout this book andother books in the Profit Suite library.

• Words in double quotation marks " " name sections or subsections in thispublication.

• Words in italics name book titles, add grammatical emphasis, introducewords that are being referenced or defined, or represent mathematicalvariables. The context makes the meaning and use clear.

• Words in bold type indicate paragraph topics or bring important phrasesto your attention. They can also indicate vector or matrices with lowercase indicating vectors and upper case implying matrices.

• brings paragraphs and table entries to your attention.

• Windows pull down menus and their options are separated by an anglebracket >. For example,Tools> Point Builders>RMPCT Point Builder

• Messages and information that you type appear in Courier font.

• Acronyms, and point names appear in UPPERCASE. The context makesthe meaning and use clear.

• File names and paths appear separated by / . For example,C:/Program Files/Profit Design Studio.

• Command keys appear as they appear on the key, but within anglebrackets. For example, press <Enter>.

• Graphic buttons appear within brackets [ ]. For example, select [OK].

• Zero as a value and when there is a chance for confusion with the letter Ois given as Ø. In all other cases, zero as a numerical place holder is givenas 0. For example, 1.0, 10, 101, CVØ1, parameter PØ.

• The terms screen and display are used inter changeably in discussing thegraphical interfaces. The verbs display a screen and call a screen are alsoused inter changeably.

• The names Profit Controller (RMPCT), Profit Controller and RMPCTmay be used interchangeably.

• The names Profit Optimizer (DQP), Profit Optimizer and DQP may beused interchangeably.

• The names Profit Design Studio, and APC Development Environmentmay be used interchangeably.

Shadin

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ReferencesThe following comprise the Profit Suite library.

Documentation Title Number

General

Profit Controller (RMPCT) Concepts Reference RM09-400Profit Controller (RMPCT) Designer’s Guide (Off-Line Design) RM11-410Profit Optimizer Designer’s Guide (Off-Line Design) PR11-400Profit Toolkit Designer’s Guide AP11-400APC Identifier User’s Guide AP09-200Profit-PID (RPID) RM11-100Profit Sensor User’s Guide PS09-100

Open

Profit Suite Installation Guide for Open SystemsViewer - Controller - Optimizer - Toolkit

RM20-501

Profit Controller (RMPCT) User’s Guide for Open Systems RM11-401Profit Optimizer User’s Guide for Open Systems PR11-421Profit Trender User’s Guide RM11 431Profit Toolkit User’s Guide for Open Systems AP11-401Profit Toolkit Function Reference AP11-410FCCU Toolkit User’s Guide for Open Systems AP13-201Fractionator Toolkit User’s Guide for Open Systems AP13-101Lab Update User’s Guide AP13-111Wrapper Builder User’s Guide AP11-411Profit Bridge User’s Guide AP20-401

TPS System

Profit Controller (RMPCT) Installation Reference for AM, AxM and Open LCN-Side RM20-400Profit Controller (RMPCT) Commissioning RM20-410Profit Controller (RMPCT) User’s Guide for AM, AxM and Open LCN-Side RM11-400Profit Optimizer Installation Reference for AM and Open LCN-Side PR20-400Profit Optimizer User’s Guide for AM and Open LCN-Side PR11-420Profit Suite ToolKit AP09-300TDC Data ConverterData CollectorStep Test Builder

Performance MonitorRMPCT CascadePV Validation

Simulation BackBuilderGain Scheduler

Fractionator Toolkit (LCN) AP13-100FCCU Toolkit (LCN) AP13-200Furnace Pass Balance Temperature Control User’s Guide AP13-600Non-Linear Level Control User’s Guide AP09-700

Embedded Uniformance

Excel Companion User’s Guide (Profit Embedded PHD) AP20-510Power Point Companion User’s Guide (Profit Embedded PHD) AP20-520Process Trend User’s Guide (Profit Embedded PHD) AP20-530

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For Technical Assistance

If You Need

Assistance

If you need technical assistance, contact your local Honeywell ServiceOrganization, as explained in the following paragraphs.

International

Customers

Outside of the United States, contact your local Honeywell Service Organization.If you are not sure of the location or telephone number, call your Honeywellrepresentative for information.

Customers Inside

the United States

Within the United States, call the Technical Assistance Center (TAC) at the tollfree number 1-800-822-7673.

Arizona Customers Within Arizona, the local number for TAC is 602-313-5558.

Services Provided Calls to TAC are answered by a dispatcher from 7:00 A.M. to 5:00 P.M.,Mountain Standard Time (6:00 A.M. to 4:00 P.M. when daylight savings time isin effect).

Outside of these hours, emergency calls—those which affect your ability tocontrol or view a process—will be received by an answering service, andreturned within one hour. TAC maintains its own TPS network, and frequentlycan duplicate problems on this equipment.

Time Saving Tip It is a good idea to make specific notes about the problem before making the call.This helps to reduce delays and expedite answers.

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Section 1 — APC Identifier / Profit Design Studio (APCDE)Overview

1.1 APC Identifier Overview

APC Identifier Honeywell’s APC Identifier is a state of the art multivariable identificationpackage that supports experiment design, data analysis and model synthesis formultivariable processes. The APC Identifier has been designed to operate in anintuitive interactive fashion and is an integral part of Profit Design Studio.

Identification has as its ultimate goal the creation of a dynamic representation of amultivariable process. To this end, the APC Identifier can be used to generatelinear-time invariant multivariable dynamic models. These models are ultimatelysaved in terms of a transfer function matrix. Creation of the matrix can beaccomplished by using raw data, by manual specification or by a host of otherinput methods. When raw data is used, indicators are provided to assess modelquality.

Variables Type and class categorize variables. There are two distinct classes of variables.One is the Var class the other is the Aux class. Variables of class Var, arevariables that can be used for identification and therefore are variables thatimplicitly define distinct rows or columns in the model matrices. Views associatedwith models will only display Var variables. Variables of class Aux are variablesthat provide a permanent home for data but do not appear as model variables. Bothclasses of variables are displayed in any views associated with data. Variables ofclass Var can of course be converted to variables of class Aux and vice a versa.

There are three distinct types of variables.

Controlled Variables (CVs) These are the variables that a controller wouldattempt to keep at setpoint or within some range. These are dependent variables

Manipulated Variables (MVs) These are the variables that a controller wouldadjust to keep the CVs within some range. These are independent variables.

Disturbance Variables (DVs) These are measured variables that are not underthe influence of a specific controller but which affect the values of the CVs. Theseare also independent variables.

Models Sub-Process Models An overall process model is composed of a matrix ofdynamic sub-process models, each of which describes the effect of one of theindependent variables (MVs and DVs) on one of the CVs. A sub-process modeldescribes how the effect of an independent variable on a CV evolves over time.

Sub-process models are null when a particular independent variable has no effecton a particular CV.

Identifying the Model When models are not known a priori, it is desired to findthe causal relationship between independent and dependent variables. Thisprocedure is known as identifying the model. The purpose of the manual is to

Section 1 — APC Identifier / Profit Design Studio (APCDE) OverviewAPC Identifier Overview

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describe this procedure

Problem Size There are no inherent limitations to problem size. Any number of CVs, MVs, andDVs can be accommodated. No restrictions are placed on the maximum number ofFIR coefficients (compression or decimation ratio must, of course, be an integernumber >= 1). There are no practical restrictions on model orders.

System model matrices can have any number of elements. Only computer speedand memory resources (RAM) limit the application.

While the off-line design package imposes no size limitations, on-line controllerdimensions ARE restricted depending on the platform. For AM implementations,CVs, MVs and DVs are limited to 20, 40, 40 respectively. For AxM and NTnodes, CVs, MVs and DVs are limited to 40, 80, 80 respectively.

Collecting Data Collecting accurate data is crucial. Be very careful to get good data. If you can getgood data, you are virtually assured to get good models.

First, conduct preliminary tests to make sure that all regulatory loops are properlytuned, and that all actuators and positioners are performing correctly. Then getinitial, but accurate, estimates on process response times, gains, nonlinearities, andnoise levels.

Once the preliminary test is complete, the full test should be properly designed toensure that the variables of interest are properly excited wherever possible.

If the data is sufficiently rich (excited over the required spectrum with appropriatesignal to noise ratios), then the identifier can extract the appropriate models. For afull discussion on the importance and issues involved with test signal design, theProfit Controller (RMPCT) Implementation Course: Identification and Control canbe helpful. In addition, the Step Builder tools, optional parts of the APC ToolKitpackage, can be used to significantly aid in the identification process.

Saving Data Data should be recorded during all plant testing. Many options exist for saving thisdata. The Honeywell Data Collector, which runs in the AM and is an optionalpart of the APC ToolKit package, can be used to collect this data automatically.

Other collection techniques can also be used. It is important to remember whenusing alternate methods that the identifier uses specifically formatted files. Beforebeginning plant tests make sure that the data can be saved with the correct format.(See Section 3.3 for a description of data formats).

Section 1 — APC Identifier / Profit Design Studio (APCDE) Overview1.2 Profit Design Studio Overview

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1.2 Profit Design Studio OverviewProfit Design Studio Profit Design Studio (APCDE), formerly know as APC Development

Environment, is a Windows based environment that allows many advancedprocess control issues to be addressed through a common interface. ProfitDesign Studio (APCDE) functionality can be easily expanded by the addition oftask specific components (libraries that are dynamically loaded as needed).Components that are currently supported by Profit Design Studio (APCDE) areshown on the About Box tabs as shown below. To view the available supportedand installed version of Profit Design Studio’s components, click the coordinatingtab. The components that are checked are installed on the current computer.

Additional functionality will be added seamlessly as an evolutionary process.

APC Identifier The APC Identifier is a proprietary advanced analysis package for identifying,manipulating, displaying, and testing process models for dynamic, multivariablesystems. With it, you can easily develop a multi-variable dynamic model of aprocess that you want to control or simulate. The Identifier includes a completeset of tools for creating and evaluating process models. Combining, rearrangingand evaluating models is accomplished with standard Windows procedures.



Honeywell Inc.

Multiple model forms are supported. Final models are saved in Laplace form.

For training on the conceptual and practical aspects of the identifier, Honeywell’sProfit Controller (RMPCT) implementation course is recommended (4516s).

Although the Identifier can be used as a stand-alone tool, it is an integral part ofHoneywell's Profit Controller (RMPCT), Profit Optimizer (DQP) Profit-PIDand Profit Loop (Future).

Profit Controller(RMPCT)

Profit Controller (RMPCT) Design software. This software can be easily usedto create an RMPCT controller, based on the model provided by the Identifier.The controller can be used on-line to control the actual process, and can also betested on a simulated process using the off-line software.

Profit Optimizer(DQP)

Profit Optimizer (DQP) Design software supports the easy development of adistributed quadratic optimizer that runs in a fully dynamic fashion. ProfitOptimizer (DQP) enables you to readily synthesize a supervisory QP controllerthat dynamically coordinates multiple Profit Controller (RMPCT) controllers. InProfit Design Studio (APCDE), this is accomplished by simply merging two ormore Profit Controller (RMPCT) models into an overall Profit Optimizer (DQP)Model.

Profit PID (RPID) Properly tuned PID loops can be maintained by using the Profit PID (RPID)library. This software determines the proper tuning constants to ensure minimumloop sensitivity based on parametric uncertainty. Tuning constants for a widerange of equation types are generated. Calculations are based either on userentered transfer functions or transfer functions derived from raw data.

Profit Loop (Future Product) High quality base level control can be accomplished by usingthe Profit Loop (SPID) library. This software generates a MISO RMPCTcontroller specifically designed for regulatory loops. Calculations are based eitheron user entered transfer functions or transfer functions derived from raw data.

Step Test Builder Effective step test design is provided by the Step Test Builder. This softwareallows you to easily create a series of one or more sequences that can be used toproperly excite the actual process. The Step Builder has been designed to workin conjunction with the APC Identifier. Sequential or/and simultaneous signalscan be readily synthesized and evaluated. Signals are designed for minimumlength and broadband uniform power. The Step Builder is available stand aloneor as part of the Profit Suite ToolKit.

Point Builder By using Point Builder, you can automatically create LCN data structures,command and configuration files for both Profit Controller and ProfitOptimizer. Point Builder is provided as part of both the Profit Controller and theProfit Optimizer packages.



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Data Converter You can use the Data Converter library to automatically convert LCN data to beProfit Design Studio (APCDE) compliant. The Data Converter is available standalone or as part of the Profit Suite ToolKit.

Model Converter The Model Converter will allow third party models to be converted to ProfitDesign Studio (APCDE) form. The Model Converter is provided as part of theProfit Controller (RMPCT) package.

Data OperationTools

Data Operation Tools will allow you to manipulate data in an interactivefashion. A host of options are available. Some of the options include:Interpolation, filtering, transformations, combinations, outlier detection andremoval, statistics, and manual editing.

Profit Toolkit Use this library to design and configure a Profit Toolkit application. Currently,the library supports both FCCU and Fractionator toolkits.

Profit Sensor Profit Sensor is an on-line, real-time, substitute for laboratory/analyzermeasurement (“soft sensor”). It offers inferential property estimation–a criticalproblem for process control and optimization. It can be used to create, train, andtest both linear and non-linear models.

Profit Sensor uses ASCII data to create three types of models: 1) Neural network,2) Ordinary Least Squares, and 3) Partial Least Squares.



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Section 2 — Installing Profit Design Studio (APCDE) and theAPC Identifier

2.1 Overview

In This Section Profit Design Studio (APCDE) is composed of many components to supportdata analysis, model synthesis and control design. The APC Identifier is oneof these integral components. As a minimum, both Profit Design Studio andIdentifier must be present for a functioning installation.

System and Software Requirements This section lists the system and softwarerequirements for loading and using Profit Design Studio and Identifier software.

Installation Quick Reference A quick reference is provided indicating the majortasks involved with installing Profit Design Studio and Identifier softwarepackage. If you have installed the Identifier before, use this quick reference toinstall the software package.

Installation Instructions Step-by-step installation instructions are provided fornew users of the Identifier. If this is your first installation of the Identifier, usethese instructions rather than the quick reference to install the software.

Section 2 — Installing Profit Design Studio (APCDE) and the APC Identifier2.2 System and Software Requirements


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2.2 System and Software Requirements

Software

Requirements

The Profit Design Studio (APCDE) package either comes on the APC IdentifierCD or may be obtained on a separate CD. You may also have received ProfitDesign Studio and/or Identifier as part of another Profit product.

Do I have to Install

the Identifier

Separately?

If you have purchased Profit Controller (RMPCT), Profit Optimizer (DQP) orProfit PID, the Identifier is automatically included in those packages. In thosecases, you do NOT have to install the Identifier separately. Please refer to thoseproducts’ User’s Guides for installation instructions.

If you have purchased the Identifier as a standalone product please follow theinstructions in this manual for installation.

PC Requirements The following table lists the recommended and minimum PC system requirementsfor using Profit Design Studio (APCDE). Depending on system size, simulationcan be computationally demanding. Although slower systems may function,maximum computational resources are recommended.

Recommended Configuration Minimum Configuration• WIN NT(all versions) or WIN 95

• Pentium 100 MHz

• 24 MB RAM

• 1MB disk space

• Profit Design Studio (APCDE)

version 220 or greater

• VGA

• Standard Windows video driver

• CD Rom Drive

• Mouse

• WIN NT (all versions) or WIN 95

• 80486 w/math coprocessor

• 16 MB RAM

• 500KB disk space

• Profit Design Studio (APCDE)

version 220 or greater

• VGA

• Standard Windows video driver

• CD Rom Drive

• Mouse

Section 2 — Installing Profit Design Studio (APCDE) and the APC Identifier2.3 Quick Reference to Installation


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2.3 Quick Reference to Installation

How to Use the

Quick Reference

Read and perform the following procedures to install Profit Design Studio(APCDE). For detailed instructions and help, see the referenced sections.

Quick Reference

Table

The following table outlines the major tasks involved with Profit Design Studioinstallation. Use this table if you have installed Profit Design Studio (APCDE)before. If this is your first installation, use the detailed instructions provided inSection 2.4.

Step Action Section Reference

1. Make sure your computing systems meet theminimum requirements.

See Section 2.2, “System andSoftware Requirements”.

2. Install Profit Design Studio (APCDE).

3. If Dongle Driver has not been installed already, installDongle Driver on the PC.

Section 2 — Installing Profit Design Studio (APCDE) and the APC Identifier2.4 Installing the Profit Design Studio and the APC Identifier


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2.4 Installing the Profit Design Studio and the APCIdentifier

PC InstallationTo run the SETUP program for Profit Design Studio (APCDE), perform thefollowing steps:

1. Make sure no other programs are running. Installation may fail or take avery long time to complete.

2. Insert the Profit CD into the CD-ROM drive.

3. Right click on Start. Choose Explorer. This will activate WindowsExplorer.

4. Click once on the CD Drive labeled Honeywell HiSpec to display thecontents of the CD Drive.

5. Double Click on the Profit Design Studio folder.

6. Double click on Setup.exe.

7. Either accept the default destination directory or change the directory byclicking [Browse] and entering the entire pathname. Click [Next].

8. All components (dlls) are now included with the installation of the designstudio. There is no longer a need or capability to select individualcomponents.



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9. Click [Next].

Installing theDongle

10. After ensuring that the physical Dongle is attached to your computer and youhave System Administration privileges, click [Yes] to proceed with the Dongleinstallation.

NOTE

The driver needs to be installed only when you first install the 32-bit version ofProfit Design Studio (APCDE). You do not need to install it again when youupgrade or add optional products to the Profit Design Studio (APCDE).

11. At this point you will receive what appears as a blank white screen. ChooseFunction > Install Sentinel Driver.

12. Click [OK] to accept the path.



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13. Installation is now complete. If at anytime during the installation procedureyou begin receiving error messages, make sure that you have the Dongle inplace AND administration privileges on the computer!

14. After exiting the installation program, restart your computer to make the newdriver take effect.

15. Select Start>Programs> Profit Design Studio 220.00

Or from the desktop, click on the Profit Design Studio icon. If all thesoftware was installed correctly, the About Dialog box will have theappropriate check marks and version information displayed.



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Check the Log

File

If you would like to check the version numbers of the dlls, check the log file. Toview a log file, open it with a text editor (Notepad). The log file can be found atc:\Winnt\APCDE32.log. An example of the information contained in the log fileis shown below.

13:45:09 24Apr01Profit Design Studio: 32 bit Version 220.00.0000

Loaded Math Library: HMATH32.DLL Version 220.00.0000Loaded Utility Library: HUTIL32.DLL Version 220.00.0000

Loaded Identification Library: HIDENT32.DLL Version 300.00.0000Loaded Controller Build Library: HBUILD32.DLL Version 200.00.0000

Loaded Controller Library: HCNTRL32.DLL Version 200.0001Loaded Run Block Library: HBLOCK32.DLL Version 200.0001Loaded Process Simulator Library: HSIM32.DLL Version 200.0001

Loaded Advanced Identification Library: HADVID32.DLL Version 110.00.0000

Profit Finder Library: HFINDER32.DLL is not present

Loaded Profit PID Library: HPID32.DLL Version 115.00.0100Loaded Process Simulator Library: HSIM32.DLL Version 200.0001

Loaded Profit Loop Build Library: HSPID32.DLL Version 100.01.0000Loaded Profit Loop Controller Library: HSCTRL32.DLL Version

100.0100Loaded Process Simulator Library: HSIM32.DLL Version 200.0001

Loaded Profit Optimizer Builder Library: HBLDQP32.DLL Version 200.00.0000

Loaded Signal Generation Library: HSIG32.DLL Version 100.02.0000

Loaded RMPCT Point Builder Library: HBLDEB32.DLL Version 200.00.0000

Loaded DQP Point Builder Library: HDBLDEB.DLL Version 200.00.0000

Loaded TDC Data Converter Library: HCONV32.DLL Version 105.00.0000

Loaded Scout File Converter Version 100.00.

Loaded Model Converter Library: HDMCNV32.DLL Version 100.00.0100

Loaded Vector Tool Library: HVTOOL32.DLL Version 110.00.0000

Loaded Profit Toolkit Designer Library: HSTOOL32.DLL Version 200.00.0000

Profit Sensor neural net builder is not present

Any problems with loading libraries will be described here. Inability to locate alibrary or library version incompatibility will prevent a library load.



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Check the ini File Default parameters that can be adjusted by the user are contained in the apcde.inifile. An example of the default information contained in this file is shown below.

Note that with release 200 of Profit Design Studio (APCDE) a new .ini structureis used. Please make a backup copy of your old file and remove it from yourSYSTEM or WINNT directory.

[Color options]PltMargBkgndClr=48300031PltMargTextClr=33554560CustomColor0=16777215CustomColor1=16777215CustomColor2=16777215CustomColor3=16777215CustomColor4=16777215CustomColor5=16777215CustomColor6=16777215CustomColor7=16777215CustomColor8=16777215CustomColor9=16777215CustomColor10=16777215CustomColor11=16777215CustomColor12=16777215CustomColor13=16777215CustomColor14=16777215CustomColor15=16777215[Recent EB File List]Field1=D:\Sample\160Test\AM\AMLARGE.ebb[Recent DEB File List]Field1=D:\Sample\DQPA_opt.ebd[Recent File List]File1=D:\Sample\DQPA_opt.ebdFile2=D:\Sample\160Test\AM\AMLARGE.ebbFile3=D:\Sample\SET1\LARGE.mdl

[Recent EB File List] - The last opened *.ebb file for Profit Controller (RMPCT)Point Builder. It will be created automatically and will be used as the default-starting file when a RMPCT Point Builder is opened next time.

[Recent DEB File List] - The last opened *.ebd file for Profit Optimizer (DQP)Point Builder. It will be created automatically and will be used as the default-starting file when a DQP Point Builder is opened next time.



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Other Options [SSC Options] - You have to manually add the option which supports a Donglebypassing code. You have to obtain the code from Hi-Spec and manually add theoption and edit the code.

[Engine Debug Flags] - You have to manually add the option. If the“CreateChgFile” is set to 1, a change file will be generated every time you run anoff-line RMPCT simulation.

[Memory Buffer] - You have to manually add the option. This option pertainsdirectly to identification and will be discussed in a subsequent section.

[User Options] - You have to manually add the option. This option pertains directlyto identification and will be discussed in a subsequent section.

[Toolbar preference] – This option is automatically added and pertains to theidentification toolbar preference of the user. Zero implies the use of the standardtoolbar while one implies the use of the detailed toolbar.

Caution Changing any .ini parameters is ill-advised without contacting a Honeywellconsultant.

Starting the PCApplication

From your program manager, click on your Profit Design Studio (APCDE) icon. Ifinstalled correctly, the About Dialog box Identifier tab will have the appropriatecheck marks and version information as shown below.



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Section 3 — APC Identifier Concepts

3.1 Conceptual Overview of the APC Identifier

Identification—A

Science and an

Art

A fundamental problem for any controller is the choice of the model that should beused to represent the system. In general, the model is one of the following:

• Linear time-invariant (lumped parameter),• Linear time varying (lumped parameter),• Linear with distributed parameters or• Nonlinear.

System identification remains both an art and a science. The science is concernedwith parameter estimation; the art is usually concerned with determiningstructure/order, the excitation requirements, and accuracy. System identificationinvolves two steps:

• First, a sequence for exciting the process to be modeled is specified.. Afamily of candidate models is then proposed. After this a representativemember of this family is selected. This is the art and is often problemspecific.

• The second step is the science. This step is a parameter-estimation problem.Parameter estimation is basically the determination of the best set ofcandidate model coefficients such that the model represents the causalinput/output relationships.

Extracting models from process data for control purposes can require severalsteps. At a minimum, the diagram shown below illustrates the overall procedure.

The Identification

Process

Start

Experimental Designand executation

Identification- Data processing- Model order/structure- Parameter estimation

Model validation

Is model ‘Good’ Use model

Steps, pulses, PRBS, etc.

Correlation, transformationplant models

Simulation, cross validation

Figure 1-1 Flow Illustration of the Identification Process

Section 3 — APC Identifier Concepts3.1 Conceptual Overview of the APC Identifier


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Identification

Environment

The APC Identifier contains a family of automated estimation tools with thefollowing characteristics:

• Provides a powerful tool to inspect/manipulate data and generate multiple inputmultiple output (MIMO) system models.

• Considers all dependent and independent variables simultaneously.• Allows Operator interaction during plant testing.• Process does not need to have steady initial or terminal conditions.• Multiple model forms and structures are accommodated.• Data segmentation is permissible.• Performance is given in terms of:

- Step response curves

- Correlation curves

- Confidence plots

- FIR null hypothesis test and ranking (statistics)

- Time series prediction per CV

- Power and Uncertainty spectrum (future)

- Residual error and cross correlation (future)• System models can be automatically chosen based on open loop prediction

performance.• Both continuous and discrete time models are generated.

Cross validation analysis is easily accommodated.

IdentificationApproach

The Identifier combines the best features of state of the art algorithms. Adopting ahybrid approach, the Identifier fits three models to arrive at a final, unifiedcontroller model for loading onto the control system:

• Finite Impulse Response (FIR) and/or general black-box models based onPrediction Error Methods (PEM)

• Reduced Order parametric models• Final system models.

Fitting FIRModels

FIR models are based on raw plant data and have these characteristics:

• Solutions result in an unbiased estimate (when plant tests are conducted in theopen loop with a properly designed signal). This is true even for colored noisedisturbances.

• Models are structure free.

• Solutions are extremely fast and exceedingly robust (the Toeplitz structure isfully exploited, and Cholesky decomposition ensures numerical robust factorizationof the normal equations).

• Solutions result in potentially high order, high variance estimates (dampingthese estimates with weights or smoothing will result in biased estimates, and in

Section 3 — APC Identifier Concepts3.1 Conceptual Overview of the APC Identifier


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addition will cause problems in the calculation of confidence limits).

• Processes with poles on the unit circle (integrators) are treated directly byautomatically modifying the standard form as necessary.

Fitting PEM

Models

PEM models are based on raw plant data and have these characteristics:

• Solutions result in a consistent estimate (when structure is compatible with theprocess and when the procedure converges to the global minimum). This is trueeven under closed-loop operation.

• For Gaussian disturbances the procedure results in a minimum varianceestimate.

• Goal is one-step (Load & Go) operation.• Relatively low order models imply minimal information loss under segmented

or contaminated data conditions• Solutions, while numerically robust (both Cholesky and QR factorizations are

supported), are computationally intense.• Requires nonlinear search in which convergence can NOT be guaranteed.

Fitting

Parametric

Models

Parametric models are based on either FIR or PEM results. The purpose of theparametric model is to take the FIR or PEM model, and fit it with a parametricmodel that reduces or eliminates the variance. In addition, the parametric modelsprovide an extremely effective mechanism for model order reduction that is easilyconfigured by the user. At this step:

• Low order parametric models are developed.• FIR or PEM models are used to get excellent initial guesses for the parametric

models. This includes an initial guess for the dead time.• Several parametric models are computed. They include:

- Laplace Domain (requires nonlinear search)- Discrete Domain

• ARX with pre-filtering• Output error (requires nonlinear search).

• Model determination is highly robust and rapidly convergent since structurehere is not a factor and the estimation is done on the FIR/PEM results not onthe raw data.

• All parametric models are ultimately converted to Laplace domain formirrespective of the original form.

Fitting Final

System Models

Parametric models are automatically selected for the final system model. Based onthe raw data (cross validation if selected), the parametric models with the best longterm open loop prediction performance are automatically selected for use in thefinal system model. Models can also be automatically eliminated based on FIRstatistics

With a click of the mouse you can manually override any default, or modify anymodel fit by default settings.

Section 3 — APC Identifier Concepts3.2 Theory Overview by Topic


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3.2 Theory Overview by Topic

Key TopicsIn the remainder of this section, the underlying concepts associated with the APCIdentifier will be discussed. Continue reading this section to find out about:

• General Identification Problem Statement

- Quadratic norm, Robust norm

• Models

- FIR, PEM, PFX, OE, Laplace

• Solutions

- Linear (FIR, PFX)

- Nonlinear (PEM, OE, Laplace)

- Initial estimates

- Transport delay

• Model Properties

- Consistency

- Bias and Variance

• FIR Statistics

- Covariance

- Confidence limits

- Null Hypothesis Test

• Factorizations (Cholesky, QR, SVD)

- Normal vs. Orthonormal

- Sensitivity and Accuracy

- An Ill-conditioned Example

- Pseudorank

- A Rank deficient Example

§ Minimum Norm Minimum Length Solutions

While not all of these topics will be of interest to all readers, a quick review isnevertheless recommended.

Section 3 — APC Identifier Concepts3.3 General Problem Statement


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3.3 General Problem Statement

Identification

Structure

In the Identifier, a very general identification structure is assumed. This structureis illustrated below:

y t z u t z e t( ) G( ) ( ) H( ) ( )= +

u t( )

ν ( )t

e t( )

G( )z

H( )z

y t( )+

+

System Identification Structure

Figure 3-1 General Identification Structure

System identification is not necessarily about fitting data, but rather about findingthe causal relationships between the inputs u and the outputs y shown in the abovefigure. This is to be accomplished in spite of the unmeasured disturbances ν.While in some instances it may be desirable to obtain a disturbance model H(z),the ultimate objective is to obtain the rational transfer function matrix G(z)whether H(z) is determined or not. In some cases a ‘good’ model may yield a poorfit of the data while in others a ‘poor’ model may yield a good fit of the data. It isthe objective of the identifier to not only extract as much useful information out ofthe data as possible, but to also indicate whether in fact the models obtained areuseful for the purposes of process control. Note that in this discussion and in allthat follows it is assumed that the dependent and independent variables (outputsand inputs) have been properly selected. This selection procedure itself may in factrequire significant analysis but in this discussion is not considered part of thesystem identification problem.

Quadratic Norm

Formulation

While many formulations are possible, the APC Identifier addresses only two: thequadratic norm and the so-called robust norm. The quadratic norm formulation canbe written as follows:

)ˆ(ˆ)ˆ(

,ˆs.t.

ˆ2

1)ˆ(

2

1min

2

2

2

2ˆ

θθθθθθθθεεεε

θθθθΓΓΓΓθθθθεεεε

yy −=

χ∈β=θ

+θ

iii

In the above expressions θθθθ , εεεε , and y are the unknown parameter, error and



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prediction vectors respectively. Here it is assumed that some of the parameterelements may be externally specified (null for example). In addition, it is alsoassumed that the user may wish to smooth the estimated parameters in a generalfashion by the penalty matrix ΓΓΓΓ. Without loss of generality the penalty matrix canbe written as:

WCCQ TT α== ΓΓΓΓΓΓΓΓ

These expressions define the starting point for the quadratic norm formulation.The smoothing term is included here only for purposes of discussion in latersections. Smoothing is never actually performed in the APC Identifier. Thereasons will be made apparent in later sections.

Robust Norm

Formulation

While the quadratic norm is the norm most commonly used for identification, thereare other norms that can be used in addition to this approach. Let )( ixl be the

positive scalar valued function of x such that the l-norm is defined as:

�=⋅i

ilxl )(

The l-function for the quadratic and Maximum Likelihood Estimates (MLE)respectively are:

2mlemle

2q

2

1)(1);(log)(

2

1)(

xxlxfxl

xxl

e =→=σ−=

=

Note: for Gaussian processes with unit variance MLE formulation satisfies thequadratic norm.

It is well known that the quadratic formulation can be sensitive to outliers. MLEon the other hand, asymptotically approaches the theoretical Cramer-Rao lowerbound on variance. The MLE formulation however, may not be the best approachin all cases. This may be due to small data sets, bias distributions or sensitivity tounknown variations in the unknown probability density function )(xf e . A

technique to reduce sensitivity in general and specifically to the unknown )(xf e is

the robust norm. In the APC Identifier the robust norm is defined in terms of thederivative of its l-function as follows:

��

��

��

−−−−<<<<−−−−>>>>

<<<<

====′′′′σρσρσρσρ

σρ

ˆˆ

ˆˆ

ˆ

)(

x

x

xx

xl r

Here σ is the estimated standard deviation of the prediction error vector e. Thisestimate is given by:

ς−

=σ))((

ˆεεεεεεεε medianmedian



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The constants ρ and ς are taken from Ruppert and Ljung to be 1.6 and .7

respectively. Clearly, the robust norm is equivalent to the quadratic norm when themagnitude of x less than rho sigma. When the magnitude of x is larger than rhosigma the norm is linear in x with a slope of rho sigma with the appropriate sign.With this definition the problem statement for the robust norm becomes:

)ˆ(ˆ)ˆ(

,ˆs.t.

)ˆ(minˆ


θθθθεεεε

yy −=

χ∈β=θ

θ

iii

lr

Section 3 — APC Identifier Concepts3.4 Model Structures


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3.4 Model Structures

Overview Identification implies a solution to the defining minimization problem. Toaccomplish this, the error and hence prediction vector y must be specified. This

specification has a profound impact on the solution methodology and in somecases in the solution itself. In the APC Identifier many models are supported.Different models are used at different times depending on the particular task athand.

Modeling is accomplished in two distinct phases. In the first phase models areused to regress the data. At this level, two different model types are supported.One is the Finite Impulse Response (FIR) model. The other is the generalizedblack-box model based on the Prediction Error Method (PEM). Rather than asingle model, PEM represents virtually all of the standard discrete-timepolynomial models.

In the second phase, models are used for order reduction and to reduce oreliminate variance. Inputs here are the models obtained in the first phase based onregressed data. At this level, three distinct types of models are supported. Two arediscrete or z-domain models and one is a continuous or s-domain model. Thediscrete models are ARX and Output Error (OE). The ARX model is actually apre-filtered ARX, where the pre-filter is used to weight the low frequency fit. Theorder of the discrete time model can be defined by the user and is unrestricted. Thes-domain model has a fixed structure form in which the order (up to 3) isautomatically determined. Discrete models in this phase are ultimately convertedto the s-domain before they are saved.

FIR Models With the proper formulation, the FIR approach can be an exceptionally effectiveestimator. Under reasonably conditions, to be discussed later, this approach resultsin a statistically unbiased and consistent estimate. To this end, the APC identifieruses the FIR model as its base form.

FIR Structure The general FIR structure used is given by:

it

mnt

min

mt

mimt

mimt

mi

ntint

it

it

i

ntint

it

it

iit

mmubububub

ubububub

ububububy

α+ν++++++

+++++

++++=

−−−

−−−

−−−

,2

,21

,1

,0

22,22

2,2

21

2,1

22,0

11,12

1,2

11

1,1

11,0

22

11

�

�

�

�



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i

t

ji

lb

tju

tiy

i

t

jil

jt

it

variabletoingcorrespondtermbias

at timenoisecolored

t variableindependenandvariabledependent

toingcorrespondtcoefficienresponseimpulse

timediscreteatt variableindependen

timediscreteatvariabledependentwhere

,

=α

=ν

=

=

=

This expression corresponds to the positional form of the FIR model. There are noinherent limitations imposed on the structure of this model, and as such, each submodel (i, j element) is free to have as many coefficients as necessary to adequatelycapture the observed response.

To obtain the velocity form of the model, all dependent and independent variablesin the above expression are differenced in time. Differencing is invoked at thediscrete sample rate of the model, which is in general different than the samplingtime of the data.

In terms of the general ID structure presented previously the FIR model can bewritten conveniently as:

)(B)(G

)()()(B)(

11

1

−−

−

=

α+ν+=�

zz

ttuzty

ii

i

ii

while the predictor in positional and velocity form respectively is given by:

α+−+−+=α+= ��−

i

inb

i

ii nbtbutubtubtuzty ))()1()(()()(B)(ˆ 101

�

)1()()(

)1()()(

)()(B)(ˆ 1

−−=∆−−=∆

α+∆=∆ �−

tututu

tytyty

tuzty

iii

i

ii

Any dependent variables that contain integrators corresponding to one or moreindependent variables are handled as special cases of the above expression. Inthese cases, the dependent and non-integrating independent variables aredifferenced while the integrating independent variables are left in standardpositional form.

Intrinsic problems that can result from data that is over sampled or improperlyscaled, are eliminated by an automated data compression and scaling routine thatis an integral component of the FIR computations.



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PEM Models This model form encompasses virtually all of the polynomial black-box models. Inits full form, this model can be used in both the open and closed loop and underreasonable conditions (to be described later) is a consistent estimator. Underslightly more restrictive conditions it yields optimal (minimum variance)estimates. These desirable features are however, not without consequence. PEMcalculations are computationally intense. In addition convergence can not beguaranteed in spite of the considerable effort expended in obtaining good initialestimates.

PEM models are provided with one goal in mind: EEaassee ooff uussee. The goal here is toprovide a mechanism that will allow truly one-step identification. One click on“Load & Go” button and that’s it. While no restrictions are imposed in the use ofPEM, it can quickly become a “fiddlers paradise” that requires expert intervention.This is NOT the intent. If the results are not satisfactory after one try, simplyrevert to the standard FIR approach. To this extent, it is useful to view the PEMmodels as a complement to the standard FIR models

While PEM models can be used in a general setting, due to computation, structureand convergence limitations, they can be less effective from a practical perspectivethan the standard FIR approach

TThhee ttaarrggeett uussee ooff tthhee PPEEMM mmooddeell iiss ffoorr rreeggrreessssiioonn sseettss oonn ssttaabbllee pprroocceesssseesswwhheenn oonnllyy oonnee oorr ttwwoo iinnddeeppeennddeenntt vvaarriiaabblleess aarree mmoovviinngg ssiimmuullttaanneeoouussllyy.

Under these conditions the PEM approach can be an effective one-step methodthat requires no a-priori user input

PEM Structure The general PEM structure used is given by:

)D()A(

)C()H(

)F()A(

)B()G(

)()D(

)C()(

)F(

)B()()A(

11

11

11

11

1

1

1

11

−−−−−−−−

−−−−−−−−

−−−−−−−−

−−−−−−−−

−−−−

−−−−

−−−−

−−−−−−−−

========

++++++++��

��

��−−−−====��

zz

zz

zz

zz

tez

zdtu

z

ztyz α

Based on this model the corresponding predictor is:

α+

��

��

�−+��

�

��

�−=

−

−

−

−−

−

−

−

−

�

)C(

)D(

)C(

)A()D(1)(

)F(

)B(

)C(

)D()(ˆ

1

1

1

11

1

1

1

1

z

z

yz

zzdtu

z

z

z

zty



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Based on the above expression, the prediction error, defined as )(ˆ)()( tytyt −=ε ,

can be written as:

��

�

�

��

�

�α−��

�

��

�−−=ε � −

−−

−

−

)()F(

)B()()A(

)C(

)D()(

1

11

1

1

dtuz

ztyz

z

zt

The polynomials in the above expressions have the following form:

ndnd

ncnc

inf

nf

inb

nb

nana

zdzdzd

zczczc

zfzfzf

zbzbzbb

zazaza

i

i

i

i

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−

++++++++====

++++++++====

++++++++====

++++++++====

++++++++====

�

�

�

�

�

22

11

1

22

11

1

22

110

1i

22

110

1i

22

11

1

1)D(z

1)C(z

)1()(zF

)()(zB

1)A(z

Within this context, the PEM structure supports the following standard forms:

• FIR• ARX• ARMA• ARMAX• ARIMA(X)• ARARMAX• BJ• OE

While the PEM model is actually solved using the completely general form givenabove, the user is currently prevented from specifying both A and F polynomialsin a given regression. There are no other restrictions on orders or structures. Withthis model there is no need to specify positional or velocity form. As a matter offact it is invariably disadvantageous to specify velocity form when using PEMunder normal conditions. The one possible exception is the ARIMA(X) model.This structure requires the velocity flag to be set.

The default model uses the multi-input single output Box-Jenkins (BJ) structure.While all the structures listed above are available, it is not in the best interest ofthe average user to select any structure but the default. If there are problems withthe default, revert to FIR.

Intrinsic problems that can result from improperly scaled data are eliminated by anautomated data scaling routine that is an integral component of the PEMcomputations. PEM calculations are fully integrated to the design studio and assuch can be used seamlessly with discontinuous, contaminated and bad data(NaN). Constraints for null models and the “Lock Model” options are fullysupported.



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Model for Order

and Variance

Reduction

In the above paragraphs, data regression related models were presented. Thefollowing paragraphs illustrate those models that are used for order reduction andvariance reduction/elimination. The discrete time models presented belowcorrespond to structures contained in the PEM model. They should however not beconfused with the PEM approach. The models presented below are much simplerby design and as such have their own solution methodology.

ARX Parametric

Models (Discrete

Time)

Parametric models are used for model order reduction and to remove the variancefound in the models regressed from raw data. While standard low order ARXmodels are typically inadequate due to biased estimates, the pre filtered form usedin the APC identifier automatically weights the low frequency fit and hence,results in high quality models.

The general form of this model is given by:

)()()B()()P(

or2211

2211

tedtuztyz

eububub

ypypypy

tdntndtdt

ntnttt

++++−−−−′′′′====′′′′

++++′′′′++++++++′′′′++++′′′′====′′′′++++++++′′′′++++′′′′++++′′′′

−−−−−−−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−

�

�

The resulting transfer function takes the form:

nn

dnn

zpzpzp

zzbzbzbz −−−−−−−−−−−−

−−−−−−−−−−−−−−−−

++++++++++++++++++++++++++++====�

�

22

11

22

11

1

)()T(

In the above expressions the prime denotes a pre-filtered value while n and dcorrespond to the order, and delay of the sub process respectively.

If the discrete time model contains an integrator, then one pole of T(z) isconstrained to lie exactly on the unit circle.

Output Error

Models (Discrete

Time)

In addition to the ARX form shown above, the identifier also generates discretetime models with an output error structure. The general form of these models isgiven by:

ttt

dntndtdt

ntnttt

ewy

ububub

wfwfwfw

++++====

++++++++++++====++++++++++++++++

−−−−−−−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−

and2211

2211

�

�

Close inspection of these expressions shows that the output y does not appear inthe regression matrix for this model. Consequently, this structure results in anunbiased estimate if the input u is persistently exciting. The above expressions canbe conveniently written as:

)()()F(

)B()( tedtu

z

zty ++++−−−−====



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The resulting transfer function takes this form:

nn

dnn

zfzfzf

zzbzbzbz −−−−−−−−−−−−

−−−−−−−−−−−−−−−−

++++++++++++++++++++++++++++

====�

�

22

11

22

11

1

)()T(

While the output error model has the desirable feature of being unbiased evenwithout pre-filtering, this structure requires that the estimation parameters appearin the regression matrix. Consequently, the estimation problem becomes nonlinear.

This implies that more computational effort is required for the output errorsolution than is required for the ARX solution.

Laplace DomainParametricModels

It is also possible to generate parametric models directly in the Laplace domain.The general Laplace domain form is given by:

delaytransport

poleprocesssecondwithassociatedconstanttime

poleprocessfirstwithassociatedconstanttime

zeroprocesswithassociatedconstanttime

Gain:where

)1)(1(

)1()T(

2

1

21

================

====++++++++

++++====−−−−

d

k

sss

esks

ds

τττ

τττ

This model is guaranteed to be over damped and open loop stable. If any pole of adiscrete time model is outside the unit circle, then the sub model is automaticallyrebuilt using this structure.

For under damped sub-processes, the discrete model form is required to capturethe complex pole structure. For the best fit, simply select the Best of both option.This uses both the Laplace and discrete forms and returns the model with thelowest prediction error.

Final Model Form All models are ultimately saved in Laplace form. Discrete models areautomatically converted from the z to s domain. The form of the saved models is:

)1(

)1()T(

11

1

11

1

++++++++++++++++++++

==== −−−−−−−−

−−−−−−−−−−−−

sasasas

esbsbks

nn

nn

dsnn

�

�

The lead s in the denominator is present only if the sub process contains anintegrator. In this case the k refers to the integration rate.

Section 3 — APC Identifier Concepts3.5 Solutions


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3.5 Solutions

Overview Due to the hybrid nature of the identifier and the variety of model structuressupported, several solution methodologies are used for parameter estimation. Someof the model types utilize linear strategies while others require nonlinear techniques.Nonlinear methods have the additional requirement of specifying an initial estimateto start the algorithm.

In the following paragraphs, each of the various solution techniques will bedescribed. Linear approaches will be given first. This will be followed by thenonlinear techniques and associated starting conditions. Finally, a brief descriptionwill be given on the delay estimation algorithm used for all reduced order models.

Linear Solutions

FIR Models

Using the FIR structure defined in the previous section, the prediction equation in

vector form can be written as θθθθ==== ˆˆ Ay . Where

��

�

�

��

�

�

=

−+−+−+−+−+

−++−+

−−−

1

1

)(2

11

31

21

1

)(12

11

111

1

)(21

21

11

nununbmtmtmtmtmt

nununbttttt

nununbttttt

uuuuu

uuuuu

uuuuu

��

��

��

��

A

and

[ ]α= nununbbbbbb )(

20

12

11

10

T��θθθθ

The subsequent prediction error becomes: θθθθεεεε ˆˆ Ayyy −=−= . With this definition

of the prediction error and the quadratic norm formulation given previously, theminimization problem becomes:

χ∈β=θ

+−θ

iii ,ˆs.t.

ˆ2

1ˆ2

1min

2

2

2

2ˆθθθθΓΓΓΓθθθθ yA

This problem can be solved by either orthonormal factorizations or by explicitminimization. Here, the latter is used. If for the present discussion, constraints areneglected, the solution to the above problem can be written in terms of thefollowing normal equations:

yAQAA TT ˆ][ =+ θθθθ

The penalty term, Q, has been included for bias and consistency discussions in asubsequent section. In the APC Identifier smoothing is not supported. Hence Q isidentically equal to zero and the normal equations actually solved are:

yAAA TT ˆ =θθθθ



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A fast correlation update algorithm, which is analytical rigorous, is used to form thenormal equations. Solution of the normal equations is accomplished by a highlyefficient numerically robust Cholesky decomposition. This rank revelingdecomposition is a reproduction, with minor enhancements, of the LINPACalgorithm. In this decomposition the normal equations are written as:

C d�θθθθ =

By inspection:

C A A d A y= =T T;

Since A AT is symmetric positive semi-definite, C can always be factored as:

RRCPP TT =

Where R is an upper right triangular matrix and P is a permutation matrix.

Multiplying the normal equations by TP gives:

dPCP TT =θθθθ

Let fdPPx == Tandθθθθ . Substituting these expressions into the above equationresults in the following relation:

fRxRdPCPxP === TTT

From this expression, the estimates can be calculated in three trivial steps. First, thefollowing equation is solved for z using simple forward substitution:

fzR =T

With z known, the following equation is solved for x using simple back substitution

zRx =

Finally, the estimates are recovered from x using the perturbation relationshipdefined above.

In some cases, accuracy of the normal equation approach might be of concern. TheCholesky algorithm used here is designed to deal directly with both poorlyconditioned and rank deficient problems. For system identification, in which doubleprecision accuracy of the plant data cannot be assumed, the approach used is asaccurate as alternative orthonormal solutions but significantly faster. Issues relatingto accuracy, rank, pseudo-rank and factorizations will be discussed in more detail atthe end of the section.



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Linear SolutionsPFX Models (Pre-Filtered ARX)

Implicit in the use of the PFX model is the assumption that the “data” is the stepresponse result of either an FIR or PEM calculation. With this information at hand,the general procedure is as follows:

• Shift “data” by delay (delay is iteratively determined in an outer loop to bedescribed shortly)

• Fold (resample) “data” if necessary

• Set filter order n

• Loop until done if required

}

;

);Convert(

);,,PfxSolve(

);,filter(

);,filter(

{

))IsStable(&)((!

−−=

=

=

=

n

PfxPfx

yunPfx

yny

unu

PfxnGoodwhile

s

ff

f

f

s

In the procedure given above, the function PfxSolve returns the conventional ARXsolution while the function Convert transforms the model from the z to the sdomain. Using the ARX structure defined in the previous section, the prediction

equation in vector form can be written as θθθθ==== ˆˆ Ay and the subsequent prediction

error becomes θθθθεεεε ˆˆ Ayyy −=−= . With this definition of the prediction error and

the quadratic norm formulation given previously, the minimization problembecomes:

θθθθαααα

θθθθ

ˆ..

ˆ2

1min

T

2

2ˆ

−β=µ

−θ

ts

yA

Constraints in the above expression apply only for integrating processes. Solutionof this problem (without constraints) is accomplished by using an orthonormalfactorization of A. The orthonormal factorization used in the APC Identifier is arank reveling QR decomposition which is essentially that of Golub and Van Loan’sMatrix Computations. In this decomposition AP = QR (P is a permutation matrix).

Let θθθθ=Px and the minimization problem becomes:

2

2

2

2

T2

2

2

2

2

2

ˆ bRxyQRxyQRxyAPxyA −=−=−=−=−θθθθ



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and the solution is:

bRx ====

In the above expression, the actual structure of R depends on the mechanism usedto establish the pseudo rank and whether or not a minimum length solution isdesired. These issues will be discussed in more detail at the end of the section.

For integrating processes no special factorizations are invoked. With the constraintsillustrated above the solution becomes:

0]00111[1

)(

)()(ˆ

T

1TT

1TTT1T

��=−=β

��

��

� −β+= −

−−

αααα

αααααααααααα

ααααθθθθAA

AyAAyAAA

Clearly, this solution can exhibit sensitivity problems when A is illconditioned. In spite of possible sensitivity, use of constraints to insure onepole is located at (1,0) has proven to be particularly effective. Earlierattempts at removing the constraints by differencing the input step responseand using a straightforward QR solution were less accurate as might beexpected due to the differencing operation. When ill conditioning exists, theintegrating pole is prescribed to be precisely on the unit circle at (1,0).Hence, the transformation to the s-domain will always result in a pole that isexactly at the origin. Rank deficient problems at this level, will return anerror message with a null model

Nonlinear

Solutions

Each of the remaining models, PEM, OE, and Laplace all require a nonlinearsolution. A general procedure is used for all models. Application of this procedureto a particular model type depends only on the mechanism used to represent theprediction error.

Solution

Procedure

Both robust and quadratic norms are used to define the nonlinear minimizationproblem. For the PEM model the user is free to select either norm. Only thequadratic norm is used to define the OE and Laplace problems. For clarity ofpresentation, use of the robust norm and constraints will not be described. Hence,the problem statement presented previously becomes:

)ˆ(ˆ)ˆ(

)ˆ(2

1min

2

2ˆ


θθθθεεεε

yy −=θ

The solution of this problem is given by the θθθθ that solves the following equation:

0J =)ˆ()ˆ(T θθθθεεεεθθθθ



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where the Jacobian matrix J, is given by.

��

�

�

��

�

�

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂ε∂

θ∂∂ε

=��

��

�

∂∂≡

n

mmm

n

n

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆ)ˆ(

21

2

2

2

1

2

1

2

1

1

1

TT

�

�

�

�

θθθθεεεεθθθθJ

For this problem the residual vector, r can be defined as:

)ˆ()ˆ()ˆ( T θθθθεεεεθθθθθθθθ Jr =

Clearly, it is desired to find the value of θθθθ such that r is as close to zero aspossible. A straightforward Newton-Raphson technique is used to accomplish this.Initially the use of a potentially superior Quasi-Newton (Broyden-Flecher-Goldfarb-Shanno (BFGS) ) algorithm was briefly investigated. Due to the natureand structure of this specific problem, attention was focused on establishing reliableinitial estimates rather than on techniques that can potentially enhance theconvergence properties of the base algorithm.

By expanding the residual in a Taylor series and neglecting high order terms, the

Newton step θθθθδδδδˆ can be written as:

)ˆ(ˆ)ˆ( θθθθθθθθδδδδθθθθ rH −=

Where the Hessian matrix in the above expression is given by:

θθθθθθθθθθθθθθθθθθθθ

θθθθθθθθθθθθ

ˆ)ˆ(

)ˆ()ˆ()ˆ(ˆ

)ˆ()ˆ( TT

T

∂∂+=

∂∂≡ J

eJJr

H

From this definition it can be seen that the Hessian consists of a specialcombination of first and second-order information. Here it will be assumed thateventually the first order term (the first term on the right hand side of the aboveexpression) will dominate the second order term. If the magnitude of the predictionerror tends to zero as the solution is approached, then the second order term in theabove expression also tends to zero. Thus the approximate Hessian becomes(dropping the notational dependence on the estimates):

JJH T≅

Using this approximation the Gauss-Newton step becomes:

eJJJ TT ˆ −=θθθθδδδδ



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Note that the approximate Hessian is semi-positive definite and the above equationsare fully compatible. By inspection, the solution to the above expression is also asolution to the following minimization problem:

2

2ˆˆ

2

1min εεεεθθθθδδδδ −

θδJ

For PEM models, this problem is solved for θθθθδδδδˆ using either a QR or Choleskyfactorization. The choice is user specified and the corresponding factorizationfollows that presented previously. For the other models a QR factorization is alwaysused.

Updates of the parameter vector θθθθ are given in terms of the Gauss-Newton stepaccording to:

iiii θθθθδδδδθθθθθθθθ ˆˆˆ1 λ−=+

A line search is used to determine the step length iλ that will insure the magnitude

of the residual decreases in a monotonic fashion.

PEM Formulation For PEM models the Jacobian matrix is obtained by analytically differentiating theprediction error with respect to the unknown parameter vector: As presentedpreviously, the prediction error is:

��

��

α−��

�

� −−=ε � )(F

B)(A

C

D)( dtutyt

It is also convenient to define the following auxiliary variables:

);()(A)(;)(F

B)( dtwtydtdtudtw −−≡−ν−≡−

From the definition of the Jacobian the (i,j) th element is:

j

iji θ∂

ε∂=��

��

�

∂∂≡

ˆJ

ˆ ,

TT

θθθθεεεε

J

Differentiating the prediction error allows the Jacobian to be defined (dropping thenotational dependence on the delay, d) by the following:



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)(CF

D)(

CFF

DB)(

)(CF

D)(

)(C

D)(

ktwktuf

t

ktub

t

ktya

t

ii

ik

i

ik

k

−=−=∂ε∂

−−=∂ε∂

−=∂ε∂

C

D)(

)(C

1)(

)(C

1)(

CC

D)(

−=α∂ε∂

−=∂ε∂

−ε−=−−=∂ε∂

t

ktvd

t

ktktvc

t

k

k

Where the index k runs over the order of the individual polynomial and the index iruns over the number of inputs. Since the right hand side of the equations presentedabove can be evaluated in terms of a fast filter (Transposed Direct Form II)operation, Jacobian evaluation is very efficient. Column shifting where possible isfully exploited.

Bad values (NaN) in the Jacobian matrix, indeed in all regression matrices arehandled in a straightforward fashion. Any rows containing bad values and anycorresponding vector elements are simply removed from the regression. Treatingbad values in the filter operations is not so straightforward. Here special filteroperations were required. While care was taken in the design of these filteroperations, they are nevertheless slightly less efficient than the standard TransposedDirect Form II filter function

OE Formulation The analytical approach described above is also used for the Output Error modelsused for model order and variance reduction. To eliminate any overhead a separatealgorithm tailored to this specific structure is used. For these SISO models theJacobian is defined by:

)(F1)(

)(F

1)(

ktwf

t

ktub

t

k

k

−=∂ε∂

−−=∂ε∂



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LaplaceFormulations

Formulation of the Jacobian as presented above cannot be used in conjunction withLaplace models. In this instance, the parameters appear in the derivative in anonlinear and complicated fashion. Therefore the Jacobian is evaluatednumerically. Laplace models at a given iteration are converted to discrete models atthe appropriate sample rate. These models are then used to generate a discreteprediction error vector. Continuous time parameters ( 21 and,, τττnk ) are perturbed

to obtain incremental changes in the error vectors. These error vectors are then usedto evaluate the Jacobian using central point finite differences.

Starting

Conditions

Initial estimates for the various models are established as follows.

• FIR - None• ARX (PFX) - Delay estimation• OE - Delay estimation and parameters from ARX solution• Laplace - Delay estimation, gain and dynamics from FIR solution• PEM

- Get initial A and B using high order ARX solution (determine orderbased on modified Akaike information criterion)

- Perform PFX model reduction step on high order models- Use A and B as filters for an Instrumental Variable refinement step to

calculate modified A, B and F (this step is not usually required unlessuser chooses to ignore high order ARX/PFX reduction option)

- Stabilize F- With refined A, B and F, calculate ν- Pre-whiten ν using high order AR model- Use Pre-whitened ν as an estimate of ε and original ν to estimate C and

D- Stabilize D- Begin search

Note the high order ARX solution followed by the PFX model reduction step forparameter initialization serves two purposes not found in conventional PEMapproaches (i.e. MATLAB). First it typically yields better initial estimates than loworder instrumental variable or bootstrap methods. Second it substantially reducesthe effect that PEM model order has on the initialization procedure.

Delay Estimation Accuracy of the reduced order models is heavily influenced by the transport delay.Unfortunately, this parameter does not lend itself to direct estimation. In additionthe current formulation requires that the delay be an integer multiple of the effectivesample rate. Note that the effective sample rate is not in general the same as thedata sample rate due to internal compression. Since the delay is constrained to be aninteger multiple of the effective sample rate, gradient based searches are notconvenient. Hence, a heuristic approach is used to estimate the delay. In the APCIdentifier a relatively brute force approach is used. Here four likely delays aredefined and each is evaluated with the solution being the delay that delivers thelowest fit error. Delay estimation is accomplished using the following procedure:



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• Determine four potential delay values• With potential delay values, estimate ARX, OE and/or Laplace

models• With these models evaluate fit in terms of prediction error

• Select delay and subsequent model with lowest fit errors

The tentative delays are selected as shown below.

t

y

0d

yε±=δ

y

1d

2d

3d

Each of the delays is determined based on the FIR or PEM step response shownabove by the solid line. Here high variance is assumed. To begin the procedure themaximum value of the step response, y is determined. The delay threshold δshown by the dash-dot lines is computed next. In this calculation ε is the user-specified threshold in percent (default value is 5%). Based on this information thefour delays are calculated as follows:

• 0d - This value always corresponds to the zero delay solution

• 1d - This value is determined by starting at the beginning of the step

response curve and finding the time when the step response first exceeds thethreshold limit.

• 3d - This value is determined by starting at the end of the step

response curve and working backward in time. The delay is given by the first timethe step response breaks the threshold limit. This value can be considered to be theminimum-phase time (delay plus any inverse response time) of the process.

• 2d - This value corresponds to the delay that would be obtained if a

first order plus dead time model were to be fit to the step response curve. The trick



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here is determining the maximum slope of the noisy (potentially high variance) stepresponse curve. In the APC Identifier an iterative least squares technique is used.Here the number of points needed to insure attenuation in the fluctuation of theslope of the straight line is determined.

While this approach is not guaranteed to be optimal, experience has shown it to bevery effective. Since optimality is not guaranteed, a mechanism is provided to allowthe user to easily override these calculations.

As a final note, a preliminary PEM option exists for delay estimation. When thisoption is invoked, the delay is estimated in a fashion similar to that described here.This estimation is performed as part of the initialization process and performedprior to the nonlinear search. Superior automated correlation-based techniques areanticipated in the future

Section 3 — APC Identifier Concepts3.6 Model Properties


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3.6 Model Properties

OverviewTwo model characteristics that are highly desirable, if not required, are that theestimated model should be

• Unbiased 0=) θθθθθθθθE(

• Consistent 0∞→= θθθθθθθθlim

n

Where the subscript 0 implies the ‘true’ process. In the discussion to follow, itwill be assumed that the observed data have been generated by the followingsingle input single output process:

(PEM))()(H)()(G)(

(FIR)1

01

0

0

tztuzty ξ+=

+=−−

ννννθθθθAy

Where G and H are as defined previously and ξandνννν are colored and white

noise disturbances respectively. It will also be assumed that the process is quasi-stationary and that disturbances are zero mean.

Discussions on bias and consistency will be limited to these properties as theypertain to the FIR and PEM models. In the PEM discussion, a full structure isassumed. Results are also valid when A=1. Other substructures need to beevaluated on a case by case bases.

FIR Bias Parameter estimates for the FIR solution can be written directly from the normalequations given previously as:

yAQAA T1T ][ˆ −+=θθθθ

Substituting for y and taking the expectation gives:

)]E([)]E([)Ê( T1T0

T1T ννννθθθθθθθθ AQAAAAQAA −− +++=

When A is uncorrelated with the zero mean disturbance, the last term on the righthand side of the above expression becomes zero and the expected value is:

)]E([)Ê( 0T1T θθθθθθθθ AAQAA −+=

Using the matrix inversion lemma, the expected value can be rewritten in thefollowing form:

11T1T

0

])[(][)(

)]([)Ê(−−− +=

−=

IQAAQAAQB

QBI θθθθθθθθ

Bias is clearly a function of the smoothing matrix Q. When there is no smoothing( 0Q = ), the FIR estimates are unbiased (given the stated assumptions).



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Alternatively, a non-zero Q results in a biased estimate.

FIR Consistency To evaluate FIR consistency, the solution presented above is expanded asfollows:

)]()()[(N

1)]()()()([

N

1ˆN

1

T

1

TN

1

T kvkkkkkkkk

+��

�

��

�+= ��

=0

−

=

θθθθγγγγγγγγθθθθ aaaa

Where )(ka is a vector composed of the kth row regressors of the original A

matrix and )(kγγγγ is a vector of the kth row of the original ΓΓΓΓ matrix. Quasi-

stationatity implies the mean and covariance converge to constant values. Thusthe limit of the summation terms given above, as N goes to infinity yields theexpected value. For example:

)E()()(N

1lim

N

1N

ννννAa =�=

∞→k

kvk

If the process is persistently excited, then

)]()()()([N

1lim T

N

1

T

Nkkkk

k

γγγγγγγγ+�=

∞→aa

is a nonsingular matrix. If this condition satisfied and A is uncorrelated with thezero mean disturbance, then in the limit as N goes to infinity the solutionbecomes;

0N

)]([ˆlim θθθθθθθθ QBI −=∞→

Clearly, for the stated assumptions, the FIR estimates are consistent when Qequals zero

PEM In what follows, it is assumed that G is stable and proper, H is stable minimumphase and monic and ξ is a white zero mean disturbance with variance R. Using

the definitions given previously for G and H, the predictor can be written in thefollowing form:

)(]H1[)(GH)(ˆ 11 tytuty −− −+=

Here it will be assumed that G and H have the correct structure. Based on theexpression given above the prediction error is:

)](G)([H)( 1 tutyt −=ε −

And the corresponding loss function for the quadratic norm is:



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��=

−

=

−==N

1

21N

1

2 )])(G)([(HN

1)(

N

1V

tt

tutytεεεε

As N goes to infinity, the loss function can be written as:

)}(E{)(N

1limVlim 2

N

1

2

NNtt

t

ε=ε= �=

∞→∞→

Using the definition of the ‘true’ process for y, the prediction error can berewritten to give:

)(HH)(]G[GH)( 01

01 ttut ξ+−=ε −−

Since H is monic, �+ξ=ε )()( tt . Thus

R)}(E{)}(E{Vlim 22

N=ξ≥ε=

∞→tt

and

RHH)}(E{]G[GH)}(E{ 20

2220

22 −− +−=ε tut

If the search for G and H converges to a global minimum, then from theexpression given above, it must be true that:

1H

H

0]G[G

2

20

20

=

=−

Therefore, if the search converges to the global minimum, then:

00N

HH,GGlim →→∞→

And the prediction error estimates are consistent.

Summary FIR models yield unbiased and consistent estimates when

• Process is quasi-stationary• All disturbances are zero mean• All inputs are uncross-correlated with all disturbances• Model is structurally compatible with the process (number of

coefficients is sufficiently large)• Process is persistently excited• Model coefficients are unsmoothed (Q equals zero)



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For reasonably designed experiments under open-loop conditions, the APCidentifier will yield FIR estimates that are unbiased and consistent. Note, Non-stationary and nonzero mean processes are effectively accommodated using thevelocity form of the predictor.

Given global convergence, prediction error estimates are consistent and thecorrect process and disturbance (noise) models are recovered. It is important tonote that no restrictions have been made on the inputs, therefore the estimates arealso consistent in the closed loop. If the disturbance is Gaussian then theestimates are maximum likelihood. It is also important to recognize however, thatif the model structure is incorrect (order is too low) the estimates will be biased.

While consistency is indeed an extremely desirable characteristic, it isnonetheless an asymptotic property and therefore does not tell the whole story.Since model errors are due to both bias and variance effects, consistent modelsmay still exhibit unacceptable errors. Models errors due to variance effects can bedescribed by the following relationship:

operation)loop-openfor1(functionysensitivitcontrollerS

signalinjectedofpowerinput

poweredisturbanc

pointsdataofnumberN

ordermodel

:where

)S(

1

)(N

)(Var

2

=≡≡Φ≡Φ≡≡

ωΦωΦ

∝

ν

ω

ν

u

iu

n

e

n

This frequency-based expression completes the story. Indeed this is probably themost important relationship for anyone involved with identification to understand.This expression clearly demonstrates that errors are proportional to the ratio ofdisturbance to input power. Errors are also proportional to the ratio of modelorder to test duration and inversely proportional to the sensitivity function (tightcontrol leads to big errors).

Implications are straightforward. If there are significant disturbances for a fixedduration test, then the errors will be high unless the input power is increased tocompensate for the disturbances. If input power is restricted (limited movement)in spite of disturbances, then the only alternative is increased test duration. Here itis assumed that n is not a strictly free parameter since improper adjustment maylead to bias. Closed-loop identification (S not equal to 1), results in increasederrors relative to a comparable open-loop test. The tighter the loop the worse theresults. If accurate models are required in a particular frequency, then the injectedsignal must have sufficient power in the desired spectrum.



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Too often when poor results are obtained people look everywhere but to thesource. These same people are prone to jump in favor of ‘magic solutions’. Mostoften problems can be addressed in terms of the expression given above. Theresimply is NO replacement for proper experiment design.

Section 3 — APC Identifier Concepts3.7 FIR Statistics


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3.7 FIR Statistics

Statistical

Properties

To establish statistical properties of the estimates, it is necessary to compute thecovariance matrix. For the purposes of covariance calculations the noise will beassumed to be white with variance s2. Next, it is necessary to define the differencebetween estimated and actual parameters. This definition is given by:

0−= θθθθθθθθθθθθ ˆ~

With this definition, the covariance matrix of the estimates is:

)~

E()~

cov( Tθθθθθθθθθθθθ ≡≡ P

Using the normal equations to define θθθθ and making the necessary substitutionsinto the expression given above, the covariance becomes:

TTT00

T

T1TT1T2 ])[(])[(

AZAAZA

ZAAAAZAAP

θθθθθθθθ+

−−= −−s

where:

1T11T1T )(])[()( −−−− += AAIQAAQAAZ

In this expression, the covariance matrix of the estimates is in fact a function ofthe actual values of the parameters that are unknown. This problem is obviouslyresolved by setting Q equal to zero. When this is done, the expression for thecovariance matrix which is used in the APC Identifier is:

1T2 )( −= AAP s

Noise variance is simply an indication of how the actual outputs vary about thepredicted outputs. Thus, the variance is estimated by:

�=

−−

=n

i

ii yyd

s1

22 )ˆ(N

1

This expression can be written in a more compact and efficient form as:

ds

−−=N

ˆ TTT2 yAyy θθθθ

Using the above expression, the final form of the covariance matrix is given by:

1TTTT ))(ˆ(N

1 −−−

= AAyAyyP θθθθd

;

Where N is the number of data samples and d is the number of estimated

parameters. Element P ,i j is the variance or standard error of the i th estimate



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when i j= and is the covariance of the ij th estimates when i j≠ .

Rather than using the inversion shown above to form the covariance matrix, TheAPC Identifier makes use of the following identity:

T111T )()( −−− = RRAA

With R known from the previously calculated Cholesky decomposition, R −1 canbe calculated in a trivial fashion since it is upper right triangular. Once thevariances are calculated, confidence limits and noise bounds for the estimates canbe determined as described below.

Confidence limits

and Noise bounds

Since the regressor matrix is deterministic and the disturbances are assumed to bewhite, the outputs are themselves random variables. This implies since theestimates are constructed from these variables, that they too are random variables.To determine the distribution of the estimates, it is further assumed that thedisturbances have a Gaussian distribution as N → ∞ . This assumption impliesthat the outputs and therefore the estimates will also have a Gaussian distributionas N → ∞ . Hence:

)(N~ˆ

0 P,∈ θθθθθθθθ

That is the estimates are normally distributed around the true values with varianceP. For sampled data sets, the above expression may be too restrictive. Thus ratherthan use the normal distribution, the APC identifier computes the Student Tdistribution. To generate the final distribution it will also be assumed that the

estimates can be independently parameterized. Thus the i th estimate will have thefollowing distribution:

)PT(ˆ, jiii ,θ∈θ 0

Note that the above expression will only be truly correct for a diagonalcovariance matrix. The implications of this assumptions will be discussed in afollowing section

For large N, the distribution of the estimate about the actual value has the familiarGaussian distribution shown below:



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In the above plot, the probability density function f is shown as a function of the

i th estimate. In the limit, as N → ∞ , the standard deviation or error is:

iii ,P)ˆS( ≡= θσ

Next, the normalized estimate is defined as follows:

)ˆS(

ˆ 0

i

iitθθθ −≡

Thus, the normalized estimate has the following distribution:

)1,0(T~∈t

Finally, for a two-sided distribution, the normalized estimate can be expected tolie between some upper and lower limit as shown below.

∗∗ ≤≤− ttt

In the above expression, ∗t corresponds to a user-specified probability orconfidence limit. As an example for large N, a 95.45% probability limit would

correspond to a ∗t of 2 (i.e. two standard deviations)



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With the above expression, the true value of the coefficient lies between a well-defined upper and lower bound as shown below.

iiiiii tt ,0 Pˆ)ˆS(ˆ ∗∗ +≤≤− θθθθ

Hence, a noise band can be defined as

iib t ,PN ∗±=

Finally, for the null hypothesis test, it is assumed that the true value of thecoefficient is in fact zero. This would imply that:

iii t ,Pˆ ∗≤θ

If the estimate is in fact larger than the bound, then the null hypothesis test fails

and the coefficient can be assumed to be non-null. In the Identifier ∗t isestablished based on the t-probability density function and is evaluated by

numerically solving the following expression for ∗t .

��

��

� +−

��

�

�

��

�

�+

Γ+Γ=

*

0

2

12

TT

1)2(

)2)1((2P~ t

dνν/

/νν

ν

π

Where

FunctionGamma

freedomofDegrees

95%)(i.e.levelyProbabilitP~

=Γ==

ν

Section 3 — APC Identifier Concepts3.8 Factorizations


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3.8 Factorizations

Background It is a simple fact that effective solutions require effective factorizations.Selection of the particular algorithm may at first seem like a simple matter.Experience has shown however that different requirements necessitate differentalgorithms. In the discussion to follow, attention will be focused only on the dataregression problem inherent in system identification. It will be assumed that thedata is of limited precision. In particular the design must accommodate singleprecision data. It will also be assumed that the data is reasonably scaled.

While it can be argued, that for system identification the conditioning of theregression matrix is to a large extent directly controllable through proper signaldesign, it is nonetheless highly desirable to accommodate poorly conditioned andrank deficient problems. Attention to proper factorization techniques will allowthe user to eliminate poor computation methods or inaccurate numerical schemesas the cause for poor models and focus for example on the design of propersignals for maximum information content. To this extent this section focuses onphilosophy rather than the details of the factorizations.

Clearly, the benchmark factorization technique is the Singular ValueDecomposition (SVD). Due to its numerical stability and revealing properties, ithas found widespread interest in a variety of fields. In many cases it is absolutelyrequired (i.e. when one or more singular values are desired). The interest herehowever is in detecting rank degeneracies and circumventing the difficulties theycause. For these problems SVD is almost never required (see the pertinentdiscussion in the LINPACK manual chapter 11).

While the SVD can be used effectively to solve the ID problem, it does requireconsiderable computational effort and it is certainly not the only method fordealing with poorly condition and/or rank deficient problems. Indeed, bothCholesky and QR factorizations can be equally effective when used withappropriate pivoting strategies and reliable condition estimators.

As presented previously, the Cholesky and QR algorithms are the basicfactorization techniques used in the APC Identifier. These rank revelingfactorizations utilize pivoting strategies (symmetric in the case of Cholesky) andreliable condition estimators. A discussion on basic performance characteristicsof these factorizations and comparison with SVD follows.

Normal vs.

Orthonormal

Solution of the quadratic norm problem, as presented previously, can be solvedby first forming the normal equations. These equations can then be solved usingan appropriate factorization. The quadratic norm problem can also be solved bydirect orthogonal factorization of the regression matrix. These solutions will bereferred to as the Normal and Orthonormal approaches respectively.



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Selection of the desired approach depends on many factors. If the discussion isrestricted to QR type of orthogonal factorizations, then indeed this is theinfamous “Normal equation Versus QR” debate described in Golub and VanLoan’s Matrix Computations and in Lawson and Hanson’s Solving Least SquaresProblems. This debate is summed up succinctly in the LINPACK manual“informed men of goodwill can disagree, even in a specific application”.

While many factors effect the final selection, sensitivity, accuracy andcomputational effort certainly play a dominant role. These factors will bediscussed briefly in the next few paragraphs.

Sensitivity andAccuracy

Accuracy and sensitivity are closely related. To discuss these topics, consider thefollowing two problems:

2

2

2

2)(ˆ)(

2

1minand

2

1min fbxEAbAx

xx+−+−

The first problem represents the true system. The second is a slightly perturbedproblem and the one that will actually be solved. The variables E and f are theerrors in A and b respectively. These errors may have a number of sources. If theelements of A and b have been measured, as is the case in identification, thenthey will be inaccurate due to the limitations of the measuring instruments. If theyhave been computed, then truncation or rounding errors will contaminate them.Even if A and b are known perfectly, they may not be perfectly representable ona digital computer.

While seemingly innocuous, this last statement is particularly important for theidentification problem. At this time the design must accommodate single-precision data acquisition devices. This implies an upper bound on the accuracyof the data and a lower bound on the size of E and f. This will impose limitationin spite of the calculation precision (double for the APC Identifier).

Subsequent computations performed on A and b can be considered as anothersource of initial error. For purposes of discussion these errors will also be lumpedinto E and f.

With this information at hand, the issues of accuracy and sensitivity can be morereasonably discussed. Accuracy in this discussion implies a solution to a givenproblem. Here the concern is the relative accuracy of the different factorizationswhen applied to the quadratic norm problem. Sensitivity on the other handaddresses the concern of error magnification on the solution. That is, for theproblems given above, what is the expected difference between x an x .

Accuracy and sensitivity are always related due to round-off errors in thecomputations. If the effects of round-off errors are neglected, then as described inchapter 19 of Lawson and Hanson’s Solving Least Squares Problems, thesolutions to a specific quadratic norm problem by SVD, QR and Cholesky areequivalent and hence have the same accuracy. Indeed, the R obtained by theorthogonal factorization of a given matrix with full column rank is identical



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(within the signs of rows) to the R obtained by the Cholesky factorization of thesame matrix. For a given arithmetic precision however the orthogonal approacheswill be more accurate than forming the normal equations and using Choleskyfactorization. This reduction in accuracy is not the result of the decompositionbut is a consequence of forming the normal equations.

With respect to accuracy, short word computations (low precision) would favorthe orthogonal approaches. With double precision computations any differencesin accuracy are effectively moot. For single precision data and double precisioncomputations, as is the case with the APC Identifier, there is no loss in accuracyincurred by forming the normal equations, hence the normal approach iseffectively as accurate as the orthonormal approach.

Sensitivity as described above is concerned with the possible magnification oferrors. Of particular concern is the extent to which the solution to a givenproblem can change as a result of errors or perturbations to the original problem.This concern can be addressed directly in terms of the condition numberassociated with the matrix A. When the condition number is large a matrix is saidto be ill-conditioned. A matrix can be ill-conditioned with respect to inversion. Amatrix can also be ill-conditioned with respect to its eigenproblem. It is possiblefor a matrix to be ill conditioned with respect to inversion but have a well-conditioned eigenproblem, and vice-versa. Here the concern is only the conditionnumber of A with respect to inversion.

Errors for a poorly conditioned problem will be greatly magnified in the solution.The 2-norm condition number is defined as:

)(

)()( 1

2

1

22 A

AAAA

nσσ

==κ −

Where σ ‘s are singular values of A. Thus the 2- norm condition number of Ameasures the elongation of the hyper-ellipsoid { }1:

2=xAx . Sensitivity

bounds of the various factorizations for the perturbation problem given above canbe stated in terms of the condition number κ. These bounds (taken from theLINPACK manual) are:

(SVD)lOrthonormaˆ

)()(ˆ

ˆ

(QR)lOrthonormaˆ

)()(ˆ

ˆ

(Cholesky)Normal)(ˆ

ˆ

2

2

22

22

2

2

2

2

22

22

2

2

2

22

2

2

A

E

xAAA

x

xx

A

E

xAAA

x

xx

A

EA

x

xx

2

2

2

��

�

��

� ρκ+κβ≤−

��

�

��

� ρκ+κα≤−

κ≤−

Where ρ is the length of the residual vector Ax-b and α and β are constantsgreater than one. Clearly, in general the sensitivity for all methods is proportional



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to the square of the condition number. It can also be seen that as the residualapproaches zero, sensitivity of the orthogonal methods becomes linearly relatedto the condition number while for the normal approach the sensitivity varies asthe square of the condition number. Hence as ρ approaches zero, the orthonormalmethods can be expected to be much less sensitive than the normal approach.Note however that strictly zero-norm residuals are academic with respect toidentification since this implies a perfect model with no external noise or acollocated model (A is n x n and b is n x 1) both of which are not at all realistic.

An Ill-conditionedexample

To illustrate more clearly the discussion on accuracy and sensitivity, a problemdefined in the MATLAB Control System Toolbox manual will be solved usingthe various factorization techniques. In this case it is desired to find the solutionto the standard quadratic norm problem given above where A and b are definedas follows.

��

��

�=�

�

��

�=

2540.

2170.and

6590.9130.

5630.7800bA

.

In this example the data can be considered to be single precision. Since thecomputations are performed in double precision, there will be no loss in accuracywhen the normal equations are formed. The known solution to this problem is:

��

��

�

−=

1

1x

Using MATLAB (with format long), A and b have the following representation:

The condition number of A is:



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This problem is clearly ill-conditioned. Since A has full column rank there is aunique solution. Since A is square this residual vector of this unique solution iszero length. That is Ax = b.

QR Solution Factorization of A using QR gives:

Consequently, the solution is bQxR T= . Let bQd T= , then



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Cholesky Solution For the Cholesky solution, bAdAAC TT and == . These values are:

In spite of the fact that 2AC )()( κ=κ , the Cholesky solution is no less accurate

than that obtained from the QR factorization. The condition number of C is:

While R from the Cholesky factorization is:

Solving dzR =T gives:



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Finally, solving zxR = gives:

It is obvious that the Cholesky factorization has suffered no loss in accuracy andin fact for this case yields the exact solution.



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SVD Solution Factorization of A using SVD gives:

Consequently, the solution is bUxSV TT = . Let bUdxVz TT and == , then

Since V is orthonormal, Vzx = and the solution is:

Inspection of the various solutions illustrates that the accuracy of both the QRand Cholesky approaches are comparable to that of the SVD.



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Sensitivity of Ill-conditionedProblem

To demonstrate the sensitivity of this problem consider the addition of a smallperturbation E to A. Let the perturbation be:

��

��

�

−−=

0010.0020.

0010.0010.E

Solution of the perturbed system for the QR, Cholesky, and SVD factorizationsrespectively are as follows.

Notice how the small perturbation was magnified in the solution. In this case, allmethods exhibit similar magnification (in spite of the fact that this is in essence azero residual solution). In general, the different approaches can have significantlydifferent magnification characteristics as illustrated by the magnification boundsgiven previously (these bounds express maximum magnification than can beexpected not necessarily the actual magnification obtained).

For model predictive control, where constraints may be active, it is possible thatat some time individual sub-models (CV/MV pairs) may totally dictate controllerperformance. Thus the true causal relationship between independent anddependent variables is desired. Therefore, it is good practice to never usesensitive models that are the result of poorly conditioned data. Attenuation ofsensitivity in this case is NOT recommended. Indeed, even using minimum length(minimum sensitivity) SVD solutions can result in FIR models are arbitrarily badrelative to control performance. Under the proper conditions gain reversal is apossibility. Proper excitation (rather than sensitivity attenuation) and hence well-conditioned data is the goal here.

It is also possible to have models that appear sensitive to perturbations even whenthe input signals are properly designed. Invariably this is the result of thedisturbance power being large relative to the input power. These models shouldalso be considered suspect as the disturbance characteristics become part of themodel.

Sensitivity problems have tremendously influenced the design of the APC



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Identifier. Indeed, it is because of these problems that the identifier supportsmultiple models for each input-output element. Here the idea is to create severalmodels based on slightly perturbed data sets and observe the sensitivity of themodels. This idea will be discussed in more detail in subsequent chapters.

At this point two questions need to be addressed. How ill conditioned does amatrix have to be before it is considered singular and can anything be done toattenuate or reduce this sensitivity to rank deficiency. These questions can beconveniently discussed in terms of the pseudorank concept.

Pseudorank In practical applications it is desirable to determine how close a given matrix is toa rank deficient or singular matrix. If it is determined that the given matrix is soclose to a rank deficient matrix that small perturbations will result in singularity,then some action should be taken. As a minimum the user should be informed. Inaddition, it is desirable to have a procedure that avoids the arbitrary instability ormagnification that occurs when a very ill conditioned problem is treated as beingfull rank.

To address these concerns, the pseudorank k of a matrix A can be defined to be

the rank of the matrix A~

that replaces A as a result of a particular procedure.Pseudorank is not a unique property of the matrix A, but depends on theunderlying computations, machine precision and the value of the toleranceparameters used in the computations. The latter is of particular importance in thisdiscussion.

In the previous example the true rank is 2. How close this problem is to singularis illustrated the second singular value, S(2,2). An indication is also given by theR(2,2) element of both the QR and Cholesky solutions. If the factorizations wereaccurate to only six places, then in fact the 2,2 elements would all beindistinguishable from zero. Thus A would be rank deficient. That is the

pseudorank of A is 1 and a rank one matrix A~

exists that is hopefully muchbetter conditioned with respect to inversion. Choice of the threshold fordetermining the pseudorank (how small is small) can be tricky and even problemdependent. If it is too big the wrong problem may be solved. In terms ofidentification this will result in a biased solution. If it is too small the solutionmaybe meaningless.

In many applications machine precision is used for the effective tolerance. Forexample, if the following test is true (1.0 + x == 1.0), then x is equal to zerowithin machine precision. This or similar tests could be used to define thepseudorank. In fact this is the approach used in MATLAB.

While this approach is reasonable in many applications, for system identification(which does not necessarily imply data fitting) it is imprudent to ignore the factthat there are inherent errors in the data itself. This implies bounds on theminimum size of E. As discussed previously, data acquisition limitations imposea single precision restriction on the data. Errors may in fact be larger but ingeneral they can not be assumed to be smaller. Hence, it is illogical to try and



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obtain a solution that is more accurate than the data upon which it is based.Therefore, the tolerance used in the APC Identifier for any data based regressionis defined as:

1and A∝λελ≡δ s

where sε is the single precision accuracy of the resident machine. While this

tolerance is used to define the pseudorank, both Cholesky and QR factorizationsare first computed to machine precision accuracy using pivoting strategies andreliable condition estimators. It is only after these factorizations are complete thatthe tolerance defined above is used to determine the pseudorank. The pseudorankk is determined by finding k+1, such that Rk+1,k+1 < δ. Final solutions are thenbased on the pseudorank k.

A Rank DeficientExample

To illustrate more clearly the discussion on pseudorank and its use in theidentifier, a problem defined in the MATLAB Users Guide will be solved usingthe various factorization techniques. In this case it is desired to minimize thestandard quadratic norm problem given above where A and b are defined asfollows.

��

�

�

��

�

�

=��

�

�

��

�

�

=

7

5

3

1

and

121110

987

654

321

bA

In this example the data can be considered to be single precision. Since thecomputations are performed in double precision, there will be no loss in accuracywhen the normal equations are formed. In this problem, the second column in Ais a linear combination of the first and third columns, therefore this is a rank 2matrix. Data has been selected such that the solution will result in a zero lengthresidual.

Because this is a rank deficient problem, column pivoting must be used in boththe Cholesky and QR factorizations. Results directly from the APC Identifier willbe used for the Cholesky discussion. By using A and b to construct input data andby selecting FIR models with only one coefficient, the solution given by theidentifier is:



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where m is the number of input data rows , order is the model order and x is thepermuted solution vector. The corresponding permutation vector is:

For this problem on this machine )6.1(is −δ eO and the corresponding

pseudorank is 2, which happens to be equal to the actual rank. Since A is rankdeficient, there are an infinite number of solutions. Based on the permutationelements, the Cholesky solution is:

��

�

�

��

�

�

=

00000000000000.0

66671666666666.0

00000000000000.0

00005000000000.0

x

This solution contains one more unknown than is found in the original problemstatement. This is simply due to the fact that the identifier automatically adds abias term in any data regressed FIR or PEM model (see section on modelstructure). The solution to the original problem is simply the first three elementsin the vector given above.

Factorization of A using QR gives:

where P is the permutation matrix resulting from the pivot calculations (In theIdentifier all permutations are saved in the JPVT array to save storage). With thefactorization complete it is easy to establish the pseudorank. As discussedpreviously, simply find k+1, such that Rk+1,k+1 < δ. Here δ can be taken to be 1.e-



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6. Thus the pseudo rank is 2 which happens to be the actual rank.

With the QR factorization there are two approaches for obtaining a solution torank deficient problems. One is the minimum length solution the other is whatwill be referred to as the zero value solution.

Zero ValueSolution

For the zero valued solution, which is typically implied when performing QR

factorization, the solution in terms of the permuted variable px is dxR =p .

Here, the dimensionality is defined in terms of the pseudorank k and 0=pix for

ki > . In addition bQd T= . The final solution is recovered by pPxx = . Theperturbed solution can easily be recovered by using only the upper k triangularportion of R and corresponding elements of d. In this instance the dimension, k is2 and the solution is obtained as follows:

and the unpermuted solution is recovered as:

Which is identical to the Cholesky solution given by the Identifier. Thesesolutions clearly satisfy the original problem within working tolerance and resultin a residual with zero length since:

In these solutions the number of non-zero coefficients is equal to the rank of theproblem. For rank deficient problems, a minimum length solution that also



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minimizes the error exists. This minimum length solution can be recovered fromthe QR factorization in a straightforward fashion.

Minimum Norm

Minimum length

QR Solution

Once the QR factorization is complete, the minimum length solution can beaccomplished by applying elementary right rotations to R based on thepseudorank of A. The objective is to simply annihilate the first n elements of thelast n-k columns of R. While elementary givens rotations are used here, anyconvenient orthonormal transformation can be used.

For this problem the first three elements of the last column need to be annihilated.For purposes of discussion, let the right rotation matrix to accomplish this be K.

Then, ∗= RRK and ∗= Kxx~ . The minimum length solution is obtained by

solving dxR =∗∗ . Thus ∗R is:

Solving for ∗x gives:

Thus the minimum length solution is ∗= Kxx~ or for this problem

Finally using the permutation matrix the solution can be written as xPx ~= or:



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MATLAB

Solutions

Solutions to this problem using standard MATLAB functions can beaccomplished as follows. First the Zero value solution is given by:

Next the minimum error solution can be obtained using SVD by invoking thePINV function as follows:

Both the zero value and minimum length solutions are essentially identical tothose given previously. It should be obvious that the MATLAB left divisionoperation results in what is referred to here a zero value solution, whileMATLAB’s pseudo-inverse function gives a minimum length solution.

Choice of which approach to use depends on the application. For MPCidentification using FIR models, there is no advantage to the minimum lengthsolution. Indeed as described previously this does not insure that the models areeven useful for control purposes. There is however a computational penalty. Inaddition it could be argued that length minimization can result in a deleteriouseffect since minimum length solutions will distribute coefficient effects overlinear dependent columns. For example if truly first order data is fit with a thirdorder model, all parameters will appear to be pertinent. Conversely, the zerovalue approach will discard non-impactive parameters. Currently, the APCIdentifier returns only zero valued solutions.



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Perturbed

Solution and

Pseudorank

Next consider the same problem subject to a small perturbation. In this case A isas defined previously but E has random errors as defined below.

For this perturbed problem R becomes:

Note that the R(3,3) element falls below the pseudorank threshold and istherefore treated a being indistinguishable from zero. The pseudorank for thisperturbed problem is still 2 in spite of the fact that that the true rank is 3. Using

2=k , the solution becomes:

Using MATLAB left division, the solution is:

Clearly, it is reasonable to account for the inherent limitations in the data. To thisextent the sensitivity can be somewhat attenuated. For identification, furtherattenuation or suppression in sensitivity by arbitrarily increasing the pseudoranktolerance is ill advised since this will lead to biased solutions or suppress inherentproblem in the data of which the user should be aware. In the Identifier, thesesensitivities are purposely displayed to indicate potential concerns withinformation content in the data.



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Timing At the data regression level, primary attention has been focused on accuracy,numerical stability and the ability to deal directly with rank deficient problems asdescribed above. With this as a requirement, some care has also been expended toensure that the delivered algorithms are reasonably fast. The delivered algorithmshave been tested in comparison with the de-facto standard MATLAB. In all casesthe computational speed was found to be comparable. This comparison was by nomeans meant to be comprehensive but simply a reasonableness check. Allcomputations were performed on the same Pentium II 366 MHz machine.

For the FIR calculations, the rank revealing Cholesky factorization routine usedin the APC Identifier was slightly faster than the chol routine used inMATLAB. Chol was approximately 30% slower than the Identifierfactorization. A 960x960 matrix takes about 9 seconds to factor using theIdentifier. A direct comparison on the formulation of the normal equations is notreally meaningful since the Identifier uses a fast correlation update. Whileintrinsic MATLAB functions are very fast, script (.m) files are not. Nevertheless,MATLAB can be used as a sanity check based on the theoretical number offloating point operations (like MATLAB, all computations in the Identifier are

double precision). Consider a 6000x720 A matrix. Using MATLAB AA T

takes approximately 410 seconds. While this full formulation makes use ofneither the symmetry nor the Toeplitz structure of the problem, the fastcorrelation update does. The operation count using this approach is a function ofthe rows and columns in A and the number of independent variables.Formulation of the normal equations in the Identifier for a 6000x720 matrix takesless than 2 seconds for 1 independent variable and less than 4 seconds for 6independent variables.

This computational speed is reasonable relative to the theoretical operation count.There is some overhead that could be reduced but the effort is hard to justifyconsidering the existing performance. Overhead is due to the interactive design(interrupts allow messaging and user intervention while the computations are inprogress), the nested indexing used to implement the fast correlation update andthe support of matrix segmentation. Only the latter can be influenced by the user.In the Identifier, the parameter, UserMemABuf, defines the maximum size ofthe A matrix. If the actual A matrix requires more than this amount of memory,then A is partitioned and the normal equations are formed looping over thesegmented A .

As a final case, consider a problem where 30 independent variables are moved ina simultaneous fashion. Let there be 6000 rows in A and let there be 30

coefficients for each independent variable. The dimensions of A and AA T are6000x900 and 900x900 respectively. Using the default settings the total time toobtain an FIR solution for this problem using the Identifier is less than 24seconds. It takes approximately 16 seconds to form the normal equations. IfUserMemABuf is increased so no partitioning occurs than it takes only 11seconds to form the normal equations and the total solution time is less than 18



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seconds.

Note that as long as the matrix if full, the cost to compute models for additionalCV’s is essentially undetectable relative to the other computations. This examplein no way implies that it is recommended to include all possible independentvariables into a single regression. In fact quite the opposite is usually true. It isusually very poor practice to simply include all variables and “see what happens”.Most often it is more effective to do block testing where the specifiedindependent variables have been designed to maximize information content in thedata. Large systems can easily be constructed simply by combining smaller sub-systems.

For PEM calculations there is almost no need to report timing. If the goal isultimate speed, then this is indeed a poor choice. PEM calculations are very slowrelative to any FIR calculations. For a reasonable class of problems however,solutions can certainly be obtained in a respectable amount of time. For theintended applications, computations should be less than 10 seconds per CV.

With this model form, the only comparisons made were for the updatecalculations. That is the time required to calculate a full Gauss-Newton step. Forthe QR update, direct comparison of the solution implemented in the Identifierand the qr MATLAB routine is not really meaningful since MATLAB’s qrroutine physically forms Q while in the Identifier only a compact representationof the Q factors are used. Thus even using the economy form, MATLAB’s qrroutine would be expected to be relatively slow. Hence, the comparison will bemade between the Identifier and MATLAB’s left division operation. For thisproblem the Jacobian is taken to be 6000x280 and the computations speeds arecomparable. Here, MATLAB is slightly faster than the Identifier. The Identifiertakes about 27-28 seconds to perform the update while MATALB takes about 22-24 seconds for the equivalent computations. Any further reduction in overheadassociated with the QR algorithm in the Identifier is difficult to justifyconsidering the stated performance. Note however, if the Cholesky option in theAPC Identifier is used, the Gauss-Newton step for the same problem takes lessthan 2 seconds to compute.

Section 3 — APC Identifier Concepts3.9 Summary


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3.9 SummaryGuidance for the technical discussion is based on the desire to present acompletely open description of the Identifier. In many instances closed or blackbox type of algorithms are undesirable from a technical perspective. For many thedetails may be more than are necessary. For some however the detaileddiscussion will help to provide a deeper understanding of the fundamentaloperation of the algorithm and hopefully address some basic issues that are oftenmisunderstood.

In the initial portion of this section, the discussion was limited to a generaloverview describing the hybrid approach. A more detailed technical descriptionwas then given with respect the general identification problem, the variousmodels available and the corresponding solution techniques. It was pointed outthat both robust and quadratic norms are supported, the former being availableonly with PEM models. Both PEM and FIR models can be used for dataregression. PEM models should be considered to complement the FIR modelsand are provided for increased ease of use. The intent is “one step” identificationin instances where there are only a few independent variables movingsimultaneously. If these models are satisfactory, move on. Otherwise use FIRmodels.

Model forms used for order reduction and variance attenuation were describednext. These include ARX, which uses a prefiltering algorithm, fixed-formLaplace and output error. The solutions used for all of the various models werethen discussed illustrating the pertinent features. Once the solutions werepresented, properties of the FIR and PEM models were delineated showing thatboth models have a sound theoretical basis. Under the stated conditions, FIRmodels, as used here, are unbiased and consistent. PEM models, when theyconverge, are consistent even in the closed loop and are minimum variance if thenoise is Gaussian. Both bias and variance effects were also discussed. Techniquesfor quantitative evaluation of FIR models were then outlined. These topicsincluded correlation, confidence limits, null hypothesis tests and statisticalranking.

Finally, the factorizations used in the various solutions were described. Exampleswere given showing the salient features of both the Cholesky and QRfactorizations. A significant portion of the discussion was focused on the ill-conditioned/rank deficient problem. For properly designed experiments, theseconditions should seldom be encountered. Nevertheless, they always remain apossibility. The intent of the factorization discussion was to convey the fact thatthe solution algorithms are numerically robust. If poor models are obtained it isnot the result of numerical problems. Consequently, switching to an alternatefactorization approach will not result in more reliable models (at least not in astatistically meaningful sense)

Section 3 — APC Identifier Concepts3.9 Summary


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FuturePerspective

Identification is a relatively involved topic and it can take many forms. Search forthe ultimate identifier can be a lifelong endeavor. These issues were known at theonset in the design of the APC Identifier. While there are some existingapproaches that are effective, there are always promising new techniques thatneed to be evaluated. Clearly any tool needs to continually evolve. This one is nodifferent.

Open-loop identification of stable and integrating process has been the objectivein the design and development of the APC Identifier. At the onset, the goal was tohave a tool that would be effective, fast, numerically stable and provideinteractive ease of use even for the relative novice. There certainly are otherapproaches, some more elegant, some less. The simple fact remains that this oneworks and has proven effective for the intended applications. This in no wayhowever, implies completion. Indeed as other approaches are proven moreeffective they need to be either integrated or used to replace existingfunctionality.

It is the intent to provide at least three new quantitative indicators: noise andconfidence bounds on the PEM models, uncertainty spectrum on the final Laplacemodels and cross correlation plots of the residuals.

It is also the intent to continuously evaluate promising techniques. Of particularinterest are canonical subspace approaches. They offer minimal order minimumvariance models, can be used in the closed loop and don’t require a nonlinearsearch. Issues on delay and short data sets may however be of concern. Also ofinterest are design approaches for poorly conditioned plants. Much of theliterature is focused on zero frequency design. Extension to mid frequencieswould aid greatly in control relevant identification.

Another enormously interesting area is closed loop identification. Much appearsin the technical literature. Yet practical success has proven to be an elusiveendeavor. A word of caution is therefore warranted regarding this topic.Irrespective of the approach, there are the so-called Identifiability conditions thatsimply can not be ignored (see foe example Ljung’s System Identification Theoryfor the User or Larimore and Seborg’s Automated Multivariable SystemIdentification: Basic principles with Control and Monitoring Applications). Evenwhen the Identifiability conditions are satisfied, the variance errors will still bedictated by the expression given previously in the discussion on bias andvariance.


Section 4 — Getting Started - The Identification Environment

4.1 Overview

In This SectionThis section explains how to read input files and begin the modelidentification process. Read this section to find out about:

• Starting an identification session

• Opening data files

• How to Create/Save model files or load existing files

• Loading Non-native files

• Hierarchical Overview

Profit Design Studio

(APCDE)

To invoke Profit Design Studio (APCDE), either click on the APC iconor double click on the APCDE32.exe file. The about dialog box shown inSection 1 is displayed illustrating the current configuration. When thedevelopment environment is launched, an APCDE32.log file isautomatically created (rewritten if one already exists). If the operatingsystem is NT, the file is placed in the WinNT directory. If the operatingsystem is WIN95, the file is placed in the WINDOWS directory. This.log file contains Profit Design Studio (APCDE) version compatibilityinformation. Any problems associated with incorrect versions etc. issummarized in this file.

Once the correct configuration has been established, the about dialog boxcan be closed. At this point Profit Design Studio (APCDE) can be used toperform any of the configured functions. An empty environment appearsas shown below.

As shown above, the environment contains an main menu, a toolbar, aresource or window area and a status bar. The toolbar contains shortcutbuttons of commonly used functions. The status bar will display promptsof any selected menu items. Both the toolbar and status bar can be turnedon or off using the appropriate View menu option.

Section 4 — Getting Started - The Identification Environment4.2 Starting an Identification Session

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4.2 Starting an Identification SessionFrom an empty environment there are essentially two options available from themain menu. The first is the File option. The other is the Tools option. Thefollowing tools are supported by Profit Design Studio (APCDE). Newer releasesmay have additional tools.

Only the Model Converter and TDC Data Converter are associated specificallywith the APC Identifier (the Point Builder is used in conjunction with ProfitController (RMPCT) design components of the Profit Design Studio (APCDE)).

The converter tools provide a mechanism for creating files that can be directlyimported into the Profit Design Studio (APCDE). Note, use of these tools does notresult in the start of an identification session. Rather, it creates one or more filesthat can be used as input for identification.

Several files are associated with the APC Identifier. A description of these filesand their corresponding extensions are shown in the following table.

File Types and

File Extensions

FileExtension File Description File Type

MPT Multiple point data file ASCII

PNT Single point data file ASCII

MDL APC MIMO model file Binary

PID APC MISO model file Binary

FIR Externally generated file of FIR models ASCII

XFR Externally generated file of transfer functions ASCII

INF APC message file ASCII

Section 4 — Getting Started - The Identification Environment4.2 Starting an Identification Session


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These various files are described in more detail in following sections.

To begin an identification session, select File. The following choices areavailable

Selecting File > New at this level results in the creation of an empty document(an empty document assumes that either data is not available and the user isgoing to enter all pertinent information by hand or the user is going to “merge”information into it from one or more existing files). The user can specify thetype of document to be created by selecting from the New dialog box shownbelow.

Only Model Dev.File and PID Dev.File are associated with an identificationsession. Each identification session is automatically associated with a specificdocument or file having the .mdl or .pid extension respectively. Any .mdl filecontains all the information necessary to represent a general Multiple-InputMultiple-Output (MIMO) identification session. While any .pid file contains allthe information necessary to represent a Multiple-Input Single-Output (MISO)identification session.

In addition to the File>New option, an identification session can also be startedby selecting File>Open. The environment displayed to the user depends directlyon the type of file or document that is opened.

Depending on the procedure followed, the options can be used to eitheropen/create an .mdl file (also referred to as an Profit Controller (RMPCT)model file) or a .pid file (also referred to as an Profit PID (RPID) model file) Adiscussion on creating/opening these files is given in the following sections.

Section 4 — Getting Started - The Identification Environment4.3 Creating a Profit Controller (RMPCT) Model File


4.3 Creating a Profit Controller (RMPCT) Model FileCreating anRMPCT Model File

To create a Profit Controller (RMPCT) model document select either File>Openor File>New from the main menu as shown below.

This allows the specification of the proper document type. To open a file, selectFile>Open, then select the desired directory. From the pull down list, choose theextension of the files that you want to display as shown below.



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This dialog box displays all files that can be read into Profit Design Studio(APCDE). This list expands as new elements are added. If a document type isselected whose functions have not been installed on the host computer, then amessage is displayed indicating that the document could not be opened.

From the file list, select the desired file. To open an existing model file, select.mdl. Selecting either .pnt or .mpt implies that a new .mdl file is created from rawdata.

Choose whether the .mdl file is based on raw data from .pnt or .mpt files.Data Source - DataFiles

File Type Action

.mpt Select a multi-point (.mpt) file to start from scratch with the variablesand test data in that file (as shown in the above figure).

.pnt Select any number (up to 2k total characters) of single-point (.pnt)files to start from scratch with the variables and data in all theselected files. The information is collected into a single model file(.mdl).

To select a single file, click on the file name in the file name box.

To select additional files, hold <CTRL> and click on the file names(<CTRL> toggles the selection state).

To select all files in a range, drag the cursor over the range of filenames. Or click on the first name and then hold <SHIFT> and clickon the last name.

At this point selected data is read into Profit Design Studio (APCDE) and anRMPCT identification document is opened. If a .mpt file was read in, thedocument is titled Filename.mdl (from .mpt), where Filename is the name of the.mpt file. If .pnt files are read in, the document is titled model*.mdl (from .pnt).An example of an .mpt file is given below.



After the file is saved, the (from .mpt) or (from .pnt) descriptor is no longerdisplayed in the title bar. With data loaded into the .mdl document, identificationmay begin. See Sections 4-7 for detailed description on the identificationprocedure.

At this stage the .mdl document can be saved for later use by selecting File>Save(or Save As) from the main menu or it can be used to begin the identification

Data Source -Manually Entered

To create models without data, choose File>New from the main menu. SelectModel Dev.File to create an .mdl file as shown below.

Choose Edit> Var Info to begin entering descriptive information about the model.Entering orChanging VariableInformation

A dialog box will appear allowing the entry of and changes to information foreach variable.



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• Use the Type field to indicate CV, MV, or DV.

• Use the Name field to name the variable. If you do not enter a name thePoint field is used. At least one CV and one MV must be entered toproceed with the identification. More will be said about this dialog boxin later sections

• A unique name is required for each variable. The unique name isestablished by the Point.Param name. If both the point and parameterfields are empty then the Name field determines the unique name. Theuser is prevented from entering a non-unique name.

• The Variable Info dialog box can be used to change existing variableinformation and if there is no raw data present to add new variables tothe work space.

• As a Variable is chosen using the next previous buttons, thecorresponding variable in the Descriptive Info view will automaticallybe highlighted.

• If any variables are selected in the Descriptive Info view prior toinvoking the Variable Info dialog box, only this subset of variables willbe accessed as the next and previous buttons are selected. The selectionstate displayed in the Descriptive Info view will temporarily be modifiedto highlight only the current variable of interest. When the dialog box isclosed the original selection states will be recovered.

• If no raw data is present, then the next button will eventually access theend of the variable list which will be reflected in the Descriptive Infoview as a highlighted empty row. This will result in an empty dialog boxsuch as that shown above. In this state a new variable will be added once



the pertinent information is entered and the OK, Next or Previous buttonis selected. Note that if any variables were selected prior to invoking thedialog box, all newly created variables will be automatically be selectedwhen the dialog box is closed.

• To modify information on a single variable simply double click on thedesired variable. Note that the Next and Previous buttons will bedisabled

After the pertinent information has been entered, switch to the Model Summaryview. Note that there are no models available at this time. To manually entermodels, double click in the empty grid area to bring up the Transfer Functiondialog box. From here proceed as described in sections 6 and 7.

Section 4 — Getting Started - The Identification Environment4.4 Creating a Robust PID Model File


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4.4 Creating a Robust PID Model FileCreating an RPIDModel File

To create an RPID model, it is necessary to select File>New from the main menuas shown below.

This allows the specification of the proper document type. Selection ofFile>Open at this point, if data were to be used to generate models, wouldincorrectly result in the creation of a MIMO .mdl document and the RPID designand simulation functions would not be available in the session. Select File>Newand PID Dev.File as shown below.

If PID Dev. File is selected and if the appropriate library has been installed, thenthe dialog box illustrated below appears.



Data Source - DataFiles

You must now choose whether your Robust PID file is based on raw data from.pnt or .mpt files or whether you want to manually enter the transfer function.Selecting the Data Files radio button and clicking on [OK] results in thefollowing dialog box.

You may select either .mpt or .pnt files as long as the total number of variablesare limited to one CV, one MV and up to 10 DVs.

• To select a single file, click on the file name in the file name box.

• To select additional files, hold <CTRL> and click on the file names(<CTRL> toggles the selection state).

• To select all files in a range, click on the first name and then hold<SHIFT> and click on the last name.

Click [Open]

At this point selected data is read into Profit Design Studio (APCDE) and a PIDidentification document titled PIDDev*.pid is opened as illustrated below.

With data loaded into the proper document (.pid), identification may begin. Seesections 4-7 for detailed description on the identification procedure.



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Data Source -Manually Entered

If you prefer to enter your transfer function manually choose the data sourceManually Entered instead of Data Files as indicated below.

Now you have a window representing your empty PID model file. Proceed asdescribed above for the empty document

Section 4 — Getting Started - The Identification Environment4.5 Reading in Data


4.5 Reading in DataGetting Test Data The starting point for identification of a multivariable process model is a file that

contains test data obtained from the process.

Test data consists of sampled values, for the independent and dependentvariables, taken over a period during which the independent variables are excitedby a test signal. The Identifier can read in test data from files having data fromone point, or from multiple points. The different data file types are describedbelow.

Single Point DataFiles

Single point files (which must have a .pnt extension) contain sampled values foronly one point. The first six records are header information for this point.

• The first record contains the point tagname.

• The second record is a general description of the point.

• The third record contains the engineering units of the point.

• The fourth record is the sample rate at which the data was taken.

• The fifth record contains the beginning time stamp marking the start of thedata record (Month/Day/Year Hours/Minutes/Seconds).

• The sixth record shows the point category (this can be either manipulated,controlled or disturbance).

Header information is followed by the actual data. One sample value per record isstored for as long as the test is run. Data not to be included in the analysis isentered as an NaN.

Single Point Data—An Example File

This is what data looks like in a .pnt file:

01AD654A.PV

JET_FLASH

DEGC

0.2

12/25/98 08/54/30

CONTROLLED

0.0000000e+00

1.2572979e-01

1.5274251e-01

NaN

9.6721695e-02



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Multiple Point DataFile

Multi-point files (which must have an .mpt extension) contain sampled values formultiple points. These files are created by the AM Data Collector and contain onevariable per column (each column is eight characters wide), and each column isseparated by a blank.

The first nine rows contain header information:

• Rows one and two allow for sixteen character tagnames for each variable

• Row three contains the parameter (OP, SP, PV, etc.) of the point

• Rows four, five, and six are used for a twenty four character pointdescription

• Row seven contains the engineering units

• Row eight contains the starting time stamp(Month-Day-Year)

• Row nine contains the point category.

This information is followed by rows of the actual data. Each subsequent rowcorresponds to a data sample. At the end of each row there is a blank characterfollowed by the time stamp of the data sample. This time information is used todetermine the sample interval.

If sample intervals are not consistent then the following message will bedisplayed.

Expected sample rates are calculated based on a frequency-bin approach. Samplerates with the highest number of occurrences are taken to be the expected value.Within a given file a slightly inconsistent sample time will have little impact.However data can not be merged when sample rates exceed a user specifiedtolerance. Merging data and models will be described in a subsequent section.

Bad data or data that should not be used in the analysis is entered as an NaN.



Multiple PointData—An ExampleFile

This is what data looks like in an .mpt file:

UTC26034 UFC26034 TC26371 PC26220

OP OP OP OP

DEBUT TE DEBUT IN DEBUT BO DEBUT PR

MP REFLU TL REFLU TTOM TEM ESSURE C

X CONTRL X CONTRL P CNTRL ONTROL

DEG F BPD DEG F PSIG

06-25-92 06-25-92 06-25-92 06-25-92

CO CO MA MA

0.00000 0.00000 0.000000 0.000000 09:46:08

0.00000 0.00000 1.000000 0.000000 09:46:38

1.26424 .316060 1.000000 0.000000 09:47:08

1.72932 NaN 1.000000 0.000000 09:47:38

1.90042 .475106 1.000000 0.000000 09:48:08

Saving an .mdl or.pid File

By selecting File>Save, Profit Design Studio (APCDE) creates a permanent copyof an appropriate .mdl or .pid model file depending on the current environment.The file is saved in the appropriate directory. All information related to the modeland its development is saved in these files.

You can save an .mdl or .pid file at any time and open it later to continue whereyou left off. Profit Design Studio (APCDE) always saves your work to theappropriate model file— Profit Design Studio (APCDE) never overwrites a rawdata file.

There are three ways to save the file:

• Select File>Save.

• Click the toolbar button that looks like a diskette (same as File>Save).

• Select File>Save As.

Section 4 — Getting Started - The Identification Environment4.6 Reading Model Files Created by Other Applications


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4.6 Reading Model Files Created by Other Applications

What the Identifier

Expects

The Identifier can read finite impulse response (FIR) models and transferfunction models created by other applications, as long as:

• The files are ASCII files

• The data is expressed in the expected format

• The file has the necessary file extension.

The Identifier needs to build the controller models, so it does not work with thesemodel forms from other applications.

Non Native FIR

Files

FIR model files created by other applications need an FIR extension, with thedata given in a single column, in this order:

• Number of CVs

• Number of MVs

• Number of DVs

• For each sub-model

(CV1-MV1, CV1-MV2, . . . CV2-MV1, CV2-MV2 . . .):

Number of FIR coefficients (Ø for a null model)

Discrete sample rate

Integrator flag (1=integrator; otherwise Ø)

• For each sub-model

(CV1-MV1, CV1-MV2, . . . CV2-MV1, CV2-MV2 . . .):

FIR coefficients (for null models make no entry — go on to the next model,leaving no blank lines)

• For each CV, MV, and DV:

Tagname

Engineering units.

Sample File See Appendix A for a sample file.

Section 4 — Getting Started - The Identification Environment4.6 Reading Model Files Created by Other Applications


Non Native XFRFiles

Transfer function model files created by other applications need an XFRextension, with the data given in a single column, in this order:

• Number of CVs

• Number of MVs

• Number of DVs

• -1 for sample rate, Laplace form

• For each CV (CVi-MV1, CVi-MV2, . . . CVi-MVn):

Number of numerator coefficients (Ø for a null model)

Number of denominator coefficients (Ø for a null model)

Transport delay in minutes (Ø for no delay)

• For each CV (CVi-MV1, CVi-MV2, . . . CVi-MVn):

Numerator coefficients for each polynomial in CV row (for null models make noentry — go on to the next model, leaving no blank lines)

Denominator coefficients for each polynomial in CV row (for null models makeno entry — go on to the next model, leaving no blank lines)

• For each CV, MV, and DV:

Tagname

Engineering units.

Sample See Appendix B for Sample file.

Section 4 — Getting Started - The Identification Environment4.7 Hierarchical Overview


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4.7 Hierarchical OverviewProfit Design Studio (APCDE) fully utilizes Microsoft’s Multiple DocumentInterface (MDI). As such the Studio enables the modification, manipulation andviewing of multiple documents simultaneously. In addition multiple views(windows) on a single document are also fully supported. Documents, in theMicrosoft sense, are associated with specific file types. The files associated withidentification were described previously. Each file has an associated main menubar. The menu bar is an interface that allows the user to select functions that aregermane to a particular task (such as identification).

One file that is not associated with any document or menu is the apcde.ini file.This file is used to tailor the Studio itself. Overviews of the APC Identifier’smain menu and relevant apcde.ini parameters are described in the next fewparagraphs. Detailed descriptions and examples will follow in subsequentchapters.

Identifier Main Menu Once an identification session has been started, the main menu will indicate allavailable high level functions that are available to the user. This menu has beenarranged in a fashion that reflects progression through a typical identificationsession. Starting at the left and progressing right as functions are completed. Thefunctions contained on the main menu are illustrated below.

In many cases, this same philosophy is used once a given item is selected. Herehowever the progression is usually from top to bottom. User options are alsomade available in a logical fashion. Access to commonly changed parameters isprovided through high-level dialog boxes. Subsequent dialog boxes can be usedto access parameters that are less frequently used.

An overview of the various menu items given above is as follows:

• File These options have already been discussed

• Edit Typical cut, copy, paste and delete type functions are availableusing this item. In addition this function also provides a shortcut to the dialogbox used to edit information associated with any variable. Access to thefunctions in this menu is highly dependent on the states of the application, thecurrent view, selection status and past events. Edit options are shown below.The cut, copy, paste and delete items can be applied to either data or acombination of data and models. These operations, to be described in detail in adedicated section, can be used to move or rearrange data and models within agiven document or to merge data and/or models into one or more documents.The copy paste functions can also be performed using the standard drag anddrop operations. Select All can be used as a short cut to select all variables ormodels depending on the current view. The procedures designated as “Special”are dedicated to operations on sub models within a given matrix. Only one



model at a time can be modified using copy and paste. Delete will work with anynumber of selected models. The User, Final, Uniform and Mixed options are forcopying results associated with different selection strategies (details will begiven in a later section). The CopyRegr2Pred and CopyPred2Reg options are forcopying selection ranges between FIR/PEM and prediction ranges.

The item below the separator bar can be used to invoke the dialog box thatsupports the editing of descriptive information about each of the variables.

• Insert Special marks that can be used to designate bad data or datathat should not be used for certain operations can be inserted or removed usingthis option as shown below.

In certain views data can be marked using special designators. These marks cansubsequently be removed. In addition, these marks can be displayed or notdepending on the user’s preference. Marking and unmarking of data can beaccomplished in a more convenient fashion using the dedicated toolbar buttonsdiscussed in a subsequent section.



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• Data Operations All manipulations to be performed on the data withthe exception of the cut, copy, paste and delete functions can be accessedthrough this menu option. The following pull down menu contains the dataoptions.

By using block manipulations, all selected variables can be manipulatedsimultaneously. Multiple ranges of selected variables can be easily modifiedusing a host of options in an interactive fashion. Undo options eliminatepotential problems

Detailed calculation applied to a single variable can be performed using theVector Calculation function. Functions include transformations, filters, statistics,manual editing, outlier detection and the ability to combine multiple variables.Operations can be stacked with source and destination variables automaticallyrecovered

• Views Different views let you display different information about youridentification—data plots for different ranges, model trials, normalized scaling,zoom and many other options. The fundamental view options are obtained byselecting View from the main menu as shown below.



Views preceding the first separator bar correspond to any information associatedwith the data. The next group of views pertains to the various models and thedifferent ways they can be presented. This group is followed by views thatpertain to qualitative and quantitative indicators that can be used to help assesdata and model quality. The next group can be used to configure how variousdata is displayed. Finally, the last group can be used to enable/disable thetoolbar and status bar respectively.

• Identify This menu option is used to perform all the functions associatedwith identification. It also supports overall setup and Load & Go operations.Selecting Identify from the main menu gives:

Typically, operations are performed top to bottom. The hybrid approachoutlined in the concepts chapter and described in detail in subsequent chapters,is supported by the second, third and forth options which perform the regressioncalculations, model reductions and model selection respectively. The finaloption performs these three steps automatically for the current selection state ofthe environment.

• Build Creation of a Profit Controller (RMPCT) or Profit PID (RPID) isaccomplished using the build option. Selecting Build gives:

In addition to building a controller, this option also supports the building of aprocess simulator which can be used for initial tuning and evaluation. When thebuild option is invoked from a .mdl environment, a Profit Controller is created.Conversely, when the build option is invoked from a .pid environment, a ProfitPID Controller is created.

• Tools Several tool functions are available during an identificationsession. In addition to those available in the empty design studio (discussedat the beginning of this section), two more options are available when asession is open. These options are shown below.



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Vector Calculations invokes the same function as that described previously inthe Data Operation pull down menu. As long as there is data available, thisfunction can always be used as a tool option irrespective of the current view oroperation. When there is no data present, Vector Calculations will support onlythe transformation options. Selecting this function from the Tools menu willautomatically switch views to the one associated with the Vector operation. Ifthe current view is not a model or data view, then the Vector Calculationfunction will not be enabled in the Data Operation menu.

Profit Toolkit allows the user to configure and evaluate a variety of selectablefunctions. See the Profit toolkit user’s guide for a full description on thisfunction.

Preferences Only two choices are currently available under thepreference menu. One is the color option. This allows the user to tailor thecolors as displayed in the identification environment. Select Preference>Colorsto obtain the following options.

At times other applications may interact with the color palette in an undesirablefashion. Select Default to correct this problem.



The other preference option is the choice of toolbar. Here, either the standard ordetailed identification (ID) toolbar may be chosen. The standard ID toolbar hasthe following form.

While the detailed ID toolbar has the additional buttons as shown below.

Toolbars can be docked/undocked or floated in the standard fashion. They mayalso be turned on or off as desired. The standard ID toolbar is the default. Thesepreferences are saved such that if detailed is chosen and the next identificationapplication is opened, then it will display the detailed ID toolbar. Meanings ofthe individual buttons are summarized below but the general function isdescribed in detail in the pertinent section of this document.

– Set Overall Options.

– Load & Go.

– Fit FIR/PEM Models.

– Fit Parametric Models (Fit parametric S – domain models to FIR/PEM

Z – domain models).

– Find (select) Final Models.

– User Models (selected in Model Summary View) to Final Models

– Clear (delete) Selected Sub-models.

– Plot Predictions for Final Models.

– Data Vector Operation.

– Mark Data Bad.

– Unmark Data Bad.

– Show/Hide Bad Data marks.

– Build Controller.

– Build Simulator.



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• Window and Help These final two selections complete the mainmenu choices.

New Window, Cascade and Tile under the Windows menu refer to theconventional options in a standard application with a multi-document interface.The Arrange Icons option, while enabled, performs no meaningful operation inthe current release and can be considered reserved for future releases.

Currently, the help menu only supports the About Profit Design Studio function.Fully integrated on-line help is planed for future releases

Keyboard Selection Any menu item can be accessed either by using the mouse or by direct keyboardaccess. To access a menu item via keyboard select <Alt * $>, where * is theunderlined character in the main menu (selecting <Alt *> will cause the menudrop down options to be displayed) and & is the underlined character in thedropdown menu. For example to save the current file select <Alt F S>.Character selection is NOT case sensitive.



APCDE32.INI Parameters that can be adjusted to alter the identification configuration are listedbelow.

[Memory Buffer]

SwapMode=0

UserMemBuf=32

UserMemABuf=8

[UserOptions]

FIRDefault=1

PositionalForm=1

StartSettleT=60

DeltaSettleT=30

FirNumCoeff=30

ConfidenceCalcs=0

StartPemOrder=2

DeltaPemOrder=1

RobustNorm=0

PemBias=1

UsePfxIC=1

AICSearch=1

NoiseModCheck=0

PfxExpRed=10

PZTol=.0001

DTTol=0.001

UsrPrecision=8

AutoAnnotate=0

MaxAnnotate=1

DisParOrder=2

MultiMean=0

AutoSelect=1

ExcludeRangeType=0



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To correctly change a parameter it is necessary to insure that it appears under itscorrect heading. The values shown above are the defaults used in the DesignStudio. If a parameter is not displayed then the value shown above will be usedto initialize the environment the next time a new file is created. Only enterparameters that need to be changed. The order in which the parameters appear isNOT important. The parameters have the following representation.

Memory Buffer

• SwapMode – This parameter was previously designed for WIN16applications

UNDER NO CIRCUMSTANCES SHOULD SwapMode BEACCESSED OR CHANGED

• UserMemBuf – Defines memory buffer for swapping operations. It alsoshould never be touched but has no meaning as long as SwapMode equals zero.

• UserMemABuf – Defines maximum memory size in MEG for theregression matrix. When using FIR models, the A matrix is automatic segmentedif memory requirements exceed UserMemABuf. When using PEM models andCholesky factorization, the J matrix is automatic segmented if memoryrequirements exceed UserMemABuf. When using PEM models and QRfactorization, if the J exceeds UserMemABuf, then the user is given the optionof continuing using Cholesky factorization or terminating and resizingUserMemABuf. This parameter also defines the maximum memory availablefor the undo command in the Block Overwrite option of the Data Operationmain menu function.

User Options

• FIRDefault – Flag defining initial regression model type. One impliesFIR. Zero implies PEM.

• PositionalForm – Flag defining default form of FIR model Oneimplies positional. Zero implies velocity.

• StartSettleT – Defines the starting FIR settling time (in minutes)for the first of potentially several models corresponding to a given input/outputpair.

• DeltaSettleT – Defines the FIR settling time increment (inminutes) for consecutive models corresponding to a given input/output pair.

• FirNumCoeff – Starting number of coefficients for all FIR models

• ConfidenceCalcs – Flag used for initializing the selection statusof confidence and statistical calculation when a new file is created.

• StartPemOrder – Defines the starting PEM order (of allpolynomials) for the first of potentially several models corresponding to a giveninput/output pair.



• DeltaPemOrder – Defines the PEM order increment (of allpolynomials) for consecutive models corresponding to a given input/output pair.

• DisParOrder – Defines the initialization order for discreteparametric models. Both ARX (Pfx) and OE models use this parameterfor initialization. Laplace does NOT

• RobustNorm – Flag defining type of norm to use in evaluating PEMmodels. One implies robust norm. Zero implies quadratic norm.

• PemBias – Flag defining use of a bias term in PEM models. Oneimplies use of a bias term. Zero implies no bias term. FIR modelsalways use a bias term

• UsePfxIC – Flag used for initializing the “Search on Start Order”option in the Overall Model Setup Options dialog box. This parameterhas effect only when a new application is created. The “Search on StartOrder” option is used to define the use of a high order ARX solutionand subsequent Pfx reduction to obtain initial PEM estimates. Oneimplies that the search option is initialized as selected. Zero impliesthat the search option is initialized as deselected. Disabling the searchresults in an initial estimate based solely on an Instrumental Variableapproach.

• AICSearch – Flag defining use of a modified Akaike InformationTheoretic Criterion for determining order of the ARX model used inPEM initialization. One implies that the search is used. Zero impliesthat there is no search. Value is ignored if UsePfxIC is false.

• NoiseModCheck – Flag used for initializing the “Auto Check NoiseMod” option in the Overall Model Setup Options dialog box. Thisparameter has effect only when a new application is created. The “AutoCheck Noise Mod” option is used to check on the effectiveness of thenoise terms in the PEM model. One implies that the option is initializedas selected. Zero implies that the option is initialized as deselected.Selecting this option results in an automatic evaluation of the need fornoise terms based on statistical considerations.

• PfxExpRed – When UsePfxIC is true and AICSearch is false,this parameter times the user-selected order defines the actual order ofthe initial ARX solution used for PEM initialization.

• PZTol – Defines the tolerance in percent for canceling poles and zerosin the Laplace domain transfer functions. Cancellation is performedonly through the Transfer function dialog box.

• DTTol – Defines the tolerance in percent difference above whichsample rates in two files to be merged can be considered different. Datawith different sample rates can not be combined into one file.



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• UserPrecision – This parameter defines the precision used todisplay data in the Data Edit dialog box.

• AutoAnnotate – This flag is used to turn on and off the auto-annotation option. Set this option at any time. Default is off (0). Formore on this option see section 12.

• MaxAnnotate – This parameter defines the maximum length of anindividual annotation item in millions of characters. If the annotationexceeds this limit a message box will be displayed prompting the userto reduce the length of the annotation. Set to zero for no display (Notrecommended).

• MultiMean – Refers to the mean removal procedure used whenscaling data for any regression calculations. Zero implies single mean.One implies that separate means will be removed for eachdiscontinuous segment of regressed data. Total data will be zero meanin either case.

• AutoSelect – Refers to the variables displayed when the “ExcludeData Ranges” button is clicked. AutoSelection implies that only thevariables selected from the “calling” dialog box will be displayed in theplot presented for excluding ranges. With AutoSelect=0, variablesdisplayed in the plot will correspond to the current selection status ofthe Descriptive Info. View.

• ExcludeRangeType – Flag used for initializing the “RegressionSelection Options” option in the Overall Model Setup Options dialogbox. This parameter has effect only when a new application is createdThis parameter defines the default used when marking data forregression calculations. When set to zero, the default is to useconventional range exclusion for all variables selected. When set equalto one, the default allows the user to mark data for exclusion on a pervariable basis. This option can be changed at any time in the overalloptions dialog box.

With the overview complete, a more detailed discussion can be presented oneach of the aforementioned topics. To do this it is convenient to present topics inan order that slightly different than that give above. Since all functions are insome way related to the notation of views, this menu function will be describedfirst. This will be followed by a detailed discussion on the Edit, Data Operationand Identify functions each of which will have one or more dedicated chapters.


Section 5 — Multiple Views and the Presentation of Data

5.1 Overview

In This SectionRead this section to find out about :

• Summary of all views and their associated functions- Data- Models- Performance

QualitativeQuantitative

• Plotting Raw Data- Time series- Scatter plots

• Configuring Plots, Selecting Ranges and Marking Data

While each view is described in this section, focus is directed primarily on the useof those views associated with observing selecting and configuring data.

Use of other views that deal primarily with models and/or performance will bediscussed in subsequent chapters as the need arises

Information displayed by Profit Design Studio (APCDE) specifies whichenvironment has the current focus. In the case shown below, the environmentsupports MIMO identification since the active document (document with thecurrent focus) is of the .mdl type. (Identification documents have the model iconshowing multiple response curves followed by the prediction symbol y). When anidentification document is initially opened the source data file and document typeare displayed in the document title as described previously. The default view is theDescriptive Info View

Section 5 — Multiple Views and the Presentation of DataBasic Views


5.3 Basic ViewsDifferent views let you display different information about your identification—data plots for different ranges, model trials, normalized scaling, zoom and manyother options. The fundamental view options are obtained be selecting View fromthe main menu as shown below.

Primary Functions As illustrated, the current view is the “Descriptive Info” view. The primary viewsand their functions are:

• Descriptive Info – Creation of a new document either from file open orfrom file new defaults to the Descriptive Info view. Basic informationpertaining to each variable is displayed in this view. Its full name, variabletype (CV, MV, DV), tagname, parameter name, variable class (Var, Aux),description and units are all displayed in this view. The category ‘Item’ is adynamically updated list that illustrates variable order or position in thecurrent model matrix. CVs are rows, MVs and DVs are columns and Auxesare not displayed in matrix views. In addition to information about thevariables, this view provides a mechanism for repositioning variables withina matrix using simple drag/drop operations. It also provides a mechanism formerging or combining variables from different files.

• Single-graph Data Plots – Use this to view all selected variables on thesame plot. Only variables selected are plotted. If no variables are selected,all variables are plotted. Select “Multiple Scale” option to plot each selectedvariable full scale on the same plot (use this option to compare variables).Select “Normalized Scale” option to plot each selected variable on its ownaxis. Select “Single Scale” option to plot all selected variables using onerange. For this option the value of the range will be displayed on thevertical axis. Configure the plot by double clicking the desired variable. Use



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right mouse button in the plot box to display time, value, and index forvariable closest to target location. Zoom and unzoom to facilitateselection/edit functions. Use this view to mark/unmark bad data at theglobal level. This data will be treated as bad for all subsequent operationsuntil the marks are removed. Use this view to cut, remove or delete data.The title for this view is “Trend Plots” and will always be displayed in thelower right portion of the vertical margin. This view can be used for theselection and data edit functions used throughout the environment.

• Show Regression Ranges – Use this view to show all ranges associatedwith regressions. Select ranges to exclude values from data used forFIR/PEM regressions. Values to be excluded are set bad entering theregression. Values can be excluded using two different approaches:

− Block Selection – With this option, ranges are selected and theseranges are applied to all variables used in the regression. All valueswithin the time range (inclusive) are set bad for any variable beingregressed. Since all variables are bad for each range selected, thedata is collapsed such that each range to be excluded is representedby a single NaN for each variable.

− Variable Selection – With this option, data can be excluded for eachvariable on an individual basis. Display of this type of selection isdifferent than that used for Block selection to avoid any ambiguity.This category supports an additional option

§ Mark only dependent variables – Independent variables areunaltered entering the regression

§ Mark both dependent and independent variables – Selection isdone on a per dependent variable basis. At regression, theselection is also be applied to the independent variables. Whenthis option is used, the effective marks (they are not displayedgraphically) are the result of the union of all marks for eachdependent variable used in the regression. This implies that theeffective bad values for an independent variable are dependenton which dependent variables (and their associated marks) areused in the regression.

Data marked as bad at the global level (in the Single Graph Data plotsView) can be displayed in this view (and any graphical view) by choosingthe Show Bad Data option. Global marks however can not be altered in thisview.

This view operates in a fashion almost identical to the Single Graph DataPlots View. Selection, zooming, scaling etc. are as described above. Youcan NOT use this view to actually cut or remove data. This can only bedone in the Single Graph Data Plots View. The title for this view is “ShowRegr. Ranges” and will always be displayed in the lower right portion of the



vertical margin. This title will have a red superscript “b”, “v1” or “v2”. Thesuperscript “b” and “v” designate block and variable selection respectivelywhile the “1” and “2” imply that marks are applied only to dependentvariables (“1”) or to both (“2”).

• Show Prediction Ranges – – Use this view to show all ranges associatedwith predictions. Select ranges to exclude values when performing anyprediction calculations. Values can be excluded only by using blockselection. These ranges are applied to all variables used in subsequentpredictions. All values within the time range (inclusive) are set bad for anyvariable being used in the prediction. Since all variables are bad for eachrange selected, the data is collapsed such that each range to be excluded isrepresented by a single NaN for each variable

Data marked as bad at the global level (in the Single Graph Data plotsView) can be displayed in this view (and any graphical view) by choosingthe Show Bad Data option. Global marks however can not be altered in thisview.

This view operates in a fashion almost identical to the Single Graph DataPlots View. Selection, zooming, scaling etc. are as described above. Youcan NOT use this view to actually cut or remove data. This can only bedone in the Single Graph Data Plots View. The title for this view is “ShowPred. Ranges” and will always be displayed in the lower right portion of thevertical margin.

• Scatter Matrix (raw data) – Use this to view raw data for each variable asa function of all other variables (excluding Aux variables) as a scatter plotin matrix form. As with all matrix views this view is fully scrollable. In spiteof optimizing the scrolling function, documents with large amounts of datamay exhibit update delay (resizing can be particular slow). This delay canbe minimized by scrolling using the page approach or by scrolling to thedesired matrix position in an alternate model and subsequently switching tothe scatter matrix view (scroll positions will be saved). Positions ofvariables in this view are reflective of those in the Descriptive Info View.

• Multi-graph Data/Scatter Plots – Use this to view each selected variableon its own individual plot. Only variables selected are plotted. If novariables are selected, all variables are plotted. Select variables from the“Descriptive Info” view in the standard Click and/or Ctrl Click fashion. Ifonly one variable is selected in the descriptive info view, then this view canbe used to display scatter plots of all variable with respect to the selectedvariable.

• Model Summary- This view is used to display summary information of thesystem model matrix and can be used to update/modify parametric models.It also provides the mechanism for merging models and/or data from onefile to another. Models to be merged from the source file must be selectedusing this view. A full discussion of the different types of models and the



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different views is presented in subsequent sections

• FIR/PEM Step Responses- This view is used to display the matrix of allFIR or PEM step responses. Sensitivity of the step responses can be used asa preliminary indicator of FIR model adequacy. Similarly, sensitivities canbe used for PEM order selection

• All Step Responses- Both FIR/PEM and parametric step responses aredisplayed in this view. Overall parametric fit and smoothing qualities can beobtained from this view.

• Final Model Xfer Function-Results of the entire identification procedureare summarized and displayed in this view. Both the FIR/PEM andparametric step responses are presented as are the final Laplace domaintransfer function and other summary information. Any subsequent builds(either RMPCT or RPID) use the models presented in this view as thestarting point for the calculations.

• Correlation (MV/MV)- Interaction or correlation between all input orindependent variables are displayed in this view. Both auto and crosscorrelations are depicted. For FIR calculations to be unbiased, the inputsmust be uncorrelated with the disturbances. Since white inputs willautomatically fulfill this requirement, ‘white-like’ correlations provideexcellent guidance for ideal signals. In addition ‘white like’ signals insuregood conditioning characteristics of the FIR regression matrix. Targetranges indicative of ‘white like’ performance are specified for both auto andcross correlations. Values significantly outside of these ranges may be causefor concern.

• Correlation (CV/MV)- Interactions between inputs or independentvariables and outputs or dependent variables are displayed in this view. Thepositive portion of the curve represents the correlation from input to output,while the negative portion of the curve represents the correlation fromoutput to input. Ideally, for open-loop data with uncorrelated inputs, thenegative portion of the curve should be zero. Target ranges are specified forthe negative cross correlation coefficients. Values outside of these rangesrepresent potential feedback in the data and may lead to bias in the FIRestimates.

• Confidence- FIR estimates in excess of the corresponding noise band areplotted in this view. Estimates are normalized. Values within the noise bandare zero. One curve for each trial is graphed. If an entire model for acorresponding trial is not statistically significant, no coefficients will bedisplayed for that trial. If no models are statistically significant for a givensub model, then a null plot box will be shown. A null plot box indicates thatthere is no causal relationship for a specific input/output pair. In thisinstance the Non-Null Hypothesis Test (NNHT) flag for the given submodel is set FALSE (i.e. the model should not be present).



• Statistical Summary- All information pertaining to the FIR/PEM statisticsand associated rankings is given in this view. Upper and lower bounds forthe step responses for each sub model are displayed. The upper bound is themaximum value of the step response plus the maximum noise bound for alltrials. The lower bound is the minimum value of the step response minus theminimum noise bound for all trials. Also depicted are the NNHT flag,model rank category, model rank for the specified category, separation andsensitivity factors and recommendations weather or not to use the model

Section 5 — Multiple Views and the Presentation of DataViewing, Selecting and Marking Data


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5.3 Viewing, Selecting and Marking Data

Working with

different Windows

Windows in Profit Design Studio (APCDE) work the same way as in any otherWindows applications. Multiple windows let you display different information, ordifferent views of specific information at the same time. You can open anynumber of files at one time. Each new file has its own window.

You can also open additional windows on the same file by using Window>NewWindow or by using the following toolbar button

The new window initially has the same view as the previously selected window,but you can now change the view. This lets you see different views of the sameinformation at the same time. This is particular helpful when marking data,building FIR/PEM models or evaluating predictions.

When marking data two views can be open to graphical views and a third can beopen to Descriptive Info. In one graph you can display a block of variables witchare of interest. From the Descriptive Info View select a subset of variables to bemarked. Drag and drop these variables into the alternate graphical view. Markvalues for these variables as desired. The marks and their temporal relationshipwith other variables will be automatically displayed in the original graphicalview.

When regressing data, one window can be opened for the fit. Another can be usedto view the step responses and a third can be opened to view the statistics.Information is automatically reflected in all views simultaneously as thecalculations are updated.

When evaluating predictions, one window can be opened for the prediction(remember to select the Store predictions option) and another can be opened toSingle Graph Data Plot. Only predicted and actual CVs along with thecorresponding residuals will be shown in the prediction view. In this view, data inany excluded ranges will not be displayed. In the other view select any variableyou wish to display including predicted variables (they will be stored as Auxvariables). Excluded ranges for the predicted variables will be represented byNaNs.

Plotting Raw Data Data can be plotted in any of four graphical views. While the Single Graph DataPlots is the primary graphical view, Exclude FIR/PEM Ranges, ExcludePrediction Ranges and Multi-graph Data/Scatter Plots can also be used to viewdata in a graphical framework. Use of each of these views will be presented in thefollowing paragraphs.



Single-Graph Plots Select View>Single-Graph Data Plots. The result is a typical Single-Graph DataPlot.

With no variables selected (none selected is the same as all selected) Allvariables will be plotted in a typical Single-Graph Data Plot as shown below.

Here the data is displayed using the normalized scale. That is each variable isdisplayed using its own axis.



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Changing the Plot

Size

Use View>Plot Options to change the plot size, or click and drag in the plot areain any of the Single-Graph Type of views to select a zoom rectangle that expandsto fill the window. Press <ENTER> to unzoom.

Viewing Single

Graph Plots

The Single Graph Plots View (as well as Show Regression Ranges and ShowPrediction Ranges) shows plots of the selected variables on one graph. Use theDescriptive Info view to select the variables that are plotted on the Single GraphPlots view (none selected is the same as all selected) Open up a second windowto Descriptive Info. and drag and drop variables from that window to the graph todisplay the dropped variables.

1. Select View>Normalized Scale to scale the ranges so that each plot occupiesits own band on the graph. Select View>Multiple Scale to scale the rangesso that each plot occupies the full height of the graph. Select View>SingleScale to view all selected variables on a common scale. The common scalewill be displayed on the vertical axis.

2. To magnify the plot (zoom in), click and hold down the left mouse buttonanywhere in the plot area and drag the cursor to open up a dashedrectangle. When you release the button, the dashed rectangle expands tofill the window. Repeat this to get finer resolution.The scroll bars become active when the view is zoomed to allow scrollingin either the time or value axes.

3. The date/times of the left- and right-most data points display at the left andright sides of the time axis box below the horizontal scroll bar.You can see the date/time and corresponding vector index of any datapoint by moving the cursor into the time axis box. A vertical dash dot lineappears in the graph above the cursor, and the data/time (index) of this linedisplays in the center of the time axis box. See for example the plot shownabove.



4. To Display values for each variable corresponding to the vertical dash dotline simply hold down the right mouse button while the cursor is in the time axisbox. The values will be displayed in parenthesis between the high and low rangesas shown below.

5. To use the spyglass feature, position the mouse anywhere in the plotbox. Click the right mouse button and hold it down. An indicator willencircle the data point closest to the cursor. A vertical dash dot line willconnect the indicator to the horizontal axis. The corresponding variable,its value and index will be displayed at the center of the time axis box asshown below.



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Reconfigure

Single-Graph Plots

To reconfigure the plot, double click anywhere in the text box on the left side ofthe plot (left margin). The following dialog box will be displayed.

This dialog box will allow you to set the plot ranges independently for each of thevariables. Initial values correspond to the variable that was double clicked. At thetop of the dialog box the ‘Item’ information from the Descriptive Info view isdisplayed along with the tagname.parameter and the variable class. The maximumand minimum values of the data are displayed as text strings. The values are usedto initialize the User Hi and Lo Values if not already set and are used to resetthese values when the Defaults button is selected. The user can enter the desiredranges and see interactive response from the plot view by simply selecting theUpdate button. Note that all range information is saved on the variable NOT theview. Hence if multiple windows are opened the range change will beautomatically reflected in all data views simultaneously.

Use the next and previous buttons to sequentially access alternate variablesdisplayed in this view. A value entered takes effect if the update button isselected or if a new variable is selected. Selecting Cancel after the fact will notundo the operation. Use the check box to prevent data that is outside user rangesfrom being displayed in the normalized mode.



Plot Modes Three modes are possible with any of the trending plots. These modes areMultiple Scale, Normalized Scale and Single Scale. All graphs displayed to thispoint have illustrated the use of the Normalized Scale. All curves are plotted suchthat each variable occupies its own band. An example of the Multi Scale modefollows.

In this mode all curves are plotted in the same graph using the individual curveshigh and low ranges. Thus the vertical axis has a different scale for each variable.For the Single Scale mode, all curves are plotted in the same graph using thesame ranges taken from the minimum and maximum value of the entire set ofdata. The data show above in this mode is as follows.

Since there is only one scale in this mode, the range is displayed on the verticalaxis. This range is automatically adjusted under zoom conditions as shownbelow.



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Selecting TimeRanges

You can select time ranges of data. If you call a view from:

• View>Single Graph Plots You can mark/delete the test data over theselected time ranges.

• View>Show Regression Ranges or Fit FIR/PEM Models>Exclude DataRanges You can mark selected time ranges for exclusion from subsequentFIR/PEM model fitting (see subsequent section).

• View>Show Prediction Ranges or Select Final Models>Exclude DataRanges You can mark selected time ranges for exclusion from subsequent modelvalidation/prediction calculations fitting (see subsequent section).

Selecting Ranges To select a range:

1. Move the cursor within the time axis box to one end of the desired timerange. The vertical dash dot line and the date/time in the center of the box showyou where you are. When you have positioned the cursor at one end of the range,press and hold the left mouse button.

2. Move the cursor to the other end of the desired time range. The secondvertical dash dot line that appears and the date/time in the center of the boxcorrespond to the other end of the range. Release the mouse button. The selectedtime range is shown with a gray background.



3. Repeat these steps to select additional ranges. The resulting operations maylook like the plot shown below.

4. Hold down [CTRL] and use the above procedure to deselect all or part of apreviously selected range.

Reading the Plots The boundary of a selected range may be notched, or the entire range may appearas a dashed rather than solid gray line ( see the plot given above). This indicatesthat there is more than one data sample that is plotted at one horizontal pixelposition on the screen.

Some of these samples are in the selected range and some are not. Zoom to afiner resolution to see which data samples are in the selected range.

Selecting ranges can be used to delete data, mark data as bad (which can besubsequently unmarked), or to exclude data from subsequent FIR/PEM fitting ormodel validation calculations. The time ranges are remembered separately forthese three cases. You can display and change the ranges at any time. Rememberhowever that excluding data MUST be done through the appropriate view or byusing the “Exclude Ranges” button on the appropriate dialog box. It is a commonmistake to use View>Single-Graph Plots to select ranges and to think that theseranges will be excluded from future calculations.



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Marking Data Badat the Global Level

Data can be marked bad at the global level in a number of ways. As an example,consider the case shown below.

Here, Three windows are opened on the same data. Window 3 shows the block ofvariables that are of concern. Window 2 will be used as the selection window. Asshown three variables have been selected and dropped into window 1, which willbe used for marking. Note that the Show/Hide NaN toolbar button is in a selectedstate (this is similarly reflected in the Insert pull-down on the main menu). Withthe ranges selected, the data may be marked bad by choosing the mark NaNtoolbar button. While data marking/unmarking can be accomplished using theInsert menu options, it is more convenient to use the following mark, unmark and

show/hide toolbar buttons



This action will result in the data shown below. Note that the selected rangeshave been cleared and only the variables displayed in the active or highlightedwindow (window 1 in this case) contain the desired marks. Data to be treated asbad at the global level has the distinct dark gray mark.

Data marks will only be displayed when the normalized scale option is in use andonly when the Show/Hide option is in the selected state. Since the Show/Hideoption is a view option, it can be set independently for each view on a givendocument. If this button is selected and the current view is a non-trending view,then the view will be automatically switched to Single-Graph Data Plots Viewusing a normalized scale. Data can be unmarked by simply selecting ranges andchoosing the Unmark button or associated menu item. Data within the selectedranges that is marked will be restored and the selection ranges will be cleared.



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Marking Data Badat the RegressionLevel

Data can be marked bad for regression purposes in a similar fashion as shownbelow.

As shown above there are four distinct methods for displaying data that is to betreated as bad. In one the data is simply removed (cut/deleted) from theenvironment. This is the case for CV13. Data from index 80 through 105inclusive has been deleted. The circles illustrate good data bracketing this range.Data marked as bad at the global level is illustrated by the dark gray bands.Global marks are the darkest of any marks. Next, data marked as bad forregression purposes is always displayed as lighter gray crosshatched bands. Thesemarks can only be seen in the Show Regression Ranges View (“Show Regr.Ranges” as illustrated above). Finally, the selected ranges themselves aredisplayed as intermediate gray bands These ranges will always be displayed overthe entire vertical height of the plot. There should never be any confusionbetween the marks even if there is just one variable. In this case the plot may looklike that shown below.



Notice that the global and regression marks do not cover the entire vertical heightof the band while the selection ranges do. Selection ranges may be used fordefining global or regression marks, as illustrated above or they may be useddirectly to exclude data for either regression or prediction. For the case shownabove the selection ranges would be ignored in a FIR/PEM fit since the v1 optionis in effect. That is the regression is to be performed using the variable selectionoption and applied to dependent variables only. Alternative, the block optioncould be used. In this case only the selection ranges would be used. Since theranges are the same for all variables being regressed, the data would be collapsedand a single NaN would be used to represent each range. These options can bechanged through the Set Overall Options dialog box described in a later section.Bad data marks at the global level are always applied when data is used for anyoperation.

While there is a great deal of flexibility in marking data, the end result may becounter-intuitive especially with respect to regression. When an unexpected resultoccurs it is best to consider what happens when data is marked as bad. This canbe done conveniently with respect to the prediction equations defined in Section3. Any prediction equation containing a bad value must be removed from theregression set.

For FIR calculations, if a dependent variable is bad, the corresponding predictionequation(s) containing that variable must be removed. This implies that the y andcorresponding row in the regression matrix be removed. When an independentvariable is bad all rows in the regression matrix containing the bad value must beremoved. This implies that a minimum of n rows is removed for a single badvalue where n is the number of response coefficients. If multiple dependentvariables are regressed simultaneously, then rows in the regression matrixcorresponding to all bad values in each dependent variable must be removed.

This implies that different but fixed NaN ranges can result in different answersdepending on which dependent variables are regressed. A dependent variableregressed by itself can yield a different answer than if the variable wereregressed with other dependent variables.

For PEM calculations all offending terms must also be removed from theJacobian matrix. In addition special considerations are made for dealing with thenoise models through NaN filtering operations. In general the number of rowsremoved for an NaN is related to the maximum polynomial order. With PEMhowever, only MISO models are supported. As such, even if multiple dependentvariables are selected for a given regression only one dependent variable isregressed at a time. Therefore NaNs for one dependent variable will never affectanother dependent variable.

Marking Data Bad

at the Prediction

Level

Data can be marked bad for regression purposes in a similar fashion as shownbelow.



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Only block ranges can be used to exclude data from prediction calculations. Assuch all ranges are collapsed and represented by a single NaN as describedpreviously. Most often, prediction ranges will be set by using the “Exclude DataRanges” button on the Select Final Models dialog box. In the prediction results,excluded ranges will be collapsed. Values marked as bad at the global level, willsimply result in the generation of corresponding bad (NaN) values. These valueswill NOT be collapsed.



Scatter Matrix Select View>Scatter Matrix. The scatter matrix appears as shown below. In thisview a scatter plot is displayed for each variable as a function of all othervariables

Multi-Graph/ScatterPlots

Select View>Multi-Graph Data/Scatter Plots. This will switch the active view toone of two modes as described next.

Multi-Graph Mode With no variables selected (none selected is the same as all selected) or more thanone variables selected, multi-graphs are displayed as shown below..

Viewing, Selecting and Marking Data


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Scatter Plot Mode Select just one variable from the descriptive info view. Then, Select View>Multi-graph Data/Scatter Plots. A typical Scatter Plot is then displayed.

Note this mode must be used to view auxiliary variables in a scatter plot sinceauxiliary variables do not appear in any matrix views. When using the Multi-Graph/Scatter Plot View, variables can be selected as described for the trendplots. For example variables can be selected from one window in the DescriptiveInfo. View and dropped into the Multi-Graph/Scatter Plot.

Viewing, Selecting and Marking Data



Section 6 — Edit, Merge and Reconfigure Functions

6.1 Overview

In This SectionRead this section to find out how to :

• Cut, copy, paste and delete data• Cut, copy, paste and delete models• Use the special copy, paste and delete commands to modify individual

models• Copy internal information from one source to another• Merge data and/or models using edit commands or drag-drop

operations• Dynamically reconfigure model matrix structure

Most functions described in this section can be accomplished in a variety of ways.In most cases you can use the Edit Menu, hot keys or drag-drop operations.

Editing data as described in this section only pertains to the modification ofVariable attributes, the rearranging or merging of data, or the deletion of data.Actual manipulation of the raw data is covered in the next chapter on DataOperations.

Similarly, editing models as described in this section only pertains to therearranging, copying, merging or deletion of models. Actual manipulation of themodels is covered in the chapters on Identification

Note, whatever reconfigure operation (cut, copy, paste, delete) is applied to themodels, is also applied to the performance indicators (i.e. statistics). That is if amodel is moved or merged, the performance indicators associated with that modelare also moved or merged.

Similarly, whatever reconfigure operation (cut, copy, paste, delete) is applied to thedata, is also applied to the data marks and ranges That is if data is moved ormerged, the data marks and ranges associated with that data are also moved ormerged.

Special Notes:

• Merging between files will result in the loss of all MV/DV auto and cross-correlation data

• Variables of class Aux can NOT be merged or repositioned

Section 6 — Edit, Merge and Reconfigure Functions6.1 Overview


Data and FileManipulation

Being able to combine files and rearrange variables/data/models, you can:

• Copy information from one file to another

• Rearrange the order of variables/data models within a file

• Delete variables/data/models from a file.

This in turn gives you the ability to combine all or some of the variables/data/and/or models from several different test data files into one model file. This datacan come from different testing periods.

Variables/Data and/or models can be moved from one model file to another via theModel Summary view

If only Variables/Data are to be moved from one model file to another, use theDescriptive Info view.

Section 6 — Edit, Merge and Reconfigure Functions6.2 Edit Functions


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6.2 Edit FunctionsDepending on the current state of the identification procedure and what variablesare selected, different editing options are available or not as is appropriate. Thebasic edit functions; Cut, Copy, Paste, and Delete are the standard windowsfunctions as applied to manipulating Profit Design Studio (APCDE) models anddata. To view the edit functions select Edit from the main menu

In addition to the standard cut copy paste and delete functions. the followingfunctions are Identifier specific:

• Select All- For views where selection states are possible, this function canbe used to automatically select all possible variables.

• SpecialModCopy- Used to copy all models and associated data for a singledependent/independent pair (sub-model). Copy can only be performed fromthe Model Summary view when a single sub-model is selected and is for thesole purpose of modifying models within a single document.

• SpecialModPaste- Used in conjunction with the SpecialCopy function. Thisfunction is available only if a SpecialCopy has transpired and only if asingle sub-model has been selected in the Model Summary view.

• SpecialModDelete- Used to delete all models and associated data forselected dependent/independent pairs (sub-models). Delete can only beperformed from the Model Summary view and applies to all selected sub-models

• User2Final- Used to copy selected sub-models to Final Models. User2Finalcan only be performed from the Model Summary view and applies to allselected sub-models. Only the models corresponding to the displayed Trialsare copied to the Final models. The copy results in an automatic residualupdate and the trials are stored as user selected (see section on selectingfinal models).



• Uniform2User- Used to copy the uniform trial solution to the user trials.Uniform2User can only be performed from the Model Summary view andapplies to all selected sub-models. The copy results in an automatic residualupdate and the trials are stored as user selected (see section on selectingfinal models) If the final model source for any selected model is “User”,then the final model is updated with the new set of user models.

• Mixed2User- Used to copy the mixed trial solution to the user trials.Mixed2User can only be performed from the Model Summary view andapplies to all selected sub-models. The copy results in an automatic residualupdate and the trials are stored as user selected (see section on selectingfinal models) If the final model source for any selected model is “User”,then the final model is updated with the new set of user models.

• CopyRegr2Pred- Used to copy selection ranges defined in the “ExcludeFIR/PEM Ranges” view to the selection ranges used for any predictioncalculations. Existing prediction ranges defined in the “Exclude PredictionRanges” view will be deleted. Only block selection ranges are copied.Regression marks have no prediction counterpart and have no impact on thecopy. If no ranges are selected then this option will be disabled.

• CopyPred2Regr- Used to copy selection ranges defined in the “ExcludePrediction Ranges” view to the selection ranges used in the “ExcludeFIR/PEM Ranges” view. Existing regression ranges defined in the “ExcludeFIR/PEM Ranges” view will be deleted. If no ranges are selected then thisoption will be disabled.

• Variable Info- Used to change variable specific information. This option isenabled in all views and is described below.

Basic Edit

Characteristics

As shown above the basic edit functions are disabled. No operations have beenperformed and nothing is selected. The basic edit functions have the followingcharacteristics.

• Cut- This function is only enabled in the Descriptive Info view and onlywhen one or more variables are selected. Cutting the selected variables willremove the variables and any corresponding models. This information willthen be copied to the internal paste buffer for subsequent retrieval. If an MVor DV is cut, then an entire column of models is removed from the modelmatrix. If a CV is cut, then an entire row of models is removed from themodel matrix. If an Aux variable is cut there will be no impact on the modelmatrix.

• Copy- This function is enabled in either the Descriptive Info or ModelSummary views when either one or more variables or one or more sub-models are selected. The copy operation does not remove any informationfrom the document. When variables are copied, all information pertaining tothe selected variables and all associated models are copied to the internalpaste buffer for subsequent retrieval. When the copy operation is performed



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in the Model Summary view all information pertaining to the selected modelsand all associated variables are copied to the internal paste buffer. Thisimplies for example that if one sub-model is copied, then the model and itscorresponding CV and MV/DV will be stored into the internal buffer.

• Paste- This function is enabled only after a cut or copy operation has beenperformed. What actually gets pasted depends on the source and destinationdocument (file). If the paste destination document is the same as the sourcecut/copy document, then contents of the internal buffer will simply be copiedback into the original document. If a copy and paste operation is performedon the same document, then no apparent changes will be displayed if nomodifications have been made since the copy operation. A warning messagesimilar to the one shown below will however be displayed in spite of the factthat the models are being are being overwritten by identical models

If a cut and paste operation is performed on the same document then the dataand models will be unaffected but the relative positions of variables andmodels in the matrix will be potentially altered depending upon the insertionpoint of the paste. This is one mechanism by which a model matrix can beeasily reorganized.

If the paste operation is performed on another document, then some or all ofthe contents of the internal buffer will be copied to the destination document.If the cut/copy operation was performed in the Descriptive Info view of thesource document, then only variables and data will be copied into thedestination document. If the cut/copy operation was performed in the ModelSummary view of the source document, then variables, data and models willbe copied into the destination document

If the paste information already exists in the destination document then thedata and/or models will be “merged” into the destination document asappropriate. If one or more of the variables to be copied already exist in thedestination document then the data will be spliced together using time stampinformation from both the source and destination documents. Time stampoverlaps are handled automatically with precedence given to the source file(destination data is overwritten). If models already exist then the user will bepresented with an overwrite option

Position or location of the information being copied from the internal bufferdepends on the insertion point in the destination file. The insertion point forthe Descriptive Info view is immediately before the focus rectangle such as



that shown below.

This focus rectangle corresponds to the last selected variable. Its index isstored even if the view loses focus and the rectangle is no longer displayed.If the focus box is unavailable then the insertion point is at the top of the list.After the paste operation is complete, the focus box will be redrawnillustrating the insertion point and the view will be automatically scrolledsuch that the focus box is clearly displayed.

At times it may be desirable to append variables/data to the end of theDescriptive Info list. When the focus rectangle is the last element in the listand the variable is of class Var, then the insertion point is defined by theusers response to the following dialog box.

Note, that if the last element(s) are of class Aux, then the dialog box shownabove will not be displayed during the paste operation since all variables ofclass Aux are required to be at the end of the list.

If the current view is a model-based view other than Descriptive Info, thenthe insertion point is established by the selection state of the ModelSummary view. It is therefore advisable to have the view of the destinationdocument set to Model Summary when models are to be merged. Considerthe selection state shown below.



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The selected model with the lowest CV index establishes the row insertionpoint. The selected model on this row with the lowest MV/DV indexestablishes the column insertion point. Models are inserted prior to thispoint. In the illustration given above the insertion point is (3,3). If no sub-models are selected then trailing rows are added for each CV, trailingcolumns are added for each MV/DV and models are inserted at theappropriate intersection.

• Delete- This function is enabled in the Descriptive Info when one or morevariables are selected. It is also enabled in the Single-Graph Data Plot viewwhen at least one range has been selected. Deleting the selected variableswill remove the variables and any corresponding models. If ranges have beenselected in the in the Single-Graph Data Plot view prior to the deleteoperation, only data corresponding to the ranges will be deleted. Variablesand models will remain intact. In this case, NaNs will replace deleted data. Ifdata ranges are deleted for all variables then the data is collapsed and asingle NaN replaces the deleted data in a given range. Since deletedinformation is lost, this operation will result in the display of the followingdialog box.

The message will change depending on the operation

These basic commands can be accomplished using either the menu edit functionsor the hot keys defined in the pull down edit menu. While the cut, copy and pastefunctions can be used to effectively merge combine and reconfigure variables,data and models these tasks can be more conveniently accomplished using thedrag-drop operations. These procedures are described in a subsequent section.



Special Edit

Functions

These functions apply only to manipulations on a particular sub-model. In thisrespect the copy, paste and delete functions have the standard connotations.Operations apply to all models and data associated with a selected CV/MV or DVpair. These functions have the following characteristics.

• SpecialModCopy- This function is enabled only in the Model Summaryview and only when a single model is selected. All pertinent data is stored ina special document structure. Since it is not written to a paste buffer, thisfunction is only applicable within a given document

• SpecialModPaste- This function is enabled only after a SpecialCopyoperation has been performed and only in the Model Summary view when asingle model is selected

• SpecialModDelete- This function is enabled only in the Model Summaryview and applies to all selected sub-models. All data associated with eachselected sub-model is deleted. The operation effectively nulls the model andinitializes all associated data to its default values. In addition this operationwill also clear the special copy buffer effectively disabling the pasteoperation until another SpecialCopy is performed.

These functions can be used effectively replicate and null models in a givendocument. To replicate a model simply select the desired model in the ModelSummary view and select Edit>CopySpecial. Next select the paste position andselect Edit>SpecialPaste as shown below.

Here the (4,1) model has been replicated in the (3,2) position. If there was anexisting model in the (3,2) location then the following message will be displayed.



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Models can be deleted in a similar fashion. From the Model Summary view selectthe desired sub-models. Then select Edit>SpecialDelete or enter <Alt Delete> oruse the toolbar button. The following message will be displayed

Copy Trial

Information

These functions apply only to selected sub-models. Operations involve copyinginformation from one Trial Source to another (See section on Selecting FinalModels for a discussion on Trial Source information). These functions have thefollowing characteristics.

• User2Final – This function is enabled only in the Model Summary viewand applies to all selected sub-models. Only the models corresponding tothe displayed Trials are copied to the Final models. The copy results in anautomatic residual update and the trials are stored as user selected (seesection on selecting final models). This function is only applicable within agiven document

To use this function, switch to Model Summary View and select the desiredsub-models. It is also convenient but not necessary to have a Final Modelwindow opened on the same document. Next, select the trials for theselected sub-models. This can be done individually by double clicking onthe Trial descriptor for each sub-model as described in a later section. Thiscan also be done more conveniently by selecting View>Trials>Change Allas shown below.



This results in the Displayed Trials dialog box.

Select the appropriate button to increase or decrease the displayed trials andcorresponding models. Adjust until the selected sub-models display themodels and corresponding trials of interest. Select the toolbar button

or select Edit>User2Final. Results of this operation are shown below.

In addition to copying the displayed trials for the selected sub-models intothe “User Trials”, the residuals for any “touched” CVs are updated. Theseresults are then loaded into the Final Models. As shown above, the trials and



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corresponding models displayed in the Final Model view reflect the userchoices. Also note that the prediction error and Final Model Source havebeen updated.

• Uniform2User – This function is enabled only in the Model Summary viewand applies to all selected sub-models. It behaves much like that discussedin the previous paragraphs. Here however the Uniform Trial solution iscopied to the User Trials.. The copy results in an automatic residual updateand the trials are stored as user selected (see section on selecting finalmodels). This function is only applicable within a given document. If theFinal Model Source for any of the “touched” CVs is of type “User”, thenthese Final Models will be updated to reflect the changes.

• Mixed2User – This function is enabled only in the Model Summary viewand applies to all selected sub-models. It behaves much like that discussedin the previous paragraphs. Here however the Mixed Trial solution is copiedto the User Trials. The copy results in an automatic residual update and thetrials are stored as user selected (see section on selecting final models). Thisfunction is only applicable within a given document. If the Final ModelSource for any of the “touched” CVs is of type “User”, then these FinalModels will be updated to reflect the changes.

Edit Variable

Attributes

To view or change information associated with Variables first switch to theDescriptive Info view. Select View>Descriptive Info. To edit the variableinformation select Edit>Var Info. As shown below.

Begin editing from the first variable. When Variable Info is selected, a dialog boxshows the basic information that is associated with each variable. This dialogbox is shown below.



Entering or

Changing

Information

You can enter or change information about the variable type. To make a change:

• Use the Type field to indicate if the variable is a CV, MV, or DV.

You can also enter or change parameters that are used to describe/define eachvariable. These parameters are as follows:

• Name – Use the Name field to give a descriptive name to the variable. Ineach model view, this name will be displayed in the row or column that isassociated with this variable. If a period is part of the name, then onlycharacters to the left of the period will be displayed. If you do not enter aname, the Point field is used.

• Point – This field is the point or ‘tagname’ of the variable and is usuallytaken directly form the DCS.

• Param. – This field is the parameter of the variable and is usually takendirectly form the DCS.

• Desc. – Use this field to give a general description of the variable

• Units – Use this field to specify the engineering units associated with thevariable.

Special Note:

Each variable in the Profit Design Studio (APCDE) must be represented by aunique name. Unique names are maintained internally and are established asfollows:

1. If a Point name exists, the unique name is given by concatenatingPoint.Param.

2. If a Point name does not exist, the unique name is given byconcatenating Name.Param.

As long as a Point name exists, you can freely modify the name field withoutaffecting the uniqueness of a particular variable. You can not enter variables



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with non-unique names. Nor can you modify any name such that it results in anon-unique name

Use the previous and next buttons to view and change data associated with theprevious and next variables. Note that the variable in question is automaticallyselected in the background Descriptive Info view. This selection statusautomatically changes as the previous and next buttons are selected. When thisdialog box is closed, the original selection state of the Descriptive Info view willbe recovered.

When no variables are selected prior to the invocation of the dialog box (such asthe case above), it is assumed that all variables are to be potentially edited. Toedit a subset of the available variables simply select the desired variables. Whenthe dialog box is opened, only this subset will be used for modification. Using thenext and previous buttons will sequentially access only the selected variables.

To modify information on a single variable just double click on that variable inthe Descriptive Info view. When this is done the next and previous buttons willbe disabled.

Document without

raw data

If no raw data is present, then the next button will eventually access the end of thevariable list which will be reflected in the Descriptive Info view as a highlightedempty row. This will result in an empty dialog box such as that shown below. Inthis state a new variable will be added once the pertinent information is enteredand the OK, Next or Previous button is selected. Note that if any variables wereselected prior to invoking the dialog box, all newly created variables willautomatically be selected when the dialog box is closed.

When the edit operation is complete, the Descriptive Info view will beautomatically scrolled to display the last variable accessed.

Empty Document If the document contains no data, then the variable information must be enteredmanually through the Variable Info dialog box. As a minimum the name and type



fields must be entered. In addition there must be at least one CV and one MV toproceed with the creation of the final model matrix.

Section 6 — Edit, Merge and Reconfigure Functions6.3 Combining Files and Rearranging Variables/Data /Models


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6.3 Combining Files and Rearranging Variables/Data/Models

Copying

Models/Data From

One File to

Another Using

Copy/Paste

1. Select View>Model Summary for the source document (file) if variables,data and models are to be moved.

2. Select the information to be copied. You can select sub modelsindividually, or you can select a row or column of sub models by selectinga CV or MV/DV. To select all models click in the upper left corner. Foreach sub model that is selected, the following is copied:

• All information for the sub model, including FIR step responses,parametric step responses, and final model selections if these exist.

• The CV and MV or DV that intersect at the sub model, including theirdescriptive information.

Based on user response to a series of prompts, data corresponding to theselected CVs and MVs or DVs may also be copied.

3. Select Edit>Copy, or click its toolbar icon (looks like two sheets of paper).

4. Select View>Model Summary for the destination file. You can determinewhere the copied variables are inserted by selecting a sub model.

Any CVs selected in the source that are not already in the destination fileare inserted just ahead of the CV of the selected sub model.

Any MVs and DVs selected in the source file that are not already in thedestination file are inserted just ahead of the MV or DV of the selected submodel.

5. Select Edit>Paste, or click its toolbar icon (looks like a sheet of paper on aclipboard).

Copying

Models/Data From

One File to

Another Using

Drag-Drop

Drag-drop can be performed with any number of windows open. All windows donot have to be associated with identification. The source and destinationdocuments must be associated with identification.

1. Arrange the windows so both the source and destination windows arevisible, and select View>Model Summary on the source window. It is alsorecommend but not required to select the Model Summary view on thedestination window. This will give direct control of the insertion point.

2. Select the models to be copied from the source file. You can select submodels individually, or you can select a row or column of sub models byselecting a CV, MV, or DV or you can select all models by clicking in theupper left corner of the Model Summary view

3. For each sub model that is selected, the following is copied:

All information for the sub model, including FIR step responses,



parametric step responses, and final model selection if these exist.

The CV and MV or DV that intersect at the sub model, including theirdescriptive information and test data.

Based on user response to a series of prompts, data corresponding to theselected CVs and MVs or DVs may also be copied.

4. Position the cursor over any part of the selection, and press and hold downthe left mouse button. Any movement of the mouse at this point will causethe cursor to change from the standard arrow to a cursor consisting of acircle with line through it. This is the “no drop” cursor which indicates thatthe selected models/data can not be dropped or inserted at this time. If thecursor is moved over any non-model based window (this includes anyperformance or statistical window) the cursor will remain in the no-dropstate. As soon as the cursor is moved over a model based window, thewindow will automatically be brought to the foreground (top of the stack)and the design studio will reflect that this window has the current focus.The cursor will remain in the no drop state until it is positioned over alegitimate model matrix. When this is done the cursor will change to a setof curves with a plus sign. This is the drag-drop cursor for models and data.If the mouse button is released the models/data will be inserted at a positionthat depends on the selection state of the Model Summary view asdescribed above. It is advised to have the destination window in the ModelSummary view

5. Drag the cursor to the destination file. As the cursor is moved over sub-models in the Model Summary view, the selection status will automaticallychange in response to the cursor position. Moving the cursor to a boundaryof the model matrix will cause the matrix to automatically scroll in thedesired direction. Release the mouse button when the desired sub-model isselected. Models and variables are pasted as described above. To append tothe end of a row or column, drag the cursor to the area after the last entry(even if it means dropping it on the scroll bar.)

When merging Models, the user can choose to also merge data by selectingthe desired option in the following dialog box.



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The test data in the source and destination files can be from different timeperiods before the copy operation. If so, the time periods of the test data inthe destination file after the operation are a union of the original timeperiods in the source and destination files.

When sample intervals are the same in both source and destination files, thesource data overwrites data in the destination file.

Rearranging

Models and

Variables Within a

Given File Using

Drag-Drop

1. Select View>Descriptive Info.

2. From this View, select the appropriate variables (select variables using thestandard mouse click/ctrl/shift options)

3. Press and hold down the left mouse button. Any movement will cause thecursor to change from the standard arrow to a cursor with a curve with a plussign. This is the drag-drop cursor for data. The focus rectangle will followthe cursor as it is moved within the Descriptive Info view. Move the cursoruntil the focus is at the desired position. Move the cursor to the very top ofthe list and the view will automatically scroll up until the first variable hasthe focus rectangle. Move the cursor to the bottom of the list and the viewwill automatically scroll down until the last variable has the focus rectangle.

4. Release the mouse button and the variables will be cut from their oldposition and inserted just before the focus rectangle. Models will beautomatically rearranged to agree with the new variable order. The “Item”descriptor illustrates variable position in the matrix.

Copy Data From

One File To

Another Using

Drag-Drop

If only data (and associated marks, ranges and descriptive information) is to bemoved from one file to another:

1. Arrange the windows so both the source and destination windows arevisible, and select View>Model Descriptive Info on the both windows.

2. From the source window, select the appropriate variables (select variablesusing the standard mouse click/ctrl/shift options)

3. Press and hold down the left mouse button. Any movement will cause the



cursor to change from the standard arrow to a cursor with a curve with aplus sign. The focus rectangle will follow the cursor as it is moved withinthe Descriptive Info view. As the cursor leaves the source window it willchange to the no-drop status. When the cursor is positioned over a windowdisplaying the Descriptive Info view, the window will automatically bebrought to the foreground (top of the stack) and the design studio willreflect that this window has the current focus. The cursor will remain in theno drop state until it is positioned over the text in the Descriptive Info view.It will then change to the appropriate drag-drop status. The focus rectanglewill follow the cursor as it is moved within the Descriptive Info view. Movethe cursor until the focus is at the desired position. Move the cursor to thevery top of the list and the view will automatically scroll up until the firstvariable has the focus rectangle. Move the cursor to the bottom of the listand the view will automatically scroll down until the last variable has thefocus rectangle.

4. Release the mouse button and the variables will inserted just before thefocus rectangle. Empty models will be inserted as appropriate to agree withthe new variable order. The “Item” descriptor illustrates variable position inthe matrix.

Typically, this merge option is performed prior to any model computations.

Merging Data As a first step in any merge operation, the sample rates in the source anddestination files are compared. If the average difference in percent is greater thanDTTol, then the following dialog box will be displayed.

If the difference is greater than DTTol, which can be specified in the .ini file, thenthe sample rates are considered different and the data can not be merged. If thedeference is less than DTTol then the sample rates are considered to be the sameand the merge operation can proceed. If the difference is less than DTTol but thesample rates are not equal then the following dialog box appears.



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If data is to be copied from one file to another and data for the same variableexists in both the source and destination files, then the data must be splicedtogether. This is accomplished using internal time stamps and the user suppliedanswer to the following dialog box.

Results depend on which option is selected as follows:

• Drop Data – The destination file contains no data

• Pad with NaN – Any missing data is represented as not a number (NaN).This can have a significant impact on any future identification or predictioncalculations since operations on NaN result in NaNs.

• Pad with Last/First Good value- Any missing MV/DV data is replacedwith the last/first valid data as appropriate. For CVs missing data is stillrepresented as NaN. This option should in general be used if identification orprediction calculations are to be performed on the merged data.



Merging Data

Marks/Selection

Ranges

When data is merged, any and all data marks and selection ranges are alsomerged. The time periods of the marks/selection ranges in the destination fileafter the operation are a union of the original marks/ranges in the source anddestination files. As with the data itself, the merge is based on the time stamps inthe source and destination files. Marks/ranges will be collapsed reflecting anycollapse in data due to consecutive NaN values in all data.

The following case illustrates this procedure. Here, file w1 is the destination fileand w3 is the source file. File w1 as shown is prior to the merge. File w4 is thedestination file after the merge operation.

File w1 has one CV and one MV. File w3 has one CV, which has the sametagname as the CV in file w1. It also has one MV, which is unique. The data infile w3 was collected a week after the data collected in filew1. There are globalNaN marks, regression NaN marks and range selections in both w1 and w3.

In addition to the marks/ranges, file w3 also has some missing data (actual baddata) in both the CV and MV. This missing data however is not consistent forboth variables. There are however overlaps of missing data in two regions.

File w4 shows the results of the merge. Since MV1 did not appear in file w3 itwas padded to the last sample time with the last good value available from theactual data in the destination file (padding option is user definable). MV2 on theother hand did not exist in the original destination file. This variable was paddedat the front end with the first good value from the data in the source file.

While a week separated the collection of these two sets of data, this time period isrepresented by a single discontinuity in file w4. This discontinuity is representedby a single NaN (for each variable) and is illustrated by the vertical dash-dot lineshown above. Small circles are always displayed around the last good valuepreceding a bad value and the first good value following a bad value. Index 333corresponds to the first good value after the discontinuity and has a time stamp of



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midnight on the 17th of September. Index 331 corresponds to the last good valueprior to the discontinuity and has a time stamp of 5:30 a.m. the 6th of September.Index 332 corresponds to the actual discontinuity. Its time stamp, like alldiscontinuities, is one sample interval after the time stamp immediately precedingit.

In file w3, there are segments of bad data but no discontinuities. This implies thatthe time stamps are all consecutive. For the CV, there are bad values in thefollowing ranges. From 1:34 a.m. – 2:08 and from 3:43 a.m. – 4:04. For the MVthere are bad values in the following ranges. From 1:39 a.m. – 2:04 and from 3:31a.m. – 3:55. Thus for all variables in the source file there are no legitimate valuesduring the times: 1:39 a.m. – 2:04 and 3:43 a.m. – 3:55. These values cantherefore be removed during any merge or reordering operation as long as no newlegitimate values are added to the data. In the example shown above there are nonew data during these intervals so the data is collapsed (note that data is paddedafter the collapse). All marks/ranges are subsequently collapsed to be consistentwith the new data as shown above. Thus the final destination file has the minimalset of informative data. Incidentally, if the variables in file w3 were rearranged inany fashion, then the data would be collapsed in a similar manor.


Section 7 — Data Operations

7.1 Overview

In This SectionRead this section to find out how to modify or manipulate raw data. A suite of toolsis available to perform automated and manual operations on any or all raw dataassociated with CVs, MVs, DVs and Auxiliary (Aux) variables.

Basic Functions Data Operations are categorized in two distinct functions:

• Block Manipulations

• Vector Calculations

Block manipulations allow the user to modify multiple variables in a simultaneouslyfashion. These manipulations are performed at a high level using interactivegraphics.

Vector calculations allow the user to perform more detailed computations. Thesecomputations are typically performed on the data associated with a single variable.In these calculations a “path history” is recorded such that the calculations can beeasily recovered at a later date.

Supported

Operations

Currently, the following operations are supported

• Transformations- Ln- Log- Exponential- Power- Special (Polynomial, Piece-wise Linear , Valve Characteristics)

• Filter- Exponential- Butterworth- Zero Phase- User

• Statistics• Outlier Detection and Removal• Edit• Combine Variables

While these options provide the user with a powerful means of manipulating thedata, the potential for misuse can not be overstated. Blind use of some of thesetechniques can lead to degraded performance. Note some techniques can not evenbe implemented in the on-line environment (i.e. zero-phase filter).

Section 7 — Data Operations7.2 Block Manipulations


7.2 Block Manipulations

Invoking Block

Manipulations

Data operation functions are enabled only if there is raw data available and only ifthe current view is model based (they will not be enabled when the current view isassociated with performance or statistics measures). To access BlockManipulations select Data Operations>Block Manipulations from the main menu.

Using this option, the current view is automatically switched to the Single-GraphData Plot view as shown below.

The normal variable selection criterion as described previously applies here.Manipulations will be made only to the ranges selected for the variables displayed.The modeless Block Manipulation dialog box is design to operate in an interactivefashion with the Single-Graph Data Plot view. While the dialog box is displayed,only those menu functions that are directly associated with its operation will beenabled. All functions associated with the Single-Graph Data Plot view asdescribed previously are available.



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Options and Their

Use

Use the radio buttons to select the replacement option. Select <Replace>. Datawithin the selected ranges will be overwritten as shown below.

Replacement options have the following connotation:

• Value- When user selects this option the edit box will be enabled. The singlevalue entered in the box will be used to overwrite the selected ranges. Thisvalue is initialized to NaN.

• NaN(s)- All data will be set bad for the ranges selected. Data will not becollapsed if all variables are selected. Under this condition a warning messagewill be displayed. For this operation use the delete function.

• Previous Value(s)- Use the value immediately preceding each range tooverwrite the selected data. A warning message will be displayed if there is nopreceding value.

• Next Value(s)- Use the value immediately following each range to overwritethe selected data. A warning message will be displayed if there is no trailingvalue.

• Interpolated Value(s)- Use the value immediately preceding andimmediately following each range to define the linear relationship used in theoverwrite. This is the case shown above.

• Original Value(s)- Data within the ranges are set back to their originalvalues.



Modifications can continue in an iterative fashion. Reselect ranges. Replace data.Zoom in. Reselect ranges again. Replace data. Continue until satisfied. Select<Undo All> to remove all manipulations. The original data will be recovered forthe entire data set irrespective of the current range selection. Select <Close> tohave your changes take effect. When this is done the data in the document will bemodified.

When the dialog box is closed, the main menu will be properly enabled and focuswill be returned to the Descriptive Info view if it had the focus when the procedurebegan. If any other view was used to initiate the manipulations then Single-GraphData Plot will remain as the current view.

Section 7 — Data Operations7.3 Vector Calculations


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7.3 Vector Calculations

Source and

Destination

Variables-

Remembering

Past events

Source and destination variables have a significant role in the Vector Calculations.As its name implies, a source variable is used as input to the calculation function.The destination variable is the result of one or more calculations that areperformed on the source variable’s data. In most cases there is a single sourcevariable. The function to combine variables and the function to perform specialtransformations supports more than one source variable. There is never more thanone destination variable.

Source variables must already exist in the document. Destination variables aredynamically created to support the vector calculations. They are initiallytemporary variables of class Aux. When computations are complete, the temporaryAux variable can be saved to a permanent variable of class Var or of class Aux.The choice is up to the user. These variables can then be used as source variablesin future vector calculations.

Internally, all pertinent information defining vector operations that have beenperformed to obtain a particular variable are stored in a VecTool object. Theseobjects contain, in addition to the calculation information, the source anddestination information necessary to reconstruct the calculation. Reconstruction isalways done from the destination variable. In this sense, the VecTool object can beconsidered loosely bound to the destination variable.

Invoking Vector

Calculations

To access Vector calculations select Data Operations>Vector Calculations fromthe main menu as shown below.

When this option is selected the current view will be automatically switched to theDescriptive Info view and the result will be similar to that given below.



Note that the Descriptive Info view is now disabled but visible. The modelessVector Calculations dialog box controls all significant functions at this stage.Virtually all menu functions are now disabled. The selected variable shown aboveis the current source variable for the vector calculations. The initial selection stateis based on the selection state of the Descriptive Info view prior to it beingdisabled. The initial source variable is the first variable on the original selectionlist. If no variables are selected then the first CV is chosen. To change the sourcevariable use the drop-down list box under Source Variable Selection as shownbelow

This selection is reflected in the disabled Descriptive Info view. Note that whenthe Vector Calculations dialog box is closed, the Descriptive Info view willbecome enabled and the selection state prior to disabling will be recovered.

The primary function of the Vector Calculation dialog box are:

• Select Source variables

• Invoke Vector Functions (performs the actual calculations)

• Save Destination variable.

Temporary destination variables are created when the Vector Function button is



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selected. This variable is initialized with a copy of the data in the source variable.Save options are enabled only when a temporary destination variable exists. Whenthe save options are enabled, you can create a permanent variable of class Aux orVar by selecting either <Save to Aux> or <Save to Var> respectively. The saveoperation will use the name displayed in the Name of Destination Variable editbox. This name is initialized to the source variable name. You can type in anyname you choose.

Vector Functions When the vector functions are invoked, the view is automatically changed toSingle-Graph Data Plot and the environment takes the following form.

Both source and destination variables are always displayed in the fully interactivegraphic view. The destination variable at this stage is temporary and is alwaysnamed “VectorCalc”. User interaction is restricted to selecting and evaluatingfunctions and the data. General operation is as follows:

Select the general function from the tabbed dialog box

1. Use the radio buttons and any auxiliary edit or option boxes to select



specific functions

2. Select <Evaluate and Plot>- Results will be stored in the VectorCalcvariable and displayed in the Single-Graph Data Plot

3. Use the plot to analyze the data. All plot options are fully functional

4. Repeat step 1-4 until satisfied

5. Select <OK> to store destination data and all current dialog settings.Select <Cancel> to loose this information. Both options will closethe vector function dialog box and return focus to the VectorCalculation dialog box and descriptive view.

More detailed descriptions of the functions and their use follow.

Transformations Data given above will be used to begin the discussion on the use of thetransformations. This data is in fact the result of a prior vector operation. It is thedifference between a predicted and actual CV. The variable was constructed byusing the function “Combine Variables.

Transformations are as listed below:

Operation and selection of the Standard transformations (first four radio buttons)should be self-evident. The only restriction imposed on a transformation in thedesign studio is that it must be a monotonic function.

It is important to note that transformations that are to be used with ProfitController must be monotonically increasing. Select <View/Set Ranges> tospecify the ranges over which you wish to use the transformation in conjunctionwith Profit Controller. On-line transformations WILL be limited to these ranges.If the Clamp Ranges check box is selected, values outside these ranges will be setequal to the range limit (clamped). Otherwise, the value will be set bad (NaN).

For the plot shown above, the data ranges from a low of -.444 to a high of .321. Ifit is desired to use the transformation over a broader range than that contained inthe data (you can’t use a range more restrictive), the use the View/Set Ranges



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button as shown below.

Here, the ranges have been extended to 1± . The stored limits are the values thatwill be used in the On-line transformation. The current limits are used when theEvaluate & Plot button is selected to insure a monotonically increasing functionover these ranges. Once successfully evaluated the stored limits are set equal to thecurrent limits.

When there is a potential problem the following message will be displayed.

Select <Evaluate and Plot> to transform the prediction error using the selectionstate given above. This results in the following data.

Values that are undefined or out of range are simply treated as NaNs. To save the



transformation and the transformed data select <OK>. If at this stage there is aproblem, the following message box will be displayed.

The EuLo/EuHi values are the ranges established by the data or the ranges inputby the user. These values define the input limits on the variable to be transformed.This message tells the user that the function is not monotonically increasing overthe stated ranges and that the transformation can NOT be used in the on-lineenvironment.



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Special

Transformations

Construction and use of the special transformation is more involved than thepredefined transformations. To use these transformations, select the Special radiobutton and use the drop-down menu to select the desired transformation as shownbelow.

Currently, three special transformations are provided:

• Polynomial – Polynomial regression fit to data

• Piecewise Linear – User defined linear segments overlaid on data.Smoothing option available.

• ValveCurve. Installed valve characteristics overlaid on data.Characteristics taken from Perry p. 22-83.

Use of the special transformation is intended for linearization purposes. Sincetemporal variations are not included in these transformations, static regression datais implied. With these transformations it is up to the user to define the specifictransformation. To do this, select one of the drop-down options. When this is donethe Define button will be enabled. You must select this button to define thetransformation. If no transformation is defined and the evaluation is performed theresult will be a zero valued vector.

Polynomial Choose polynomial then select <Define>. The following dialog box will bedisplayed

While somewhat involved, the fundamental purpose of this dialog box is to specifya polynomial for use as a transformation. The intent is to take input/output data atdifferent operating points and plot the output or dependent variable against the



input or independent variable. A polynomial can then be fit to this data

At this level no restrictions are placed on the definition of the dependent andindependent variables. Use the drop-down list box to select any or all variablesfrom the document. Any variable that is selected from the drop-down box will beadded to the variable list box. To remove a variable from the list box select<Delete Variable>. The selected variable will be removed.

All data displayed in the dialog box will be based on the two variables displayed inthe list box. Use the up or down arrow key to change the variables displayed in thelist box. Scatter plots will automatically be plotted for the two variables displayedin the list box. The selected variable is the independent variable (typically, but notrestricted to, the variable to be transformed). When scatter plots are displayed theFitPoly button will be enabled as will the User max/min y edit and MonotonicCheck boxes. The Max and Min ranges are the data ranges of the displayedvariables. The User ranges define the axes on the plot box.

It is important to note that with all of the special transformations, the data rangesdisplayed on the plot define the maximum allowable data ranges for use in theOn-line operation of the transformation. The View/Set Ranges dialog box shownpreviously does NOT apply to special transformations.

Polynomials are usually defined by selecting the FitPoly button after theappropriate order has been selected. Polynomials can also be entered manually byselecting the order and specifying the coefficients in the edit box. In fact, if there isonly one variable present or there is no data, then this is the only way to define thepolynomial. The polynomial always has the following form.

�=

+=Order

i

ii xccy

1

0

In the above expression y is the dependent variable and x is the independentvariable. Set the following.

0.1,0,3 3210 ===== ccccOrder

(At this stage the polynomial coefficients are not scaled. Once a fit is performedthe coefficients are automatically scaled. Select the Clear button to set allcoefficients to zero and eliminate scaling). Select <x-y Plot>. This will force anupdate of the plot. The following results will be displayed.



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The blue curve is the polynomial transformation and the green curve is its inverse.The inverse would result if the axes were switched.

At this point the Polynomial transformation has been manually defined. You couldselect the accept button and continue from here. Usually it is desired to use data todefine the transformation. To do this another variable can be selected using thedrop-down list box. If another variable is selected and a fit done, the result of thisoperation is shown in the following picture.



In this case the polynomial was fit to the observed data. For a good fit a higherorder polynomial was required. While the fit shown above is relative good, theresults can’t be used with the on-line RMPCT controller since the polynomial isnot monotonic. When there is a problem with the fit or its inverse, a message boxsuch as that shown below will be displayed

Higher order polynomials are notorious for their erratic behavior. To circumventthis problem, which many times will result in unacceptable non-monotonicresponse, a Monotonic option has been added to the polyfit routine. This defaultoption results in a polynomial fit that satisfies a set of gradient constraints. Thegradients are calculated using a grid distributed uniformly over the data range ofthe independent variable. With the Monotonic option selected the results shownabove become.



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To see the inverse fit switch the axes by selecting the second variable in the listbox. This results in the following plot.

It is a reasonable practice to use as low an order as possible. If the variables areswitched back to their original position (which variable is independent and whichis dependent is extremely important) and the order is reduced to 7, then thefollowing results are obtained.



If this transformation is to be used on-line and the variable to be transformedactually varies between –.5 and 2.5 then these values MUST be entered at thispoint otherwise the input value will be clamped between 0 and 2.5. To do this,simply enter the desired value in the appropriate edit box. In this case enter –.5 inthe User min x edit box.

Note: Any time a value is changed in any edit box it is strongly recommended toimmediately update the data and plot by selecting the <x-y Plot> button.

For this example the following results are obtained.

Here it is obvious that the extrapolation results in a non-monotonic function (Inspite of the fact that the function is monotonic over the actual data range). Thiscurve is not a monotonic function and therefore can not be used with the on-lineRMPCT controller since its inverse is not unique in the specified range. It ispossible to tailor this curve by adding fictitious “user” data to the scatter plot. Todo this, change the range so you can shape the curve. Then hold the left mousebutton down and move the cursor over the plot. When it is over the plot the cursorwill change to a cross hair. Release the left mouse button and a cyan circle will bedrawn at the position of the cursor. Continue clicking the left mouse button to addmore data at the desired positions. Repeat the fit of the data. Iterate through theprocess of adding data and refitting until the curve is monotonic with the desiredshape. For the data given above the procedure results in the following plot.



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This curve is now monotonic and can be used with the on-line RMPCT controllerif desired. To remove user data hold the right mouse button down and move thecursor over the plot. When it is over the plot the cursor will change to a cross hair.Release the right mouse button and the user data point closest to the cursor will beremoved. Continue this process until all desired user points are removed. You cannot remove actual data points in this fashion.

Note. User data can be added only when there are just two variables in theselection list. While user data is present, variables can neither be added to norremoved from the selection list

At this point the data can be saved or further modifications can be made. To seehow the inverse transformation fits the data over the extended range, select thedependent variable from the selection list (this causes a swap in the dependent andindependent variables as shown below).



Changing ranges or adjusting user data in the swapped state is NOTaccommodated. Switch back to the original independent variable or refit the databefore trying to change data.

If the ranges are adjusted in the swapped state and the update plot button isselected, the following message will appear.

Similarly, if the ranges are adjusted in the swapped state and another variable isselected, the following message will appear.

When desired results are achieved and the Accept button is enabled, thepolynomial can be saved. When a polynomial is saved, the user data is stored inthe associated VecTool object for possible reconstruction purposes. This data isnever added to any source or destination variable

In the creation of the polynomial transformation, no restrictions are placed on theselection of the dependent and independent variables. However, if thetransformation is to be used with Profit Controller then the independent variableshould always be the same as the source variable.

Next, consider further extension of the ranges for the same data. Here theminimum value of x is set at -.75 while the maximum and minimum value of y are4 and -.3 respectively. In this case the transformation exhibits a slight rise at thehigh end and a slight dip at the low end as shown in the following picture.



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To remove the dip and rise in the transformation more data can be added but it isusually far easier just to redefine the ranges. The following curves are displayed byspecifying the maximum and minimum value of x to be 2.3 and -.1 respectively.



Thus, in the on-line operation of this transformation (when ranges are clamped),values of the input variable less than -.1 will be taken to be -.1 and have a forwardtransformation of approximately 0. Similarly, values of the input variable greaterthan 2.3 will be taken to be 2.3 and have a forward transformation ofapproximately 3.8. For the inverse transformation, values less than -.3 will betaken to be -.3 and will yield an inverse of -.1. Similarly, values greater than 4 willbe taken to be 4 and will yield an inverse of 2.3.

As this procedure illustrates, the user is free to define ranges inside actual data.

Note, the ‘FitPoly’ function uses ALL data irrespective of the current data ranges

To recover data ranges (both raw and user input) simply select <Data2Usr> asshown below.

Actual data ranges (raw and user input) are displayed by the in the Data Rangesbox by the Max x, Min x, Max y and Min y descriptors. These parameters areunaltered (when data is present) by the modification to the User max and minvalues define the graphical display and the limits used in the forward and reverse(inverse) transformations. What is displayed graphically is precisely what is usedfor the forward and reverse transformation. For the plot shown above, the forwardtransformation will have a minimum and maximum value of -.068051 and 3.9793respectively.

To evaluate the polynomial select <Accept>. When this button is selected, thePolyFit dialog box will be closed and focus will be returned to the VectorFunctions level with access to the Single-Graph Data Plots. Select <Evaluate andPlot>. Observe the source and destination variables. Modify the transformation as



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necessary. When satisfied with the results, select <OK>. This will allow you tosave the destination variable and all related information. If the cancel button isselected, then the temporary destination variable and all associated informationwill be lost. If the destination variable is saved, then the effectiveness of thetransformation can be observed by using the Scatter Plot view from the mainmenu. For this example, the scatter plots are as shown below.

Here Aux1 is the transformed variable. The first plot is essentially the same as theinverse transformation shown previously in the FitPoly dialog box. Theeffectiveness of the transformation is illustrated by the almost linear characteristicsdisplayed in the second plot.

Piecewise Linear Choose Piecewise Linear then select <Define>. The following dialog box will bedisplayed



This transformation is fundamentally different from the polynomial transformationin that here the function is defined irrespective of the data and in normalizedcoordinates. While all data (raw data, transformations and smoothing functions)are stored in normalized coordinates, data can be displayed/entered in engineeringunits by simply deselecting the Display Normalized Data checkbox.

Note: Ranges are always entered in engineering units.

Since this transformation is defined and stored in normalized coordinates, itimplies that both input and output (transformed) variables are scaled. If the inputand output variables are defined as u and y respectively and the correspondingscaled variables are u ′ and y ′ . Then the transformation becomes:

)(ufy ′=′

Where the input to the transformation is given by:

minmax

min

uu

uuu

−−

=′

And the resulting output in engineering units is given by:

minminmax )( yyyyy +′−=

This dialog box is used to define )(uf ′ and the scaling. . The scaling is simply

defined by entering the expected ranges into the user defined edit boxes. Thesevalues should be the maximum and minimum values that are expected to occur inthe on-line process. These values will have NO effect on the entered function. Ifthe physical process is self-similar then this is precisely the desired effect. Forexample two valves of the same type and characteristics but of different sizeswould likely have self-similar characteristics with respect to normalized flow vs.normalized stem travel. The same normalized function will work for both cases.The ranges simply need to be redefined.

In the normalized mode all function values are displayed/entered between 0 and100. To display/enter values in engineering units, deselect the Display NormalizedData checkbox.

Note: Make sure the ranges are set PRIOR to entering values in engineering units.Since the transformation are stored in normalized coordinates, a change in theranges will result in an appropriate modification of the functional points inengineering units to maintain a self-similar shape.



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A flexible graphical interface is provided to define the piecewise linear function(any part or all of the function can also be entered manually) The function isdefined by adding or deleting line segments. The function is always initialized by asingle segment. Its two endpoints can not be changed. To add a segmentgraphically, hold the left mouse button down and move the cursor over the plot.When it is over the plot the cursor will change to a cross hair. As the cursor ismoved (without lifting up the left mouse button) a red circle will track the crosshair. The x-y coordinates of this target position will be displayed in the dialog box.In addition connecting cyan segments between the red circle and its nearestneighbor points on the existing function will indicate the potential results if the leftmouse button is released. The red circle and its two associated cyan segments willtrack the cursor as long as the mouse is in a legitimate position (the entire functionis guaranteed to be monotonically increasing). A picture of this procedure isshown below without the cursor.

The blue line represents the existing piecewise linear segments and the blue circlesrepresent the segment endpoints (it is actually endpoints that are added to thefunction). The cyan curve represents the new shape of the function if the leftmouse button is released over a legitimate point. Release the left mouse button andthe target curve becomes real. Continue adding segments until the functionexhibits the desired shape. After adding a few points the function may look asshown below.



Select the smooth function check box to smooth the transformation. As long as thebox is checked and the function can be properly smoothed, a magenta curve willbe displayed such as that shown in the following picture.

With the smooth box checked, the function will be continually smoothed assegments are moved, added or deleted.

If a smoothed curve is present when the Accept button is selected, then thesmoothed function will be used to perform the transformation. To use the actualpiecewise linear profile, turn the smoothing function off before accepting.

With two or more variables available, all fields are enabled. At this point theHide/Show Data button is enabled. For now Hide Data is selected. To manuallyadjust endpoints use the vertical scrollbar as illustrated below.



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Scroll up or down to increment or decrement the segment index and correspondingx and y positions. The cyan circle will track the corresponding data displayed inthe Piece-wise Data box. Enter the x and y values in the corresponding edit boxedand select <x-y Plot> to update the plot with the entered information. If theinformation is valid the plot will be updated otherwise an appropriate message willbe displayed. To display/enter engineering units, deselect the Display NormalizedData checkbox.

To move a point and associated segments graphically, select <Ctrl> and hold theleft mouse button down and move the cursor over the plot. When it is over the plotthe cursor will change to a cross hair. As the cursor is moved (without lifting upthe left mouse button) a red circle will track the cross hair. The x-y coordinates ofthis target position will be displayed in the dialog box. In addition connecting cyansegments between the red circle and the neighbors nearest to the point to be movedon the existing function will be displayed. The cyan segments will indicate thepotential results if the left mouse button is released. The red circle and its twoassociated cyan segments will track the cursor as long as the mouse is in alegitimate position (the entire function is guaranteed to be monotonicallyincreasing). An example of moving a point and its associated segments is shown inthe next figure (Here the cross-hair cursor is not displayed).



To remove a point and its associated segments, hold the right mouse button downand move the cursor over the plot. When it is over the plot the cursor will changeto a cross hair. As the cursor is moved a single cyan segment will connect the twoneighbor points of the point to be removed as shown below.



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The point to be removed is the point that is closest to the current cursor positionwhich although not shown in the picture is located at the coordinates given by theparameter “Target position” (here given in engineering units). Release the rightmouse button and the point closest to the cursor will be removed. Continue thisprocess until all desired user points are removed. For this example the plot willtake the following form.

This technique can be used to enter arbitrary functions of the user’s choice. Tospecify the function based on data select <Show Data> from the dialog box. Thefunction can then be shaped for the desired data. For the previous data the functionwill look as follows.



In this case either the piecewise liner or smoothed function could be used. To usethe piecewise linear function the Smooth Function checkbox must be deselected. Ajudicious choice of segments can yield an effective smoothed transformation asshown below.

In this case there are not enough segments to accurately represent the data.However, the smoothed function gives a reasonable approximation.

To specify a function that is unrelated to the data select <Hide Data> from thedialog box. It is possible to specify almost any function. An example follows.

In this particular case the smoothed fit is not very accurate. To improve theaccuracy, simply add more points as appropriate. Add, delete or move points untilthe desired shape is achieved. After adding several points to the above function,the following curve is obtained.



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Use of the smoothing function involves a significant calculation (it uses the sameconstrained minimization algorithm that the polynomial fit uses to insure amonotonically increasing function. Here however the order is determined in aniterative fashion). As such, for some complex curves the calculations may take upto a second. When the hour glass is displayed (indicating calculations inprogress), do NOT try to add, move or delete points.

To see the function without data simply select <Hide Data>. The Accept,Data2Usr and Evaluate functions work the same for this function as describedpreviously for the polynomial.

Make sure to select <x-y Plot> when changing scale factors otherwise thechanges will NOT be in effect. When there is data in the environment this shouldbe obvious. Without data however there will be no graphical queue (the text inthe Scale Ranges box will reflect this change).

It is important to realize that when dealing with normalized functions (such aspiecewise linear and valve characteristic to be shown next) that the ranges do NOThave the same effect as they do with non-normalized functions such as describedin the previous polynomial discussion. With non-normalized functions the rangescan be considered to act in a clamping fashion as described previously. This is notthe case with normalized functions. Here the ranges can be used to translate,expand or contract the function relative to the data.

Consider the case where using the data presented above the minimum andmaximum values for the independent variable are constrained to be .75 and 2.0respectively. The corresponding curves for the polynomial and piecewise lineartransformations are shown in the following graph.



Clearly, these curves illustrate the differences between the two approaches. Withthe polynomial, all values of x less than .75 will result in a transformed value of.539 while all values of x greater than 2 will result in a transformed value of 3.8.For the piecewise linear transformation maintains self-similar function form, henceall values of x less than .75 will result in a transformed value of -0.0681 while allvalues of x greater than 2 will result in a transformed value of 3.87. Correspondingscatter plots are shown below.

Next, consider the case where x is not constrained but the minimum and maximumvalues of y are taken to be 1 and 2.5 respectively. The corresponding curves forthe polynomial and piecewise linear transformations are shown in the followinggraph

With the polynomial, all values of x less than .925 will result in a transformedvalue of .1 while all values of x greater than 1.45 will result in a transformed valueof 2.5. For the piecewise linear transformation maintains self-similar functionform, hence all values of x less than 0 will result in a transformed value of 1 whileall values of x greater than 2.5 will result in a transformed value of 2.5.Corresponding scatter plots are shown below.



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Note that in this case the normalized transformation still linearizes the data but thegain (slope) is directly modified by the change in the range.

Installed Valve

Characteristics

Choose Valve Curve then select <Define>. The following dialog box will bedisplayed

It is the intent of this dialog box to provide a simple mechanism to represent thecharacteristics of installed valves. Operation is similar to that discussed in theprevious paragraphs. Here however the transformation is specified by defining theparticular valve characteristics. The characteristics are taken from Perry’sChemical Engineers’ Handbook Sixth Edition. The characteristics are given by:

( )( ) 2121( L

LQ

αα −+=

For linear valves and for parabolic or equal percentage valves by:

( )( ) 214

2

1( L

LQ

αα −+=

In the expression given above Q and L are the fractions of maximum flow and



stem travel respectively in percent. The parameter α is the ratio of valve headdifferential at maximum flow to the valve head differential at zero flow. Thetheoretical range of alpha is between 0 and 1. For the transformations alpha ispermitted to vary between .005 and 10.

To define the transformation, simply select the valve type and the desired value ofalpha. Remember to select <Update Valve Curve> to refresh the plot and to storethe most current user entered information. When satisfied, evaluate and saveresults as prescribed previously.

For the valve characteristics dialog box there is no Data2Usr button. In thisinstance the ranges can never be set inside the actual data ranges. If a value isentered inside a range a warning will be displayed and the entered value will bereset to the appropriate minimum or maximum data value.

Transformations

without Data

In some cases it may be desirable to define transformations even when there is nodata. For these cases the Data Operations will have the following form.

Since there is no data the Block Manipulation option is disabled. Select <VectorCalculations> and then <Vector Functions> in the normal fashion to obtain.



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This dialog box defines the only functions available when there is no data presentin the environment. All transformations work as described previously except nodata will be displayed. Hence, for the polynomial no fit is performed. To use thisoption the polynomial will have to be manually entered. When the transformationis defined simply select OK. In this mode the vector calculation dialog box will be.

Note that the only save button available is Save to Src (Source). Thus thetransformation can only be saved back to the source variable. Creatingtransformations when there is no data is only meaningful for building on-linetransformations for Profit Controller.



Filter Several types of filter calculations are available. Select the Filter tab in the VectorFunctions dialog box. The following options will be displayed.

All filter operations can be performed on discontinuous data or data contaminatedwith bad values (NaNs). Characteristics of these filters are as follows

• Exponential- This standard filter is the default option. Its transferfunction has the following form.

nss

)1(

1)(T

+τ=

Here n is the filter order and τ is the filter time constant in minutes.

• Butterworth- This lowpass filter is design such that its transferfunction:

)())(()(T

21 npspsps

ks

−−−=

�

is characterized by a magnitude response that is maximally flat in thepassband and monotonic overall. In this case n is the filter order andτ defines the cutoff frequency. If the cutoff frequency is greater thanthe Nyquist frequency, then a warning message will be displayed.When this occurs, results may be erratic or even meaningless.

• Zero Phase- This option utilizes a forward/backward filterprocedure that results in a sequence with zero phase distortion. Sincea forward/backward pass is performed this filter is non-causal andcan not be performed on-line. Initial transients are handled by using aflipped reflected copy of the input signal. A special procedure is usedto deal explicitly with NaNs.



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• User- This option is provided to allow the user to explicitly specifythe filter transfer function. Data can be easily shifted forward orbackward in time since both positive and negative delays areadmissible. As such this can be a non-causal filter. Both proper andsemi-proper transfer functions can be specified.

When the Evaluate and Plot button is selected to filter the data, the filter transferfunctions for both the analog and discrete designs will be printed to the messagewindow (This window will be discussed in a subsequent chapter).

Use of the exponential filter is illustrated below. In this case the source variablehas been marked bad over an intermediate region (index 91 – 114) and at the point251. The destination variable (named VectorCalc) is the result of a copy operationon the source variable. As such the painted values appear as actual bad values inthe destination variable.



Results of the filter operation obtained by selecting <Evaluate and Plot> are asfollows. Note that with Vector Operations bad value data marks are displayed evenin a non normalized plot mode. This is an exception to the general rule.

When displaying filter results it is convenient to switch to either the Multiple orSingle Scale options so that the source and destination (filtered) variables areplotted on top of each other. Here the single scale option has been chosen since theranges are displayed on the vertical axis. Plot ranges for the VectorCalc variabledepend on which operation is currently being performed.

For Filter and Edit Data operations, the plot ranges are not modified. For all otheroperations the plot ranges are reset to the data ranges each time the function isevaluated

Phase distortion due to the exponential filter is self evident in the graph shownabove. To remove this distortion, select <Zero Phase> and select <Evaluate andPlot>.The results are shown below.



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To specify the filter manually, select <User>, then select <Enter Tf>. The UserFilter Transfer Function dialog box shown below will be displayed.

Discussion of the transfer function and its use is given in detail in a subsequentchapter. Suffice it to say that the entered transfer function is equivalent to that usedin the Exponential filter. To shift the data back in time 6 minutes enter a 6 in thedelay edit box and select <Calculate>. This gives.

Take the default option and the transfer function is displayed as:



Select <Exit> to close the User Filter Function dialog box and select <Evaluateand Plot> to obtain the following results.

As a final case illustrating the filter operation, data will be filtered with a usersupplied transfer function with a non-unity gain. In this case the transfer functionis:

)13)(13(

5.)(

++=

sssT



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The input and output data for this case are presented in the next plot.

At first glance it looks like data is missing from the second part of the graph. If theplot is changed to normalized view, the result becomes apparent.

Notice that the two horizontal lines are identical. What happened to the filter inthis case? The top curve is the input to the filter while the bottom curve is theoutput. The first half of the plot looks as expected. The input however suffers adiscontinuity. After this time there is no movement in the input. Hence there willbe no movement in the output. The value of the output is arbitrary. Its bias value isbased on the input for the associated segment. Note that it is not related to thevalue of the output during prior segments (time before the discontinuity). Hence,the output value is free to change across discontinuities. This characteristic isrequired to handle filtering over general discontinuities as illustrated previously.



Statistics Mean removal, normalization and information on range, mean and variance can beobtained by selecting the Statistic tab on the Vector Functions dialog box. Whenthis tab is selected the following information is displayed

When the Evaluate Plot button is selected the source data is copied to thedestination variable and the functions corresponding to the selection state of radiobuttons are applied to the destination variable. Current values reflect the state ofthe destination variable

Use of this function will overwrite the destination variable. This function is notintended to provide statistics on the results of prior calculations.



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Edit Data A mechanism is provided for outlier detection and removal and to manuallymodify data. To perform these functions, select the Edit Data tab on the VectorFunctions dialog box. This results in the following display.

At this point two options are available; Outlier Detection and Removal andManual Data Manipulation.

Auto Outlier Detection and Removal- This option can be used to removestatistically unreliable data. Adjust the confidence level used in the detection byselecting the up or down arrows. The default is usually adequate. Next select thereplacement option. An example using the default options is shown below.



Manual Data Manipulation- This option allows you to edit data in an interactivefashion.

This function is the one exception to the vector function calculation sequence. Inall other functions the source variable is the input and the destination variable isthe output. Here the operations are performed directly on the destination variable.This means that you can modify the results of a prior vector calculation by usingthe Manual Data Manipulation function

To modify the results of the previous outlier removal calculations select <ManualData Manipulation>. Then select <Alter Data>. The following dialog box will bedisplayed.

In the picture presented above, the Manual Data Manipulation dialog box is showntogether with a zoomed in view of the pertinent data.



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For this example it is desired to extend the prior interpolation from index 176 tothe encircled point which has a value of .3682 and an index of 189.

To do this it is useful to understand the design and operation of this dialog box.Index, is the position of a particular data point. This index corresponds to thevalue displayed in the time axis box of the Single-Graph Data Plot. The firstcolumn of data corresponds to the source variable. It can not be changed. Thesecond column of data corresponds to a local copy of the destination variable. Thisis the variable that will be modified. Since it is a copy, you can modify it andselect cancel without changing the destination variable.

Replacement options are identical to those described in the section on blockmanipulations and will not be repeated here. The replacement strategy is:

1. Select the appropriate rows in the data window. This is a standardmultiple selection list box. Use standard click/drag/shift etc.

2. Select the desired replacement option.

3. Select <Replace>. All selected rows will be modified using thedesired replacement option.

Search options have been provided to facilitate the data selection process. Theseoptions are as follows.

• Find Index- Enter the index of the data point you wish to find. Getthis index by using the right mouse button in the Single-Graph Dataplot. Select <Find Index> and the data window will be automaticallyscrolled to display this index and a focus rectangle will be drawnaround this row

• Find Next- Enter the next value you wish to find. Select Find Next>.A string search is used such that if you enter .383 and the actualvalue was .383246, the search will be successful and the datawindow will be automatically scrolled to display this value and afocus rectangle will be drawn around this row. Both current andsource variables are searched. Repeated values are skipped. If .383appears 10 consecutive times, the first occurrence will be found onthe first search while the last occurrence will be found in the secondsearch. NaN is a legitimate search value. When the end of the list isreached, the search will prompt to begin again at the top of the list

• Find Previous- Same as above but in the opposite direction.

• Stop Find- Terminates the search.

Once the current values have been modified, select <Plot Data> to display theresults. At this stage you can still cancel without making any real modifications.Continue making modifications until you are satisfied, then select <OK> Thecurrent values are now used to overwrite the destination variable.



For the example problem, enter 177 (we want to use index 176 as an endpoint inthe interpolation) in the index edit box. Select <Find Index>. Select the focus row.Enter 188 into the index edit box. Select <Find Index>. Ctrl/Shift click in thefocus row. Select <Interpolate>. Select <Replace> and the dialog box has thefollowing appearance.

Select <Plot Data> and the result is shown below.



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Combine

Variables

In some instances it may be necessary to combine two or more variables. To dothis select the Combine Variables tab on the Vector Functions dialog box. Thenselect the variables you wish to combine. This operation may look something likethat shown below.

Add variables to the list using the drop-down selection box. Delete variables fromthe list by first selecting the variable in the list box. Then select <DeleteVariable>. As shown above, all Variables in the list box are displayed in the Plotwindow. To change an operation, first select the variable in the list box. Thenselect the desired Operation Type.



Variables are combined using the following operational sequence

Bias))Con())Con())Con())Con((((( 44332211 +⊕⊕⊕⊕ �� xxxx

÷∗+⊕ ,,-,operatorsstandardtheofanyimpliesWhere

When a new variable is added, it is appended to the list. When a variable isdeleted, the list is collapsed. For the problem shown above the destination variableis calculated as:

0.0CV0.1))CV0.2())CV0.1())CV0.1(((( 13517 ++÷∗−=y

User Notes It is also possible to annotate your work. To do this select the User Notes tab onthe Vector Functions dialog box. This gives you access to the following interface.

Simply enter a description of any notes you may wish to keep. You can also lookat notes you made for other variables.



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Saving and

Recovering Vector

Calculations

As described previously, temporary destination variables are created when theVector Function button is selected. If the Vector Function dialog box is closed byselecting the Cancel button, then this variable, its data and associated VecToolobject will be destroyed. If the OK button is used to close the dialog box, then theVector Calculation dialog box will have the following appearance.

Whenever the temporary destination variable exists, the Save buttons will beenabled.

Hint. You can make a copy of any variable in the environment by doing thefollowing

1. Select the variable

2. Select <Vector Functions>

3. Do nothing in the Vector Functions dialog box and select <OK>

4. Select <Save to Var> or <Save to Aux>

When you save the results of a Vector Calculation both the destination variableand an associated VecTool object will be saved. The VecTool contains allinformation associated with the vector operation and allows you to easily modifyor reconstruct a prior calculation.

Use the Save Option choices to configure the save procedure. “Name ofDestination Variable” is the name of the permanent variable to which thetemporary destination variable will be copied. For new vector calculations, thisname is initialized using the name of the source variable. You are free to change



this name to whatever you desire.

Two save options are available when data is present.

• Save to Aux- With this choice the data will be copied into a variable with thename specified in the edit box of class type Aux. If the variable does not existit will be created and a message will be displayed telling you that a variablewith the specified name has been added to the workspace. If it already exist,and is not the source for another calculation, the following message will bedisplayed.

Remember that Aux variables can not be merged or reordered.

• Save to Var- With this choice the data will be copied into a variable with thename specified in the edit box of class type Var. If the variable does not existit will be created and a message will be displayed telling you that a variablewith the specified name has been added to the workspace. If it already existsthen the following dialog box will be displayed.

These options are as follows.

- Overwrite Var and copy to Aux- This option first copies the datafrom the existing Var variable to the Aux variable. If the Aux variabledoesn’t exist it creates it. If it does exist then it asks if it is ok tooverwrite. It then copies the destination variable to the existing Varvariable. It also modifies the VecTool objects as appropriate to reflectthese changes.

- Overwrite with no copy- In this case, the existing variable is simplyoverwritten with the data in the destination variable. Information



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associated with the calculations is destroyed. Links to any othervariables are removed. The associated VecTool object is destroyed andhence the calculations can NOT be reconstructed. The end result ofusing this option is to simply modify the data with no record of anyoperations.

- Cancel overwrite procedure- When this option is chosen no changesare made to the workspace.

Once a save is performed, the temporary destination variable is deleted and thesave buttons are disabled.

Since connections and operations are stored in the VecTool object, it is easy toreconstruct a Vector calculation. Simply select the variable from the descriptiveInfo view and select Data Operations>Vector Calculations. For the CombineVariable example presented above this will give the following results.

Anytime a variable is selected, the appropriate VecTool object is queried to see ifit is the result of a prior calculation.

Tip – Use the drop-down selection box shown above to see if a variable is theresult of a prior calculation.

At this point you can either reconstruct a prior calculation by selecting yes to themessage box or you can use the selected variable as a source in a new calculationby selecting no to the message box. By selecting no multiple calculation sequencescan be performed



Selecting yes for the case shown above, results in the following

In this case the four variables that were previously combined are automaticallyselected. The first variable in the combination list (CV7) is displayed as the sourcevariable since there is only one field for this parameter (Usually there will only beone source variable). Select <Vector Function> to obtain the following results.

Which is the last state for this set of calculations. Note that all data contained inthe Vector Functions dialog box is recovered. This implies for example that, if apolynomial transformation was performed, the polynomial and its last plot statewill be recovered when the FitPoly button is selected.

Since links between variables can become involved, internal checks are made toprevent the user from inadvertently destroying source variable information. If theuser tries to delete a source variable, then the following message will be displayed.



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A similar situation exists if for example you edit the name of a variable thathappens to be a source variable. In this case the message is.

If a source variable is in fact deleted and you try to reconstruct the vectorcalculation, then you will get this message.

In some instances you may want to remove the effect of a calculation. Take thecase where a Var variable is filtered and its source is saved as an Aux variable.You can restore the original variable to its initial state by doing the following.

1. Select the Aux variable

2. Select <Vector Functions> Do no calculations and select <OK>. Younow have a copy of the original variable

3. Enter the original variable name in the edit box

4. Select <Save to Var>

5. Select <Yes> to the overwrite message box

At this point the original data will be restored but there will be a linkto the Aux variable (it will consider the Aux variable as a source).To delete the link

1. Select the Var variable.

2. Select <No> to the source message box

3. Select <Vector Functions> Do no calculations and select <OK>.

4. Select <Save to Var>

5. Select <Overwrite with no copy>

This Var variable is now returned to its initial state. If it is selected no source isrecognized. If Vector Functions are invoked, then all tabbed dialog boxes are intheir initial state.



A similar situation can exist for Aux variables. When an Aux variable is the resultof a prior calculation and the prior settings are ignored, the Aux becomes the newsource. If after the calculations are complete and the results are selected to bestored back into the same Aux variable used for the source, then the followingmessage box will be displayed.

Since the original VecTool object will be destroyed, information associated withthe calculations is destroyed. Links to any other variables are removed. Hence thecalculations can NOT be reconstructed. The end result of selecting yes is to simplymodify the data with no record of any operations. Note that if the source variablewas not the original Aux variable, then only the data would be overwritten and theVecTool object would remain intact. In this case the message box would be thatshown previously in the discussion on Save to Aux.

If you try to save any variable to another variable which is a source variable inanother calculation, then the following message will appear.

Selecting ‘Yes’ at this point will cause the existing data to be overwritten. Thefollowing dialog box will then be displayed.

This options allows the user to keep or destroy the VecTool Object.



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Merging Vector

Calculations

Merging the result of Vector Calculations or more appropriately merging VecToolobjects is automatically accomplished in an implicit fashion. Anytime a variable ismerged its associated VecTool object is also merged. Since a single calculationusually relates at least two variables, a merge may result in an incompatibility inthe destination file if not all the appropriate variables are merged. Consider a casesimilar to the combined variable example given previously. If only one sourcevariable is merged with the combined variable into a destination file and if thecombined variable is selected for a Vector Operation, the following message willbe displayed.

This message tells the source file of the original calculation. It will be displayeduntil a new set of calculations are performed and saved in the destination file. If allthe variables involved in the original Vector Calculation are not present in thedestination file, then the following message will be displayed



If yes is selected, then the missing variables will be displayed as given below.

Missing variables can be merged into the destination file at any time. The displayshown above will reflect the current state of the destination file as variables areadded to or removed from the file. Vector Calculations can be performed at anytime regardless of the state of the destination file.

Be careful when merging data to a variable that is a result of a VecTooloperation. The data merged will NOT be automatically updated with thecalculation. In this instance, the vector calculation must be redone.

Saving

Transformation

For Use with

Profit Controller

Any of the transformation described above can be used with the profit controller aslong as they are monotonically increasing (only CV transformations are currentlysupported in the on-line environment). These functions and their ranges ofapplicability (defined by the user limits entered as discussed above) can beautomatically instantiated into an on-line module via the Profit Controller PointBuilder.

The Profit Controller Point Builder allows an engineer to enter control design dataor import an .mdl file into a graphical user interface on the PC. Depending on theoptions chosen, the Point Builder then generates files that can be transferred to aTPS system to be used to automatically construct the profit controller points. Itgenerates the necessary configuration files, too. The application also determinesthe scheduling of the various points, and ensures that AM loading is balanced bydistributing the points among various execution cycles. For a full description ofthe Point Builder see section 9 of the Profit Controller Designer’s Guide.

Any CV transformation saved in a .mdl file is automatically available to the pointbuilder. While there are many ways to save transformation information for usewith Profit Controller, the simplest is as follows.

1. Select Data Operations>Vector Calculations

2. From the Vector Calculation dialog box select the CV to be transformed asthe source variable

3. Use the Vector Function button to define the specific transformation

4. When finished specifying the transformation, make sure the destination



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variable name is the same as the source variable name (default)

5. Select the Save to Var button

6. Select the default (Copy to Aux and Overwrite Var) option in the OverwriteOption dialog box

7. Close the Vector Calculations modeless dialog box

8. Save the .mdl file

You can also update transformations from previously releases by simply openingthe existing .mdl file in the latest release. Observe the transformation and bounds.You can modify the bounds and/or transformations and evaluate the results in thestandard fashion. You MUST save the .mdl file before rebuilding the EB/Configfile

Bound definitions for transformation have been changed between PDS release200 and 220. Old bounds are inconsistent and must be updated.

Simply load and save the old .mdl file as described above. Inconsistent boundswill generate the following message

While it is good practice to check the bounds and transformation, it is not strictlyrequired.


Section 8 - Overall Identification Functions

8.1 Overview

In This SectionRead this section to find out about the main identification functions and specificallyhow to set overall options and how to run Load & Go.

Main Functions To access the main identification functions, Select>Identify from the main menu. Adrop down selection list as shown below displays the five main identificationfunctions: Set Overall Options, Fit FIR/PEM Models, Fit Parametric Models, SelectFinal Trials and Load & Go.

Alternatively, you can use the associated toolbar buttons. The main identificationtoolbar buttons which appear in both the standard and detailed toolbars are:

Only the Set Overall Options and the Load & Go functions are described in thissection. These functions correspond to the first two buttons in the above group.The other options shown in the pull down menu are described fully in later sections.

Overall Options In the APC Identifier there is a large list of options that the user can set. Optionshave been logically grouped first according to function then according tocomplexity

Each function has its own associated set of user configurable parameters. For agiven function, some parameters need to be configured more often than others.Dialog boxes are set up to deal with this structure in an intuitive fashion. Eachfunction has a main dialog box. Sub-dialog boxes can be invoked to allow the usermore and more flexibility for a specific application depending on the usersexperience and knowledge.

Some parameters or options apply to more than one function or in some cases applyto all identification functions. These parameters are accessed from the main menuby selecting Identify >Set Overall Options or by selecting the toolbar button.

Section 8 - Overall Identification Functions8.1 Overview


Load & Go As described previously, the APC Identifier is a hybrid approach consisting of threeprimary steps. These three steps; Fit FIR/PEM Models, Fit Parametric Models andSelect Final Trials, are usually accomplished in a sequential, interactive fashion.

At times it may be desirable to automatically perform all necessary functions, that isto go from start to finish, without any user interaction. This can be accomplishedfrom the main menu by selecting Identify>Load & Go or by selecting the

toolbar button.

Section 8 - Overall Identification Functions8.2 Overall Model Setup


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8.2 Overall Model SetupSetting OverallOptions

To set or observe overall options, Select>Identify>Set Overall Options asillustrated in the following figure or select from the toolbar.

This invokes the dialog box shown below. The Overall Model setup dialog boxcontains the highest level options. These options allow the user to specify thenumber of trials, model structure (FIR/PEM), model form and initial conditiontreatment. It also allows the user to access less used high level options.

Data Rate / TrialSpecification

Data collection frequency or scan rate is displayed only in this dialog box. Thisvalue may or may not correspond to the sample rate of the discrete time modelobtained by data regression. The sample rate of the discrete time model is always



an integer multiple of the scan rate and this integer value will be referred to as thecompression ratio. Models are eventually saved in the Laplace domain and as suchare not associated with the original data rate.

Detection of model sensitivity is of fundamental concern. Use of more than onemodel for a given CV-MV/DV pair provides a reasonable mechanism foraddressing this concern. The number of trials corresponds to the number ofdiscrete (FIR or PEM) and continuous models that can exist for a given CV-MV/DV pair. Through this document, a CV-MV/DV pair will be referred to as asub-model of the overall model matrix. Increasing the number of trials, results inmore models for a given sub-model. The number of trials is the same for all sub-models in the entire matrix. Be aware that decreasing the number of trials willresult in the loss of those models corresponding to the deleted trials. A warningmessage s will be displayed if any of these models are in the solution matrix.

FIR/PEM Step responses corresponding to different trials are color-coded. Up toten (10) trials can be specified. The color coding for each of these trials is as givenbelow.



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Here there are ten trials. As shown above the trial number corresponds to thesettling time and gain (i.e. Trial 1 has a settling time of one minute and a gain of 1,Trial 2 has a settling time of 2 minutes and a gain of 2 etc). Each response willalways have the assigned color designation irrespective of the settling time andgain. Colors corresponding to the trials are as follows.

Trial 1 – Green

Trial 2 – Red

Trial 3 – Blue

Trial 4 – Neon Green

Trial 5 – Neon Red

Trial 6 – Neon Blue

Trial 7 – Purple

Trial 8 – Olive

Trial 9 – Cyan

Trial 10 – Magenta

These Trial/color combinations apply to all response curves shown in theFIR/PEM Step Response and All Step Responses views.

MIMO DiscreteModelSpecification

Both FIR and PEM model structures can be used for data regression. While FIR isthe default, you can select either by choosing the appropriate radio button. Forinformation regarding these model structures see the concepts section of thisdocument.

TThhee ttaarrggeett uussee ooff tthhee PPEEMM mmooddeell iiss ffoorr rreeggrreessssiioonn sseettss oonn ssttaabbllee pprroocceesssseesswwhheenn oonnllyy oonnee oorr ttwwoo iinnddeeppeennddeenntt vvaarriiaabblleess aarree mmoovviinngg ssiimmuullttaanneeoouussllyy.

PEM models are provided with one goal in mind: EEaassee ooff uussee. The goal here is toprovide a mechanism that will allow truly one-step identification. One click on the“Load & Go” button and that’s it (Two or three if you are particularly ambitious).If the results are not satisfactory after one try, simply revert to the standard FIRapproach. To this extent, it is useful to view the PEM models as a complement tothe standard FIR models. Setup for these models will be described shortly

Initial Conditionsand Model Forms

While these choices apply to both FIR and PEM models, they are much moreimpactive when used in conjunction with the FIR models and to some extent theyare even required. When using PEM models it is strongly recommended to staywith positional form. In fact, if velocity form is used in this case a message boxwill recommend a switch back to positional form. Ignore this message if you wantARIMA models. To obtain this structure the form must be set to velocity.

Section 8 - Overall Identification Functions8.3 FIR Setup


8.3 FIR SetupConfiguring FIRModels

To configure the FIR models, select <FIR Setup>. This results in the followingdialog box.

.

Use the drop down list box to configure and observe model characteristics for thevarious trials. Significant parameters are described below.

Max Settle T(Settling Time)

This is the one variable that needs to be set on a process by process basis. It is notimportant that the value be accurate. In fact, it is recommended to enter several(three to five) settling times that span the expected range for one or more CVs.

When the resulting step responses are plotted on the same graph, self-similarprofiles indicate that the models are reasonable. If a few responses are similar butone or more diverge, then this is an indication that either some of the settling timeswere too long or that the process was not sufficiently excited for the full range ofsettling times entered.

If no responses are similar or if the step response changes sign at times greaterthan half the settling time, this is an indication that there is probably no model, orthat this model has not been sufficiently excited.

In any case, in the final model selection step (described in a subsequent section),the identifier picks the model with the best long-term open loop prediction relativeto the raw data. At this stage models based on inappropriate settling times areautomatically rejected. (Note, if models corresponding to all trials are bad, thenthe final step picks the best of the bad models. The result being a bad model.These models should be rejected or nulled out before building the controller.)

By default, three trial fits are made, each with a different maximum settling time.You can select more or fewer trials. If the settling times turn out to be incorrect,



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you can try others later.

The settling time that you choose per trial is the maximum settling time for theCVs that are to be used for FIR model generation. Settling times of CVs notselected for the build will not be affected. In a later step, you can select shortersettling times per sub model.

For each trial, select a maximum settling time. Typically the value chosen for oneof the trials is your best guess, and the values for other trials bracket the best guesson either side.

# of Coefficients Select the number of coefficients required to accurately represent the curvature ofthe step response curve. The default 30 is a good number. It is not, in generalnecessary to change this number as the settling time changes. In most instances, 30is equally effective for settling times of a half hour, 3 hours, or 30 hours. Changethis value to more accurately represent step response curves with large curvature.(i.e. Long delay with relatively short response times.) There is no internal limit onthe number of coefficients.

FIR Model Form While this option is not as important as the settling time, it is still a parameter ofwhich the user should be aware. Remember that the Positional Form gives goodlow frequency performance (accurate steady state gain), but is not well suited tonon-stationary processes (i.e. processes with drift). In contrast, the Velocity Formcan result in some low frequency information loss, but gives good performance fornon-stationary processes. The Positional Form is the default. If there arediscontinuities in the data that possess significantly different means, or if there issignificant drift, the Velocity form can be used to potentially improveperformance.

When using Positional Form, pay particular attention to the last value in the stepresponse curve. The FIR coefficients are unaltered and represent the unsmoothedsolution. Since smoothing is not done, the last coefficient may serve as anindicator of the proper form. (for non-integrating models)

If the last coefficient changes dramatically, then either the settling time is too shortor, as in the case shown below, it may indicate the need to switch from Positionalto Velocity form.

Prior to release 150, Velocity form was internally disabled for integrators. Thisoption is now enabled. If Velocity form is selected for integrating CVs, thenperformance may be improved by slightly extending the settling time.



FIR InitialConditions

Three initial condition options are provided. The default option is Unsteady. Forthis option it is assumed that the data is not in equilibrium at the start of the test orat any breaks (discontinuities) in the data. This option should be used if the initialconditions are unknown.

If the process is at rest at the beginning of the test, then select Steady at start only.If the process is at rest at the start of the test and at all breaks (discontinuities) inthe data, then select Steady at start/NaN breaks. Use of the initial condition optioncan be helpful especially in cases where the data set is severely limited. When theinitial conditions are steady, the solution can be modified such that no data iswasted dealing with unknown initial conditions.

Section 8 - Overall Identification Functions8.4 PEM Setup


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8.4 PEM SetupGeneralGuidelines

With PEM Models the goal is simply EEaassee ooff uussee. To this end it would bedesirable to require no setup. Unfortunately, structure plays an extremelyimportant role in the use of PEM models. If the structure or order of the PEMmodel is not sufficient to represent the process, then the model will be biased (seethe concept section for a discussion on the characteristics of the PEM models).Bias may result in a completely useless model. Therefore, with PEM models orderbecomes the key parameter

This parameter’s effect on model quality can be loosely described in the followingmanner. When the order is too low, the model will be biased and yield poorperformance. As the order is increased the performance of the model will improve(given that there is reasonable information in the data). At a certain point the orderwill be sufficient to capture the response of the process. Increasing the order pastthis point can lead to over-fitting the data and eventually may lead to convergenceissues. For example consider the data shown below.

In the plot given above there is one CV one MV and one DV. If this data is fit withPEM models and the models are first, second and third order respectively, then thefollowing set of step responses will result.

These curves clearly indicate the effect that changing order has on the resultantmodel. The sensitivity of the response curves to model order is obvious. The green(short), red (medium) and blue (long) curves correspond to first, second, and thirdorder models respectively. Fitting the same data again using fourth, fifth and sixth



order models results in the responses given below.

Clearly, the sensitivity is virtually eliminated and the models give self-similarresponse. Quality of models such as these is high as indicated by the predictiveperformance shown in the following plot.

Thus with respect with the above discussion, a third or forth order model would beappropriate for this problem. Note that PEM models tend to be much moresensitive to model order than FIR models are to settling time.

As described above, there is a preferred order. Selection of this order can beautomated by using, for example, an Akaike Information theoretic Criterion (AIC).While this is a sound theoretical approach it is not the one used here. In manypractical cases the data sets are short and not particularly informative. In thesecases there is a likelihood that fit quality (loss function) is fairly insensitive tomodel order while at the same time the model characteristics are very sensitive tomodel order. This implies that significantly different models give similar fitperformance. Automated techniques can be insensitive to this phenomenon. It willhowever be immediately exposed when the models are viewed as shown above.

In practice, model order itself is not of concern. What is important is to choose areliable model. The graphical approach illustrated above is an effective way to dothis. General guidance for PEM model selection is as follows.



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1. Chose starting order (second)

2. Load & Go using three trials

3. Observe step responses as shown above. Sub-model(s) with two or more self-similar step responses indicate that for these models the PEM fit is finished. If thepredictions are satisfactory the model(s) should appear in the Final model matrix

4. If sub-models are not self-similar, increase starting order (fifth)

5. Repeat steps 2 and 3

6. If sub-models are not self-similar after 2 tries, use FIR (if you areconvinced a model really exists)

These steps are meant only as general guidelines. In fact, in the true EEaassee ooff uusseespirit they are too complex. To make the procedure even easier, the followingapproach can be used.

1. Set starting order to 5

2. Load & Go using three trials

3. Observe step responses as shown above. If sub-models have two ormore self-similar step responses, fit is done. Otherwise use FIR (ifyou are convinced a model really exists)

Auto Setup There are two ways to specify PEM model orders. One is to first select the PEM(Auto Setup – Order override) radio button in the Overall Model Setup dialog box.Then simply scroll the start order to the desired value. This will automaticallymodify the orders of all polynomials in the PEM models for each trial. If there arethree trials and the start order is 2, then all polynomials for the PEM modelcorresponding to trial 1 will be second order. All polynomials in the PEM modelcorresponding to trial 2 will be third order. All polynomials in the PEM modelcorresponding to trial 3 will be fourth order. Thus the order selection is extremelysimple using the auto setup method.

Detailed Setup Orders can also be specified by selecting the PEM Detailed Model Selection radiobutton in the Overall Model Setup dialog box. This radio button will enable thePEM Setup button.

Use of the PEM Setup Button should not be necessary in practical applications. Ifyou need to come here and are looking for expedient results, you should switch toFIR models.

If on the other hand you have a curious nature select <PEM Setup> to display thedetailed dialog box shown below.



This dialog box allows access to all elements of the PEM model (see conceptschapter of this document for a discussion on the PEM model). The options andtheir meanings are as follows

• Include Noise Terms- When this is checked (the default), the term

)()(

)(te

zD

zCwill be included in the model. If unchecked this term will

be ignored

It can be advantageous to turn off the noise model in cases where the data sets areshort or when there are only a few moves in the independent variable(s).

• Index i- When set to zero, Values entered for nB, nF and nK will beapplied for all MV/DVs. For values other than zero, this is the indexof the MV/DV for which nB, nF and nK are set

• nB(i)-Number of terms in the B(i) polynomial

• nF(i)- Number of terms in the F(i) polynomial

• nK(i)- Number of delay intervals for the ith MV/DV. This value isequal to the delay divided by the compression ratio. Set this value tozero to obtain a semi-proper model

• AR Terms- While the general PEM solution accommodates the fullPEM model shown above, a practical application should not needboth A and F polynomials. (F is the default)

• nA- Number of terms in the A polynomial

• nC- Number of terms in the C polynomial



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• nD- Number of terms in the D polynomial

• Trial Selection- Trial for which the models are set

• Reset Order- When this value is scrolled all orders are reset to thisvalue

• Compression Ratio- Defined previously. In the FIR models thisvalue is automatically determined once the settling time and numberof coefficients are specified. With PEM the default is one and it mustbe set explicitly.

In most applications, there should not be a need to set these parameters. The oneexception is the noise term option. A brief discussion follows

PEM InitialConditions andModel Form

These options are intended primarily for use with FIR models. However, sincethey are used in data preparation, they also apply for PEM models. While theinitial condition option has little effect on PEM models, the Model Form optioncan have a serious detrimental effect when used with PEM models. In general,selecting velocity form will result in reduced model performance. Let the noiseterm deal with disturbances. To prevent inadvertent use of velocity form, amessage box will be displayed when this form is selected for use with PEMmodels. Note however to obtain ARIMA models this option must be set tovelocity.

Section 8 - Overall Identification Functions8.5 Overall Model Setup Options


8.5 Overall Model Setup OptionsClicking on the Option button in the Overall Model Setup dialog box will displaythe next level of options. Selecting this button results in the dialog box shownbelow.

Parameters in this dialog box allow the user to configure the overall identificationprocedure. The nine general categories that can be modified are described below.

CalculationOptions

This first category contains parameters that allow the user to specify informationrelating to the calculation of the quantitative measures indicating model quality.With the exception of “Correlation”, these parameters are initially all deselected.The parameters are:

• Correlation – This check box enables both the MV/MV and CV/MVcorrelation calculations.

• Confidence – Select this check box to enable confidence, noise bounds, nullhypothesis calculations and model ranking. When this check box is selected,the following items will be enabled; <Confidence limit>, <Rank option>,<UseConfidenceOnTset> and <Auto null uncertain models>. SetConfidenceCalcs=1 in the .ini file to initialize this parameter asselected in all new documents.

• Confidence limit – Specify the desired confidence or probability limit inpercent using this option. Scrollable values range from 0 (no noise orinfinite confidence bands, all models are valid) to 99.9%. The default valueof 95% is highly recommended since it corresponds to a two sigma band.

• Rank option – Several internal rankings are performed. The result of each



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ranking is a recommendation to keep or reject the subsequent parametricmodel. Use this item to define the default results presented in the statisticalsummary view. The following rank options are provided:

− 0 No Rank – All computations are performed and results displayedbut models are not ranked

− 1 NNHT Rank – Models are ranked based on the non-null hypothesistest. The rank is linearly related to the number of trials that pass thetest. If all trials pass the test then the rank is set to 1. If no trials passthe test then the rank is set to 5 (recommendation to reject). If the ranklevel is < 5, then NNHT = PASS, otherwise NNHT = FAIL.

− 2 Separation Rank – Here models are ranked based on the separationfactor. The separation factor is the average maximum noise band forall trials. Values range between 0 and 1 which correspond to a rank of1 and 5 respectively.

− 3 Sensitivity Rank – Models are ranked based on the sensitivityfactor. The sensitivity factor is a heuristic indicator based on thesensitivity of the FIR step response models to perturbations in settlingand/or discrete sample rate. Self similar step responses have a lowsensitivity factor while dissimilar step responses have a high sensitivityfactor. Values range between 0 and 1 which correspond to a rank of 1and 5 respectively.

− 4 Combined Rank – As the name implies, models are ranked using acombination of available information. The rank of the previous threecategories are combined in a linear unweighted fashion.

• UseConfidenceOnTset – Based on the noise estimates, it is possible todetermine the length of the response curve that is statistically significant.When selected, this additional information will be used to evaluate models.

• Penalize Oscillations – Use this option to penalize oscillations in the FIRstep response. Confidence levels will be automatically degraded whenoscillations are present. Deselect this option to eliminate degradations.

• Auto null uncertain models – When selected, this option will use therecommendation of the user specified rank option to automatically set theparametric model flag to null or auto depending if the recommendation is toreject or keep the model.

• Uncertainty spectrum – Select this check box to perform the uncertaintycalculations. This function is planed for a future release.

• Power spectrum – Select this check box to perform the input powerspectrum calculations. This function is planed for a future release.

• Residual Correlation – Select this check box to perform both auto and



cross correlation on the prediction error. This function is planed for a futurerelease.

Data Options Options that are user configurable pertaining to data that serves as input to theregression routines are as follows:

• Use average of sub-interval data – The settling time, number ofcoefficients and data scan rate determine the internal compression ratio. Acompression ratio greater than one implies that not all of the data is used inthe regression. When this is the case a choice exists. Either use individualdata at the effective sample rate (compression ratio times the actual scanrate) or uses a moving average over the effective sub-interval.

FIR Only Options • Proper models only – Proper FIR models are those in which all terms onthe right hand side of the prediction equation correspond to sample timesprior to the output. Semi-proper implies that one or more terms on the righthand side of the prediction equation correspond to sample times equal to thetime of the current output. Choose this option to force proper models to beused in the regression.

• Allow Semi-proper models – While essentially all dynamic industrialprocesses are intrinsically proper, data compression effects may cause theprocess to appear semi-proper. Choose this default option to account for thispossibility.

• Semi-proper models only – Choose this option only if the original sampleddata is semi-proper. While this condition should seldom occur, it is possiblewhen the data is severely undersampled (i.e. a process that appears to haveno dynamics or a process with a small time constant that is sampledrelatively slowly).

Data Scaling In general, data used in the regression and decomposition calculations must bescaled in some fashion. The basic idea of the scaling algorithm is to adjust thedata such that data for all variables passed to the FIR calculations ranges between+ 1. The following three options are provided:

• Auto Scale/Mean Removal – Data for each variable passed to theregression calculations is zero mean with a range between + 1.

• No scaling – Only to show effect of scaling. NEVER choose this optionwhen reliable models are desired.

Null ModelTreatment

In many instances the lack of a causal effect between CVs and MV/DVs may beknown apriori. In these instances, the models can be set to null from the verystart of the identification process. The following two options for null modeltreatment are provided:

• Constrain to zero in regression – With this option, the identificationproblem is cast as a two-norm minimization problem subject to equalityconstraints. The constraints being that the coefficients corresponding to the



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null models must be identically equal to zero.

• Set to zero after regression – With this option, the identification problemis cast as an unconstrained two-norm minimization problem. After theproblem is solved the results for the null models are simply set to zero.

RegressionSelection Options

There are two basic options available for selecting segments of data to be treatedas bad values when used for regression calculations. These options described in aprevious section of this document are:

• Block for all Dep Vars – With this option, ranges are selected and theseranges are applied to all variables used in the regression. All values withinthe time range (inclusive) are set bad for any variable being regressed. Sinceall variables are bad for each range selected, the data is collapsed such thateach range to be excluded is represented by a single NaN for each variable.

• One for each DepVar – With this option, data can be excluded for eachvariable on an individual basis. Display of this type of selection is differentthan that used for Block selection to avoid any ambiguity. This categorysupports an additional option

1. Only Mark Dep Var – Independent variables are unalteredentering the regression

2. Mark Dep Var and Ind Vars – Selection is done on a perdependent variable basis. At regression, the selection is also beapplied to the independent variables. When this option is used, theeffective marks (they are not displayed graphically) are the resultof the union of all marks for each dependent variable used in theregression. This implies that the effective bad values for anindependent variable are dependent on which dependent variables(and their associated marks) are used in the regression.

PEM Only Options Several options pertain only to the PEM models. Of these options, some will beused infrequently, if ever. Others may need to be used more often. The generalOptions are:

• Search on Start Order- This is a flag to enable the search for the optimalorder of the high order ARX model used as a first step in generating initialestimates. This flag only has meaning if the UsePfxIC parameter in the .inifile is set to 1. When this option is unchecked the order of the ARXinitialization is based PfxExpRed. Initialization of this option in newdocuments is controlled by the AICSearch parameter in the .ini file.

• Auto Check Noise Mod – Noise terms enable the PEM models to be a veryeffective identification tool. There is however no magic here. Use of thenoise terms does NOT insure that immeasurable disturbances will beautomatically accommodated in all cases. In fact the search must convergeto a reasonable noise model for the deterministic portion of the model to be



acceptable.

When there is reasonable information content in the data, then the noiseterms can be used to significant advantage. If however the informationcontent is low, then the noise terms may in fact be ineffective. While theseare typically cases where the models are prone to being suspect, it may bethe only data that is available. In these cases, the most appropriate modelcan be easily selected by evaluating models both with and without noiseterms.

While the user is free to turn the noise terms on and off, this is somewhatcontrary to the EEaassee ooff uussee spirit with which the PEM models are intended.To automatically address this issue, it is recommended that the user selectthe AutoCheckNoiseMod option and still use Load&Go. When this optionis checked, the search automatically evaluates the effectiveness of the noiseterms and chooses the most appropriate model. This option can beinitialized as checked in new documents by setting NoiseModCheck=0 inthe .ini file

• Auto Integrator Flag- Integrators may be handled in the conventionalmanor described in the next chapter. With PEM models however, it ispossible under reasonable conditions to detect their presence based on thepoles of the model. This flag enables the detection and automatic setting ofthe corresponding sub-model integrator flags. Effective use of this flag isbest accomplished by combining all potentially integrating CVs into adedicated file. When the identification is complete, the models can bemerged to their final destination. The heuristics used in this set ofcalculations can certainly fail in the presence of slow poles. Even not soslow poles that are over samples. Because of this it is not recommended tohave this parameter selected for the general case.

• Auto Delay Flag- Long dead times will likely cause problems for PEMmodels. Set this flag true to perform an initial dead time estimation prior toPEM calculation. The current estimator is overly simplistic and the intent isto extend it using a correlation approach. Do NOT set this flag true for thegeneral case. Only set it if you are sure there is a long dead time.

• Detailed Output- This controls the messages output to the messagewindow. Detailed displays are quite lengthy are intended primarily fordebugging purposes.

FactorizationOptions

Options associated with the factorization are:

• Cholesky- When this option is selected (default) the Cholesky factorization,as described in the concept section of this document, is used. Thisfactorization is usually significantly faster the QR factorization

• Orthonormal- When this option is selected the QR factorization, asdescribed in the concept section of this document, is used. If memory



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requirements of the Jacobian matrix J exceed UserMemABuf, then amessage will be displayed allowing the user to enable a automatic switch toCholesky

Search Options Options associated with the nonlinear search are:

• Max Iter- This defines the maximum number of iterations that are allowedin the search procedure. When set to zero no search will be performed andthe resultant model will be the initial estimates. Some model forms, such asARX, don’t require a search. For these forms this parameter has nomeaning. If the procedure has not converged after Max Iter iterations thenan appropriate message will be displayed in the message window. Note inmany cases the resultant model will be effectively converged.

• Search tol- When the cost function drops below this value, the searchterminates.

Section 8 - Overall Identification Functions8.6 Running Load & Go


8.6 Running Load & Go

Load & GoLoad & Go is a one-click-does-all function for creating FIR/PEM, parametric,and final models in one pass rather than creating these models one at a time. Afterplotting the results, you can modify the models, or take them as they are and buildthe controller. Make sure that you critically review the models before using themin a subsequent operation.

Select Identify>Load & Go or select from the toolbar. This displays the

following dialog box for performing Load and Go calculations. As shown below,the detailed toolbar has been enabled. Enabling of the toolbar and status bar is atthe users discretion. Both modes will be illustrated in this document.

Default Model

Settings

The Identifier has pre-set defaults for FIR/PEM, parametric, and final models.Load & Go uses these defaults. Before accepting the defaults, Check to see ifyou’re using an FIR or PEM model. Then check the FIR/PEM, parametric, andfinal model drop down menus to make sure the settings are appropriate. If theyare not, the Load & Go procedure should not be used. Please see sections 9, 10and 11.


Section 9 - Creating Finite Impulse Response or PredictionError Models

9.1 Overview

In This SectionRead this section to find out how to set FIR/PEM options, identify the sub models,and view the FIR/PEM model summaries.

About the FIR

Model

FIR step response models are obtained by integrating the finite impulse responsecoefficients. These models represent the response of a dependent variable (CV) toa step change made to an independent variable (MV or DV).

If input signals have been designed properly, then FIR models can result inunbiased estimates, even in the presence of colored noise in the test data, and donot require structural information about the process dynamics.

FIR results typically have a high variance, evidenced as kinks or wiggles in theFIR step response that would not be reproduced if a different set of test data wereused for another identification calculation.

This high frequency behavior is eliminated by a second set of calculations(described in Section 10) in which parametric models are fit to the FIR stepresponses.

While there are several FIR options available, there are only two parameters thattypically ever need to be adjusted:

• The maximum likely settling time for a given CV, and

• The model form.

These parameters have been discussed in the previous section describing OverallOptions. Set these parameters for the application at hand. Review and if desiredset any other overall option before doing identification.

After the overall options have been reviewed/set, identification can begin. Theseoptions (especially the settling time and model form) can be adjusted at any timeduring the identification process. This offers essentially unlimited flexibility.Settling times and model forms can be changed every time an FIR model is built.By building individual CVs, each CV can have different model forms and/orsettling times. By using Lock Model and other options (described in this section)individual sub models can have different forms and/or settling times.

About the PEM

Models

These models represent the response of a dependent variable (CV) to a stepchange made to an independent variable (MV or DV).

TThhee ttaarrggeett uussee ooff tthhee PPEEMM mmooddeell iiss ffoorr rreeggrreessssiioonn sseettss oonn ssttaabbllee pprroocceesssseesswwhheenn oonnllyy oonnee oorr ttwwoo iinnddeeppeennddeenntt vvaarriiaabblleess aarree mmoovviinngg ssiimmuullttaanneeoouussllyy

As described previously, the goal here is EEaassee ooff uussee. Under the above conditions,Load & Go is the preferred option for building PEM models. These models can ofcourse be built using the “Fit FIR/PEM Models” approach described in thissection. In fact selection of CVs and MV/DVs must be performed prior to using

Section 9 - Creating Finite Impulse Response or Prediction Error Models9.1 Overview


the Load & Go function.

PEM results typically have relatively smooth step responses. In some cases,however, the responses may exhibit high frequency behavior. This may be theresult of using too high order model or any number of conditions in the test data.

Any high frequency behavior, however, is eliminated by a second set ofcalculations (described in Section 10) in which parametric models are fit to thePEM step responses.

With this model the only parameter of interest is the Start Order. This parameterhas been discussed in the previous section describing Overall Options. You shouldnever have to adjust it more than once or twice. If so use the FIR model. Forgeneral ease of use, it may be desirable to have the AutoCheckNoiseMod optionselected prior to PEM fitting especially for pre-test and model ID associated withregulatory-loop tuning.

After the overall options have been reviewed/set, identification can begin. Theseoptions can be adjusted at any time during the identification process. This offersessentially unlimited flexibility. By building individual CVs, each CV can havedifferent any combination of FIR or PEM models. By using Lock Model and otheroptions (described in this section) individual sub models can have different formsand/or settling times

Section 9 - Creating Finite Impulse Response or Prediction Error Models9.2 Procedure


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9.2 Procedure

Fitting the FIR/PEM

Mode

To begin the procedure of fitting FIR/PEM models, select Identify>Fit FIR/PEMModels from the main menu or select the toolbar button. This changes the

view to the FIR/PEM Model matrix view (defined in the upper left hand corner ofthe model matrix as ‘Select Vars. For FIR/PEM Fit’) and displays the FitFIR/PEM Models dialog box as shown below.

The Fit button will reflect the structure currently selected in the Over ModelSetup dialog box described in the previous chapter. If FIR has been selected then

the button will be . If PEM has been selected it will be .

Fit Fir/PEM Models

Dialog Box and

Associated View

Like all model views, the FIR/PEM model matrix view shows information foreach sub model in a two-dimensional matrix of sub-model boxes. The MVs andDVs are the columns of the matrix and the CVs are the rows.

Fit the FIR/PEM models to the data using the default options by clicking [Fit FIRor Fit PEM] on the Fit FIR/PEM Models dialog box. To modify the defaultoptions select the appropriate buttons on the main dialog box or double click inthe appropriate areas in the FIR/PEM model matrix as described in the followingparagraphs.

Show & Select Vars This button can be used to return to the FIR/PEM model matrix view as shownabove at any time. If the Fit FIR/PEM Models dialog box is displayed and theuser selects another view (i.e. Single-Graph Data Plots), then the FIR/PEM modelview can be restored by simply clicking the Show & Select button.



Set Overall Options When this button is selected an Overall FIR/PEM options dialog box isdisplayed. The Overall FIR/PEM options dialog box shown below allows the userto change information for all sub models selected in a simultaneous fashion.Remember that if no models are selected, then this implies that all models areselected.

Set parameters shown above for all sub-models selected. The parameters shouldbe selected for the following conditions:

Null Sub-Process – Check this box, if there is no physical way that theindependent variable (MV or DV) of the selected sub model can affectthe CV of the selected sub model.

Any sub-model corresponding to a selected CV and an unselectedMV/DV will be effectively nulled after the fit procedure. Results willbe identical to those obtained using the above option (if theConstrain to zero in regression parameter is selected in OverallOptions). However, when the MV/DV is not selected, the sub-modelwill appear empty instead of blank.

Integrating Sub-Process – Check this box, if the sub model includes anintegrator (ramping sub process).



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Set Options perSub Model

When this button is selected two actions occur. An Options per Sub model dialogbox is displayed and a focus box (colored outline) is drawn around the sub modelcorresponding to element (1,1) of the matrix (i.e. the sub model whoseparameters are to be potentially changed.

Alternatively, the Options per Sub model dialog box can be invoked by doubleclicking anywhere in the text field (except on the trial descriptor) for any desiredsub model. In this instance the focus box is drawn around the sub model fromwhich the dialog box was invoked. The Options per Sub model dialog box shownbelow allows the user to change information for one sub model at a time.

As illustrated above, the selected sub model is indicated with a highlighted framein the two-dimensional FIR/PEM model matrix view. Its CV and MV/DV indicesare also shown on the dialog box.

Move to a different sub model by using the Next MV/DV or Next CVbuttons.The focus box and CV, MV/DV indices changes accordingly.

Null Sub-Process – Same as described above

Integrating Sub-Process – Same as described above

• Lock Model – This is an option that should be used only by anexperienced user and only when necessary. This option allows the userto ‘lock’ an FIR/PEM sub-model for a specific trial. To use this option,the selected sub-model must have at least one existing FIR model andits corresponding parametric model (see next section for discussion ofparametric models). If more than one trial exists for the selected sub-model, all models corresponding to all trials other than the trial selectedare deleted. That is, only the FIR/PEM and its corresponding



parametric model for the selected trial are retained. Once the FIR/PEMsub-model is locked, it is NOT altered upon rebuilding the CV models.

When a CV that has one or more locked sub models is refit, thefollowing occurs:

I. Effects of the locked models are deconvoluted from theraw data by using predictions based on the parametricmodels corresponding to the locked FIR/PEM models andmeasured inputs.

II. Deconvoluted data and inputs corresponding to all non-locked models are used to regress non-locked FIR/PEMmodels.

Even though an FIR/PEM sub-model may be locked, its correspondingparametric model is NOT. That is, you are free to modify parametricmodels corresponding to any locked FIR/PEM model (thesemodifications are reflected in the deconvolution the next time theFIR/PEM models are updated).

Use of the locked model option is intended for situations where inputvariables are highly correlated over some of the data but uncorrelatedover other sections of the data. The procedure is as follows:

I. Select region of data over which inputs are uncorrelated.

II. Select a CV (one at a time) and only those MV/DV ofinterest.

III. Fit FIR/PEM model

IV. Fit Parametric Model

V. Lock appropriate sub-models

VI. Proceed in normal fashion

Locked models are indicated by the ‘Locked’ flag displayed in all modelviews next to the trial descriptor. In addition, several dialog boxes (asappropriate) will indicate which models are locked.

Options per MV/DV Use this button to independently adjust settling times for a given CV (this optionpertains only to FIR models). Select [Set Options per MV/DV] to invoke theOptions per MV/DV dialog box shown below. For each trial for a given MV/DV,reduce the settling time, if this sub process has a shorter settling time than themaximum specified for the trial set in the Overall Options dialog box.

As illustrated below by the focus boxes, changing this parameter potentiallyeffects an entire column of the model matrix. Different reduced settling times forsub elements for each CV can be easily accommodated by building the CVsindependently (i.e. fitting the CVs one at a time.)



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Excluding Data

From the

Regression

For a detailed discussion on selecting, marking/unmarking and viewing data seesection 5. What follows is a summary of pertinent topics. To exclude data fromthe regression (mark data bad), choose [Exclude Data Ranges]. This changes theview from the FIR/PEM Model matrix view to the Show Regression Ranges viewas shown below. The variables displayed will correspond to those selected forregression in the FIR/PEM model matrix view. The main FIR/PEM dialog boxwill still be displayed.



At this point, data can be excluded from the FIR/PEM regression calculationsusing two different approaches:

Block Selection – With this option, ranges are selected and these ranges areapplied to all variables used in the regression. All values within the timerange (inclusive) are set bad for any variable being regressed. Since allvariables are bad for each range selected, the data is collapsed such thateach range to be excluded is represented by a single NaN for each variable.

Variable Selection – With this option, data can be excluded for eachvariable on an individual basis. Display of this type of selection is differentthan that used for Block selection to avoid any ambiguity. Here each cross-hatched range will be “painted” as bad values at the time of regression. Thiscategory supports an additional option

• Mark only dependent variables – Independent variables areunaltered entering the regression

• Mark both dependent and independent variables – Selection is doneon a per dependent variable basis. At regression, the selection isalso be applied to the independent variables. When this option isused, the effective marks (they are not displayed graphically) are theresult of the union of all marks for each dependent variable used inthe regression. This implies that the effective bad values for anindependent variable are dependent on which dependent variables(and their associated marks) are used in the regression.

Data marked as bad at the global level (in the Single Graph Data plots View) isalso displayed in this view (and any graphical view) whenever the Show BadData option is selected. Global marks however can not be altered in this view.These marks are however applied at the time of the regression.

This view operates in a fashion almost identical to the Single Graph Data PlotsView. The title for this view is “Show Regr. Ranges” and will always bedisplayed in the lower right portion of the vertical margin. This title will have ared superscript “b”, “v1” or “v2”. The superscript “b” and “v” designate blockand variable selection respectively while the “1” and “2” imply that marks areapplied only to dependent variables (“1”) or to both dependent and independentvariables(“2”). The actual ranges used in the regression will correspond to thevalue of the superscript at the time of the regression. To change the method usedfor excluding data, modify the Regression Selection Option in the Overall ModelSetup Options dialog box described in the previous section

To Select Ranges, do the following:

• Move the cursor within the time axis box to one end of the desired timerange. The vertical dash dot line and the date/time in the center of the boxshow you where you are. When you have positioned the cursor at one end ofthe range, press and hold the left mouse button



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• Move the cursor to the other end of the desired time range. The secondvertical dash dot line that appears and the date/time in the center of the boxcorrespond to the other end of the range. Release the mouse button. Theselected time range is shown with a gray background

• Repeat these steps to select additional ranges

• Hold down CTRL and use the above procedure to deselect all or part of apreviously selected range.

• Remember that the data that is grayed is excluded from the regressioncalculations.

Once the ranges have been selected, the data can be marked/unmarked for

individual dependent variables and displayed using the toolbar

buttons. (See section 5 for a detailed discussion on selecting ranges andmarking/viewing data). Alternatively, the ranges can be used directly forexclusion applied to all variables used in the current regression. The choice isentirely up to the user and depends on the specific application. The methodchosen can however have a significant impact on the results. More on this topic atthe end of this section.

Fit FIR/PEM Models At this point either choose [Show and Select Vars] from the FIR/PEM dialog box(this will automatically switch the view to the FIR/PEM models matrix view) or ifthe dialog has been deleted Select Identify>Fit FIR/PEM Models from the mainmenu.

Select the CVs and MV/DVs for which models are to be built. Do this by clickingon the desired CVs in the normal fashion. Click in the far left column where theCVs are described. Similarly, for MV/DVs click on the top row where theMV/DVs are defined. The rows corresponding to the selected CVs and columnscorresponding to the selected MV/DVs are highlighted. Click in the upper leftcorner of the model matrix to select the entire matrix.

Next click [Fit FIR or Fit PEM depending upon the current selection state]. Thisinitiates the FIR/PEM model identification calculations.

Model Example During the FIR/PEM calculations, a window pops up to display progress,information, warning and error messages [NOTE the warning message (messagedisplayed in yellow text) “rank reduced” during computations impliesinsufficient information in the regression matrix. Do not continue until thiscondition is resolved. For FIR models, this is usually an indication that the userhas specified a settling time that is too long for the given data set or has specifiedtoo many coefficients].

Several messages (informative only, as indicated by green text) are presentedrelative to memory usage. The APC Identifier relies on the windows memorymanager to deal with memory allocation. If the memory messages are reported inred (error messages) or if excessive disk access occurs (indicated by inordinately



slow operation), contact your Honeywell representative.

When the calculations are complete, the message window is switched to thebackground mode. After reviewing the message window, close it if desired. TheFIR/PEM dialog box can also be closed at this time by selecting [Exit] or it canbe moved to view results.

After the FIR/PEM calculations are complete, the FIR/PEM model matrix viewwill be automatically switched to the FIR/PEM Step Response view if anotherwindow is not already open to this view. It is also recommended to have anadditional window open to concurrently display model statistics.

Selecting any button on the Fit FIR/PEM Models dialog box, will cause theassociated view to be automatically switched back to the relevant FIR/PEM modelmatrix view. After a fit, this view will look something like that shown below.

Model DescriptorsThe FIR/PEM model matrix view shows the model information for each subprocess, including a plot of the step response for the selected trial. Severaldescriptors are displayed for each sub model.

• Trial - Indicates the model index corresponding to the user specified settlingtime or in the case of PEM the internally calculated settling time for whichmodel information is displayed. (i.e. If the user specified three settling timesof 60, 90 and 120 minutes, then trial 1 would correspond to all modelinformation related to the 60 minute settling time. Trial 2 would correspond tothe 90 minute settling time, etc.) This descriptor can be doubled clicked toselect different trial information.

• ARX/LAP/OE Order - Source parametric model form and its correspondingorder are displayed by this descriptor. (Default ARX implies pre-filteredARX). If parametric models have not been built this field contains only



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default information.

• FIR/PEM Order – For FIR, this field defines the number of FIR coefficientsused in the model. For PEM, this field defined the PEM order. If Auto setup isused this is the order of all model terms, otherwise it is the order of the termwith the largest polynomial.

• Stable/Integrator - Defines if sub model is self regulating or an integrator.

• Dead T - Transport delay of the parametric model. Set to zero if parametrichas not been built.

• Gain - Steady state gain of the parametric model. Set to the last value of theFIR/PEM step response curve at completion of FIR/PEM calculations whenno transfer function exists.

• Settle T - User specified settling time for FIR model or internally calculatedsettling time for PEM model for indicated sub-model.

• TfSettle - Settling time of parametric model. TfSettle > 1.5 * Settle T impliessignificant extrapolation and indicates potential deficiencies. In these casesboth Settle T and TfSettle will be backlit in blue to bring this extrapolationproblem to your immediate visual attention. If TfSettle > 2 * Settle T in theFinal model, then this matrix can not be used in subsequent controller buildoperations.

• FIR Form (PEM Form) - Gives the form of the source FIR or PEM model.Can be either Positional (Pos), Velocity (Vel) or Unknown (UK).

Checking Trial

Dependent

information

To check model information pertaining to different trials, from the main menu ClickView>Trials>Change All to bump the trial number for all sub models up or down.



Similarly, trial information for a specific sub model can be accessed by DoubleClicking on the “Trial” descriptor. This invokes the Displayed Trial dialog boxshown below. In addition the model box that was double clicked has a focus box.

Use the pull down menu in the Displayed Trial dialog box to change the displayedmodel information for the sub model of interest. The displayed trial is‘remembered’ by all model views. Thus if the value is changed in one view, then allother concurrent views will reflect this change.



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FIR/PEM StepResponses

One of the most useful views is the FIR/PEM step Response View. SelectView>FIR/PEM Step Responses as shown below. This is the same view that willautomatically be displayed at the conclusion of the FIR/PEM calculations.

This selection displays the step response plots for the FIR/PEM models for alltrials as follows.



InterpretingResults

When using PEM models, multiple trials can be used to effectively selectmodels/orders as described in the previous chapter on Overall Identificationfunctions. When using FIR models, multiple trials correspond to multiple settlingtimes and the FIR Step Responses can be used to indicate the goodness of the sub-models. Since the different settling times for a given sub-model effectively resultin a perturbation to the solution matrix, these curves reflect sensitivity problems asdiscussed in the concept section of this document.

All self-similar curves (exclusive of variance) indicate little sensitivity and are afirst indicator of a reasonable model. Some self-similar and some divergent curvesindicate a potentially reasonable model but some sensitivity. Many times this iscaused by a settling time specified outside the power range of the input signal (i.e.continually increasing the settling time for a fixed input signal of finite power bandwith noisy data eventually results in a divergent step response). Sometimes this iscaused by a settling time that is too short. If adjustment of the settling time resultsin self-similarity, then this indicates a potentially reasonable model.

If the sensitivity is low and the model prediction good (see section on selectingfinal trials), then the model usually can be used with a high degree of confidence.

If the sensitivity is high, then the model should NOT be used. Under someconditions, nulling of the model is all that is required. Under others, moreinformation (data) may be necessary. Sensitivity is usually caused by poor signaldesign or by adverse test conditions and in both cases indicate that the model isnot reliable.

Poor signal design is usually the result of correlated inputs and/or insufficientpower spectrum. Unfortunately, the sensitivity of the regression matrix is relateddirectly to both of these variables. As the correlation increases, the covariancematrix becomes more poorly conditioned and the sensitivity of the regressionmatrix increases. Similarly, if the power is too low over the frequency range ofinterest, then the covariance matrix will again be ill-conditioned. If the signal isnot persistently exciting, then the covariance matrix becomes singular. Sensitiveresponse curves imply that large changes in the models have relatively smalleffects on prediction errors and hence these models are unreliable.

Even with proper signal design, the FIR/PEM step responses may exhibit sensitivebehavior. If there is no causal relationship for the CV and MV/DV pair, thenessentially random response curves would be expected. In this case, no modelexists. Hence the model can and should be nulled and no further issues need to beaddressed.

Another more troubling possibility is the result of adverse test conditions. Theconcern here is that there is a causal relationship for the CV and MV/DV pair, yetthere is still a sensitivity problem. This condition is possible even for properlydesigned experiments and even when all modeling assumptions (i.e. linearity,stationarity, etc.) are satisfied. Under these conditions, a theoretically unbiased oraccurate (insensitive) model would be expected only in the limit as the length ofthe test goes to infinity. Since this is not a practical possibility, the issue here is the



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limitation of a finite duration test. Errors in the model are proportional to thepower of the disturbances and inversely proportional to the input power.Sensitivities due to adverse test conditions, indicate that the model should not beused and that more data is required. Under these conditions, attention should befocused on minimizing or eliminating disturbances to the extent possible andmaking sure the amplitude of the input signals are large enough to move the CVsoutside the noise and/or disturbance bands.

Section 9 - Creating Finite Impulse Response or Prediction Error Models9.3 Statistics


9.3 StatisticsBackground At the end of the previous section guidelines were presented for interpreting FIR

results. To complement/enhance this information, the APC Identifier also providesstatistical information relative to the MIMO models. Information is provided intwo general areas; signal content and confidence data on the individual FIRestimates.

Signal design is by far the most important aspect of the identification process. TheIdentifier automatically provides key information as to the quality of the signalsused to create process models. This information is presented in terms of easy tointerpret plots depicting both auto and cross correlation functions and powerspectrum. These plots should always be reviewed before generating models to beused in any controller. Problems with information content (which will be evidentin the correlation and power plots) will invariably cause problems or at leastconcerns with the resultant models.

While signal information is germane to any type of model identification, with theFIR structure it is possible to generate additional information pertaining directly tothe model itself. This additional information is provided in terms of statisticalestimates of noise bands associated with individual model coefficients. Based onuser specified probability levels, standard null hypothesis tests can be evaluated todetermine if coefficients are in fact distinguishable from the noise present in thedata set. This information, summarized in the Null Hypothesis or ConfidenceView, can be used to detect causal relationships between inputs and outputs in astraightforward analytical fashion.

At first, one would expect that the confidence information discussed above is allthat is needed for accepting or rejecting models. Unfortunately, this informationonly specifies if the coefficients are in fact statistically significant. It must beremembered that the identification objective is NOT to simply fit the data but toobtain the causal effect between inputs and outputs in spite of both deterministicand stochastic disturbances. To this end, heuristics capturing practical experiencehave also been incorporated into the analysis.

As discussed previously in Section 9.2, the self similarity of unsmoothed FIR stepresponse curves is correlated in a qualitative sense to the goodness of the model.This concept is extended by utilizing noise estimates on the individual coefficientsto generate noise bands on the step response curve. Hence in the StatisticalSummary View, the models are represented in terms of step response bands ratherthan an individual step response curve. The bands visually display the degree ofseparation. While it is true, that in this framework, a single trial will have aseparation band, it is still highly recommended to use more than one trial. Whenmore than one trial is used, the band will expand to encompass all models. Theupper bound is the maximum value of the step response plus the maximum noisebound for all trials. The lower bound is the minimum value of the step responseminus the maximum noise bound for all trials. A separation factor indicating thedegree of separation is also calculated and displayed.



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Experience has shown that in some situations grossly over or under estimating thesettling time can cause undue separation even with relatively small noise estimates.In these instances it may be possible to in fact have two or more step responsesthat are relatively self similar, which might imply the existence of a causalrelationship. To preclude the omission of this condition, a sensitivity factor is alsocalculated. This factor is calculated based on the two trials which exhibit aminimum separation. Step responds bands (excluding noise) for the two trialscorresponding to the sensitivity factor can also be displayed in the StatisticalSummary View.

Guidelines To begin, a few basic guidelines will be presented. After these are discussed, aninterpretation of the model rank will be summarized. Following thissummarization, an overview will be given to highlight the use and interpretation ofthe statistical results.

When setting overall parameters it is recommended to consider the following:

• Model Form- The default Positional form is recommended. In general, thisform is more restrictive or conservative than the Velocity form. That is it willusually result in noise estimates larger than those obtained using the Velocityform. The reasons for this are primarily twofold.

1. For a given input signal, the auto correlation function will always beworse for the positional form than it is for the velocity form. Forproperly designed signals this difference can be minimized.

2. The estimated noise variance, s2 , will always be larger for thepositional form than it is for the velocity form.

Due to the factors given above, it might be tempting to simply use the velocityform in all cases. While this will in general improve the noise bounds, it willalso usually result in some information loss and hence, some modeldegradation can be expected.

It is recommended to always start with positional form. When this givessatisfactory performance, confidence in the results should be high. Alwayscheck the correlation curves. For properly designed signals, these should bewithin target ranges. If the correlation curves are not within or close to thetarget ranges, then the velocity form must be used to obtain reliable bounds.

In some cases, when the signal design is tentative, it is possible to establish thecausal relationships between input/output pairs using velocity form first andthen rebuilding using positional form. In these instances it is necessary toperform conventional checks (i.e. spikes in step response, predictions andresiduals) to insure that the positional form is justified.

• Settling Time- While the user specified settling times does not have to beparticularly accurate, it should range form a low of 2 Tau to a high of 6-8 Tau.It is better to over, rather than under, estimate the settling time. This is



particularly true when using the positional form.

When the settling time is significantly over estimated, it is common that thereis not enough power range in the input to distinguish the response from noise.This effect is typically exhibited by separation in the step response curves.The approximate range of statistically valid response is graphically evident inthe Confidence View. Use this information to refine settling time estimates ifdesired.

• Coefficients- Choose the number of coefficients as discussed in Section 4.2.It is a common mistake to arbitrarily increase the number of coefficients toimprove performance. In fact, arbitrarily increasing the number of coefficientscan actually decrease the performance. Values in the 20-30 range should beadequate for most applications.

While the statistical information is not strictly necessary for effectiveidentification, it can provide a very valuable resource. At the simplest level it canserve as a guide or sanity check for model validity. With additional effort, it can beused as an investigative tool to improve results. In this context, it would bedesirable to have all models ranked as low (reliable) as possible.

SpecialConsideration

An underlying objective in the design of the quantitative indicators was to use asfew heuristics as possible. The primary noise heuristic, based on past experience,is the correlation between model rank and separation/sensitivity. Based on datareviewed to date, these relationships appear quite reasonable. Nevertheless, someslight modifications may appear in future releases.

There are however two areas which required the addition of special heuristics. Oneis the ranking involving integrating systems and the other is the need to adjust thesensitivity factor for processes that have complex poles.

• Integrators- Noise calculations for integrators are exactly the same as forstable process. Hence noise bands are calculated for each impulse responsecoefficients. Since the FIR form requires special differencing, the noise bands tendto be larger for integrators than for stable processes. Therefore, in many instancesthe non null hypothesis test will fail and the overall rank will be correspondinglypoor (Note, this situation can invariably be reduced or eliminated by rich inputsignals).

For mediocre input signals, the standard heuristic tends to be overly severe. Thusfor integrators, the overall ranking is modified based on the overall averagesensitivity of all trials. The ranking will be modified by at most one unit (see nextsection for interpreting the model rank).

Switching from positional to velocity form to improve results with integrators isNOT recommended. While there have been incidences when this approach hasbeen advantageous, the recommended approach is to design information rich inputsignals and to use the positional form if at all possible.

• Complex Poles- In some instances the FIR step response model may exhibit



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oscillatory behavior. While it is possible the actual process does in fact have thischaracteristic, it is much more likely, especially for industrial processes, that thephenomenon is due to process noise or poor signal design

Usually, oscillatory models require no special considerations since the oscillationsare either bounded by the noise estimates or they are not consistent between trials.If, however, the oscillations exceed the noise bounds and/or are consistent betweentrials, then the calculated sensitivity factor is adjusted. The factor is penalizedbased on overshoot and number of cycles. When this condition occurs, a warningmessage will be displayed in the message window.

This heuristic may not be acceptable to all users. For example the oscillations maybe real and the model accurate. To circumvent this heuristic simply choose eitherthe NNHT or Separation Rank Option.

As a final word of caution on oscillatory behavior, remember that even if theoscillatory behavior is real, well modeled and accurate, it still may not benecessary or helpful to include this characteristic in the controller model. This isparticularly true of predictive controllers since they are in many cases bandwidthlimited.

Interpretation ofModel Rank

To interpret the results, it is necessary to understand the meaning of the modelrank. In the context of the statistical estimates, rank relates only to the confidencethat the model exists. The rank levels and their interpretation shown below are thesame for all rank options.

Rank = 1 This implies that a model clearly exists. In addition, the model isprobably of very high quality particularly if built using positional form. Thesuggested recommendation is to keep (use) the corresponding parametricmodel.

Rank = 2 While not quite as good as a Rank 1 model, this level implies thata model exists and that it is of potentially high quality. The suggestedrecommendation is to keep (use) the corresponding parametric model.

Rank = 3 At this level it is likely that a model exists, but it is probably ofmediocre quality. As a rule of thumb, this is the lowest level model thatshould be used for control without other overriding information. When theRank Option = 4 (Combined), this level is further divided into a + categoryas follows:

Rank = 3+ If any one of the constituent ranks is at level 1, then thismodel is probably more reliable than a typical rank three model. Whenthis occurs, check the NNHT rank option. If this is level 1 then thisshould probably be treated with level 2 confidence.

Rank = 3- If any one of the constituent ranks is at level 5, then thismodel is probably less reliable than a typical rank three model.

The suggested recommendation is to keep (use) the corresponding parametric



model.

Rank = 4 While the model may exist at this level,. The quality could be poor. Inmany instances level 4 models should not be used without rework. Typically, thislevel is the result of high noise and/or relative weak input signals. In someinstances these models may exhibit a favorable sensitivity factor. If this is the case,the models may still be adequate. The suggested recommendation is to keep (use)the corresponding parametric model.

Rank = 5 No reliable model exists. The suggested recommendation is to null(reject) the corresponding parametric model.

While in many instances model rankings are correlated to model quality,remember that this does not have to be the case. In addition, it is tempting to relatemodel rank to its predictive ability. This relationship does not always exist. In factfor data with immeasurable disturbances or drift, prediction errors can be expectedeven for perfect models since only the causal relationship between input/output iscontained in the model. Finally, it is tempting to interpret the noise or confidencebands on the step responses as uncertainty. While they may be related, they are notthe same. The issue of model uncertainty needs to be addressed directly throughthe uncertainty spectrum calculations.

Overview A good starting point is the Demo data presented in Section 9.2. Here Velocityform is used and the correlation and confidence check boxes are selected. Defaultsare used for all other options.

After fitting the FIR models, switch to the MV correlation view. To do this . SelectView>Correlation (MV/MV) as shown below.



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This will display the correlation view for independent variables. Unlike modelviews, the Correlation matrix view for independent variables shows informationfor each independent variable in a two-dimensional matrix of correlation plotboxes. This matrix, as shown below, will always be square. The MVs and DVsform both the columns and rows of this anti-symmetric matrix.

Correlation ViewMV/DV to MV/DV



Diagonal elements of this matrix correspond to the auto correlation function whileoff-diagonal elements correspond to the cross correlation function. It is desirablefor these functions to be within the target ranges specified by the dashed red targetlimits. Results shown above illustrate ideal behavior. If the correlation functionssignificantly exceed their limits, then there may be a sensitivity problem. In thiscase it may be possible to obtain better results by making appropriatemodifications. If the positional form is used then the models can be either rebuiltusing the velocity form or additional data could be collected.

While less impactive than the independent variable correlation view, thedependent variable correlation view still presents useful information. To invokethis view, Select View>Correlation (CV/MV). The following matrix will bedisplayed.

Correlation ViewCV to MV/DV

In this correlation view, the correlation matrix shows information for eachcorrelation function in a two-dimensional matrix of correlation plot boxes. TheMVs and DVs are the columns of the matrix and the CVs are the rows.

It is the objective of this view to visually display potential feedback effects in thedata. The positive portion of the curve represents the correlation from independentto dependent variable. The negative portion of the curve represent the correlationfrom dependent to independent variable. Hence for ideal open-loop data thecorrelation function should theoretically be zero in this region. Practically, thecorrelation function should be within the target ranges indicated by the red dashedlines. Results shown above illustrate completely acceptable behavior

When the inputs themselves are auto correlated, then non zero values of the crosscorrelation function can be expected in the negative region. To compensate for thiseffect, the endpoints of the target ranges are dynamically adjusted. Thus for inputsthat have little or no auto correlation the target ranges will encompass the entirenegative axis.



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Certainly, one of the most important pieces of statistical information available isthe confidence intervals associated with the individual models. This information,along with a visual display of the null hypothesis test is contained in theConfidence view. To access this information, Select View>Confidence. Thefollowing matrix will be displayed.

Confidence/NullHypothesis View

This matrix shows information relating to each impulse response model in a two-dimensional matrix of confidence plot boxes. The MVs and DVs are the columnsof the matrix and the CVs are the rows.

While information presented in this and other statistically related views, does notstrictly pertain to models per se, input/output pairs may still be referred to as submodel elements.

In the Confidence View, only the elements of the impulse response model thatexceed the noise band are displayed. The value displayed is the normalizeddifference between the coefficient value and its corresponding noise band orconfidence limit. If no coefficients corresponding to a given trial are displayed,then the model is not statistically reliable. If no models are reliable for a given submodel, then a completely empty plot box, such as that shown above, is displayed.The empty or null plot box implies no causal relationship between input/output. Anull plot box also implies that the non null hypothesis test has failed and the submodel is of rank 5 for the NNHT rank option. Always check this view for agraphic summary of the confidence results

In addition to simply displaying obvious causal relationships, the confidence viewcan be used to ascertain information pertaining to the temporal quality of the data.Element (1,1) and (1,2) clearly illustrate rich information content up toapproximately 60 minutes. Beyond this, the coefficients are indistinguishable from



the noise. By inspecting the FIR step response curves, it can be seen that this timerange in fact captures essentially the entire model. It can therefore be concludedthat the corresponding input signals were sufficiently powerful over the spectrumappropriate for these models.

Element (1,3) illustrates rich information content up to about 25 minutes. As in theprevious cases, this time range encompasses the entire response curve for trial 1and trial 2. Thus these models are clearly statistically significant.

Note however, that Trial 3 (120 minute settling time) is not displayed. While theinput signal has adequate power in the relative high frequency range, the signaldoes not have enough bandwidth to reliably excite the desired low frequencymodes. Hence, the model is free to drift or fit low frequency noise. Model (1,3,3)has the characteristic that only a small portion of the response is statisticallysignificant. In addition the insignificant portion contributes in a substantial way tothe overall model.

To directly address this characteristic, which can occur relatively frequently, astatistically significant settling time is internally computed and used to detect andreject unreliable responses. This option is controlled by the<UseConfidenceOnTset> check box in the Overall Model Setup Options dialogbox described in Section 8.5

The Confidence view can also be used to quickly establish proper settling times,which if desired can be used to enhance performance. For example the defaultsettling times of 60, 90 and 120 minutes are too long for element (3,1) sinceresponses longer than 20 or 30 minutes are probably not statistically reliable.

Overall statistical results are provided in the Statistical Summary View. To invokethis view, Select View>Correlation (CV/MV).

StatisticalSummary View

This matrix shows the statistical summary for each sub model in a two-dimensional matrix of summary plot boxes. The MVs and DVs are the columns ofthe matrix and the CVs are the rows.

Several descriptors and a plot of the separation or sensitivity bands are displayedfor each sub model for the selected Rank Option.



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Descriptors Descriptors shown in the Statistical Summary view have the following definitionsfor all rank options.

• NNHT- Indicates the results of the non null hypothesis test. Status is eitherPASS or FAIL. Its Value is independent of the selected Rank Option. If thestatus is FAIL then the corresponding rank (for Rank Option = 1(NNHT)) is5.

• Rank Option- Indicates which option has been used to rank the given submodel and corresponds to the information displayed.

• Rank- This is the actual rank corresponding to the displayed Rank Option.

• Separation Factor- Value upon which the rank is based. This value isdisplayed for all Rank Options except Rank Option = 3 (Sensitivity). Thefactor indicates the degree of separation relative to the mean response and willcorrespond to the displayed step response bands.

• Sensitivity Factor- Value upon which the rank is based. This value isdisplayed only for Rank Option = 3. It indicates the smallest sensitivity of thestep response curves when noise estimates are not included and willcorrespond to the displayed step responses.

• Suggested Action- This item is the recommendation based on the modelRank for the selected Rank Option. The recommendation will be to eitherkeep or null the subsequent parametric model. When the recommendation isNULL, an empty plot box will be displayed to visually reinforce the absenceof a particular sub model.

• Pending Action- This item reflects the status of the parametric source flag.When null is displayed, parametric models will not exist for this sub model.

It is possible to automatically perform the Suggested Action. This is accomplishedby selecting the <Auto null uncertain models> check box in the Overall ModelSetup Options dialog box described in Section 8.5. This will automatically loadthe suggested action into the pending action as soon as the FIR models are built.

To select different Rank Options and/or to manually modify the Pending Actionstatus, double click anywhere in the Summary matrix. When this is done, the submodel double clicked will automatically be selected and the Statistical ViewOptions dialog box shown below will be invoked.



Only those models that are selected can be modified. If additional sub models areto be selected , click on the CV name to select the entire row of sub models forthat CV, click on the MV or DV name to select the entire column for that MV orDV, click the upper left box to select all sub models or hold <CTRL> and click toselect any desired combination of sub models. <CTRL> also acts as a toggle.Deselect an item by clicking it again.

Since the Statistical View Options dialog box is modeless, selections can be madeat any time. This dialog box will automatically be closed if the view is changed(since it pertains only to this view), and it can obviously be manually closed beselecting the close button.

Use the pull down list box to select the desired Rank Option. All information inselected models will reflect this change. Select the <Load suggest action> buttonto overwrite the parametric model source flag with the suggest actions for theselected models. This overwrite will be reflected in the Pending Act. descriptordisplayed in the Summary matrix.

Actions can also be manually specified. Select the <Load user action> button tooverwrite the parametric model source flag with the User action source defined bythe selected radio button for the selected models. This overwrite will be reflectedin the Pending Act. descriptor displayed in the Summary matrix.

Positional Form /1 Trial

Having presented a brief overview, a few supplementary remarks will be given toprovide additional insights. Using the same debutanizer data as present above, theFIR models will be refit with positional instead of velocity form. In this instance asingle 90 minute settling time will be used. All other parameters remain the same.Correlation results for the independent variables are shown below.



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By using positional form with this data, the inputs become highly auto correlated.When the correlation functions significantly exceed the target ranges, sensitivityand/or model confidence may be suspect

Typically, there will be correlation concerns when positional form is used and thesignals are not well designed. These deleterious effects can certainly be eliminatedor at least reduced by switching to velocity form as previously illustrated with thisdata. While this is a tempting approach, it is almost always better to try andproperly design the input signals. Doing so will eliminate the need forunnecessarily differencing the data and thereby loosing some low frequencyinformation.

At this point it is possible to use the correlation information to ascertain potentialproblems with respect to the confidence estimates. In Section 3.7, the covariancematrix (upon which the confidence estimates are based) was shown to be directlyrelated to the inverse of the regression matrix. This matrix is basically a scaledversion of the correlation matrix. Hence the conditioning if the covariance matrixwould be expected to increase as the auto correlation (or cross correlation)function degrades. Indeed, in the limit, as the inputs become perfectly auto orcross correlated, the covariance will become singular.

To graphically illustrate the effects of partially correlated inputs, consider theStatistical Summary shown below. This matrix corresponds to the Correlationmatrix presented above.



These results are very revealing considering that the step response bands shownare for a single trial. Thus the bands are due solely to the large noise estimates andreflect a complete lack of confidence. Indeed, all models shown have failed thenon null hypothesis test (To observe these bands it is necessary to switch to RankOption = 0 (No Rank), otherwise empty plot boxes will be displayed).

It is worth mentioning, that for this data, which is discontinuous and exhibits non-stationary behavior, velocity form would most likely be necessary irrespective ofthe input signal design.

Impact ofExclude dataOptions

As discussed previously, the manor in which data is excluded can have an impacton the regression results. In this sub-section a brief example will be presentedshowing some of the significant results. The first case is given below.

Both files have the same 3 CVs and one MV. In file b1, the block selection optionhas been chosen. In file b2, the variable selection option has been chosen with theexclusion applied to all variables. The exclusion ranges are the same in both cases.Results are as follows.



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As can be seen the answers are identical. This is to be expected, since in thevariable selection file, the range was the same for all dependent variables and theselection was applied to the independent as well as dependent variables. Thus infile b2 the values of all variables within the selected range are set bad entering theregression. This data can therefore be collapsed and represented as one NaN foreach variable. This is also precisely what occurs by definition for the block rangeshown in file b1. Hence the results should be identical. The advantage of the “v2”option is that a different range can be defined individually for each dependentvariable. This may be useful when only a subset of the dependent variables isincluded in the regression at any given time. Next, consider the following case.

Here, the data is identical to that given previously. The difference is that the “v1”selection option is used in file b3 and no selection range is specified in file b4.Notice that data corresponding to the selection range has been cut from CV1 in fileb4. Results for these two files are given next.



In both cases shown above the results are identical. Note however that they aredrastically different than the results presented previously. Indeed, the gains are infact of opposite sign for most models. The first set of selection ranges actuallyresulted in rank deficient solutions and completely degenerate models. This wascaused by the removal of additional rows in the regression matrix corresponding tothe bad values of the independent variable. In essence this removed the effect ofthe second step thereby resulting in insufficient information content.

In file b3, the “v1” option was used. Thus only the marked data for the dependentvariables was set bad. This resulted in the removal of only the corresponding rowsof the regression matrix and unnecessary data was not lost. Even though thedesired data was removed, the regression matrix was of full rank and the resultantmodels were of reasonable quality.

In file b4, data was physically removed from CV1. The other CVs however areunaltered. Since the data removed is physically set bad in the .mdl file, any futureregression will obviously see only bad data for these values regardless of anyselection strategy. Since this data and all corresponding rows in the regressionmatrix are removed, identical results such as those shown above should beexpected for CV1. Why do CV2 and CV3 exhibit identical results? The reason issimply because they have been built simultaneously with CV1. As such, the badvalues in CV1 require removal of corresponding rows in the regression matrix,



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which impact CV2 and CV3. This impact yields results that are identical to thecase where CV2 and CV3 are themselves marked with bad values. Hence thesolutions are the same as those obtained in file b3. Note that if CV2 and CV3 wereregressed independently of CV1, then no bad value rows would be removed andthe results would be correspondingly different.

Finally, the discontinuity shown in the prediction plot is NOT due to anyregression range selection or internally bad values. Rather it is the result of aprediction range exclusion. Here the poor data shown previously was excludedfrom the predictions. Had these values not been excluded, the following resultswould have been obtained.


Section 10 - Creating Parametric Models

10.1 OverviewIn This Section This section tells you how to build the parametric models. You can use

the automatic build capability to build parametric models, and you canmanually:

• Select different model forms

• Rebuild individual sub models

• Enter transfer functions.

What Are ParametricModels Used For?

Parametric models are used primarily for model order reduction and toremove the variance of the FIR/PEM models. FIR step response modelsare generated by integrating the impulse response coefficients. Stepresponse models, are fit by the parametric models. While default modelsare low order, no limit is imposed on the order of the parametric models.

Any high frequency behavior of the FIR/PEM model can be captured bythe parametric fit. The defaults almost always capture all control relevantcharacteristics.

Each FIR/PEMmodel is fit by a parametric model. This includes each ofthe models corresponding to the various trials.

Parametric models are used in an open loop fashion with raw data(described in the next section) to select only those models correspondingto the trial that yield the best long term open-loop predictiveperformance. This eliminates the need to be concerned about the choiceof a specific FIR/PEM step response model.

Section 10 - Creating Parametric Models10.2 Procedure


10.2 ProcedureFitting theParametric Models

Select Identify>Fit Parametric Models or select the toolbar button. In the

diagram shown below, the current view corresponds to the FIR Model matrix(Note the Select Vars. For FIR/PEM Fit descriptor in the upper left corner of themodel matrix). In this instance an initial set of FIR models have been created.

Fit ParametricModels Dialog Boxand AssociatedView

Selecting the Fit Parametric Models option automatically changes views. Anyparametric models that are not current are automatically selected (backlit). Amodel is not current if its corresponding FIR model has been modified. As shownbelow, no parametric models are current (since none exist at this time) and whilethe view looks similar to that shown above, this view corresponds to theparametric model matrix (Show Sub-models for Par Fit in the upper left corner ofthe model matrix).



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It is important to remember that different views have different text sensitive areasthat when double clicked, invoke views of specific dialog boxes. This topic isdiscussed in more detail in the paragraphs that follow.

Like all model views, the parametric model matrix view shows information foreach sub model in a two-dimensional matrix of sub model boxes. The MVs andDVs are the columns of the matrix and the CVs are the rows.

Only those models that are selected are updated in the fitting procedure. If submodels are not selected, click on the CV name to select the entire row of submodels for that CV, click on the MV or DV name to select the entire column forthat MV or DV, click the upper left box to select all sub models or hold <CTRL>and click to select any desired combination of sub models. <CTRL> also acts as atoggle. Deselect an item by clicking it again.

Models that are automatically selected (not current), can not be manuallydeselected (These models are automatically deselected when the model isupdated).

Fit the parametric models to the FIR/PEM models using the default options byclicking [Fit Models] on the Fit Parametric Models dialog box. Typical resultsare shown below.

This view now illustrates the results of the parametric fit. No models are selectedas all models are current. The plot boxes show the parametric step responsessuperimposed on the FIR step responses indicating the quality of the fit. The textin the sub model boxes defines the pertinent parameters for the displayed models.Since sub-model (2,2) trial one has a TfSettle that is too long relative to Settle T,both these descriptors are displayed in blue.

When TfSettle is > 1.5 * Settle T, the text for these descriptors will be displayedin blue. Use this text sensitive display in any model view to visually identifymodels with potential deficiencies



Change the method of fit by selecting the appropriate radio button:

• Laplace- Using this option, the FIR model is fit using the Laplace formdefined in Section 1.

• Discrete- With this option, the FIR model is fit using either a pre-filteredARX model or an output error model. ARX is the default discrete option. Tochange this option see the following paragraphs. Whichever z-domain method isused, the final answer is automatically converted back into the s-domain before itis saved. Note that all parametric models are saved in the s-domain (Laplace)irrespective of the source method.

• Best of Both- As the name implies, both Laplace and discrete models areused to fit the FIR models. The model with the best fit is returned as the finalanswer.

To modify the other default options select the appropriate buttons on the maindialog box or double click in the appropriate areas in the parametric model matrixas described in the following paragraphs.

Show & SelectSub-models

This button can be used to return to the parametric model matrix view as shownabove at any time. If the Fit Parametric Models dialog box is displayed and theuser selects another view (i.e. FIR/PEM Step Responses), then the parametricmodel view can be restored by simply clicking the Show & Select button.

Overall Options Use this button to set options at the highest parametric level. Before choosing thisbutton one or more sub models must first be selected. If no models are selectedprior to choosing this button, a message box will be displayed prompting the userto first select one or more sub models before selecting this option.

Remember, parameters are first set, then a function (i.e. Fit Models) performed.The APC Identifier keeps track of which parameters are current (those shown inany view are current) and which are pending (those shown in any dialog box arepending i.e. they are to be used in the next fit). The parameters may or may not bethe same.

To set overall options for sub models (1,2) and (2,1), select the models as shownbelow and click on the [Set Overall Options] button.



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At this point any options that are set apply to ALL models selected. That is, theoptions apply to each of the sub models for all corresponding trials.

Overall options are set from the Overall Parametric Option dialog box shownbelow. This dialog box can be invoked only by selecting the Set Overall Optionsbutton on the Fit Parametric Models dialog box. Since this is a modal dialog box,it must be closed before fitting the model.

Overall Parametric options and their use are described below (as usual, bold textapplies to parameters that can be set by the user).

Discrete ModelInformation

These options allow the user to specify the desired characteristics of the z-domainmodels.

• FIR Extension Sometimes the FIR step response does not settle out (cometo equilibrium). In these instances it is possible that the parametric fitexhibits significant extrapolation. (if the TfSettle parameter is much largerthan the Settle T parameter then there is probably too much extrapolation).To significantly reduce or eliminate extrapolation select this check box.When checked, it automatically pads the FIR step response, that is used forfitting the parametric model, with constant future values. This parameter hasno effect on integrating sub models.

• Auto Pre-Filter This check box allows the user to turn the pre-filtercalculations on or off. When checked, the pre-filtering calculations are doneautomatically. When not checked, the user can control the pre-filtercalculations by changing the pre-filter order described next. Pre-filteringapplies only to the ARX model.

• Pre-Filter When the Auto Pre-Filter check box is deselected the Pre-Filteroption becomes enabled. This parameter enables the user to specify the orderof the pre-filter. A zero implies no pre-filtering calculations. Increasing theorder shifts the fit emphasis from higher to lower frequencies.

• Order This parameter refers ONLY to the order of the discrete timeparametric model (both ARX and Output error). It does not effect the order



of the model used in the Laplace fitting calculations. While no effectivelimits are place on this parameter, highly accurate fits can be achieved withrelatively low orders. For most cases the default (2) should be adequate.Orders of 3 or 4 fit essentially all physically based processes. To force firstorder plus dead time models, it is necessary to fit the parametric model usingOrder = 1 and Method = Discrete. Do not use Best of Both.

• Model Type When a discrete model is built, the model type defines thestructure and solution procedure that is used in the fitting process.

- ARX Selecting this radio button results in the use of the pre-filteredARX structure defined in Section 1. With the appropriate order, thisstructure will effectively result in an unbiased estimate. Models willbe converted to the Laplace domain before being saved.

- Output-Error Selecting this radio button results in the use of theoutput error structure defined in Section 1. Since this is an unbiasedestimator, pre-filtering is not necessary. Models are converted to theLaplace domain before being saved.

• Parametric Model Source- It is possible to specify the source of theparametric model calculations through this parameter. These selections applyto ALL trials for any selected sub model.

− Auto Calculation When this radio button is selected, parametric modelsare automatically fit to the FIR/PEM step response models for the selectedsub models when the Fit Models button is clicked.

− Null Override It is possible to completely null out or eliminate thetransfer function model by selecting this button and clicking [Fit Models].This option has no effect on the FIR models. It is therefore possible to nullor eliminate parametric models from the model matrix in an independentfashion. Note the Fit Model function must be invoked for the option to takeeffect.



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It is also possible to change other parameters by selecting the More Optionsbutton. It is usually not necessary to change the parameters at this lower level.

Since the More Overall Options dialog box (shown below), is modal and isinvoked from the Overall Options dialog box, it is necessary to close this dialogbox before proceeding.

A summary of the parameters in the More Overall Options dialog box is asfollows:

• Level of Laplace Search When the model method is Laplace, the user canfurther restrict the form of the model used in the search.

− Full Search This option results in the use of all terms defined in theLaplace model description given in Section 1.

− Drop short lead terms Reserved for future releases. Whenavailable it allows the user to restrict the placement of the transferfunction zero.

− No lead terms Select this option to completely eliminate processzeros from the transfer function. When selected, the resultanttransfer function is guaranteed not to have positive lead or inverseresponse characteristics.

• Threshold for Delay Calculation or specification of the transport delay(both are provided for in the Identifier) is an important part of the parametriccalculations. Incorrect delays can result in biased estimates. This parameteris used to establish bounds for the delay calculations. It is assumed that anyresponse within the threshold could be effectively considered as delay. Thethreshold is given as a percent of the maximum magnitude of the stepresponse. While in general, it is true that nosier step responses require largerthresholds, experience has shown that the default value almost never needs tobe adjusted.



Once the parameters are set as desired, click [OK] to save the settings. If [Cancel]is clicked, then the settings are not saved. Select [Fit Models] to perform thecalculations using the newly set parameters.

If at this stage, the Set Overall Options dialog box is again invoked, an apparentdiscrepancy may be observed. Initialization of the overall dialog boxes describedabove is provided based in internal defaults NOT from current or pendingsettings. Thus for example, if the discrete model order was set to 3 originally, it isdisplayed as 2 in the Overall dialog box. It is done in this manner since multiplemodels (CVs, MV/DVs and Trials) are almost always selected simultaneouslyand each may have different current or pending parameters. Note, all other dialogboxes display the actual current or pending parameters for the appropriatemodels. Thus if the Individual Parametric Options dialog box (described inparagraphs that follow) is invoked, the order is correctly displayed as 3.

Individual Options Use this button to set options for individual parametric models. At this level,options are specified for a specific sub model corresponding to a specific trial.Unlike in the case described above no sub models need to be selected beforechoosing the Individual Options button. This is a modeless dialog box and assuch, other operations can be performed while it is still opened.

When this button is selected two actions occur. An Individual Parametric Optionsdialog box is displayed and a focus box (colored outline) is drown around the submodel of interest (i.e. the sub model whose parameters are to be potentiallychanged). If no sub models are selected when the Individual Options button isclicked, then the focus box defaults to sub model (1,1) such as that shown below.If one or more sub models are selected then the focus box is drawn on theselected sub model with the smallest row index and the smallest column index.



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Alternatively, the Individual Parametric Options dialog box can be invoked bydouble clicking on the desired sub model.

Note, that if the “Trial” text is double clicked, the Displayed Trial dialog box, asdescribed previously, is invoked. To display the Individual Parametric Optionsdialog box as shown below, double click anywhere else in the text field.



As illustrated above, double clicking on the sub model in the parametric modelview also selected (backlit) the sub model that was double clicked (element (2,2)in this case). A description of the information contained in this and subsequentdialog boxes is presented in the following paragraphs. Parameters that have beendefined once will not be redefined.

Dialog BoxInformation

Navigation of the dialog box is controlled by the previous and next buttons. Usethese buttons to move the focus box to the sub model whose parameters are to bechanged or reviewed. The indices of the focus box are displayed in the dialog boxby the “CV” and “MV/DV” parameters. Note, the selected (backlit) sub model(s)do NOT change as the focus box changes. Therefore, if a model fit is done onlythe models that are selected (backlit) are updated with the modified parametricinformation. Modified parametric information in other models is retained untilthe model is updated. Information in this dialog box is summarized as follows:

• When data is present in the environment, check boxes for the first threeparameters; Null Sub-Process, Integrating Sub-Process and FIR/PEM ModelLocked are for information only. This information reflects parameters thatare current and that have been specified at the FIR/PEM level of theidentification procedure. These parameters apply to all trials. To changethese parameters you must go back to the FIR/PEM level. If however, thereis no data, then these parameters will be enabled and can be modifieddirectly through this dialog box.

• Radio buttons in the Parametric Model Source area apply to all trials for thesub model that has the current focus. The Auto Calculation and NullOverride buttons have the same meaning as described previously

• Values selected are only pending. Models are not changed until a function(Fit Models or Do It) is executed

• Parameters in the Info Per Trial area apply only to the trial number selected.Click on the pull down menu to change the trial number. The settling timecorresponding to the trial number is displayed in non-editable text. Themodel matrix view reflects the selected trial (i.e. model informationdisplayed in the model view corresponds to the selected trial). Trial specificinformation is as follows:

• Lock Dead Time Check this box to specify the desired dead time. Then usethe scroll bar to enter the desired dead time or type it directly in the edit box.The dead time must be in minutes. When checked, the delay estimationroutine is not invoked and the parametric models are fit with the specifiedtransport delay. Values returned after the fit may be slightly different thanthose originally entered since the dead time must be an integer multiple ofthe effective sample rate of the FIR model.



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• Settle Time This is a non-editable field and is shown only to display thesettling time corresponding to the selected trial number. Remember, theparametric model is fit to the FIR model with the specified settling time.

• Current Model Source This is a non-editable field and is shown only todisplay the model source corresponding to the selected trial number. The sourceis “Auto” if the model is automatically fit; “User” if the model is manuallyentered (described below); or “Null” if transfer function model has deleted.

Parametric OptionsPer Trial

Other parameters at the trial level can be changed/reviewed by selecting theoptions button. This invokes the Parametric Options Per Trial dialog box shownbelow.

Parameters in this dialog box correspond to the model defined by the CV,MV/DV and Trial indices. Information not previously defined includes the MoreOptions and Do It buttons.

• More Options Select this button to change options at the lowest parametriclevel. The corresponding dialog box shown below is similar to the MoreOverall Options dialog box described previously, the difference being, herethe options apply only to the model with the displayed CV, MV/DV andTrial indices.



• Do It This button executes the parametric fit routines depending on theselected options. At this level the fit is for only one CV and MV\DV pairfor the selected trial. With this option, the progress window is automaticallyswitched to the background mode and you can repeatedly rebuild a modeland immediately observe the results in the focus window. Selecting the DoIt button implies that the pending parameters are to be used and saved andthe selected model updated.

All three dialog boxes described above are modeless and have been designed towork in conjunction with each other. Parameters changed in one areautomatically reflected in the others. For example, a change in the sub model ortrial from the Individual Parametric Options dialog box results in an automaticupdate of the indices displayed in subsequent dialog boxes. Closing a higher leveldialog box automatically and properly closes all subsequent boxes. Selecting OKor moving the focus box to another sub model saves any modified parameters.

Viewing theTransfer Function

To view, manually enter, or change the transfer function of a sub-model, doubleclick on the plot box of the desired sub model from any model matrix viewexcept the Final Trials view (described in the next section). The following dialogbox is displayed.



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The transfer function of the sub-model for the listed trial is displayed. Thetransfer function is presented in “standard” form. The red trace on the plot is thestep response based on the displayed transfer function. Any time the calculatebutton is depressed, the red trace extends to the steady state of the stable portionof the transfer function (equivalent to about 4 -5 times the time constant for a firstorder system). The green trace is the step response of the FIR model and is shownfor reference only.

Switch to another trial or sub model by using the buttons and pull down listshown in the dialog box. The selected model is indicated by the focus outline onthe model summary view.

The transfer function can be changed by editing the gain, numerator polynomial,denominator polynomial or dead time. After making modifications, click[Calculate] to update the plot of the transfer function and to redisplay the transferfunction in standard form. Click [Accept] to save the new transfer function. If youclick [Exit] or change to a different model without clicking Accept, the model isnot saved.



Example of LegalPolynomials

• Polynomials are entered in the Laplace operator s unfactored, or as anynumber of factors multiplied together. Multiplication is implied when twoquantities appear together (do not use *). A power of s must be preceded by ^.Blanks are ignored.

Examples of legal polynomials are:

2 31 1

3 2 1

4 2 2 2 1

58

2 3 4 5 32 2 2 12

2 3 3 2 3 2

2 6 4 3 2 2

.

^

(5. )( ^ )

.

( . )( . )( . ^ )( . )

( )( ^ ^ )

( . ^ )( )

s

s s

s s s

s s

s s s s

s s s s

++ ++ + +

+ ++ + +

− − + +

• If there is more than one factor, each factor must be enclosed in parenthesiseven if the factor consists of only a single term.

• The transfer function does not have to be entered in standard form, but it willbe automatically standardized when the calculate button is clicked.

• The error box shown above displays the average absolute error between thetransfer function step response and the FIR step response. The error is onlyevaluated over the settling time displayed.

Step ResponseOverview

To observe the FIR/PEM and parametric step responses simultaneously, selectView>All Step Responses as shown below.



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All Responses Since all step response models for all trials are presented in this view, overallperformance can easily be evaluated by observing responses such as those shownin the following figure.

Performance should be judged with respect to sensitivity (separation) andgoodness of fit (parametric vs. FIR). Once the FIR and parametric models havebeen tentatively established (usually in an iterative fashion), it is necessary tocreate a system matrix that contains the best or final models. This procedure isdescribed in the next section.


Section 11 - Selecting Final Models

11.1 OverviewIn This Section This section describes how to find and select the best set of models. This best set

is referred to as the Final models.

Final ModelsDefined

Final models are derived from all possible non-null parametric models. Since amodel may exist for each trial (which corresponds to a user specified settling timefor FIR models or to a particular structure for PEM models), it is necessary tochoose one of potentially several models.

Parametric models are built for each dependent/ independent relationship and forevery trial. By specifying a range of settling times or a range of orders,identification can proceed without regard to model structure.

When dealing with FIR models, accurate ranges for the settling time are notrequired, it is however, possible to specify values that are significantly short orlong. (It may at first appear that long settling times are always good.Unfortunately, depending on excitation signals, specifying settling times that aretoo long can result in poor models.) It then becomes a question of which settlingtime results in the best model.

Similarly, when dealing with PEM models, exact orders are not required, it ishowever possible to specify values that are too low or others which are too high.Either of which can result in a less than desirable model.

Comparing step responses of the FIR models is one way to determine thereasonableness of the specified settling times. This information can be obtainedfrom the FIR/PEM step response summary. Unfortunately, this approach can beambiguous and may not result in the most effective models. Similarly, thestatistics can be used for a more unambiguous estimate. However, even forstatistically valid models with refined settling times, it may be possible to havemodels with somewhat different characteristics.

Comparing PEM step responses is an effective way to evaluate reasonableness ofthe PEM models. However, even for cases when the responses are satisfactory, itmay be possible to have models with somewhat different characteristics.

To avoid these difficulties, the last step in the identification procedure is atechnique that automatically searches all models to find the final set that give thebest long term open loop prediction relative to raw process data. This searchprocedure effectively rejects any models that are ill suited for predictionpurposes. This technique strives to select the best of the available models (Note,that it does NOT guarantee that the selected model is necessarily good. It maysimply select the best of a poor set of models). To prevent the use of poor FIRmodels use the statistical results to null the appropriate parametric models beforeperforming this final step. To prevent the use of poor PEM models use theguidelines given previously on PEM step responses to null unreasonable models.

Section 11 - Selecting Final Models11.1 Overview


Searching for theBest Final Models

Open loop prediction forms the basis of the search procedure. Prediction is doneon a Multiple Input Single Output (MISO) basis. That is, only one CV at a time isevaluated but all possible MVs/DVs are used in the evaluation.

The MISO model is evaluated based on its open loop predictive performancerelative to raw plant data. The default data is the same as that used by theregression, but any segment can be used including data never regressed (crossvalidation).

The actual MVs/DVs are used as inputs to the MISO model. The output of theMISO model is the predicted value of the process CV. Since this predicted CV isnever updated by the actual CV data, the results illustrate the long term open loopperformance of the model.

The figure of merit for these models is the absolute value of the average residualerror (difference between actual and predicted values).

Two Procedures In evaluating the MISO model, the APC Identifier uses two analytical strategies.

In one strategy, the Identifier uses models corresponding to uniform trial indices(each index corresponds to a user specified settling time). The Identifier attemptsto find the models that result in the lowest average prediction error, given that allmodels for a CV are based on the same trial index.

In the other strategy, the Identifier uses models corresponding to mixed trialindices. Its starting point is the uniform trial solution. Starting with the firstMV/DV, each model not corresponding to the uniform solution is evaluated inthe overall MISO model holding all other models constant.

If the current model results in a reduction in the average prediction error, then themodel is added to the mixed trial solution. The procedure continues until allmodels have been evaluated. Although this search is not exhaustive, it almostalways finds an optimal or near-optimal solution.

Section 11 - Selecting Final Models11.2 Procedure


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11.2 ProcedureSelecting FinalTrials/FindingFinal Models

Select Identify>Select Final Trials or select the toolbar button. In the diagram

shown below, the current view corresponds to the parametric Model matrix (Note theShow Sub models for Par Fit descriptor in the upper left corner of the model matrix).In this instance an initial set of all FIR/PEM and corresponding parametric modelshave been created.

Selecting the Select Final Trials option automatically changes views and invokes theSelect Final Models dialog box as shown below. The default view is the Final Trialsview as shown in the upper left corner of the model matrix.



This view presents the model matrix which corresponds to those models that aredeemed the final solution to the identification problem. When a controller (eitherProfit Controller (RMPCT) or RPID) is built, only final models are used in thecalculation procedure. The “Final Trials” descriptor is used to indicate that each finalsub model is one of potentially several possible models. When the identificationprocedure is completed, this view will show the final parametric models in Laplacedomain form along with other pertinent information.

It is highly recommended to inspect this view before continuing with any controlbuilding operation. Note that this view can also be accessed by selecting View>FinalModel Xfer Function from the main menu.

Inspection of the final models shown above, illustrates that all models are “Invalidfinal models”. Invalid final models indicate that final models have not yet beenselected (as is the case here) or that there is something wrong with the final model.The “Invalid final model” state will preclude the building of either a ProfitController (RMPCT) or RPID controller.

Many options are available for selecting/defining the final models. Taylor theselection procedure by choosing the desired options from the Select Final Modelsdialog box as Discussed below.

Trial Source By far the most important option is the choice of the trial source. The APC Identifiermaintains four separate and distinct sets of models for use as effective long termopen-loop predictors. These models are characterized by their trial indices. The fourtrial sources are:

Auto Best Uniform - Trials whose submodels produce the smallest averageabsolute error between the predicted results and the test data from the process,given that the submodels for any row in the matrix are all from the same trial

Auto Best Mixed - Trials whose submodels produce the smallest averageabsolute error between the predicted results and the test data from the process,given that the submodels for any row in the matrix can be from different trials

User Selected - This choice allows the user to manually select the trial foreach submodel. Since this set of trials is initialized by either the uniform ormixed trial set, it is highly recommended that one of these be updated (asdescribed below) before manually selecting the trial for any submodel

Final - Trial for each sub model that corresponds to the final model.



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Both the uniform and mixed buttons are associated with a calculation procedure thatstrives to minimize the prediction error for a selected CV subject to the restrictionsgiven above. As such, referral to these buttons is made with respect to either theuniform or mixed solution. That is, the trials for these buttons are the result orsolution to the search for the minimum prediction error. The solution for the AutoBest Mixed trials has as its starting point the Auto Best Uniform solution. Thus auniform solution is always performed prior to the calculation of the mixed solution.

Final model selection should always begin by choosing either the uniform or mixedbutton. Selection of any of the Trial Source buttons automatically changes to theappropriate view. The text displayed in the upper left corner of the model matrix(defining the view) for the various buttons is as follows:

• Auto Best Uniform - “Auto Best Uniform Trials”

• Auto Best Mixed - “Auto Best Mixed Trials”

• User Selected - “User Selected Trials”

• Final - “Final Trials

Information displayed in these views corresponds to the models with the trial indicesdefined by the selected radio button. If the Auto Best Mixed button is selected in thediagram presented above, then the result is as follows.



In this figure, the view has been automatically changed to Auto Best Mixed Trials. Inaddition, all CVs have been automatically selected (backlit). Note: CVs with anyuniform or mixed trial solutions that are not current are automatically selected(user defined trials that are not current are NOT automatically selected). Theuser can not select/deselect from any of the trial views (To select/deselect see thediscussion on the Show & Select Sub models button). A solution is not current if anycorresponding parametric model has been modified since the last update. A CV witha solution that is not current can not be deselected.

As shown above, no solutions are current and the Final and Pending Errors areundefined (since none exist at this time). To obtain a best mixed solution usingdefault options, click [Update Trials]. A message window shows the progress of thesearch. To view the progress messages click Window>Messages

After the search is complete, the Auto Best Mixed Trials view shown below isdisplayed.



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This view shows the models that correspond to the Auto Best Mixed solution. It alsodisplays the final and pending prediction errors. Since final models have not yet beencreated, the final error is still undefined. The pending errors are those errors that areassociated with the mixed solution. Final Source designates the trial source of thefinal model. The source can be Uniform, Mixed or User. . Since final models havenot yet been created, there is no final source.

Further modification of the final model selection process is accomplished throughthe use of the appropriate buttons on the main dialog box shown above. Thesebuttons have been designed to work in conjunction with each other to offer themaximum degree of flexibility in selecting a final model.

There are essentially three categories of operations. The first is the Trail Sourcewhich has just been described. The next two are the selection and functionoperations respectively. The selection buttons apply to both variables and data. Thefunction buttons apply to whatever is selected (variables, data, trial sources). Use ofthese buttons is described below.

Show & SelectSub-models

This button has two primary functions. If the Select Final Models dialog box isdisplayed and the user selects another view from the main menu (i.e. Single-GraphData Plot), then the original view can be restored by simply clicking the Show &Select button.



Its other primary function is to allow the user to independently select variables foruse in any of the functions supported in the Select Final Models dialog box. Note:At this level, only CVs can be selected. Individual sub models can not be selectedsince all functions involve operations on raw data and this implies the use of MISOmodels.

This button must be used for the user to select/deselect any variable or to show avariable that has been previously selected by the user. When the button is clicked,the view is automatically changed to the Select Sub models for Final Model view.Models displayed correspond to the Trial Source radio button selected.

If there are no CVs highlighted, click the CVs of interest, or click the upper left boxto select all. <CTRL> toggles the selection state.

If the Trial Source is changed to either Uniform or Mixed after variables areselected, then only CVs that are not current will show as selected. Clicking the Show& Select button again redisplays the user selected CVs.

Excluding DataFrom thePredictionCalculations

For a detailed discussion on selecting, marking/unmarking and viewing data seesection 5. To exclude data from the Prediction Calculations (mark data bad), choose[Exclude Data Ranges]. This changes the view from the current view to the ExcludePrediction Ranges view (titled “Show Pred. Ranges”). The variables displayed willcorrespond to those selected in the Select Submodels for Final Models view.

Data that is to be excluded applies to what ever calculations are to follow. Theseranges remain in effect until the next time they are changed. Note that these rangesare independent of the ranges selected in the FIR/PEM calculations (see Section9.2). Only Block Selection is supported for excluding data when performingprediction calculations. This allows a simple mechanism for cross validation. Inaddition, it is easy to evaluate the search (for the best trials) over one set of data,then re-select the ranges to see the predictive performance over another potentiallydifferent set of data.

Selection or changing of a selected range does NOT affect whether CVs are currentor not. Prior to release 115 of Profit Design Studio (APCDE), the [Update Trials]option would not result in a true update based solely on a selection rangemodification. Subsequent releases will perform full updates for all selected CVs. Ininstances when no CVs are selected a dialog box such, as that shown below will bedisplayed.



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Select TrialManually

Use this option to manually specify the desired trials. This button is disabled unlessthe trial source is [User Selected]. When this button is selected, the User TrialSelection dialog box is displayed with the focus box on matrix element (1,1)

The same dialog box can be invoked by double clicking in the non Trial text area ofany desired sub model (double clicking on the Trial descriptor invokes the displayedtrial dialog box as described previously). Remember that the Trial Source must be[User Selected] for this to work.

To set the user trial for matrix element (2,1) to trial 3, choose the [User Selected]Trial Source and double click in the text area of CV 2, MV 1. Then select TRIAL 3from the Select User Trial pull down menu as shown below.

Dialog BoxInformation

Navigation of the User Trial Selection dialog box is controlled by the previous andnext buttons. Use these buttons to move the focus box to the sub model whose trialsare to be changed or reviewed. The indices of the focus box are displayed in thedialog box by the “CV” and “MV/DV” parameters.

Information in this dialog box is summarized as follows:

Trial Value - This value is the current trial index for each of the four trial categoriesfor the sub model with the current focus. In the case shown above the Uniform,Mixed and User all have indices of 2 while the Final trial value is empty. The threeentries are the result of the one Auto Best Mixed solution. As described previously,the uniform solution is determined prior to calculating the mixed solution. In thiscase the solution was the same for both searches. That is the model corresponding totrial 2 resulted in the minimum prediction error. The User trial is initialized with themixed trial solution if it exists. Otherwise the uniform solution is used forinitialization



Info Per Trial – FIR/PEM settling time and parametric model source are displayedunder this category. Use the pull down menu to change the displayed trial. Thesettling time and model source change accordingly as does the model informationdisplayed in the User Selected Trials view

Selected User Trial - Use this pull down menu to actually specify the desired trial.All displayed information automatically reflects this selection.

Since this is a modal dialog box, it must be closed before another operation can beperformed. Selecting OK or moving the focus box to another sub model saves thespecified trial information.

Update Trial When any trial information is not current, this button can be used to invoke thevarious searches and/or update the prediction errors. This button works inconjunction with the first three trial categories and applies to whichever variables areselected. The procedure depends on the trial source as follows:

Auto Best Uniform - It performs the uniform trial search for the minimum predictionerror as described above and update the pending error with the resultantminimum prediction error

Auto Best Mixed - It first performs the uniform trial search. With this as a startingpoint, it subsequently performs the mixed trial search for the minimumprediction error as described above and update the pending error with theresultant minimum prediction error

User Selected - No search is performed when this radio button is selected. Themodels corresponding to the user selected trials are merely used to update theprediction error. Since user modifications to the trials are NOT tagged in theautomated selection process, it is recommended that the trials be updated assoon as they are modified. This ensures that the prediction errors remain current.

Stop Use the Stop button to prematurely terminate the search procedure. Note, the searchfor the best mixed solution can be time consuming for large data sets especially whenthere are many trials.

Plot Predictions Performance for any set of models over any range of data can be obtained byselecting this button. Long term open-loop predictions are displayed in terms ofpredicted, measured and residual values as a function of time. These values aregenerated for any CVs that are selected using the transfer function modelscorresponding to the indices of the current Trial Source.

Two types of prediction calculations are supported.

Positional - With this default option, raw data is used unaltered in the predictioncalculation. Bias shifts and drift effects cause discrepancies between predicted andactual data. Use this option for standard evaluations.

Velocity - With this option, raw data is differenced prior to the predictioncalculations. While bias shifts and drift effects are reduced or eliminated, noiseeffects are amplified. Use this option in support of the default. Since responses are



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impulse ‘like’, this option can be very useful for integrating models.

Note: These prediction options have nothing in common with the FIR/PEMmodel forms that use the same names.

Use the prediction information to evaluate model performance before selecting finalmodels. If for example, the performance of the mixed trial solution for CV 1 is to beobserved, select [Auto Best Mixed]. Then chose [Show & Select Sub models] andclick on CV 1 (it will be highlighted). Next, select [Plot Predictions]. The followingresults are displayed.

Store Predictions – Use this option to store any predictions into an Aux variable.This variable can then be observed at a later time in the Single Graph Data Plotsview and as such can be plotted against any other variable.

In addition to cross validation, use of range selection when performing predictioncalculations can be very effective in evaluating models subject to significantdisturbances. Consider the case shown below. Here there is one CV and 2 MVs.



As illustrated above, both sub-models contain integrators. As shown, severalwindows have been opened on the same file. The dialog box is associated withwindow 3 (it was created from this window). Windows 2 and 4 are Single GraphData Plots and Exclude Prediction Ranges views respectively. Two ranges have beenselected as shown by the vertical bands in window 4. A range of both MV1 andMV2 has been marked bad. Window 3 shows the predicted verses actual valuesalong with the residual. Note that in this view, excluded ranges are collapsed to asingle point which is represented by a single NaN. Hence, circles mark the beginningand end of each discontinuous segment with a single blank point in between. Notealso that CVs that are either marked as bad or are the result of a computation using avariable that is marked as bad, will NOT result in the collapse of data. In this caseeach bad value will have a corresponding blank space in the associated plot.

Since the Store Prediction option has been selected, the predicted value is saved as aspecial Aux variable. The maximum and minimum range value used for display istaken directly from the actual CV. This makes future comparisons more convenient.Window 2 shows the MVs, CV and its predicted value. Note that in this view, thepredicted values are expanded (if ranges were specified that resulted in the collapseof data) to be consistent with the time axis.

If at this point, the exclude data range button were selected, Window 3 would beswitched to an Exclude Prediction Ranges view. In the case shown above this is notnecessary since window 4 is already set to this view. Use this window to selectranges for exclusion. Then select <Plot Predictions>. The prediction and residualview (window 3) will be updated as will the trend plots in window 2 (update of thiswindow is required since the Aux variable which represents the prediction can



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potentially change when the prediction is updated).

Clearly the results shown above are horrific. Consider what happens when 3additional ranges are selected for exclusion. In this case, each range consists of asingle time slice (here, data points 337, 402 and 866 are excluded). The idea is NOTto remove data per se, but rather to reinitialize the bias. Remember that the bias isrecalculated for each discontinuous data segment. The results for this case are shownbelow.

These results illustrate a dramatic improvement in the predicted valued over all butthe initial segment. Note also the comparison of the predicted and actual values inthe trend plot shown in window2. The importance of using the same scale for thepredicted values is self-evident.



Configuration of the prediction plots (and all other views in which plots occur) canbe modified by adjusting the plot options. To do this select View>Plot Options fromthe main menu. Set the Magnification and Height/Width ratio as shown below.

With these settings, the prediction plots (from the original Demo example) take thefollowing form.

Adjust the settings as desired and continue evaluating the performance of all themodels. When the results are satisfactory, select the final trials as discussed below.

Load Source toFinal

Use the [Load Source to Final] button to create final models. To load the entiremixed trial solution into the final models for the case shown above; select [Auto BestMixed] as the Trial Source, click on Show & Select Sub-models, click on upper leftcorner of the model matrix and select [Load Source to Final]. The results areillustrated below.



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As can be seen above, the view is automatically switched to the Final Trial view,which displays the final models. Also displayed in this view are the final errors andthe final source. Final errors are now defined (in this view the pending error has nomeaning and is therefore not displayed). As illustrated the mixed trial solution is thesource for all final models.

This view is unique in that the Laplace domain transfer function of the finalparametric model is displayed. In addition the step response curves are displayedwith the time axis corresponding to the maximum of TfSettle and Settle T.

At this point, the first pass of the identification procedure is complete. The file canbe saved for later use or it can be used for control design.

Modification and/or adjustments to the final models can be achieved in a relativelystraightforward fashion. For example, consider the case where it is desired tomanually adjust the transfer function of element (2,1) for trial 3 (the user selectedtrial shown above). In addition, it is desired for the final model to contain the bestmixed solution for CV1 and CV3, the user solution for CV 2 and the uniformsolution for CV4. To do this proceed as follows.

Change the transfer function as desired. The Transfer Function dialog box formanual entry can be invoked by double clicking in the plot box for all views exceptthe final trial view. In this case select [User Selected] and double click in the plotbox for element (2,1). Then enter the transfer function as shown below.



After the transfer function is entered click [Calculate] to show the full step responseof the transfer function. When satisfied click [Accept] then [Exit]. The user enteredtransfer function is now save in element (2,1,3)

When the dialog box is closed, the view is still User Selected Trials. If at this point,the Trial Source is changed to either Uniform or Mixed, the CV2 is backlitindicating that solutions for this CV are not current. The solution for this CV istherefore automatically calculated the next time the Update Trials button is chosenwith either the uniform or mixed button selected.

In the User Selected Trials view, CV2 is not selected. Similarly, if the Show &Select button is chosen (Select Submodels for Final Model view), CV2 is notselected. That is because these views only show user selected CVs. Typically at thisstage, it would be advisable to update the prediction error for the user model. Select[Show & select Submodels], click on CV2, then click [Update Trials]. If there isconcern that the manually entered transfer function is a better predictor than the onecalculated automatically, then the search can be re-evaluated. Select either [AutoBest Uniform] or [Auto Best Mixed] (CV2 is automatically selected) then click[Update Trials]. All information is now current.

To construct the final models, select [Auto Best Mixed], click [Show & selectSubmodels], click on CV1 and CV3, then click [Load Source To Final]. Next, select[User Selected], click [Show & select Submodels], click on CV2, then click [LoadSource To Final]. Finally, select [Auto Best Uniform], click [Show & selectSubmodels], click on CV4, then click [Load Source To Final].



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Null Final Model In some circumstances it may be desirable to see the impact that one or more finalmodels have on overall prediction performance. This can be easily accomplished bytemporarily nulling final submodels.

In the final model (Final Trials) view, double click on any model box. The Null FinalModel dialog box shown below will be displayed.

This is the only dialog box that can be invoked from this view. Only one submodelcan be nulled at a time. If more than one submodel for a given CV is to be nulled,null all but the last without residual update (this saves the time associated with theprediction calculations). For the last model, select [With residual update]. Theprediction error now reflects the effect of the null model(s). The null model isdisplayed as shown below.



Compare the errors with and without the models. Observe the prediction as describedabove.

Restoration of the null models is simple. If the Select Final Models dialog box is notpresent, select Identify>Select Final Trials from the main menu. Select theappropriate Trial Source, choose [Show & Select Submodels], click on the desiredCV, then click [Load Source To Final]. If at this stage it is attempted to build acontroller, the following message will be displayed.

Be sure to inspect the final matrix. As shown above, the backlit settling timeinformation indicated there may be a problem. If the response is not reasonable, it isalways best to correct or eliminate any sub-models before building any controllers.

When finished with the identification, it is a good idea to save the file. After thedocument is saved, the title reflects the new name as filename.mdl (or filename.pid ).The source descriptor (from .mpt or .pnt) will no longer be displayed. This file canbe opened at another time to merge and/or rebuild models.

Section 11 - Selecting Final Models11.3 Final and Model Summary Views


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11.3 Final and Model Summary ViewsAt the completion of any identification session, it is recommended that allpertinent model information be reviewed. This is true whether the models aresimply to be saved or be used to synthesize a controller. The Identifier providestwo convenient methods for reviewing/observing overall model information.This information is presented in terms of the Final Model and Model Summaryviews, which are summarized below.

Final Model View Creation of the final models based on the final trial selection procedure has beendiscussed in detail in the previous section. The final model view can be invokedby either selecting Identify>Select Final Trials or by selecting View>FinalModel Xfer Function. The former approach will invoke the Select Final Modelsdialog box and since the default Trial Source is Final, the view will beautomatically switched to the final model view. Note, the view referred to asfinal model view has the Final Trials descriptor in the upper left corner of themodel matrix. As shown below.

Information displayed in this view has been described in the previous section. Inaddition to the Final Model view, another view that is very useful for reviewingthe various model information is the Model Summary view. This view isdiscussed below.



Model SummaryView

To switch to the model summary view, select View>Model Summary from themain menu. Like all model views, the model summary view shows informationfor each sub model in a two-dimensional matrix of sub model boxes. The MVsand DVs are the columns of the matrix and the CVs are the rows.

This view is extremely similar to the parametric model view discussed in Section10. In fact all text sensitive areas and displayed text are the same with theexception of the view descriptor found in the upper left corner of the modelmatrix. The only real difference between this and the Show Sub-models forParametric Fit view described in Section 10, is the type of dialog box invoked bydouble clicking on non-Trial, text sensitive areas. This action results in the dialogbox shown below.

Information displayed in this dialog box is very similar to the IndividualParametric Options dialog box discussed in Section 10. The primary difference isthat the Model Summary Parameters dialog box contains the current trial indicescorresponding to the various trial sources as described in Section 11.2. Thisinformation is displayed under the Trial Value category.

Operation of this dialog box is identical to that for the Individual ParametricOptions dialog box. In fact, individual parametric models can also be fit/refit hereby using the Options button in a fashion identical to that used with the IndividualParametric Options dialog box.



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Copy Trials fromOne Source toAnother

These functions apply only to selected sub-models. Operations involve copyinginformation from one Trial Source to another (See section 6.2 for a discussion onthe edit procedure). These functions have the following characteristics.

• User2Final – This function is enabled only in the Model Summary viewand applies to all selected sub-models. Only the models corresponding to thedisplayed Trials are copied to the Final models. The copy results in an automaticresidual update and the trials are stored as user selected. This function is onlyapplicable within a given document. Select the toolbar button or select

Edit>User2Final.

In addition to copying the displayed trials for the selected sub-models into the“User Trials”, the residuals for any “touched” CVs are updated. These results arethen loaded into the Final Models. The trials and corresponding models displayedin the Final Model view reflect the user choices. Also note that the predictionerror and Final Model Source are automatically updated.

• Uniform2User – This function is enabled only in the Model Summaryview and applies to all selected sub-models. It behaves much like thatdiscussed in the previous paragraphs. Here however the Uniform Trialsolution is copied to the User Trials.. The copy results in an automaticresidual update and the trials are stored as user selected (see section onselecting final models). This function is only applicable within a givendocument. If the Final Model Source for any of the “touched” CVs is oftype “User”, then these Final Models will be updated to reflect thechanges.

• Mixed2User – This function is enabled only in the Model Summary viewand applies to all selected sub-models. It behaves much like that discussedin the previous paragraphs. Here however the Mixed Trial solution iscopied to the User Trials. The copy results in an automatic residual updateand the trials are stored as user selected (see section on selecting finalmodels). This function is only applicable within a given document. If theFinal Model Source for any of the “touched” CVs is of type “User”, thenthese Final Models will be updated to reflect the changes.


Section 12 - Annotation

12.1 OverviewProfit Design Studio (APCDE) supports annotation at many levels. Both user andautomatic annotation are available. The following items can be annotated:

• Applications (Overall file)

• Variables

− Var

− Aux

• Sub-models

• Graphs

− Single Graph Data Plots

− FIR/PEM range selection

− Final Trials/Predictions range selection

• Vector Calculations

Variables and graphs can be automatically annotated. This feature can be turnedon or off at any time by setting the AutoAnnotate variable equal to one or zerorespectively in the .ini file.

Access to the annotation for any item can be easily accomplished from virtuallyany appropriate view. To access an item simply lift up on the right mouse buttonwith the cursor over the desired item. The next section describes annotationaccess and update in some detail.

Section 12 - Annotation12.2 Annotation Access and Update


12.2 Annotation Access and UpdateAccess Overview Annotation dialog boxes can be invoked in many different ways. Annotation for

each item is available as summarized below.

Applications – To access this item, lift up on the right mouse button when thecursor is over the upper left-hand corner of any matrix view. The sameannotation will be accessed irrespective of the current view.

Variables – These annotations can be accessed by lifting up on the right mousebutton when the cursor is over virtually any tag name that is not in a dialog box.Use the left or top margins in any matrix view. Use the Descriptive Info. Viewor any Single Graph Data Plot View.

Sub-models – With this item, the annotations apply to the row-column elementof any matrix. The same annotation will be accessed irrespective the particularmatrix view. The one exception is the MV/DV – MV/DV correlation view. Thisview has its own annotation items. Lifting up on the right mouse button whenthe cursor is over any sub-model can access these annotations.

Graphs – Graphical annotations can be accessed by

1) Selecting View>Single Graph Data Plots and lift up on the rightmouse button when the cursor is in the text margin. You will be given the optionto annotate the variable closest to the cursor, all displayed variables or the plotcorresponding to the general data.

2) Selecting View>Exclude FIR/PEM Ranges or by selecting ExcludeData ranges from the Fit FIR/PEM dialog box and lift up on the right mousebutton when the cursor is in the text margin. You will be given the option toannotate the variable closest to the cursor, all displayed variables or the plotcorresponding to any data ranges that have been selected for exclusion withrespect to FIR/PEM regression.

3) Selecting View>Exclude Prediction Ranges or by selectingExclude Data ranges from the Select Final Trials dialog box and lift up on theright mouse button when the cursor is in the text margin. You will be given theoption to annotate the variable closest to the cursor, all displayed variables orthe plot corresponding to any data ranges that have been selected for exclusionwith respect to final trials/prediction calculations.

Vector Calculations – Annotations for these items can only be accessed byselecting Vector Calculation> Vector Function>User Notes from either the DataOperations or Tools main menu

Once annotations are made they can be viewed and or modified at any time bysimply reselecting as described above. An annotation descriptor (superscript A)will appear in all matrix views for any variable or sub-model that is currentlyannotated.



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Detailed Access andUpdate

Annotations can be updated either manually or automatically. For the former,the user can enter any and all descriptions that are desired. The only limitationsis that a warning message will be displayed for an individual annotation item ifthe number of characters exceeds the parameter MaxAnnotate times 1000000defined in the .ini file. Once opened the annotation dialog box can not be closeduntil the character limit is satisfied.

As its name implies, the AutoAnnotate flag initiates auto annotation.Currently, auto-annotation is provided under five different scenarios asdescribed below.

1) Data Cut – When all variables are displayed in the Single Graph Data Plotsview and one or more ranges are defined and the user selects delete, then thedata in the ranges (inclusive) is eliminated from the environment and a singleNaN replaces each range for each variable. This implies a reduction in the datalength for all vectors. When this occurs and the AutoAnnotate flag is on, thefollowing annotation will occur

• Annotation for Single Graph Data Plots view will list the number of datacuts (ranges) and the start and end time of each cut. It will then list the time atwhich each NaN is inserted.

• Annotation for each variable will list the NaN insertion time correspondingto each data cut

• Annotation for FIR/PEM range exclusion will be updated if any ranges areselected. Start and end indices corresponding to the ranges prior to the cut willbe listed followed by the start and end indices after the cut.

• Annotation for Final Trials/Prediction range exclusion will be updated ifany ranges are selected. Start and end indices corresponding to the ranges priorto the cut will be listed followed by the start and end indices after the cut.

2) Data Deletion - When one or more but not all variables are displayed in theSingle Graph Data Plots view and one or more ranges are defined and the userselects delete, then the data in the ranges (inclusive) for each displayed variableis set bad (NaN). In this case the data vectors are NOT collapsed. When thisoccurs and the AutoAnnotate flag is on, the following annotation will occur

• Annotation for Single Graph Data Plots view will list the number of dataranges selected for deletion and the start and end time of each deleted range. Itwill then list all variables for which data has been set bad (NaN).

• Annotation for each displayed variable will be updated to reflect the rangesover which data has been set bad. The start and end times of each range will berecorder

3) Data Modification – When data is modified using the Block Manipulationoption under the Data Operations main menu, annotations will be updated for



each variable that is modified. The start and end index for each modificationrange and corresponding replacement option will be recorded

4) Block Range Change – When FIR/PEM block range selection is modified,the corresponding annotation will be updated the next time a FIR/PEM model isregressed. The start and end indices of each range will be recorded. When dataused for selecting Final Trials or predictions is modified, the correspondingannotation will be updated the next time an update or prediction is performed.

5) Data/Model Merge – When data is merged, annotations in the destinationfile for each variable will be updated reflecting when the merge took place andthe source file from which the data was merged. When models are merged,annotations in the destination file for each model “touched” will be updatedreflecting when the merge took place and the source file from which the modelswere merged.

All auto-annotations will be marked accordingly at the beginning of theannotation. When any annotation is made, a time/date stamp is automaticallyinserted at the end of the annotation.

For user annotations, it is recommended to start all new text on a new line in thedialog box. When exiting the annotation dialog box, you do not need to insert anew line as this is automatically done prior to the insertion of the time/datestamp. Since it is anticipated that annotation for sub-models will occur from avariety of different views, these annotations will be automatically appended withthe particular view from which the annotation was made. The comment willimmediately proceed the time/date stamp. The next section presents use of theannotation features through the demo example



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Annotation Example To start, the auto-annotate feature is turned on. Some data is excluded and anFIR fit is performed. From the Fit FIR/PEM dialog box Select>Exclude DataRanges. Then move the cursor over DV3 and lift up in the right mouse button toobtain the following.

The text margin in the picture given above can be used to select annotationitems corresponding to either variables or plots. Only this area can be used toinvoke annotations (Use of the right mouse button in the plot or in the time axisbox is reserved for displaying data). When the right mouse button is lifted up, apop-up menu is displayed at the cursor position. If Annotate This Var is selectedthen the annotation dialog box for the variable corresponding to the cursorposition will be displayed. If Annotate Displayed Vars is selected then theannotation will be applied to all variables listed in the left margin. SelectingAnnotate Plot as shown above invokes the following dialog box.



All annotation dialog boxes display a heading corresponding to the particularannotation item of interest. In this case it is the Plot associated with FIR/PEMregressions. As a next step data will be cut from the workspace. To do thisSelect>Single Graph Data Plots. Select the desired ranges then hit <Del>. If theright mouse button is used from the Single Graph Data Plots view then selectingannotate plots as before will give the following.

Note that this annotation applies to the Single Graph Data plots while theprevious annotation was for the FIR/PEM range selection. Which annotationitem appears depends on the current view and weather or not ranges are to beselected for FIR/PEM or Final Trials/Predictions. Now, Data will be deleted forthe first three MVs and the second DV. When this is done the annotationbecomes.

As more text is added the dialog box supports scrolling. By default, the mostcurrent annotation is scrolled into view. As shown above any annotation text canbe selected. This selected text can be cut, copied and pasted in the normalfashion. Text can be pasted into other annotation dialog boxes or into you



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favorite text editor. Note that when data is cut (removed from the workspace)the data length is altered. As such, information in the above dialog box is givenin terms of time stamps (since they are invariant) rather than indices. At thispoint the annotation on the FIR/PEM range selection has the followingappearance.

This information is given in terms of indices since indices are more convenientfor resetting of ranges. As such when data is cut both before and after indices ofall ranges are presented. Note that an annotation is made any time data is alteredindicating that the FIR/PEM model needs to be updated. The last two remarks,of which only one is shown above, were made when the data was cut and thenwhen some data was deleted respectively. Since no new range information isdisplayed after these comments, it is apparent that the FIR/PEM models havenot yet been updated. After all models are refit, the annotation will be asfollows.

Note – Auto annotation for range selection only applies for range selection



using the block option.

The last message will be displayed even if no ranges are selected. As a finalmodification, DV2 and CV2 can be altered using block manipulations.Annotations on DV2 are given below.

These annotations can be invoked by using the right mouse button wherever thetag name is displayed. As described in the dialog box two ranges were selectedand overwritten using the interpolation option. Following this an independentrange was selected and the range was overwritten with the value immediatelypreceding the range.

Note – Data modifications made using Vector Calculations will NOT be auto-annotated. It is up to the user to annotate these modifications.

To enter or modify an annotation simply select the annotation item using theright mouse button. Type any desired text. Use the enter key to start a new line.When satisfied select <OK>. To ignore modifications select <Cancel>.Toremove any segment of text, select the text to be removed in the standardfashion and select <Clear>. If no text is selected then <Clear> will remove allcontents in the dialog box.

As mentioned previously, annotations can be accessed at many different levels.To access sub-models a matrix view must be present. Sub-models can beaccessed from every matrix view. Every view except MV- Correlation willaccess the same sub-model annotation. Variable annotations can be accessed byusing the right mouse button in the appropriate margins (Left for CVs and topfor MV/DVs).



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Overall annotations can be accessed by lifting up the right mouse button whenthe cursor is in the upper left corner of the model matrix. In the matrix views asmall superscript A as shown below will indicate annotation for any variable orsub-model.

Visual inspection of the picture given above indicates that MV2 and sub-model(2,1) are annotated. Scroll around to observe any other annotated items. Allviews will display the same annotation information. Hold the right mouse buttondown and move it over sub-model (2,1). Nothing will happen until the mousebutton is released. When this is done the following annotation dialog box will beinvoked.



Finally, consider the case where a sub-model is merged into the previous demoexample. In this case the source model data sample rate was different than thatcontained in the destination file. Hence the data is automatically dropped but themodels are merged in the normal fashion. Results are shown below.

A new row and column (CV3 and MV2) have been added to the matrix. Sub-model (3,2) is the new element and has been automatically annotated. Theannotation for this element is given in the following graphic.



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Note that if data were included in the merge, then all variables would beautomatically annotated to reflect this operation. When data is dropped, noannotation is made to the corresponding variables.

12.2 Annotation Access and Update



Section 13 - Tutorial

13.1 OverviewIn previous sections of this document, the main emphasis was to present asequential approach to the use of the APC Identifier. While relevant backgroundand guidelines were furnished in many instances, the focus was nevertheless moreon the mechanical aspects of using the Identifier than on illustrating actualidentification.

This section has been added to briefly show some identification examples, whichillustrate a few of the more practical aspects, involved in model synthesis. Thisinformation is presented as a high level overview and is not intended as aninstructional device. For those interested in proficient use of this tool, the ProfitController (RMPCT) Implementation class is highly recommended. For thoseinterested in a more detailed use of this tool and a better understanding ofadvanced identification topics and procedures, the new Advance ID class isrecommended.

This chapter will be split into two major themes. The first will deal with thegeneral use of the tool using FIR models as the main regression function. Thesecond will illustrate basic use of the PEM approach

While there are many ways in which the FIR information can be presented, it willbe arbitrarily categorized based on data quality. The categories are splitaccording to data that was generated using:

• Rich Input Signals

• Typical Input Signals

• Limited Input Signals

A few sets of data in each of the above categories will be presented, as will thenuances of practical model synthesis.

PEM applications will also be presented using various data sets. Basic operationwill first be given using synthetic data for which there is a known answer. Therest of the discussion will be based on plant data. Pressure data will be used toshow simple use and performance. Furnace data and data with large disturbanceswill be presented next. The demo example used through this document will besolved as will a high purity (very slow dynamics) column. Finally, applicationsinvolving integrators and long delay will be presented.

Section 13 - Tutorial13.2 Rich Input Signals


13.2 Rich Input SignalsTwo case studies will be presented in this section. In the first, synthetic data withknown models will be used. In the second, actual data taken from a silicon wafermanufacturing facility will be used

RichDoc1 This data is characterized by a relatively high order process that is subject tolarge immeasurable disturbances and high levels of output noise. There are threeCVs and three MVs. The signals are given below

Note The inputs have been designed specifically for this process. FIR models arefit using positional form and settling times that range from 10, 12 and 15 minuteswith between 25, 30 and 25 coefficients respectively. Corresponding correlationplots follow



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Comparing the diagonal elements illustrates that the designed signals are close toideal (the pseudo-white). Cross correlations are near perfect. With thesesatisfactory results, the next step is to check FIR and confidence data.

Based on both the FIR and confidence views, it is obvious that elements (1,1) and(1,3) exist while element (1,2) does not. Similarly, elements (2,2) and (2,3) existwhile (2,1) does not.



A less clear case is presented by CV3. The FIR data shown indicates that all sub-models may exist. FIR results worse than these have been interpreted by some toindicate model existence. Confidence results on the other hand indicate nomodels exist. The answer can be obtained by a closer inspection of the FIR stepresponse curves. Observe the spike in the last coefficient. As describedpreviously, this indicates non-stationary behavior. In deed, this variableexperiences a large drift during the test. Hence, CV3 should be built usingvelocity form. When this is done the following results with respect to FIR andconfidence data will be obtained.

Now, the correct answer is readily apparent. Models (3,1) and (3,3) do not exist,while model (3,2) does. It would also be possible to rebuild all models usingvelocity form. This will only result in a relatively small loss in accuracy for CV1and CV2. These results are shown below.



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With the appropriate forms selected, the Statistical Summary View will illustrateboth the correct density pattern and the fact that the models that exist are highquality..

Using these results, the predictive performance for CV1 is illustrated in the nextgraph.

The performance speaks for itself. Indeed the model obtained for CV2 is within2% of the analytical solution. In fact even the high frequency lead term wascorrectly captured. The model for CV3 has been identified with the same level ofaccuracy. This is to be expected based on the quality levels presented previously.

Predictive performance for CV3 is illustrated next. While the performance lookspoor, the model is in fact correct. This case demonstrates the effect of a largeunmeasured disturbance



As discussed previously, it is important to determine the correct causalrelationship between input and output and NOT necessarily to fit the data. In thisinstance the quality of the causal relationship has already been established.

While predictive performance is an important metric, it does not always relatedirectly to model quality. Here it has been shown that a high quality model canexhibit relatively poor predictive performance. Later, the converse will bedemonstrated. That is models that are relatively uncertain may be reasonablepredictors.

As a final comment on predictive performance, the following possibilities exist.

1. High quality models/Good predictions - This is the ideal situation and shouldinspire high confidence in the models

2. Low quality models/Poor predictions - This too, is an ideal situation in that itis consistent. Here, the indication is that there is a serious problem with themodel and it should not be used.

3. Low quality models/Good predictions - In this case the user has conflictinginformation. In all data observed to date, this is caused by limitedinformation content. Usually caused by large noise/signal ratios and /orinsufficient number of steps. Use caution here. Better data is the bestsolution, but in some cases the models may be adequate.

4. High quality models/Poor predictions - This is the case shown above.Typically the models are desirable but true performance is not manifested inthe prediction due to immeasurable disturbances.



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Anomalies associated with item 4 above are more annoying than they are serious.Invariable, they can be detected by judicious use of the Identifier. A powerful butseldom used function is the ability to select appropriate data ranges whenperforming predictions. Results are shown below for the same data as presentedpreviously.

In the above plot, only 8 single data points have been deselected. The usefulnessof this capability is strikingly apparent.

In spite of the aforementioned capability, the need to make manual adjustments issomewhat time consuming. In cases such as these, this need can be eliminated bythe use of a noise model. The capability is planned to be included as a PEMoption in the future release of the advanced ID module.

WafrDoc1 This plant data shows the response of silicon wafer temperatures to radiant heatlamps. The thermal transport mechanism is predominately radiation. As such thetemperature response exhibits integral behavior. There are three CVs and threeMV. The signals are given below



Integrator flags are set for all sub models and FIR models are fit using positionalform and settling times of 2, 3 and 4 minutes. The overall Rank Option is set toNNHT and the <Auto null uncertain models> check box is selected. TheCorresponding correlation plots follow

The input correlations are relatively good. The negative auto correlation at + 1minute is of concern. However, since it recovers rapidly, it will most likely beacceptable.

As shown in the second set of plots, the output correlations indicate potentialproblems. Diagonal elements are very acceptable. Off diagonal elements,however, indicate significant feed back in the data. This occurrence is actually bydesign. While the signals were properly designed , this integrating processrequired some closed loop control to keep the temperatures in an acceptable



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range during the duration of the test. Hence the feedback must simply beaccommodated. The FIR and confidence views are presented next

FIR responses indicate that models (2,1), (3,1) and (2,3) are questionable.Existence is shown more clearly in the Confidence View. Only the diagonalmodels are statistically relevant.

With the default Rank Option = 1 (NNHT), the Corresponding StatisticalSummary View takes the form shown below.



For integrators these high quality models are outstanding. Usually it is verydifficult to obtain a level 1 rank for integrators. In fact in many instances level 3integrators can be considered good. Final corresponding models and subsequentpredictions are shown next.

Section 13 - Tutorial13.3 Typical Input Signals


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13.3 Typical Input SignalsTwo case studies will be presented in this section. Both utilize plant data. In thefirst, tests from an atmospheric tower will be reviewed. In this test data, the inputsignal can be considered marginal. In the second, data for a high purity distillationcolumn with a long settling (20-24 hrs.) will be presented. Here the input signalscan be considered to be of relatively good quality.

TowrDoc1 This data is characterized by an input signal that has a fairly limited power band.The band is adequate for some variables and lacking for others. There are fourCVs and one MV. The signals are given below

As a quick first pass models are first fit using all default options (positional form,60, 90, 120 minute settling time with 30 coefficients). In this case the correlationhave the following characteristic.

This should be deemed suspect (values that are significantly off scale, asillustrated in this plot, may result in undesirable behavior). Rebuilding usingvelocity form gives the improved results depicted below.



While still not ideal, they are acceptable. Proceeding with these settings, allowsthe generation of the FIR step responses and the Confidence estimates to becalculated. Results for these quantities are given next. Results will be presentedfirst for CV1 and CV2.

In the above plots, the first column corresponds to the FIR step responses whilethe second column corresponds to the confidence estimates. Rows correspond toCV1 and CV2 respectively.

FIR Results indicate potential separation concerns. Intuition would dictate thepresence of a model at least for CV1. Inspection of the confidence estimates givesa clear indication of model existence for both CVs. In addition, the confidenceestimates indicate that the FIR coefficients become unreliable at settling timesgreater than 60-90 minutes. To check this, the models can be rebuilt using 40, 50and 60 minute settling times. Results for CV1 are given below.



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These curves show essentially ideal behavior. FIR responses are self similar andthe confidence data shows that most of the model is in fact captured in the first 30-40 minutes of the response.

Similar results can be obtained for CV2 as presented below. With CV2 however, amore judicious choice of the settling times is required to obtain such satisfactoryresults. That is, as settling times exceed 60 minutes, the results deteriorate rapidly.This anomaly, which is somewhat characteristic of this entire case study, is causesby a lack of power band in the input signal.

Further inspection of the results given above, indicate that there is probably someamount of non-minimum phase behavior (time delay in this case) associated withCV2. In addition it is obvious that there is limited steady state information in thismodel. As such, this is an ideal candidate for using the <FIR Extension> flag whenfitting the parametric model.

That CV1 and CV2 are quality models is self evident as illustrated by theStatistical Summary results given next.



It was relatively easy to extract reasonable models from this data for CV1 andCV2. For these variables model existence is clear an unambiguous. The remainingCVs illustrate the case where model existence becomes an issue.

Next consider CV3 and CV4. These CVs have a longer response time andconsiderably more noise than the previous CVs. However, the same input signalwill be used to build these models. To start, default settling times are also used.Results in terms of FIR and confidence plots follow.

For CV4 the FIR and Confidence answers are consistent. This is clearly a casewhere there is no reliable model. For CV3 however, the FIR and confidenceanswers apparently conflict. FIR results indicate that the 120 minute settling timeis too long, while the 60-90 minute curve are relatively self similar. Theconfidence curves on the other hand indicate that the shorter settling times are infact not significant.

Confidence curves, such as those presented above for CV3, would in generalindicate that the shorter settling times are either statistically unreliable or are justtoo short. Based on FIR intuition, however, it seems reasonable to rebuild usingshorter settling times. Results for 45, 60 and 90 minutes are illustrated in thefollowing plots.



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For CV4 it is still clear that no reliable model exists. For CV3, it can be seen thatthe noise level is too high to determine reliable models for the shorter settlingtimes. The estimates just start to exceed the confidence threshold for the 90 minutetrial. Unfortunately, as the settling times are further increased with insufficientinput power (as illustrated by the 120 minute settling time presented previously),the model begins to fit the noise in a statistically meaningful fashion. Hence theshorter settling times do in fact result in statistically unreliable models, however,the longer settling times are also dubious.

Inspection of the FIR step responses for CV3 illustrates that the 90 minute trial isnot quite able to capture the steady state behavior of the process. The inability tocapture steady state is caused by lack of input range. Corresponding summaryresults are given below.



In the final analysis, CV3 is seen to be of questionable quality while CV4 has nomodel at all. While CV3 does have relatively reasonable step response curves, itstill should not be considered to be statistically reliable. To understand this morefully, consider the predictive performance shown below for all CVs.

While the fit is very good in all cases (even for CV4), the amplitude of movementfor CV3 and CV4 relative to the magnitude of the noise should be of concern.Indeed, this is one of the major limitations in this data set. In addition, the inputpower frequency is rather limited. Since the duration of the input ‘steps’ appearsmore as a pulse for CV3, information in both the low (steady state gain) and highfrequency ranges is compromised. Indeed, the transfer function settling times forthis variable exceeds the specified settling time by more than 50%.

As discussed previously, it is possible for a model to exhibit good predictiveperformance, yet not be statistically reliable. Here the lack of reliability is dueprimarily to the noise and to a lesser extent to the limited duration of the steps.While there is no doubt that models far worse than these have been used inpractice, the textbook recommendation would be to either gather moreinformation, or exclude them from the controller design.



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To more clearly illustrate the problems associated with noisy data, it can be moreinformative to generate the predictions in velocity form. To do this select the<Velocity> radio button then select <Plot Predictions>. Result shown below arethe velocity equivalent of the predictions given previously.

This information is more reflective of the data actually used in the regressioncalculations. Since velocity form is used, the predictions are the impulses thatresult from the changing input. While causal relationships are clearly demonstratedfor the first 2 CVs, it would be difficult to state with any certainty that arelationship exists for the remaining CVs if the above data were all that wereavailable.

It is precisely this information that is reported in the statistical summary. Thus, thisinformation reflects the confidence that the model is not attributable to or undulyinfluenced by noise effects. Even though models for CV3 and CV4 fit the datawell (in the least squares sense), their reliability remains in question. In fact thereis little difference between the reliability of CV3 and CV4. Both models should beconsidered unreliable due to the noise level. The fact that CV4 has a slightlyhigher noise level than CV3, results in a cross over from level 4 to level 5 rank. Assuch, it is clear that this model should not be used. Note, that the level 4 results arenot that much different. Therefore, while model retention is the level 4recommendation, these models should still be considered with some degree oftrepidation.



At this point it should be realized that the statistical information addresses twoconcerns that are often encountered. The first concern is the use of ‘small gain’models. In the current framework, a model is either reliable or it is not. Modelsthat have gains that are small relative to noise will automatically be rejected. Ifthere was sufficient authority in the input signals to move the process outside itsnoise band and the models is reliable, then the model is useful regardless of thenumerical gain value. Of course, other considerations such as MV movementlimitations may in the end be the determining factor.

With respect to the second concern, the need to capture high frequency dynamicsis sometimes in question. Irrespective of controller bandwidth limitations, if thestep response bands encompass the high frequency dynamics, then there is no needto have this level of detail in the final model since it can be considered to be withinthe noise level. This issue is addressed directly in the uncertainty estimates.

As a practical point it is worth mentioning, that step tests should be designed toinsure signal to noise ratios of about 3. That is, the output (CV) movement shouldexceed the noise present by a factor of three. This rule of thumb is consistent withthe statistics provide by the Identifier. Note, that for integrators it is desirable tomove the inputs such that the impulse exceed the noise level. Simply making asmall move and having the integrating nature of the process move the CV outsidethe noise is in general not sufficient.

ColDoc1 This data is characterized by a process that exhibits a very long response time.Hence the inputs need to be sufficiently exciting over a relatively wide spectrum.Input signals here are of reasonable quality. There is one CV and three MVs. Thesignals are given below



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With long settling time such as this, it might be tempting to adjust the number ofcoefficients accordingly. This however, is not necessary. The length of the settlingtime imposes NO restriction on the number of coefficients. Only the curvature ofthe response function influences the required number of coefficients. In this casethe data is sampled every minute and the controller is to run every 2 minutes. Inspite of this the default number of coefficients give excellent results. Using defaultsettings with settling times of 11.5, 15 and 20 hours gives the following correlationplots.

From this figure, it is clear that there is a relatively large auto correlation. Inaddition, there is a strong cross correlation between flow1 and the feeddisturbance. Results using Velocity form are presented next



Observe the marked improvement in the correlation functions. By using thevelocity form, the data has been transformed such that the variables are effectivelypre-whitened. Corresponding FIR and Confidence data are presented next.

In spite of moderate separation, the confidence plots clearly indicate existence ofall three models. Elements (1,1) and (1,3) are statistically valid up to around 600to 800 minutes. Element (1,2) is valid only to about 300 minutes. Review of theStatistical Summary view, shown below, illustrates that all models are of verygood quality

At this stage the validity of the models has been established. However, since thereis a moderate amount of separation, the best model still need to be selected. It isprecisely under these condition for which the final pass of the APC Identifier hasbeen specifically designed. Results of the automated selection process andsubsequent predictions follow.



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It is interesting to note that the selected solution corresponds to those trials that arecompletely within the statistically valid band. Even though the FIR responsescorresponding to these trials for MV1 and DV1 had not completely settled,confidence should remain very high due to the duration of the test and the qualityof the predictions.

Section 13 - Tutorial13.4 Limited Input Signals


13.4 Limited Input SignalsIn this last section, two case studies will be presented. Both deal with plant data.The first, is indicative of a single step used in conjunction with an integratingprocess. The second illustrates results of a pre-step test of a process with anexceptionally long dead time.

LevDoc1 This data represents the response of a level to valve opening. The data has a mildamount of noise and only a single step (down then up). There is one CV and oneMV. The signals are given below

Single steps, such as those shown above should never be conducted in actualpractice. Nevertheless, the corresponding correlation results for Positional andVelocity forms respectively are presented next.

As has been the case in previous results, poor correlation performance can bedrastically improved by using positional form. Unfortunately, for this problemthere is no magic wand. In fact, it will be shown shortly that the velocity formactually degrades performance.



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Performance, in terms of step responses for the two model forms is highlighted inthe plots presented next.

Note that the velocity form is more prone to separation. This is true in generaland not restricted to integrating data. Since differencing the data will result insome low frequency information loss, this can be expected. The information losstypically results in a (small) reduction in steady state gain accuracy. This loss inaccuracy can be either to over or underestimate the gain (or integration rate ifappropriate). It is precisely for this reason that it is recommended to at least startwith positional form. In either case, the non-null hypothesis test fails andtherefore the Confidence view is null. The corresponding Statistical SummaryView and prediction plots ( for the positional model only) are presented in theplots that follow



Results presented in the Statistical Summary view reflect the special heuristicsused for integrators. Without these heuristics both plots would display a level 5ranking. With velocity form, the degraded step response sensitivity, results in areduced rank relative to that given for the positional form. To check thesensitivity, the Rank Option can be changed to 3 to view the followinginformation.

The corresponding ranking and sensitivity for the positional form are 1 and .117respectively. Combined Level 3 ranking for integrators are most likely indicativeof decent models. For Integrating processes with questionable statistics payparticular attention to the prediction results. The goal here, should be results suchas shown above. For difficult cases try the <Velocity > Plot Prediction option formore insight.

BlecDoc2 This data represents the response of two key variables in a bleach plant. The datahas a mild amount of noise and only few steps. There are two CVs and one MV.The signals are given below



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As is obvious from the data, the response for CV1 has a long dead time and bothCVs exhibit quick dynamics. This data indicates one of the few legitimate caseswhen the default number of coefficients needs to be increased. Here CV1 andCV2 will be built using different settling times. To capture the long dead time inCV1 settling times of 90, 110, and 140 will be used. For CV2 settling times of10, 15, and 20 will be used. In this case positional form will be used. Use ofvelocity form here results in slightly poorer results.

Correlation results for the long and short set of settling times are displayedbelow.

Confidence and Statistical Summary views are presented next.



Results indicate a low confidence in CV1. In spite of the proper settling times anda large number of coefficients, accuracy of the delay estimation should beconsidered suspect. To accurately estimate delay, it is necessary to have sufficientpower in the high frequency portion of the response curve. In addition to theproper discrete time resolution (number of coefficients) is also required.

Even though the step response band for CV2 is fairly large, the quality of thismodel is very good. Note, that the confidence data for all three trials are sosimilar, they appear as one curve. These results are typical for very fastresponding processes even for relatively limited data.

Finally, as shown below, even though the models are not necessarily reliable,they both give excellent predictive performance.

In this instance the dead time for CV1 was 50 minutes while the time constantwas approximately 10 minutes. To capture this type of response the increase inthe number of coefficients was not only justified it was required.

Section 13 - Tutorial13.5 Creating PEM models


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13.5 Creating PEM modelsAs mentioned several times in this document, use of the PEM models is intendedprimarily for problems with only one or two inputs that are movingsimultaneously.

Synthetic Data To start a problem with a known solution will be used. The data for this problemis shown below.

A rich input signal and significant drift characterize this one input-one outputproblem. This is a subset of the data shown at the beginning of this section(RichDoc1). For the first try, the Start Order will be set to 1 (in general youshould not use an order of less than 2 or 3). This choice will result in thefollowing dialog box.

This dialog box will be displayed any time the algorithm detects a potentialproblem with the model. With the PEM approach, the settling time is determinedfrom the model itself. For this problem, the first order model results in a biasedestimate that has as an enormous effect on the model. Usually you would take thedefault (for ease of use) and still use the rule of thumb that there should be atleast 2 self similar trials. Here we are curious so we won’t null the model



The results for the three models are:

In this case the known order is 3, the settling time is approximately 12 minutesand the gain is 1. Set the Start order to 3 and Load & Go to get.

Note the self-similar responses for the three trials. Inspection of the transferfunction shows the gain to be .989 and TfSettle is 12.4. In addition one of theroots of the D polynomial is 1.0002 which corresponds to the pure drift exhibitedby the data. Not only is B and F correct so is the noise term.

Next the same data will be used but in this case we will change some of the setupparameters. Here we will turn the Pfx initial search off, the instrumental variableapproach will be used for initialization, the robust norm will be used, the PEMbias term (α) will be tuned off, QR factorization will be used and there will be noscaling. Only one trial will be used and that trial will correspond to a third ordermodel which can result in the correct answer as shown above.. The results will becompared to the MATLAB solution.



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Results from the message window in the APC identifier are as follows.

Corresponding results from MATLAB are.



The output above illustrates that under the right conditions both MATLAB andthe APC Identifier will yield the same results. Note however that these answersare not the same as the third order case run previously. As a word of caution, donot modify .ini parameters lightly.

Pressure Data In the next case data from a pressure loop will be used. Here, there is one MV,one DV and one CV. The data for this loop is shown below.

In this case, the start order is set to five and Load & go. The step responses are:

It is clear that the DV model exhibits behavior that is due to the use of an orderthat is probably higher than is necessary. The effects are however easilyattenuated by the parametric fit. So in general a slight ringing of the PEM modelshould not be of concern as long as it is not too significant. Especially if it isattenuated by the model reduction step.



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Predictions for this case are shown below.

Furnace Data This data is taken from a furnace application. Here there is one MV, and twoDVs, only one of which will be used.

Note the large deviation in CV1 at the beginning of the test set. This data hasbeen excluded. For this case DV2 is also nulled. Start order is 5, Load & go.

For the pressure data, the predictions are stored in an Aux variable and the Auxvariable is plotted in the Single-Graph Data plot view against the inputs and CV.



It looks pretty reasonable even after the disturbance hits towards the end of thetest. Consider the case where the front portion of data is not removed. The stepresponses for the condition are.

The large initial drift in the CV is not due to the initial moves in the DV and it isnot handled well by the noise model. Thus the models tend to be degraded. Notethat the noise model is NOT a cure all. Even with the noise model, it is alwaysbetter to remove suspicious data.

Large disturbance Here is a case where the disturbance starts small and continues to grow as the testcontinues. The input signal has a good spectrum however, the power can not dealwell with the disturbance. The data is shown below.

Set start Order to 3 and Load & Go give the following step responses.



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This is not a pretty picture. If the back end of the data is excluded as shownbelow.

Then performing the same calculations results in the step responses shown next.



The predictions, using the stored Aux variable, are

When the disturbance is removed, the model does a fairly good job. It is clear inthis case that the noise model could not accommodate this disturbance. It is worthnoting that when fitting the entire data set with an FIR model using velocity forma better, though relative poor model, was obtained. In some instances a prioriknowledge can be used to advantage. This in no way is meant to imply that youshould ever include disturbances such as those in the regression no matter whattechnique you are using.

When comparing FIR and PEM step responses remember that PEM models tendto be more sensitive to model order than FIR models are to settling time. Alsoremember that when messages warning of too short data sets are displayed,better results may sometimes be obtained by turning off the noise terms in thePEM model.

Demo Data While the two input guidelines should be adhered to, there are no physicalrestrictions on inputs when using the PEM models. In fact the PEM approach hasbeen used with demo data used throughout this document. The step responses aregiven below.



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And the corresponding final models are.

While it is possible to use the PEM models on problems like this it is simplyimpractical. This is a relative small problem and the amount of resources wasunacceptable.

ColDoc1 Next, PEM will be used with the coldoc1 data, which was already discussed.Using Load & GO in this case generates a warning message on undersampling.Nevertheless the following step responses are generated.

These curves show an extremely long settling time. Remember for PEM thedefault is no compression. The means the regression is at the data rate which inthis case is one minute. These settling times with a one-minute sample rate are thereason for the warning message. The predictions for these models are quite good.



WafrDoc1 To demonstrate use of PEM with integrators, a subset of the WafrDoc1 data willbe used. In this instance, the AutoInteg flag will be used to detect integrators.Note when this flag is set NO special consideration is given to the data (i.e. nospecial differencing), the algorithm will simply try to identify the presence ofintegrating dynamics from the poles of the PEM model. Using the defaults andselecting AutoInteg and Load & Go, the following message box is displayed.

This message box tells you that a potential integrator has been found. Ifintegrating characteristics are found for all trials and you select yes, then the localsub-model integrator flag will automatically be set to insure perfect integrators atthe parametric level. If some trial for a given sub-model have integrator likecharacteristics but others don’t then the following message box will be displayed.



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Note that this message will not be displayed if the sub-model integrator flag isalready set. Continuing with the calculations will result in the following Stepresponses.

Predictions for this case are as illustrated below.

BlecDoc2 As a final case a subset of the BlecDoc2 data described previously will be used.For this case the approach will fail if the auto delay flag is not set. Even when it isset, the approach requires several iterations. And as such can not be consideredvery effective. Work must be done to develop a better delay estimator. With astart order of 5 the step responses are.

Note that the model corresponding to seventh order failed in this case (initialestimate had problems). Again you can see the ringing phenomenon that oftenaccompanies high order fits. This is almost never a problem. Note the smooth



transfer function shown below in the final model.

The final predictions however, as shown below are good.


Appendix A - Sample of a FIR Model File

Sample FIR input data for a 3 CV, 2MV and one DV model matrix is as follows.

321411.00631.00511.00631.00171.000.01.00.03001.013001.01151.010-0.233-0.4349-0.61-0.7617-0.8933-1.007-1.106-1.192-1.266-1.331



-1.386-1.435-1.477-1.513-1.545-1.572-1.596-1.616-1.634-1.649-1.663-1.674-1.685-1.693-1.701-1.707-1.713-1.718-1.722-1.726-1.729-1.732-1.734-1.736-1.738-1.74-1.741-1.742-1.743-1.7440000.043980.084440.12170.15590.18740.21640.24310.26760.29020.3110.33010.34770.3638



0.37870.39240.4050.41660.42730.43710.44610.45440.46210.46910.47560.48150.4870.4920.49670.50090.50490.50850.51180.51480.51770.52020.52260.52480.52680.52870.53040.53190.53340.53470.53590.53710.53810.53910.53990.54070.54150.54220.54280.54340.54390.54440.54480.54520.54560.546



0.546300.033310.063440.090710.11540.13770.15790.17620.19270.20770.22120.23350.24460.25460.26370.27190.27930.28610.29210.29770.30260.30710.31120.31490.31820.32130.3240.32650.32870.33070.33260.33420.33570.33710.33830.33940.34040.34130.34220.34290.34360.34420.34480.34530.3457



0.34610.34650.34680.34710.34740.3476000-0.05997-0.1151-0.1659-0.2126-0.2556-0.2951-0.3315-0.3649-0.3957-0.4241-0.4501-0.4741-0.4962-0.5164-0.5351-0.5523-0.5681-0.5827-0.596-0.6083-0.6197-0.6301-0.6397-0.6485-0.6566-0.6641-0.671-0.6773-0.6831-0.6884-0.6934-0.6979-0.7021-0.7059-0.7094-0.7127-0.7156



-0.7184-0.7209-0.7232-0.7254-0.7274-0.7292-0.7308-0.7324-0.7338-0.7351-0.7363-0.7374-0.7384-0.7393-0.7402-0.7409-0.7417-0.7423-0.7429-0.7435-0.744-0.7445-0.744900.31180.53520.69530.810.89220.95110.99331.0241.0451.0611.0721.081.0861.091.0931.095000000.001242



0.0048630.010720.018660.028570.040320.05380.068890.08550.10350.12290.14350.16530.18820.21220.23710.26290.28950.3170.34520.37410.40370.43390.46460.49590.52770.560.59280.6260.65950.69340.72770.76230.79720.83240.86790.90360.93950.97571.0121.0491.0851.1221.1591.1971.2341.271



1.3091.3471.3851.4231.4611.4991.5371.5751.6141.6521.6911.7291.7681.8071.8451.8841.9231.9622.0012.042.0792.1182.1572.1962.2352.2742.3132.3522.3922.4312.472.5092.5492.5882.6272.6662.7062.7452.7842.8242.8632.9022.9422.9813.0213.06



3.0993.1393.1783.2183.2573.2963.3363.3753.4153.4543.4943.5333.5733.6123.6523.6913.733.773.8093.8493.8883.9283.9674.0074.0464.0864.1254.1654.2044.2444.2834.3234.3624.4024.4414.4814.524.564.5994.6394.6784.7184.7574.7974.8364.876



4.9154.9554.9945.0335.0735.1125.1525.1915.2315.275.315.3495.3895.4285.4685.5075.5475.5865.6265.6655.7055.7445.7845.8235.8635.9025.9425.9816.0216.066.16.1396.1796.2186.2586.2976.3376.3766.4166.4556.4956.5346.5746.6136.6536.692



6.7326.7716.8116.856.896.9296.9697.0087.0487.0877.1277.1667.2067.2457.2857.3247.3647.4037.4437.4827.5227.5617.6017.647.687.7197.7597.7987.8387.8777.9177.9567.9968.0358.0758.1148.1548.1938.2338.2728.3128.3518.3918.438.478.509



8.5498.5888.6288.6678.7078.7468.7868.8258.8658.9048.9448.9839.0239.0629.1029.1419.1819.229.269.2999.3399.3789.4189.4579.4979.5369.5769.6159.6559.6949.7349.7739.8139.8529.8929.9319.97110.0110.0510.0910.1310.1710.2110.2510.2910.33



10.3710.4110.4410.4810.5210.5610.610.6410.6810.7210.7610.810.8410.8810.9210.961111.040-0.0001532-0.0006002-0.001323-0.002306-0.003532-0.004987-0.006656-0.008528-0.01059-0.01283-0.01524-0.0178-0.02051-0.02336-0.02634-0.02945-0.03267-0.03599-0.03942-0.04294-0.04656-0.05025-0.05403-0.05788-0.0618-0.06578-0.06983



-0.07393-0.07809-0.0823-0.08656-0.09086-0.09521-0.09959-0.104-0.1085-0.113-0.1175-0.122-0.1266-0.1312-0.1358-0.1405-0.1452-0.1499-0.1546-0.1593-0.164-0.1688-0.1736-0.1784-0.1832-0.188-0.1928-0.1976-0.2025-0.2073-0.2122-0.2171-0.2219-0.2268-0.2317-0.2366-0.2415-0.2464-0.2514-0.2563-0.2612-0.2661-0.2711-0.276-0.2809-0.2859



-0.2908-0.2958-0.3008-0.3057-0.3107-0.3156-0.3206-0.3256-0.3305-0.3355-0.3405-0.3455-0.3504-0.3554-0.3604-0.3654-0.3704-0.3753-0.3803-0.3853-0.3903-0.3953-0.4003-0.4052-0.4102-0.4152-0.4202-0.4252-0.4302-0.4352-0.4402-0.4452-0.4502-0.4552-0.4602-0.4652-0.4701-0.4751-0.4801-0.4851-0.4901-0.4951-0.5001-0.5051-0.5101-0.5151



-0.5201-0.5251-0.5301-0.5351-0.5401-0.5451-0.5501-0.5551-0.5601-0.5651-0.5701-0.5751-0.5801-0.5851-0.5901-0.5951-0.6001-0.6051-0.6101-0.6151-0.6201-0.6251-0.6301-0.6351-0.6401-0.6451-0.6501-0.6551-0.6601-0.6651-0.6701-0.6751-0.6801-0.6851-0.6901-0.6951-0.7001-0.7051-0.7101-0.7151-0.7201-0.7251-0.7301-0.7351-0.7401-0.7451



-0.7501-0.7551-0.7601-0.7651-0.7701-0.7751-0.7801-0.7851-0.7901-0.7951-0.8001-0.8051-0.8101-0.8151-0.8201-0.8251-0.8301-0.8351-0.8401-0.8451-0.8501-0.8551-0.8601-0.8651-0.8701-0.8751-0.8801-0.8851-0.8901-0.8951-0.9001-0.9051-0.9101-0.9151-0.9201-0.9251-0.9301-0.9351-0.9401-0.9451-0.9501-0.9551-0.9601-0.9651-0.9701-0.9751



-0.9801-0.9851-0.9901-0.9951-1-1.005-1.01-1.015-1.02-1.025-1.03-1.035-1.04-1.045-1.05-1.055-1.06-1.065-1.07-1.075-1.08-1.085-1.09-1.095-1.1-1.105-1.11-1.115-1.12-1.125-1.13-1.135-1.14-1.145-1.15-1.155-1.16-1.165-1.17-1.175-1.18-1.185-1.19-1.195-1.2-1.205



-1.21-1.215-1.22-1.225-1.23-1.235-1.24-1.245-1.25-1.255-1.26-1.265-1.27-1.275-1.28-1.285-1.29-1.295-1.3-1.305-1.31-1.315-1.32-1.325-1.33-1.335-1.34-1.345-1.35-1.355-1.36-1.365-1.37-1.375-1.38-1.385-1.39-1.395-1.4-1.405-1.41-1.4150000



0-0.03-0.06-0.09-0.12-0.15-0.18-0.21-0.24-0.27-0.3TI002.PVNoneTI003.PVNoneLI001.PVNoneFC001.SPNoneTC001.SPNonePC001.PVNone

After reading this data, the corresponding model file will have the following form.


Appendix B - Example of an XFR FileSample Transfer Function input data for a 3 CV, 2MV and one DV model matrix is as follows. Thisdata corresponds to the transfer function models of the FIR data given previously.

321-1111222131110220310121433005-1.750.550.357.01.012.01.010.01.0-0.751.1

Appendix B - Example of an XFR File


12.01.03.01.00.03950.0049-0.005-0.0390.420.51.00.016.01.00.00.0011.00.0TI002.PVNoneTI003.PVNoneLI001.PVNoneFC001.SPNoneTC001.SPNonePC001.PVNone



After reading this data, the corresponding model file will have the following form.