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3/17/14 Edited by D. H. Chen 1Edited by D. H. Chen 1

Model Predictive Control

Focused on

Dynamic Matrix Control (DMC)

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Multivariable Controller

y1

y2u

2

u1 G

11(s)

++

G21(s)

G12(s)

G22(s)

++

Multivariable

Controllery2,sp

y1,sp

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Model Predictive Control (1)

MPC technology was developed by Engineers at ShellOil in the early 1970's, with an initial application in

1973.

Cutler and Ramakerpresented details of an unconstrained

multivariable control algorithm named Dynamic MatrixControl (DMC) at the 1979 National AIChE meeting and at

the 1980 Joint Automatic Control Conference.

Prett and Gillettedescribed an application of DMC

technology to an FCCU reactor/regenerator in which the

algorithm was modified to handle nonlinearities and

constraints.

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Model Predictive Control (2) Most popular form of multivariable control.

Effectively handles complex sets of constraints.

Has an LPon top of it so that it controls against the

most profitable set of constraints.

Several types of industrial MPC but DMC & RMPCT(Robust Multivariable Predictive Control Technology)

are the most widely used. ADMC(Adaptive DMC) is

the latest addition.

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Model Predictive Control - Overview

Justification

MPC reduces

processvariance

Operating

point can be

moved closer

to process

constraint

MPC Justification

0

0.1

0.2

0.30.4

0 5 10 15 20Controlled Variable

Pro

babilityDensity

Function

Original

MPCBenefit

Specification

Original

Operation

MPC

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Model Predictive Control - Overview

Justification

Constraints defined by

process equipmentphysical limits

Comfort zone defined

by operators ability to

maintain process near

constraints

Comfort

Zone

OperatingZone

Outside

Constraints

Independent Variables

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3/19/12 Edited by D. H. Chen 7

DMC Introduction

DMC was developed by Charles Cutler, a Lamar

University Chemical Engineering alumnus.

Implemented Dynamic Matrix Control (DMC) in

the 1970s at Shell Oil. Cutler later started his own company - Dynamic

Matrix Control Corp. (DMCC)

Help develop and implement many successful

industrial applications

Sold DMCC to ASPEN TECH in 1996

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Key features of the DMC control algorithm

Linear step response modelfor the plant Linear/Quadraticperformance objectiveover

a finite prediction horizon

Future plant output behavior specified bytrying to follow the set point as closely as

possible

Optimal inputs computed as the solution to aLeast-Squares problem


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DMC Concepts

Adjustments are made on

a minute-by-minute basis

Middle Level Multi-

variable control Uses dynamic response

data in order to create

step response models

May be implemented in arelatively short amount

of time

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DMC Open-Loop Tests

Finite Step Response (FSR) Model

t

MV

i=0 i=1 i=k

...CV

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DMC Model File Structure

CV1

MV2

SSGAIN

A POSITIVE GAIN

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DMCPrediction Equation

CV^1

CV^2

CV^3.

.

.

CV^i

=

a1 0 0

a2 a1 0

a3 a2 a1.

.

.

.

.

.

.

.

.

ai ai-1 ai-2

MV^1

MV^2

MV^3

CV1 MV13 columns

3 MV moves

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Model Predictive Control - Controller Design

Manipulated variables (MVs) setpoint to PID blocks and cascades

setpoint to FFand ratiocontrol structures

Disturbance variables (DVs)

independentvariables affecting process CVs

Controlled variables (CVs)

product specifications

important constraints

Economic dependent variables (EDVs)

Product Throughput, Fuel Gas Consumption

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Variables

Independent Variables include MVs and

DVs

Dependent variables include CVs and EDVs

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Dynamic Matrix Control Example (2a)

Consider the followingfurnace example(Cutler & Ramaker)

MV

Fuel flow FIC

DV

Inlet temperature TI CV

Outlet temperature TIC

TICFIC

FpTi

Fuel

Process

Heater

Process Flow

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Dynamic Matrix Control Example (2b)

The furnace DMC

model is defined by its

dynamic coefficients CV Response to step

change in fuel, a

CV Response to step

change in inlet

temperature, b

658.0

653.0

640.0

622.0

590.0

540.0

465.0

340.0

240.0

0.0

,

986.0

949.0

904.0

836.0

736.0

600.0

414.0

214.0

086.0

014.0

ba

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Dynamic Matrix Control Example (2c)

DMC Dynamic

coefficients

Response to stepchange in fuel, a

Response to step

change in inlet

temperature, b

Fuel Coefficients ai

0

0.5

1

1.5

0 5 10

Fuel Coefficients ai

Inlet Temperature Coefficients bi

0

0.5

1

0 5 10

Inlet TemperatureCoeff icients bi

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Dynamic Matrix Control Example (2d)

The DMC prediction may be calculated from

those coefficients and the independent variable

changes

3

2

1

3

2

1

2121

123123

1212

11

3

2

1

00

0000

DV

DV

DV

MV

MV

MV

bbbaaa

bbbaaa

bbaa

ba

CV

CV

CV

CV

iiiiii

i

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DMC Application (3a)

MV

Reflux flow SP

Tray 28 Temp

DV

Feed Flow

Feed Temp

CV

C3 in Isobutane

C4 in Propane

(Steam Flow)

C3/C4Splitter

From Cutler &Finlayson

FC

AI

LC

TC

TC

FC

Steam

AI

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DMCCV Transform (3b)

CVs are predicted to

be linear with respect

to MV changes Column reflux rate

versus C4 in propane

composition

A log transform ismore linear over a

range of operations -1

-0.5

0

0.5

1

1.5

2

500 700 900 1100 1300

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Ln(C4)

C4 in

Propane

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DMC CV Transform (3c)

A CV which was very

nonlinear with respect

to the MV is nowalmost linear

Column reflux rate

versus C3 in butane

composition The more linear the

better the prediction-2

-1.5

-1

-0.5

0

0.5

11.5

2

2.5

500 700 900 1100 1300

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Ln(C3)

C3 in

Butane

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MPC Tuning - C3/C4 Splitter Example

MV

Reflux flow SP

Tray 28 Temp

DV

Feed Flow Feed Temp

CV

C3 in Isobutane

C4 in Propane

(Steam Flow)

C3/C4Splitter

From Cutler &Finlayson

FC

AI

LC

TC

TC

FC

Steam

AI

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DMC - Controller Design

TTSS TTSSTTSS/2

Control Horizon

Prediction Horizon

Current Time

Manipulated

Variable

Past Future

Controlled

Variable

Time to Steady State (TTSS)

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DMC control strategy

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Matrix Form for Predictive Control (1)

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Dynamic Constraint Monitoring

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DMC Controller Characteristics The prediction horizon is 1.5 * time to steady state (TTSS)

Future moves (FMOV) are calculated half time to steady state,i.e., the control horizon is only 0.5 TTSS.

14 FMOV calculated, but only 10 can be sent to DCS.

Only first FMOV is implemented. And all moves recalculated

each execution. Sum of the moves need to equal LP target.

LP stands for Linear Programming, an optimization algorithm

for a linear objective with linear constraints. Here the LP target

means the optimized, future MV target.

LP target can also mean a future CV Target in other occasions.

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Impulse Response Form

Rearrange for Identification

312

12

1

23

23

2

1

1

12

12

1

1

01

MVaMVaaMVaaCVCV

MVaMVaaCVCV

MVaCVCV

231

12

2

1

323

12

1

1

212

1

101

aaMVaaMVaMVCVCV

aaMVaMVCVCV

aMVCVCV

Finite Impulse Response (FIR) Model

CV1

CV2

CV3

CV2 = CV2-CV0

CV1 = CV1-CV0

CV3= CV3-CV0

Change from a base line

Successive changes

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Merits of Finite Impulse Response

This form is arranged for solving a1, a2-a1, a3-a2,

etc. Then synthesize a2= a1+ (a2-a1); a3= a1+ (a2-a1)

+ (a3-a2);etc.

Only CVs are used, so the results will not be

spoiled when a section of the original data issliced (due to corrupt data). Since we do multipletests,we still have enough data points forregression even after discarding certain bad data.

Inaccuracy is introduced due to taking differencein CVs and synthesis of ai.

MPC t l d i d t i d

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MPC control design and tuning procedure

From the stated control objectives, define the size of the problem,

and determine the relevant CV's, MV's, and DV's Test the plant systematically by varying MV's and DV's; capture

and store the real-time data showing how the CV's respond

Derive a dynamic modelfrom the plant test data using an

identification package Configure the MPC controller and enter initial tuning parameters

Test the controller off-lineusing closed loop simulation to verify

the controller performance.

Download the configured controller to the destination machineand test the model predictions in open-loopmode

Commission the controllerand refine the tuning as needed.


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MPC Identification - Pretest

Determine the following

settling times or time to steady state (TTSS)

regulatory tuning (imbedded in controller)

quality of process signals

excitation required (6*noise band)

control issues to be resolved (e.g., valves

sticking)

identical equipment (representative responses)

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MPC Identification - Test

No correlation between step movesbut normal

plant adjustments to maintain specifications OK

Test on continuous basis(DMC recommends

24 hour coverage)

One minute data for all variables related to unit

being tested

Watch valves for saturation during test data during valve saturation discarded

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MPC Identification - Test

Data for cost factors (EDVs)included

Testing time estimate (#independents * 8

moves * TTSS)

one day for a charge gas controller

three weeks for an ethylene fractionator

Step each MV multiple times during the test

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MPC Identification - Modeling

Knowledge of process used to determine

which models make sense

Below noise level and those models not

understood should be eliminated

Only those independents with valid models

should be included in an identification run

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S=5

S=0

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Cycle Time or Control Interval

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DMC Model File Structure

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A Possible

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4/6/09 Edited by D. H. Chen 473/20/07 Edited by D. H. Chen 47

A Possible

SolnNot

Implemented

in DMC

Implemented

in DMC

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MPC Tuning - C3/C4 Splitter Example

Equal Concern Error (e.g., Concentration

of propane in isobutane given higherweight or sm. equal concern error)

most difficult to control

isobutane most valuable

Move suppression(the Major MPC

Tuning Parameter)set so that 20%

fluctuation in disturbance (Column Feed)

would not cause more than 10% move in

refluxor 0.5C move in tray temperature

FC

AI

LC

TC

TC

FC

Steam

AI

The weight is the inverse of the ECE

CV Tuning

MV Tuning

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MPC Tuning - CV weighting (I)

If the weight is equal to the inverse of anequivalent measure (equal concern error)theneach residual will receive the appropriate weight

The equal concern error(ECE) represents thestandard deviation the operation can tolerate.

The weight is the inverse of the ECE The smaller the ECE, the more important is the

CV (i.e., the weight is bigger).

iieD 1

1

i

i

eD i

i

w

1

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DMC Tuning - MV weighting (II)

As the move suppression is increased

Stability increases(more gradual MV changes)

More sluggish behavior results

Usually determined by trial and error

for a given step change in disturbance or CV

target the controller should implement an

acceptable MV move (that are not too large,especially in the first few moves, that operators

feel can upset the operations)

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CV

Future CV Current CV

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Example 2: Steady State Gain Matrix

CV1= 2.0*MV1+ 4.0 *MV2

CV2= 1.5*MV1+ 6.0 *MV2

CV1 CV2

MV1 2.0 -1.5

MV2 4.0 6.0

Express CV as a function of MV

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3/30/09 Edited by D.H. Chen 61

CVSS

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S.S.=(4,6) in terms of MV; S.S. = (0,0) in terms of MV

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Express Constraint in terms of MV Changes

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Express Constraint in terms of MV Changes

101 = 105 + 2.0*MV1+ 4.0 *MV2

MV2= 1 0.5*MV1

137 = 105 + 2.0*MV1+ 4.0 *MV2

MV2

= +8 0.5*MV1

These equations have a slope of 0.5 and anintercept of 1 and +8, respectively

The equations need to be viewed with the originshifted to MV1= 0 and MV2= 0; i.e., the currentsteady state

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Express a Constraint in terms of MV Changes

29 = 29

1.5*MV1+ 6.0 *MV2 MV2= 0.25*MV1

76 = 29 1.5* MV1+ 6.0 *MV2

MV2= 47/6 + 0.25*MV1

These equations have a slope of 0.25 and anintercept of 0 and +7.833, respectively

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69

MVs refers to the MV moves from the current

steady state MV values, i.e., (MV1, MV2) = (4,6)

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LP Cost (1)

LPi= G j,i*$Cost of CVj+ $Cost of MVi

+ Gk, i*$Cost of Prod/Utilities/Feed

LP1= G j,1*$Cost of CVj+ $Cost of MV1

+ Gk, 1*$Cost of Prod/Utilities/Feed

LPiis the overall economic impact of changing MViby 1 unit

LP1 = G j 1*$Cost of CVj+ $Cost of MV 1

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j,1 j 1

+ Gk, 1*$Cost of Prod/Utilities/Feed

Price Data F 100; U 10; P1 400; P2 120 Cost Data F 100; U 10; P1 400; P2 120

LP1 = 1.0 * 100+0.5*10-0.5*400-0.5*120= 155

LP2 = 2.0 * 100+0.2*10-1.5*400-0.5*120= 458

F U P1 P2

MV1 1.0 0.5 0.5 0.5

MV2 2.0 0.2 1.5 0.5

SS Gain Matrix

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DMC Linear Program - MVCost

Calculations

Resulting equation

Increasing the MVs will reduce this cost

function

Gradient of cost function maximized

slightly less than 3 to 1

SSSS

MVMV 21 )458()155(

Minimize

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LP Objective function

OBJ

Min LPi* MVi

OBJ

Min LP1* MV1+ LP2*MV2

For 2 MVs

LP Cost (2)

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LP Cost (2)

Price Data P1 100; P2 80; MV1 0; MV2 2 Cost Data P1 100; P2 80; MV1 0; MV2 2

LP1 = (.23)*(100)+ 0.3*(80) + 0 = 1

LP2 = 0.16 *100+0.1*80 + 2= 6

OBJ = MV16 *MV2 P1 P2

MV1 -.23 0.3

MV2 .16 -0.1

Unit Cost is -$100 for product A, -$80 for Product B, +$2 for MV2

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MV1= 2; MV1= 6; MV2=7; MV2= 13

CV1= 32; CV1= 137; CV2=39; CV2= 68

For Point A

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4/04/11 Edited by D.H. Chen 80

For the new optimal Steady state,

MV1 is 2, MV1 is 4+2=6MV2 is 7, MV2 is 6+7=13

CV1 is 32, CV1 is 105+32=137

CV2 is 39, CV2 is 29+39=68

RMPCT

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RMPCT

RMPCT (robust multivariable predictive control technology) is offered

by Honeywell Hi-Spec Solutions

Multiple optimization levels to address prioritized control objectives.

Additional flexibility in the steady-state target optimization.

Direct consideration of model uncertainty

Improved identification technology based on prediction error methodand sub-space ID methods.

Automatic identification test

Multivariable and closed-loop test

Use of compact/parametric models

Use of error bounds in model validation.


RMPCT features

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RMPCT features Aperformance ratio is provided in RMPCT which is defined as the ratio of the

required closed loop settling time to the weighted-average open loop settling

time.

The performance ratio is used to determine the length of the funnel, which is

somewhat similar to the settling time of a set point trajectory.

A performance ratio equal to one means that the closed loop settling time is equal to the open

loop settling time.

A performance ratio less than one results in a more aggressive controller.

Only one tuning parameter per CV needs to be specified.

Independent tuning is available in RMPCT for feed forward control, which

allows the user to achieve faster response in feed forward control than in set

point tracking.

The RMPCT package provides a min-max design procedure in which the user

enters estimates of model uncertainty directly. Tuning parameters are computed to optimize performance for the worst case model mismatch.

Robustness checks for the remaining MPC controllers are performed by closed loop simulation


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85

DMC vs RMPCT

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DMC vs. RMPCT

1. Two tuning parameters are used- move suppressionfactors - weights on delta u and ECE factors-inverse ofoutput weights

2. If a non-critical CV fails, DMC controller completely

removes it from control calculation while the RMPCTalgorithm continues control action by setting the failedCV measurement to the model predicted value-i.e. nofeedback for the failed CV.

3. If a critical CV fails, the DMC controllers turn offimmediately.

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Feed : Ethane, Propane, Butane and Naphtha (primary feed stock)

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Total plant system (TPS) DCS with two app-nodes one for the hot side and another

for the cold side.

App-nodes is an NT workstation with built-in AM software personality

Two app-nodes were connected directly to the existing LCN

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An increase of 4% in plant production rate after implementing RMPCT

Objective: Olefins Product maximization

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Objective: Olefins Product maximization

Typical constraints for a furnace Profit Controller include

Severity/Conversion, Furnace load,

Inlet temperature,

Flow deviations for flow balancing,

Tube skin temperature, and

Combustion constraints such as fuel pressure, stack

temperature, excess O2, damper positions etc.


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Typical constraints for a column Profit Controller include

Overhead purity, Bottoms purity,

Approach to flooding,

Reflux drum level,

Bottoms level,

Selected tray temperatures

Valve positions etc


Typical compressor constraints include

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Typical compressor constraints include

Compressor speed,

Discharge pressure, and Turbine steam flow etc,


Typical converter constraints include

Delta temperature and pressure,

Maximum bed temperature

Outlet composition etc.

Model Predictive Control Scheme

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Adaptive control

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dapt ve co t o

Adaptive controlinvolves modifying the controllaw used by a controller to cope with the fact thatthe parameters of the system being controlled areslowly time-varying or uncertain.

Adaptive control is different from robust controlinthe sense that it does not need a prior informationabout the bounds on these uncertain or time-varying

parameters; robust control guarantees that if the

changes are within given bounds the control lawneed not be changed, while adaptive control isprecisely concerned with control law changes.

Applications of Adaptive control
http://en.wikipedia.org/wiki/Robust_controlhttp://en.wikipedia.org/wiki/Robust_control

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Applications of Adaptive control

Adaptive control of linear controllers for nonlinear or time-varying processes

Adaptive control or self-tuning control of nonlinearcontrollers for nonlinear processes

Adaptive control or self-tuning control of multivariable

controllers for multivariable processes .

Usually these methods adapt the controllers to boththe process statics and dynamics. In special cases theadaptation can be limited to the static behavior alone,

leading to adaptive control based on characteristic curvesfor the steady-states or to extremum value control,optimizing the steady state.

Problems with DMC

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Changes made at the plant requiredexpensive and time-consuming testing andremodeling

Small changes cause the controller to

degrade in its ability to maintain fullefficiency

Noise in model

Not able to use full range of control valve Sensitive to control valve stick

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ADMC (1)

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( )

Adaptive Dynamic Matrix control , is the worlds first andonly adaptive multivariable controller.

It uses open loop (all valve) model of the process,eliminating

PID controllers from the hierarchy of control.

The controller uses the field proven DMC alogorithm for its

basic control.

ADMC (2)

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ADMC (2)

There are two hurdles to clear in order to use dynamic process model rather than aMPC control model:

1.Identify the dynamic process model.

2.Provide sufficient regulatory control of the process such that PID regulatory schemecan be bypassed.

Solutions:

1. Generation of matrix where PID dynamics are embedded in the process data,possible to open the classic form by switching the set points in the independentvariable list with corresponding process value and/or valve positions in thedependent list.

2.Design of new controller that has features

1. that allow it to provide PID control which is running not as fast as PID loops butsix times faster than typical one-minute execution frequency of the classic DMCcontroller).

2.Next controller includes real time on-line adaptive valve transformations withunmeasured disturbances such as sticking valves, hystersis, Pump switches).

3.that drive toward the economic optimum operating point.

ADMC (3)

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( ) The Controllers uses the DMC algorithm and

operate at a high frequency on an open loop model

of the process Use the all valve model

Not affected by tuning or configuration changes in PIDcontrollereliminate PID controllersfirst level ofadaptation

Permits the controller to operate with valves all theway open or closeswitch to manual/cascade orsaturating

The dependent variables response to the valves can bedetermined directly

Eliminate major problem encounter in the

identification analysis. Ex: no data is lost by valvesaturating

Eliminate noise

ADMC Benefits

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ADMC enables PID controllers to be removed from the control hierarchy.

Unleashes at least two to three percent of the processes potential value due to running with

valves wide openwith no control degradation.

ADMC eliminates PID retuning and reconfigurationfrom affecting MPC performance.

ADMC allows the upgrading of existing MPC controllers by starting with the existing

dynamic models as well as test data and converting to all valve models.

ADMC includes closed loop step testing capability. As a result, step testing requirements

are significantly reduced on new projects ADMC includes patented technology for handling of sticky valves and also reduces process

noise since PID controllers tend to amplify noise.

ADMC automatically adapts its all valve models every time it runs. It also eliminatesinteractions between PID controllers.

The higher quality model from adaptation and elimination of wandering PID controller

valve positions will permit much tighter control closer to all constraints, which improvespayout.

DMC vs. ADMC (1)

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DMC vs. ADMC (1)

Noise in Model

Degrades with time

High Maintenance

Not able to use full

range of control valve

Sensitive to valve stick

Lower Profits and

Higher Maintenance

Remove noisy PID

Analog to Digital

High Speed Computer

needed

Eliminates need of PID Self Tuning

Better Control

More Profits Trains & Advises at

100 times real time

DMC vs ADMC (2)

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MVs are set points Some outputs are CVs

Model Includes process

dynamics and regulatory

control action Regulatory Control Scheme

remains in place

MVs are Actuators/Valve

positions

Process values are CVs

Model is the dynamic

process model

Regulatory Control

replaced by the MPC

controller

DMC vs. ADMC (2)

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References

http://www.cutler-tech.com/

http://www.scielo.br

Dynamic Matrix Control (2)
http://www.cutler-tech.com/http://www.scielo.br/http://www.scielo.br/http://www.cutler-tech.com/http://www.cutler-tech.com/http://www.cutler-tech.com/

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y ( )

Dynamic Matrix Control (3)

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In DMC and OPC two types of

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In DMC and OPC, two types of

tuning parameters are used: Move suppression factors, which are weights on the MVs The move suppression factors change the

aggressiveness of the controller,

Equal concern error factors, which are the inverse of output weights.

the equal concern error factors normalize the

importance of each CV..


DMC

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DMC What are the typical manipulated variables

(MVs) in DMC?

SP for the regulatory loops

Are disturbance variables (DVs) dependent or

independent variables? Independent variables

Write down the equation for solving futureMVs using regression.

AT*A* MV = AT* E where E = CVtarget-CVopen

What are the typical lengths of prediction horizon and

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What are the typical lengths of prediction horizon andcontrol horizon in terms of time to steady state(TTSS)?

1.5 TTSS and 0.5 TTSS

What are the benefits of a finite impulse response(FIR) model over a finite step response (FSR) modelin terms of plant testing?

FIR model allows for data slicing which is almostinevitable for plant testing

FIR also limits the effect of unmeasured disturbancesto one SS

If the equal concern error (ECE) for CV1 is less thanthe ECE for CV2 and both have the sameengineering unit, which CV is more important?

CV1

If i bl h ti t d th t d

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If a variable has a negative cost, does the steady-state optimization tend to minimize or maximize that

variable? Maximize

Write down the matrix formulation for identifying,Predicting, and controlling a linear process in DMC.

Which variables are known during identification,during prediction, and during control, respectively?

MVT*MV *A = MVT* CV Identification

A* MV = CV Prediction

AT*A* MV = AT* (CVTarget- CVPred) Control

Can move suppression factors be different for

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Can move suppression factors be different fordifferent MVs?

Yes Is DMC a decentralized controller?

No

What are the advantages of a one-way

decoupler over a two-way decoupler? More robust, i.e., less interaction due to

model mismatch

What is an over-determined system?

More equations than unknowns such as inregression

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How does DMC determine the CV targets?

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How does DMC determine the CV targets?

Through LP optimization given constraints

and cost objective function In the DMC plus model program, what would

cause the data to be marked bad?

Instrument failures, power failures,

communication failures, maintenance shutdowns, emergency shut downs, etc.

What is the disadvantage of a finite impulseresponse (FIR) model?

Additional noise results from taking thederivative of CV.

What are the characteristics that define a

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process being linear?

Additive and Proportional

How many MVs and CVs are involved in thetop slide, p. 5 of the 3rd set of DMC notes,marked Multi variable Least SquaresMinimization? What does each submatrix (a),(b), (c), (d) stand for? What does the vector(e) stand for?

Are independent variable constraints always

honored in the controller's calculations? Yes

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