adaptive controller design directly from … · steady-state curve of van de vusse reactor 33. ......

ADAPTIVE CONTROLLER DESIGN DIRECTLY FROM PLANT

DATA

YAN LI (B. Eng., National University of Singapore, Singapore)

(M. Sc., Mines ParisTech, France)

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2010

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my research supervisor, Dr.

Min-Sen Chiu, for his excellent guidance and valuable suggestions during my studies

in the National University of Singapore.

I am also thankful to Dr. Lakshiminarayanan for his valuable advices to my

research work. Special thanks and appreciation are due to my lab mates, Martin

Wijaya Hermanto, Xin Yang, Qinglin Su and Vamsi Krishna Kamaraju, for the

stimulating discussions that we have had and the helps that they have rendered to me.

I would also wish to thank technical and administrative staffs in the Chemical and

Biomolecular Engineering Department for the efficient and prompt help.

I cannot find any words to thank my parents for their unconditional support,

affection and encouragement, without which this research work would not have been

possible.

i

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

TABLE OF CONTENTS ii

SUMMARY iv

LIST OF TABLES v

LIST OF FIGURES vi

NOMENCLATURE x

CHAPTER 1. INTRODUCTION 1

1.1 Motivations 1

1.2 Contributions 3

1.3 Thesis Organization 4

CHAPTER 2. LITERATURE REVIEW 5

2.1 Adaptive Control for Nonlinear Processes 5

2.2 Direct Data-based Controller Design Methods 11

2.2.1 The VRFT Design Framework 12

2.2.2 Adaptive VRFT design 14

2.3 Nonlinear Internal Model Control (NIMC) 15

CHAPTER 3. ADAPTIVE PID CONTROLLER DESIGN USING EVRFT

METHOD 19

3.1 Introduction 19

ii

iii

3.2 PID Controller Design by VRFT method 20

3.3 Enhanced VRFT Design Method 23

3.4 Examples 24

3.5 Conclusion 42

CHAPTER 4. ADAPTIVE INTERNAL MODEL CONTROLLER DESIGN

USING EVRFT METHOD 43

4.1 Introduction 43

4.2 IMC Controller Design Using VRFT Method 44

4.3 Enhanced VRFT Design Method 47

4.4 Examples 48

4.5 Conclusion 62

CHAPTER 5. CONCLUSIONS AND FURTHER WORK 63

5.1 Conclusions 63

5.2 Suggestions for Further Work 64

APPENDIX A. DERIVATION OF THE 2ND-ORDER REFERENCE MODEL 65

REFERENCES 66

SUMMARY

Controller design for nonlinear dynamic processes has been of great interest in

the chemical industry. Various nonlinear controller design strategies have been

studied in the literature. Among them, adaptive controller is a well-established

solution for this issue. In this thesis, a new adaptive controller design method is

proposed based on the virtual reference feedback tuning (VRFT) method which was

originally developed for linear controller design. This new method is termed as

enhanced VRFT (EVRFT) design to account for the difference from the linear VRFT

method.

In the proposed method, not only a second-order reference model is employed

instead of the first-order reference model commonly used in the linear VRFT design,

but also the parameters of reference model are updated at each sampling instance to

ensure the adaptive nature of the design strategy. In addition, to complete the on-line

adaptation process, the database is updated at each sampling instance by adding the

current process data into it and a relevant dataset is selected from the current database

according to the k-nearest neighborhood criterion.

Two different adaptive controllers are developed implementing the EVRFT

strategy, i.e. an adaptive PID controller and an adaptive Internal Model Controller.

Simulation results show that both proposed controllers give improved control

performance than the linear PID controller designed using VRFT method. They are

also shown to be quite robust in the presence of modeling error and can tolerate

reasonable process noise through simulation studies.

iv

LIST OF TABLES

Table 3.1 Model parameters for polymerization reactor 26

Table 3.2 Steady-state operating condition of polymerization reactor 26

Table 3.3 Tracking errors of servo responses obtained by various

design methods

28

Table 3.4 Tracking errors of servo responses obtained by various

design methods

35

Table 3.5 Tracking errors for EVRFT and VRFT designs for time

delay case

40

Table 4.1 Tracking errors obtained by various design methods 49

Table 4.2 Tracking errors obtained by various design methods 56

Table 4.3 Tracking errors of VRFT and EVRFT designs for time

delay case

58

v

LIST OF FIGURES

Figure 2.1 Diagram of adaptive control scheme 6

Figure 2.2 Reference model 13

Figure 2.3 Feedback control system 13

Figure 3.1 Polymerization reactor 25

Figure 3.2 Input-output data used for constructing the database

(example 1) 27

Figure 3.3 Responses for +50% (top) and -50% (bottom) set-point

changes. Solid line: EVRFT; dotted line: VRFT using 2nd-

order model (A = 0.77); dashed line: VRFT using 1st-order

model (A = 0.95) 29




model (A = 0.5) 29

Figure 3.5 Updating of tuning parameters in EVRFT design (+50%

set-point change) 30

Figure 3.6 Updating of tuning parameters in EVRFT design (-50%

set-point change) 30


changes by EVRFT design in the presence of modeling

error 31


changes by EVRFT design in the presence of process noise 32

vi

Figure 3.9 Steady-state curve of van de Vusse reactor 33


(example 2) 34

Figure 3.11 Responses for set-point changes from 1.12 to 1.25 (top)

and to 0.62 (bottom). Solid line: EVRFT; dotted line:

VRFT using 2nd-order model (A = 0.65); dashed line:

VRFT using1st-order model (A = 0.65) 36


and to 0.62 (bottom). Solid line: EVRFT; dotted line: VRF

using 2nd-order model (A = 0.35); dashed line: VRFT

using 1st-order model (A = 0.37) 36

Figure 3.13 Updating of controller parameters in EVRFT design for

set-point change to 1.25 37


set-point change to 0.62 37


and to 0.62 (bottom) in the presence of modeling error 38


and to 0.62 (bottom) in the presence of process noise 39


and to 0.62 (bottom) for time delay case. Solid line:

EVRFT; dotted line: VRFT 40


set-point change to 1.25 in the presence of time delay 41

Figure 3.19 Updating of controller parameters in EVRFT design for 41

vii

set-point change to 0.62 in the presence of time delay

Figure 4.1 Block diagram of IMC structure 45



order model (A=0.77); dashed line: VRFT using 1st-order

model (A=0.95) 50




model (A = 0.5) 50

Figure 4.4 Updating of parameter A and IMC parameters (+50% set-

point change) 51

Figure 4.5 Updating of parameter A and IMC parameters (-50% set-

point change) 51

Figure 4.6 Responses for 50% (top) and -50% (bottom) set-point

changes in the presence of modeling error 52

Figure 4.7 Responses for 50% (top) and -50% (bottom) set-point

changes in the presence of process noise 53

Figure 4.8 The catalytic continuous stirred tank reactor 54

Figure 4.9 Steady-state curve of catalytic CSTR 55


(catalytic CSTR) 55

Figure 4.11 Closed-loop responses for set-point changes to 21.5 (top)

and to 19.5 (bottom). Solid line: EVRFT; dotted line:

VRFT using 2nd-order model; dashed line: VRFT using 57

viii

ix

1st-order model

Figure 4.12 Updating of parameter A and IMC parameters for set-point

change to 21.5 59


change to 19.5 59

Figure 4.14 Responses for set-point changes to 21.5 (top) and to 19.5

(bottom) in the presence of modeling error 60


(bottom) in the presence of process noise 60


(bottom) in the presence of time delay. Solid line: EVRFT;

dotted line: VRFT 61

Figure 4.17 Updating of parameter A and IMC parameters set-point

change to 21.5 in the presence of time delay 61


change to 19.5 in the presence of time delay 62

NOMENCLATURE

A Tuning parameter of reference closed-loop model

C Controller

AC , , , , , AfC BC bC 1bC 2bC Concentrations

IC , inIC Initiator concentrations

mC , inmC Monomer concentrations

id Distance between ix and x

e Error between set-point and output

F , IF Flow rates

f Low-pass filter

*f Parameter of polymerization reaction

G Process

G~ Model of the process

h Liquid level

1J , 2J Objective functions

PK , , IK DK PID controller parameters

k Number of nearest neighbours

Ik , , , , mfk pk

cTkdTk Parameters of polymerization reaction

1k , , 2k 3k Kinetic parameters

mM Molecular weight of monomer

N Process time-delay

P Process

Q IMC controller

r Set-point

r~ Reference set-point signal

T Reference closed-loop transfer function

u Process input

u~ Virtual input

V Reactor volume

w Weight parameter

x

1w , 2w Flow rates

ix , x Information and query vector

y Process output y Model output in IMC

y Predicted output

Greek Symbols

α ,β Parameters of reference closed-loop model

1α , 2α , 1β ARX model parameters

η Adaptive learning rate

λ IMC filter time constant ρ Vector of discrete time-transfer function

Abbreviations

AIBN Azo-bis-isobutyronitrile

ARX Autoregressive exogenous

CSTR Continuous stirred tank reactor

FA Function approximator

IFT Iterative feedback tuning

IMC Internal model control

ITAE Integral of Time multiplied by Absolute Error

MAE Mean absolute error

MMA Methyl methacrylate

NAMW Number average molecular weight

NN Neural network

PID Proportional-integral-derivative

RMRAC Robust model reference adaptive control

VID2 Virtual input direct design

VRD2 Virtual reference direct design

VRFT Virtual reference feedback tuning

xi

1

Chapter 1

Introduction

1.1 Motivations

In recent years, there is a significant growth in demand for improved process

efficiency due to the competitive market environment. This has driven engineers and

researchers to develop more efficient and reliable techniques for process control.

These techniques, when applied, will not only improve the operating profit of the

controlled process but also ensure its safety during operation and limit its

environmental impacts. In chemical industry, this is especially important since most

of the waste by-products are extremely hazardous to the environment and require

extra treatment before their release. As a result, the study of process control has

become an important subject in chemical engineering research.

In chemical industry, hundreds or even thousands of variables, such as flow

rate, temperature, pressure, levels and compositions are routinely measured and

automatically recorded in historical databases for the purposes of process control,

online optimization or monitoring. Despite of the significant potential benefits that

Chapter 1 Introduction

2

may be obtained from the database, it is generally not a trivial task to extract useful

information and knowledge from the databases. Therefore, “data rich but information

poor” has become a well-known problem in chemical processes. Thus how to extract

relevant information from data and to use this information for controller design

becomes a significant research topic for chemical industry.

Toward this end, several data-based methods for controller design were

developed in the last fifteen years. Spall and Cristion (1998) has developed a

stochastic design framework in which the controller is represented by a function

approximator (FA), such as a polynomial or a neural network, whose parameters are

determined stochastically based on the process measurement. Another direct design

method is the iterative feedback tuning method developed by Hjalmarsson et al.

(1994). However, this method is computationally expansive and it risks to be trapped

in a local optimum when obtaining a solution for the proposed minimization problem,

not to mention its dependence on the trial and error procedure for initialization.

Furthermore, its computation needs unbiased estimates of some variables, which

impose much more stringent requirements during data collection. To overcome this

problem, the virtual input direct design method (VID2, Guardabassi and Savaresi,

1997; Savaresi and Guardabassi, 1998) was the first direct controller design method

without any gradient calculation. Campi et al. (2000) improved and reorganized the

idea of VID2 and renamed the new method as the virtual reference feedback tuning

(VRFT) method. Guardabassi and Savaresi (2000) also developed their new version

called virtual reference direct design (VRD2) which basically follows the same design

principles as VRFT. The VRFT design and its variants share a common feature that

controller parameters are obtained off-line by solving a quadratic optimization

problem based on a set of process input and output data. However, these methods are


3

developed for linear systems and their applications to nonlinear systems are limited.

An adaptive version of the VRFT method (Kansha et al., 2008) is thus proposed to

extend its application to nonlinear processes. This method includes the online update

of the database and selection of relevant dataset to implement its adaptive nature.

However, it still uses a pre-specified reference model which may hinder the

performance of resulting adaptive controller. Therefore, in this thesis, attempts will be

made to develop an enhanced version of the VRFT method to achieve better

performance for nonlinear systems.

1.2 Contributions

In this thesis, an enhanced VRFT (EVRFT) method is developed with

application to adaptive PID controller and adaptive Internal Model Control (IMC)

designs. The main contributions of this thesis are as follows.

(1) Adaptive PID controller design using EVRFT method: In the proposed

EVRFT design, a second-order reference model is employed instead of the

first-order reference model commonly used in the literature. In addition, other

than the update of database and relevant dataset, the parameters in the

reference model will also be updated at each sampling instance to further

improve the resulting control performance.

(2) Adaptive IMC controller design using EVRFT method: IMC is a powerful

controller design strategy for the open-loop stable dynamic systems. However,

the performance of IMC controller will degrade or become unstable when it is

applied to nonlinear processes with a range of operating conditions. In the

proposed IMC design, the EVRFT design is applied to update the IMC

controller at each sampling instance. Again, the reference model is updated at


4

each sampling instance, in addition to the update of database and relevant

dataset.

1.3 Thesis Organization

The thesis is organized as follows. Chapter 2 comprises the literature review

of nonlinear process control. In Chapter 3, an enhanced version of the VRFT method

is developed to design an adaptive PID controller. Moreover, by incorporating the

enhanced VRFT design framework into IMC design, an adaptive IMC controller for

nonlinear processes is proposed in Chapter 4. Finally, the general conclusions from

the present work and suggestions for future work are given in Chapter 5.

Chapter 2

Literature Review

This chapter examines the research work that has been conducted in the field

of nonlinear process control. An overview of adaptive controller design method is

presented. Following that, the detailed development of the VRFT method will be

discussed. Finally, various nonlinear IMC designs will be reviewed.

2.1 Adaptive Control for Nonlinear Processes

In chemical and biochemical industries, majority of the processes are

inherently nonlinear, however most controller design techniques are based on liner

control techniques to deal with such systems. The prevalence of linear control

strategies is partly due to the fact that, over the normal operating region, many of the

processes can be approximated by linear models, which can be obtained by the well-

established identification methods and the available input and output process data. In

addition, the theories for the stability analysis of linear control systems are quite well

5

Chapter 2 Literature Review

developed so that linear control techniques are widely accepted. However, due to the

nonlinear nature of most chemical processes, linear control design methodologies may

not be adequate to achieve a good control performance for these processes. This has

led to an increasing interest in the nonlinear controller design for the nonlinear

dynamic processes. Adaptive controller is a well-established solution for nonlinear

process control and its concept will be used throughout this thesis.

Figure 2.1 Diagram of adaptive control scheme

Research in adaptive control has a long and vigorous history. The

development of adaptive control started in the 1950’s with the aim of developing

adaptive flight control systems. With the progressing of control theories and computer

technology, various adaptive control methodologies were proposed for process control

in the last three decades. Åström (1983), Seborg et al. (1986) and Åström and

Wittenmark (1995) gave detail reviews of the theories and application of adaptive

control. Most adaptive methodologies integrate a set of techniques for automatic

adjustment of controller parameters in real time in order to achieve or to maintain a

6


desired level control performance when the dynamic characteristics of the process are

unknown or vary in time. The diagram of adaptive control concept is depicted in

Figure 2.1. There are three main technologies for adaptive control: gain scheduling,

model reference control, and self-tuning regulators. The purpose of these methods is

to find a convenient way of changing the controller parameters in response to changes

in the process and environment dynamics.

Gain scheduling is one of the earliest and most intuitive approaches for

adaptive control. The idea is to find process variables that correlate well with the

changes in process dynamics. It is then possible to compensate for process parameter

variations by changing the parameters of the controller as function of the process

variables. The advantage of gain scheduling is that the parameters can be changed

quickly in response to changes in the process dynamics. It is convenient especially if

the process dynamics in a well-known fashion on a relatively few easily measurable

variables. Gain scheduling has been successfully applied to nonlinear control design

for process industry (Åström and Wittenmark, 1995). One drawback of gain

scheduling is that it is open-loop compensation without feedback. Another drawback

of gain scheduling is that the design is time consuming. A further major difficulty is

that there is no straightforward approach to select the appropriate scheduling variables

for most chemical processes.

Model reference control is a class of direct self-tuners since no explicit

estimate or identification of the process is made. The specifications are given in terms

of “reference model” which tells how the process output ideally should respond to the

command signal. The desired performance of the closed-loop system is specified

through a reference model, and the adaptive system attempts to make the plant output

match the reference model output asymptotically.

7


The third class of adaptive control is self-tuning controller. The general

strategy of this controller is to estimate model parameters on-line and then adjust the

controller settings based on current parameter estimate (Åström, 1983). In the self-

tuning controller, at each sampling instant the parameters in an assumed dynamic

model are estimated recursively from input-output data and controller setting is then

updated. The whole control strategy can be divided into three steps: (i) information

gathering of the present process behavior; (ii) control performance criterion

optimization; (iii) adjustment of the controller parameters. The first step implies the

continuous determination of the actual condition of the process to be controlled based

on measurable process input and output and appropriate approaches selected to

identify the model parameters. Various types of model identification can be

distinguished depending on the information gathered and the method of estimation.

The last two steps calculate the control loop performance and the decision as to how

the controller will be adjusted or adapted. These characteristics make self-tuning

controller very flexible with respect to its choice of controller design methodology

and to the choice of process model identification (Seborg et al., 1986).

In the past two decades, many research efforts have focused on the

development of intelligent control algorithms that can be applied to complex

processes whose dynamics are poorly modeled and/or have severe nonlinearities.

(Stephanopoulos and Han, 1996; Linkens and Nyongesa, 1996). Because neural

networks (NNs) have the capacity to approximate any nonlinear function to any

arbitrary degree of accuracy, NNs have received much attention in the area of

adaptive control. Perhaps the most significant work of the application of NNs in

adaptive control is that of Narendra and Parthasarathy (1990) who investigated

adaptive input-output neural models in model reference adaptive control structures.

8


Hernandez and Arkun (1992) studied control-relevant properties of neural network

model of nonlinear systems. Jin et al. (1994) used recurrent neural networks to

approximate the unknown nonlinear input-output relationship. Based on the dynamic

neural model, an extension of the concept of the input-output linearization of discrete-

time nonlinear systems is used to synthesize a control technique under model

reference control framework. te Braake et al. (1998) provided a nonlinear control

methodology based on neural network combined with feedback linearization

technique to transform the nonlinear process into an equivalent linear system in order

to simplify the controller design problem. Recently, some researchers have

constructed stable NN for adaptive control based on Lyapunov’s stability theory

(Lewis et al., 1996; Polycarpou, 1996; Ge et al., 2002). One main advantage of these

schemes is that the adaptive laws are derived based on the Lyapunov synthesis

method and therefore guarantee the stability of the control systems. While neuro-

control techniques are suited to control an unknown nonlinear dynamic process, it is

generally difficult to present the control law in simple analytical form. Also, a

nonlinear optimization routine is required to determine the control input, which may

lead to the problems of large computational efforts and poor convergence. To

alleviate this problem, an innovative real-time model reference neural control system

is developed by Pérez et al. (2009). This neural controller allows one to assimilate the

complex system dynamics to a simple first-order linear system, which can be easily

controlled by a conventional PID controller.

The PID controllers have received widespread use in the process industries

primarily because of its simple structure, ease of implementation, and robustness in

operation. Due to these advantages, several adaptive PID controller designs have been

developed in recent years. For example, Riverol and Napolitano (2000) proposed an

9


adaptive PID controller whose parameters are adjusted on-line by a neural network,

while Chen and Huang (2004) designed adaptive PID controller based on the

instantaneous linearization of a neural network model. Altinten et al. (2004) applied

the genetic algorithm to the optimal tuning of a PID controller on-line. Bisowarno et

al. (2004) applied two adaptive PI control strategies for reactive distillation. Andrasik

et al. (2004) made use of a hybrid model consisting of a neural network and a

simplified first-principle model to design a neural PID-like controller. Yamamoto and

Shah (2004) developed an adaptive PID controller using recursive least squares for

on-line identification of multivariable system. Shahrokhi and Baghmisheh (2005)

designed an adaptive IMC-PID controller based on the local models estimated by the

recursive least squares method to control a fixed-bed reactor. Similar approaches for

adjusting PID controller parameters on-line were investigated based on the multiple

linearized models obtained by factorization algorithm and lazy learning identification

method at each sampling instance (Ho et al., 1999; Alpbaz et al., 2006; Pan et al.,

2007). Another multi-model adaptive strategy for PID controllers was proposed based

on a set of simple linear dynamic models where each model has the same structure but

different values of the model parameters (Böling et al., 2007). Yu et al. (2007)

extended the adaptive PID control strategy to multivariable nonlinear systems with

unknown dynamics by proposing a stable self-learning PID control scheme based on a

neural network (NN) model of the plant. In these works, basically, the parameters of

the process model are updated with respect to the current process condition and then

PID parameters are computed by the corresponding adaptation algorithm and

implemented.

In this thesis, an adaptive PID controller design strategy using directly process

data will be developed in Chapter 3 with EVRFT as the adaptation algorithm.

10


2.2 Direct Data-based Controller Design Methods

Designing controllers directly based on a set of measured process input and

output data, without resorting to the identification of a process model, is an attractive

option for process control application. Such ‘direct’ data-based design techniques are

conceptually more natural than model-based designs where the controller is designed

on the basis of an estimated model of the process, because the former directly targets

the final goal of tuning the parameters of a given class of controllers. However,

despite the appeal of direct data-based design methods, very few genuine direct

design techniques have been proposed in literature

Hjalmarsson et al. (1994) developed iterative feedback tuning (IFT) method

with promising result for real application (1998). However, IFT may require

considerable computational time to obtain a solution with a risk of being a local

optimum in the proposed minimization problem, not to mention its dependence on the

trial and error procedure for initialization. Furthermore, its computation needs

unbiased estimates of some variables, which impose much more stringent

requirements on the experiment. As a result, the experiment required for IFT is

typically complicated.

Spall and Cristion (1998) proposed a stochastic approach for adaptive control

using a function approximator (FA) to calculate the action needed from the controller.

FA can be a polynomial or an artificial neural network, whose parameters are updated

repeatedly in accordance with the minimization of a cost function. However, since a

plant model is not available, the gradient of this cost function has to be evaluated by

simultaneous perturbation stochastic approximation instead of quadratic methods.

Thus, the computational burden of this method is very high due to the iterations and

the convergence of the trained parameters may not be guaranteed.

11


To alleviate the aforementioned drawbacks, Campi and Lecchini (2000, 2002)

proposed the virtual reference feedback tuning method (VRFT). VRFT stems from the

idea of virtual input direct design (VID2) (Guardabassi and Savaresi, 1997; Savaresi

and Guardabassi, 1998), but in a better-organized form. This methodology is simple

and directly calculates the feedback controller parameters from the available process

input and output data without the need of model identification. Under this tuning

framework, only the specification of desired reference model is required. Nakamoto

(2005) extended this controller design technique to multivariable systems and showed

a chemical process application.

An adaptive version of the VRFT method (Kansha et al., 2008) is proposed to

extend its application to nonlinear processes. In this adaptive VRFT design, the off-

line database employed in the conventional VRFT design is continuously updated by

adding the current process data into the database. Furthermore, PID parameters are

determined by the VRFT design at each sampling instance using the relevant dataset

selected from the current database based on k-nearest neighborhood criterion. An

enhanced version of this method will be developed in Chapter 3 by using a second-

order reference model with update in its parameter as well.

2.2.1 The VRFT design framework

The VRFT method approximately solves a model-reference problem in

discrete time as depicted in Figure 2.2, where the reference model ( )1−zT describes

the desired behavior of the closed-loop system consisting of a linear time-invariant

process ( )1−zP and a parameterized controller ( )θ;1−zC as shown in Figure 2.3. Let

us assume that ( )1−zP

( )

is unknown and only a set of process input and output data,

( ){ }kku = n~1 and{ }kky 1= n~ , have been collected from e experime th nt on the plant and

12


that a reference model ( )1−zT has been chosen. The design goal is to solve θ , a

vector consisting of the controller parameters, such that the feedback control sys em

in Figure 2.2 behaves as closely as possible to the pre-specified reference model

t

( )1−zT .

Figure 2.2 Reference Model

Figure 2.3 F

u

eedback Control System

Given the meas red output signal ( ){ } nkky ~1= , the corresponding reference

( )} nkkr ~1{~ = in Figure 2.2 is obtained by signal

( ) ( ) ( )1111~ − = Tzr −−− zyz (2.1)

where ( )1~ −zr and ( )1−zy are the Z-transforms of discrete time signals ( ){ } nkkr ~1~

= and

( ){ }kky spect ( ){ } nkkr ~1n~1= , re ively. ~= is called ‘virtual’ reference signal because it does

s not used in the generation of ( )ky . However, it

plays a pivotal role in the VRFT framework in that the fundamental idea of the VRFT

framework is to treat ( ){ } nkky ~1= as the desired output of the feedback system when

the reference signal d by

not exist in reality and in fact i

s specif

t wa

iei ( ){ } nkkr ~1~

= . As a consequence, given error

( ) ( ) (ky− )signal krk = ~ , the controller output ( )ku~e is calculated as:

13


( ) ( ) ( ){ ( )}1111 ~;~ −− zy −−− = zrzCzu θ (2.2)

where

( )1~ −zu is the Z-tr e siansforms of discrete tim gnal ( ){ } nkku ~1~

= .

It is noted that, even though the process dynamics ( )1−zP is not known, when

the process is fed by ( )ku , i.e. the measured input sign era ate , i.e. the

corresp

l, it gen s ( )ky

onding measured output signal. Therefore, a good controller generates ( )ku

when the error signal is given by ( )ke . The idea is then to search for ( )θC whose

output ( )ku~ matches ( )ku as closely as possible. Hence, the controller design task

reduces to the following minimization problem:

( )

;1−z

( ) ( ){ }∑ −=n

kukuJ 21

~1nθ (2.3) =kn 1

miθ

If the controller is given by ( ) ( )θρθ 11; −− = zzC T where ( )1−zρ is a vector of

discrete nsfer function, it can -time tra en that Eq. (2.3) is quadratic inbe se θ .

Consequently, the controller parameter es Eq. ( n be explicitly

obtained by the classical least-square technique. As a result, the VRFT des n

framework effectively recasts the problem of designing a model-reference feedback

controller into a standard system-identification problem. More detailed discussions on

the VRFT can be found in Campi et al. (2000, 2002).

2.2.2 Adaptive VRFT design

*θ which minimiz 2.3) ca

ig

In the conventional VRFT design, the database collected from an off-line

as a result the resulting controller is expected to

perform well in the vicinity of operating

open-loop experiment is utilized and

space close to the operating condition where

this dataset is generated. To extend the VRFT design to nonlinear systems, one

possible approach is to augment the original off-line database by adding the current

14


process data at each sampling instance so that the expanded database can cover new

operating space where its dynamics is not available in the construction of original

database. This expanded database is subsequently used to obtain PID parameters by

VRFT design at each sampling instance. In doing so, the relevant data in the expanded

database that corresponds to the current process condition is first determined by using

the k-nearest neighborhood criterion based on the following distance measure:

( ) ii xkxd −−= 1 (2.4)

where ⋅ denotes the Euclidean norm, ( ) ( )[ ]Tiuiyx = is a pair of input and output i

data in the present dataset, and ( )1−kx is a vector with similar definition for the input

nstance. and output data at the (k-1)-th sampling i

By using Eq. (2.4), those ix cor

hich the constrained least squares

problem

ear Internal Model Control (NIMC)

al Model Control (IMC) proposed by Garcia and Morari (1982) is a

ic systems (Morari

and Za

responding to the k smallest id are selected as

the relevant data in the current database, by w

discussed in the subsection 2.2.1 is solved to calculate PID parameters for

the current sampling instance. This design procedure repeats at the next sampling

instance when the database for VRFT design is further updated by the corresponding

process data.

2.3 Nonlin

Intern

powerful controller design strategy for the open-loop stable dynam

firiou, 1989). This is mainly due to two reasons. First, integral action is

included implicitly in the controller because of the IMC structure. Moreover,

plant/model mismatch can be addressed via the design of the robustness filter. IMC

design is expected to perform satisfactorily as long as the process is operated in the

15


vicinity of the point where the linear process model is obtained. However, the

performance of IMC controller will degrade or even become unstable when it is

applied to nonlinear processes with a range of operating conditions.

To extend the IMC design to nonlinear processes, various nonlinear IMC

schemes have been developed in the literature. For instance, Economou et al. (1986)

provide

n in Bhat and McAvoy

(1990)

d a nonlinear extension of IMC by employing contraction mapping principle

and Newton method. However, this numerical approach to nonlinear IMC design is

computationally demanding. Calvet and Arkun (1988) used an IMC scheme to

implement their stat-space linearization approach for nonlinear systems with

disturbance. A disadvantage of the state-space linearization approach is that an

artificial controlled output is introduced in the controller design procedure and cannot

be specified a priori. Another drawback of this method is that the nonlinear controller

requires state feedback (Henson and Seborg, 1991a). Henson and Seborg (1991b)

proposed a state-space approach and used nonlinear filter to account for plant/model

mismatch. However, their method relied on the availability of a nonlinear state-space

model, which may be time-consuming and costly to obtain.

Another popular design method for implementing nonlinear IMC schemes is

based on the neural networks. In the earlier methods give

and Hunt and Sbarbaro (1991), two NN were used in the IMC framework,

where one NN was trained to represent the nonlinear dynamics of process, which was

then used as the IMC model, while another NN was trained to learn the inverse

dynamics of the process and was employed as the nonlinear IMC controller. Because

IMC model and controller were built by separate neural networks, the controller

might not invert the steady-state gain of the model and thus steady-state offset might

not be eliminated (Nahas et al., 1992). Moreover, these control schemes do not

16


provide a tuning parameter that can be adjusted to account for plant and model

mismatch. To ensure offset-free performance, Nahas et al. (1992) developed another

NN based nonlinear IMC strategy, which consists of a model inverse controller

obtained from a neural network and a robustness filter with a single tuning parameter.

In this control strategy, a numerical inversion of neural network process model was

proposed instead of training neural networks on the process inverse. Aoyama et al.

(1995) proposed a method using control-affine neural network models. Two neural

networks were used in this approach: one for the model of the bias or drift term, and

one for the model of the steady-state gain. As the process is approximated by a

control-affine model, the inversion of process model is simply obtained by

algebraically inverting the process model.

However, the above nonlinear IMC designs sacrifice the simplicity associated

with linear IMC in order to achieve improved performance. This is mainly due to the

use of computationally demanding analytical or numerical methods and neural

networks to learn the inverse of process dynamics for the necessary construction of

nonlinear process inverses. To overcome these difficulties, a promising approach has

been proposed to yield a flexible nonlinear model inversion (Doyle et al., 1995; Harris

and Palazoglu, 1998). This controller synthesis scheme based on partitioned model

inverse retains the original spirit and characteristics of conventional (linear) IMC

while extending its capabilities to nonlinear systems. When implemented as part of

the control law, the nonlinear controller consists of a standard linear IMC controller

augmented by an auxiliary loop of nonlinear “correction”. The fact that only a linear

inversion is required in the synthesis of this controller is the most attractive feature of

this scheme. However, Volterra model derived using local expansion results such as

Carleman linearization is accurate for capturing local nonlinearities around an

17


18

operating point, but may be erroneous in describing global nonlinear behavior (Maner

et al., 1996). Harris and Palazoglu (1998) proposed another nonlinear IMC scheme

based on the functional expansion models instead of Volterra model. However,

functional expansion models are limited to fading memory systems and the radius of

convergence is not guaranteed for all input magnitudes. Consequently, the resulting

controller gives satisfactory performance only for a limited range of operation. This

limitation restricts the implementation of these models in practice (Xiong and Jutan,

2002).

Shaw et al. (1997) used recurrent neural network (RNN) within the partitioned

model inverse controller synthesis scheme in IMC framework and showed that this

strategy

e the parameters of the

IMC co

sed in Chapter 4.

provided an attractive alternative for NN-based control application.

Maksumov et al. (2002) investigated the first experimental application of this control

strategy using NN as a nonlinear model and a linear ARX model. However, one

fundamental limitation of these global approaches for modeling is that the on-line

update of these models is not straightforward when the process dynamics are moved

away from the nominal operating space. Evidently, this will interrupt the plant

operation when these models are used in the controller design.

To circumvent the aforementioned drawback, an adaptive IMC controller is

proposed based on just-in-time learning (JITL) technique wher

ntroller as well as its filter parameter are updated based on the information

provided by the JITL (Cheng and Chiu, 2007). Kalmukale et al. (2007) developed

another partitioned model-based IMC design strategy for a class of nonlinear systems

that can be described by the JITL modeling technique.

In this thesis, a new adaptive IMC design strategy using Enhanced Virtual

Reference Feedback Tuning (EVRFT) method is propo

Chapter 3

Adaptive PID Controller Design

Using EVRFT Method

3.1 Introduction

Model-based controller design techniques have been receiving much research

attention in the past several decades. Among them, a number of PID tuning formulas

such as ITAE performance index, direct synthesis design method and Internal Model

Control (IMC) design, are well established in the literature. These model-based

controller design methods generally follow a two-step procedure: the first step is to

identify a process transfer function model with a pre-specified model structure, for

example a first-order-plus-delay model, which gives a reasonably good modeling

accuracy; the second step is to design the controller based on the model thus obtained.

Another interesting alternative for controller design is the data-based methods.

For example, the Virtual Reference Feedback Tuning (VRFT) method developed by

Campi et al. (2000, 2002) is a direct data-based method that determines the

19

Chapter 3 Adaptive PID Controller Design Using EVRFT Method

parameters of a controller by using a set of input and output data from the process

without resorting to the identification of a process model. However, this design

framework is originally developed for linear systems. As a result, controllers designed

under this framework have limited performance for nonlinear processes. An adaptive

version of the VRFT method (Kansha et al., 2008) is thus proposed to extend its

application to nonlinear processes. In this adaptive VRFT design, the off-line database

employed in the conventional VRFT design is continuously updated by adding the

current process data into the database. Furthermore, PID parameters are determined

by the VRFT design at each sampling instance using the relevant dataset selected

from the current database based on k-nearest neighborhood criterion.

In this chapter, an enhanced version of the adaptive VRFT method is

developed to design an adaptive PID controller. In the proposed enhanced VRFT

design, a second-order reference model is employed instead of the first-order

reference model commonly used in the previous work. In addition, other than the

update of database and relevant dataset, the parameters in the reference model will

also be updated at each sampling instance. Simulation results are presented to

illustrate the proposed design method and a comparison with conventional VRFT

design is made.

3.2 PID Controller Design by VRFT Method

In this section, the detailed formulation for PID controller design using VRFT

method with a second-order reference model is discussed.

Consider a PID controller given by:

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( )[ 21211−+−−+ ]+−−+−=

kekekeKkeKkekeKkuku

D

IP (3.1)

20


where is the manipulated value at the k-th sampling instance, is the error

between process output and its set-point at the k-th sampling instance, and ,

and are PID parameters. The corresponding

( )ku

DK

( )ke

PK

IK ( )θ;1−zC will be:

( ) ( 11

1 11

; −−

− −+ )−

+= zKz

KKzC DI

pθ (3.2)

In VRFT design framework, the reference model ( )1−zT is specified by the

following second-order closed-loop transfer function derived in Appendix A:

( )⎩⎨⎧

−−=+−=

+−+

== −−

−−−

−

−−

AAAAAAA

wherezAAz

zzzrzyzT

N

lnln1

21)()()( 2221

11

1

11

βαβα (3.3)

where A is the tuning parameter related to the speed of the response and N is the

process time delay.

To design a PID controller by the VRFT method, the virtual input ( )1~ −zu is

calculated by Eqs. (2.1), (2.2), (3.2) and (3.3) to obtain

( ) ( )( ) ( 1

11

112211

11 211

1)(~ −

−−−

−−−−−−

−−

++−+−

⎥⎦⎤

⎢⎣⎡ −+

−+= zy

zzzzzAAzzK

zK

Kzu N

N

DI

P βαβα )

)

(3.4)

Define ( 1~ −Φ z as follows:

( ) )()( 111 −−− +=Φ zuzz βα (3.5)

The following equation can be obtained from Eqs. (3.4) and (3.5)

( ) ( ) ( 11

112211

11 211

1)(~ −

−−

−−−−−−

−− +−+−

⎥⎦⎤

⎢⎣⎡ −+

−+=Φ zy

zzzzAAzzK

zK

Kz N

N

DI

Pβα ) (3.6)

Equation (3.5) can be rewritten as:

( ) ( ) ( )1−+=Φ kukuk βα (3.7)

Equation (3.6) can then be rewritten as:

( ) ( )kKk Ψ=Φ~ (3.8)

where

21


( ) ( ) ( ) ( )[ TDIP kkkk ΨΨΨ=Ψ ]

]

(3.9)

[ DIP KKKK = (3.10)

( ) ( ) ( ) ( ) ( 1)1(21 2 −−−−+++−++=Ψ kykyNkyANkAyNkykP βα ) (3.11)

(3.12) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )⎪

⎪

⎩

⎪⎪

⎨

⎧

≥++−++−−++

=++−−+=+−+

=Ψ

∑−

=

1

1

2 21121

111220)ln1(1

N

i

I

NforkyikyANkyANky

NforkykyAkyNforkyAAky

k

β

β

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( 212

121212

2

−+−−+−−+−

−+++++−++=Ψ

kykykyNkyA

NkyAANkyANkykD

ββαα ) (3.13)

Equation (2.3) is then expressed by

( ) ( ) ( ){ }

2

2

11

1min

1min

Ψ−Φ=

Ψ−Φ= ∑=

Kn

kKkn

KJ

K

n

kK (3.14)

subject to the following constraints of PID controller parameters:

0,,0,, >< DIPDIP KKKorKKK (3.15)

where

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΨΨΨΨΨΨΨΨΨ

=Ψnnn

DDD

III

PPP

L

L

L

212121

(3.16)

( ) ( )[ n ]ΦΦ=Φ L1 (3.17)

Consequently, PID parameters are obtained by solving the constrained least

square problem as stated above. It is evident that this solution not only depends on the

database for VRFT design but also the design parameter A in the reference model.

22


3.3 Enhanced VRFT Design Method

The proposed enhanced VRFT design method (EVRFT) is focused on the

updating of the reference model while keeping the basic VRFT framework described

in section 3.2. The objective of this updating procedure is to enhance the capability of

VRFT design to cope with the variation of process dynamics resulting from the

process nonlinearity. This is achieved by updating the tuning parameter A in the

reference model at each sampling instance as it has a direct impact on the PID

parameters calculated using the VRFT method.

The objective of adjusting reference model parameter A is to minimize the

following quadratic function:

( ) ( ) ( ) ( )[ ] ( ) ( )[{ }222 11ˆ11

21

−−++−+−= kukuwkykrwkJ ]

]

(3.18)

where is a weight parameter and [ 1,0∈w ( )1ˆ +ky is the predicted output using the

reference model at the (k+1)-th sampling instance which can be calculated by:

( ) ( ) ( ) ( ) ( )1121ˆ 2 −−+−+−−=+ NkrNkrkyAkAyky βα (3.19)

By the steepest descent method, the updating algorithm is derived as:

( ) ( ) ( ) ( )( )kA

kJkkAkA∂∂

−=+ 21 η (3.20)

where ( )kη is the learning rate.

In the proposed design, the following rules are used to update the learning rate:

(i) if the increment of is more than a threshold, the tuning parameter A remains

unchanged and the learning rate is decreased by a factor , i.e.

2J

decl ( ) ( )klk decηη =+1 ;

(ii) if the absolute value of the change of is smaller the threshold, only the

parameter A is updated; otherwise, (iii) the parameter A is updated and the learning

rate is increased by a factor , i.e.

2J

incl ( ) ( )kk lincηη =+1 .

23


The following outlines the computational algorithm for the proposed EVRFT

design of adaptive PID controller.

Step 1 The process input (u(k)) and output (y(k)) data that characterize the dynamics

of nonlinear system are assumed to be available and the off-line database for

EVRFT design is constructed as ( ) niix ~1= ;

Step 2 At the k-th sampling instance, based on the current database for EVRFT

design, the relevant dataset is selected according to Eq. (2.4) and then the PID

parameters are calculated by solving the minimization problem given by Eq.

(3.14) subjected to the constraint Eq. (3.15). The manipulated variable ( )ku is

obtained by Eq. (3.1).

Step 3 The database for EVRFT design is augmented by appending the current

process data and ( )ky ( )ku , while the tuning parameter A is updated using Eq.

(3.20) whenever required according to the empirical rules mentioned above.

Step 4 Set and go to Step 2. 1+= kk

3.4 Examples

Example 1 Considering a continuous polymerization reaction that takes place

in a jacketed CSTR (Doyle et al., 1995), as depicted in Figure 3.1. In the reactor, an

isothermal free-radical polymerization of methyl methacrylate (MMA) is carried out

using azo-bis-isobutyronitrile (AIBN) as initiator and toluene as solvent. The control

objective is to regulate the product number average molecular weight (NAMW) by

manipulating the flow rate of the initiator ( ), i.e., NAMW is the process output

and is as process input u . Under the following assumptions: (i) isothermal

operation; (ii) perfect mixing; (iii) constant heat capacity; (iv) no polymer in the inlet

IF y

IF

24


stream; (v) no gel effect; (vi) constant reactor volume; (vii) negligible initiator flow

rate (in comparison with monomer flow rate); and (viii) quasi-steady state and long-

chain hypothesis, the dynamics of the reactor can be described by the following

equations:

VCCF

PCkkt

C mmmfmp

m in)(

)(d

d0

−++−=

VFCCF

Ckt

C in IIIII

I

dd −

+−=

VFD

PCkPkkdt

dDmfTT mdc

00

20

0 )5.0( −++=

VFDPCkkM

dtdD

mfpm m1

01 )( −+=

0

1

DDy = (3.21)

where 5.0

II0

*2

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=

cd TT kkCkfP . The model parameters and steady-state operation

condition are given in Tables 3.1 and 3.2.

Figure 3.1 Polymerization reactor

25


Table 3.1 Model parameters for polymerization reactor

cTk = 10103281.1 × h)/(kmolm3 F = 1.00 /hm3

dTk = 11100930.1 × h)/(kmolm3 V = 0.1 3m

Ik = 1100225.1 −× L/h inIC = 8.0 3kmol/m

pk = 6104952.2 × h)/(kmolm3mM = 100.12 kg/kmol

mfk = 3104522.2 × h)/(kmolm3inmC = 6.0 3kmol/m

*f = 0.58

Table 3.2 Steady-state operating condition of polymerization reactor

mC = 5.506774 3kmol/m 1D = 49.38182 3kmol/m

IC = 0.132906 3kmol/m IFu = = 0.016783 /hm3

0D = 0.0019752 3kmol/m y = 25000.34 kg/kmol

To obtain the database for EVRFT design, an open-loop test is conducted by

introducing random steps around the nominal value of process input and the

corresponding input and output data are given in Figure 3.2. The process input and

output are scaled by

IF

016783.0016783.0~ −

=uu and

34.2500034.25000~ −

=yy respectively.

To proceed the EVRFT design of adaptive PID controller, the tuning

parameters specified are as follows: the initial value of parameter A = 0.6, initial

learning rate 002.0=η , weight parameter used in the objective function Eq. (3.18)

given by , the updating parameters for learning rate 7.0=w 02.1=incl and ,

and the number of data points in the relevant dataset is set to 1200.

55.0=decl

26


Figure 3.2 Input-output data used for constructing the database (example 1)

To evaluate the performance of the adaptive PID controller designed using

EVRFT method, +50% and -50% set-point changes are conducted, as illustrated in

Figures 3.3 and 3.4. Figures 3.5 and 3.6 show the updating of tuning parameter A and

PID parameters in EVFRT design for the abovementioned servo responses.

Since polymerization reactor is operated in a wide range of operating space

between the set-points 12500 and 37500, a linear PID controller cannot provide

satisfactory control performance as a result of process nonlinearity. To better illustrate

this point, four PID controllers designed using VRFT method with first-order and

second-order reference models are developed. To match the performance of EVRFT

design for +50% set-point change as closely as possible, two PID controllers are

obtained by VRFT design using tuning parameters A = 0.95 and A = 0.77 for the first-

order and second-order reference models, respectively. From Figure 3.3, it can be

27


seen that both adaptive PID controller designed using EVRFT method and the PID

controller designed using VRFT method with second-order reference model

outperform the PID controller designed using VRFT method with first-order reference

model. Although VRFT design using second-order reference model is able to match

the EVRFT design for +50% set-point change, it gives slower performance than the

EVRFT design for -50% set-point change.

Similarly, to match the performance of EVRFT design for -50% set-point

change as closely as possible, another two PID controllers are obtained by VRFT

design using tuning parameters A = 0.5 and A = 0.3 for the first-order and second-

order reference models, respectively. Figure 3.4 shows that, while comparable

performances for -50% set-point change are observed for both VRFT and EVRFT

designs, EVRFT design gives better performance than the two VRFT designs for

+50% set-point change. Table 3.3 summarizes the tracking errors of the

abovementioned servo responses.

Table 3.3 Tracking errors obtained by various design methods

VRFT design using 1st-order model

VRFT design using 2nd-order model

% improvement (based on VRFT design using 2nd-order model)

Set-point change

A=0.95 A=0.5 A=0.77 A=0.3

EVRFT design

A=0.77 A=0.3+50% 1836 2920 868 2479 921 -6.1% 62.9% -50% 2984 356 1465 387 362 75.3% 6.4%

28


Figure 3.3 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A = 0.77); dashed line:

VRFT using 1st-order model (A = 0.95)

Figure 3.4 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A = 0.3); dashed line:

VRFT using 1st-order model (A = 0.5)

29


Figure 3.5 Updating of tuning parameters in EVRFT design (+50% set-point change)

Figure 3.6 Updating of tuning parameters in EVRFT design (-50% set-point change)

30


Next, to evaluate the robustness of the proposed control strategy, 10%

modeling error in the kinetic parameter and 20% modeling error in the

coefficient are considered. Figure 3.7 illustrates that the proposed EVRFT design

gives satisfactory performance in the presence of modeling error.

Ik

mM

Finally, to study the sensitivity of the proposed method with respect to noise,

both process input and output data are corrupted with 1% Gaussian white noise. It can

be seen from Figure 3.8 that the proposed controller can yield reasonably good control

performance in the presence of process noise.

Figure 3.7 Responses for +50% (top) and -50% (bottom) set-point changes by EVRFT design in the presence of modeling error

31


Figure 3.8 Responses for +50% (top) and -50% (bottom) set-point changes by EVRFT design in the presence of process noise

Example 2 Considering the van de Vusse reactor with the following reaction

kinetic scheme: , , which is carried out in an isothermal CSTR.

The dynamics of the reactor are described by the following equations (Doyle et al.,

1995):

CBA →→ DA →

)(231 AAfAA

A CCVFCkCk

dtdC

−+−−= (3.23)

BBAB C

VFCkCk

dtdC

−−= 21 (3.24)

where the model parameters used are: , , 11 50 −= hk 1

2 100 −= hk )hL/(mol103 ⋅=k

,mol/L0.3=AC

BC

,

, and . The nominal operation condition is

, and . The concentration of component B, , is the

process output and the flow rate, , is the process input.

mol/L10=AfC

mol/L12.1=BC

L1=V

34=F L/h3.

F

32


A salient feature of the above reactor is that the sign of its steady state gain

may change as the operation condition changes (see Figure 3.9). The database for

EVRFT design is generated by introducing random steps around the nominal value of

process input F as shown in Figure 3.10, where 1500 input/output data are used to

build the database. The process input and output are scaled by 31.34

31.34~ −=

uu and

12.112.1~ −

=yy respectively.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

F

C B

Figure 3.9 Steady-state curve of van de Vusse reactor

33


Figure 3.10 Input-output data used for constructing the database (example 2)

To proceed with the EVRFT design, the tuning parameters specified are as

follows: the initial value of parameter A = 0.61, initial learning rate 5.8=η , weight

parameter used in the objective function Eq. (3.18) given by , the updating

parameters for learning rate

8.0=w

02.1=incl and 6.0=decl , and the number of data points

in the relevant dataset is set to 1200.

To evaluate the performance of the adaptive PID designed using EVRFT

method, two set-point changes from 1.12 to 1.25 and from 1.12 to 0.62 are conducted.

as illustrated in Figures 3.11 and 3.12. The corresponding updating of tuning

parameter A and PID parameters is shown in Figures 3.13 and 3.14, respectively.

Similar to Example 1, four PID controllers are designed using VRFT method

for the purpose of comparison. To match the performance of EVRFT design for the

set-point change to 0.62 as closely as possible, two PID controllers are obtained by

34


VRFT design using the tuning parameter A = 0.65 for both reference model. Likewise,

to match the performance of EVRFT design for the set-point change to 1.25 as closely

as possible, another two PID controllers are obtained by VRFT design using tuning

parameters A = 0.37 and A = 0.35 for the first-order and second-order reference

models, respectively. As can be seen from Figure 3.11, the adaptive PID controller

designed using EVRFT method gives much faster response for set-point change from

1.12 to 1.25 than the two PID controllers that are deliberately designed to give

comparable performance to the EVRFT design for the set-point change from 1.12 to

0.62. Figure 3.12 shows similar observation for the other two PID controllers that are

designed to match the performance of EVRFT design for the set-point change to 1.25.

The tracking errors for various design methods are summarized in Table 3.4.




EVRFT design

% Improvement (based on VRFT design using 2nd-order model)

Set-point change to

A=0.65 A=0.37 A=0.65 A=0.35 A=0.65 A=0.35 1.25 21054.3 −× 21021.2 −× 21051.3 −× 21093.1 −× 21055.1 −× 55.8% 19.7% 0.62 21008.5 −× 21079.5 −× 21092.4 −× 2 1025.8 −× 21078.4 −× 2.9% 42.1%

35


Figure 3.11 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom). Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A = 0.65); dashed line: VRFT using1st-order model (A = 0.65)

Figure 3.12 Responses for set-point changes from 1.12 to 1.25 (top) and to

0.62 (bottom). Solid line: EVRFT; dotted line: VRF using 2nd-order model (A = 0.35); dashed line: VRFT using 1st-order model (A = 0.37)

36



set-point change to 1.25


set-point change to 0.62

37


Next, to evaluate the robustness of the proposed control strategy, it is assumed

there exist 20% modeling error in the kinetic parameter and . Figure 3.15

illustrates that the proposed EVRFT design gives satisfactory performance in the

presence of modeling error. Furthermore, to study the sensitivity of the proposed

method with respect to noise, both process input and output data are corrupted with

1% Gaussian white noise. It can be seen from Figure 3.16 that the proposed controller

can yield reasonably good control performance in the presence of process noise.

1k 2k


0.62 (bottom) in the presence of modeling error

38



0.62 (bottom) in the presence of process noise

Finally, to further evaluate the performance of the proposed method, it is

assumed that there exists time delay in the output measurement of two sampling time.

In this case, most of tuning parameters for EVRFT design are identical to those used

for the case in the absence of time delay, except that the initial values of A and

learning rate η are changed as A = 0.53 and 5.3=η .

Figure 3.17 shows the resulting performances of ERFT design for the same

set-point changes described previously. The corresponding updating of controller

parameters is shown in Figures 3.18 and 3.19. For the purpose of comparison, a PID

controller designed using VRFT method with second-order reference model is

developed. The tuning parameter A = 0.63 is selected to obtain minimum tracking

error for the two set-point changes. The tracking errors for both EVRFT and VRFT

designs are summarized in Table 3.5. It can be seen from Figure 3.17 that the adaptive

39


PID controller designed using EVRFT method gives better overall performance than

that obtained by the PID controller designed using VRFT with second-order reference

model.

Table 3.5 Tracking errors of EVRFT and VRFT designs for time delay case

Set-point change VRFT EVRFT % improvement 1.12 to 1.25 0.0382 0.0119 68.9% 1.12 to 0.62 0.0452 0.0413 8.6%

Figure 3.17 Responses for set-point changes from 1.12 to 1.25 (top) and to 0.62 (bottom) for time delay case. Solid line: EVRFT; dotted line: VRFT

40


Figure 3.18 Updating of controller parameters in EVRFT design for set-point

change to 1.25 in the presence of time delay

Figure 3.19 Updating of controller parameters in EVRFT design for set-point

change to 0.62 in the presence of time delay

41


42

3.5 Conclusion

In this chapter, an adaptive PID controller design method is proposed. The

proposed method makes use of the VRFT framework and achieves the adaptive nature

by updating the database, selecting a relevant dataset and most importantly, updating

the reference model parameter A at each sampling instance. Moreover, a second-order

reference model is employed for the VRFT design instead of the normally used first-

order reference model to achieve better performance. Simulation studies illustrate that

PID controllers designed using VRFT with a second-order reference model gives

better or comparable control performance than that obtained with a first-order

reference model. This justifies the use of second-order reference model in the

development of the proposed method. The simulation results also demonstrate that the

proposed EVRFT design gives marked improvement over the conventional VRFT

method.

Chapter 4

Adaptive Internal Model Controller

Design Using EVRFT Method

4.1 Introduction

Internal Model Control (IMC) is a powerful controller design strategy for the

open-loop stable dynamic systems (Morari and Zafiriou, 1989). A satisfactory

performance can be expected from IMC design as long as the process is operated in

the linear range around the point where the linear process model is obtained. However,

IMC controller may produce unsatisfactory or even unstable performance when it is

applied to nonlinear processes. To extend the IMC design to nonlinear processes,

various nonlinear and adaptive IMC schemes have been proposed in the literature as

reviewed in Chapter 2. However, all of the above nonlinear and adaptive IMC designs

bring in computationally demanding analytical or numerical methods and neural

networks in order to exhibit improved performance. This undermines the simplicity of

linear IMC design.

43

Chapter 4 Adaptive Internal Model Controller Design Using EVRFT Method

To circumvent the aforementioned drawback, the EVRFT method developed

in Chapter 3 is extended to the design of adaptive IMC controller in this chapter. In

the proposed method, the database for EVRFT design is updated at each sampling

instance by adding the current process data into the database and a relevant dataset is

selected according to the nearest neighbors’ criterion. In addition, the parameter of

reference model shall are updated for improved control performance. Finally, the

parameters of IMC controller are determined by solving an optimization problem

formulated in the VRFT design framework. Simulation results are presented to

illustrate the proposed design method and a comparison with conventional VRFT

design is made.

4.2 IMC Controller Design Using VRFT Method

The VRFT design using a second-order reference model discussed in Section

3.2 is applied here to design an IMC controller as shown in Figure 4.1. Without loss

of generality, a second-order ARX model is used to represent the IMC model:

( ) ( ) ( ) ( )121 121 −−+−+−= Nkukykyky βαα (4.1)

where ( )ky is the model output at the k-th sampling instance, is the

manipulated input at the (k-N-1)-th sampling instance, N is the process time delay,

and

( )1−− Nku

1α , 2α and 1β are IMC model parameters.

The corresponding transfer function of Eq. (4.1) is:

( ) 22

11

111

1~

−−

−−−

−−=

zzzzG

N

ααβ (4.2)

By using a first-order IMC filter, the IMC controller is obtained as )( 1−zQ

( ) 11

22

111

111

−

−−−

−−

⋅−−

=z

zzzQλλ

βαα (4.3)

44


r

Figure 4.1 Block diagram of IMC structure

where λ is the time constant of IMC filter. The corresponding feedback controller

( )θ;1−zC can be derived as

( ) ( )( ) ( ) ( ) 11

1

22

11

11

11

1111

~1; −−−

−−

−−

−−

−−−−

⋅−−

=−

= Nzzzz

zQzGzQzC

λλλ

βαα

θ (4.4)

In the VRFT design framework considered in this thesis, the reference model

( )1−zT is specified by the following second-order transfer function:

( )⎩⎨⎧

−−=+−=

+−+

== −−

−−−

−

−−

AAAAAAA

wherezAAz

zzzrzyzT

N

lnln1

21)()()( 2221

11

1

11

βαβα (4.5)

where A is the tuning parameter related to the speed of the response.

To design the IMC controller by the VRFT method, the virtual input ( )1~ −zu is

calculated by Eqs. (3.1), (3.2), (4.4) and (4.5) to obtain

( ) ( )( ) ( )( )[ ]( ) ( 1

1111

11221

1

22

111

112111~ −

−−−−−−

−−−−−−−−

+−−−+−+−

⋅−−−

= zyzzzz

zzzAAzzzzu NN

N

βαλλβα

βλαα )

)

(4.6)

Define ( 1~ −Φ z as follows:

( )[ ]( ) ( )11111 11)( −−−−−− +−−−=Φ zuzzzz N βαλλ (4.7)

The following equation can be obtained from Eqs. (4.6) and (4.7).

+ -

G

G~

Q+ -

y

yu

45


( )( ) ( ) ( 11

11221

1

22

111 2111

)(~ −−−

−−−−−−−− +−+−

⋅−−−

=Φ zyz

zzzAAzzzz N

Nβαβ

λαα ) (4.8)


( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )211121

−−−−−−−−−−−−+=Φ

NkuNkukukukuk

λβλαλβαλβα

(4.9)


( ) ( )kKk Ψ=Φ~ (4.10)

where

( ) ( ) ( ) ( )[ ]Tkkkk 321 ΨΨΨ=Ψ (4.11)

⎥⎦

⎤⎢⎣

⎡−−=

1

2

1

1

1

1βα

βα

βK (4.12)

with

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]11211 21 −−−−+++−++−=Ψ kykyNkyANkAyNkyk βαλ (4.13)

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]212121 22 −−−−−++−+−+−=Ψ kykyNkyANkAyNkyk βαλ (4.14)

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]3232211 23 −−−−−++−+−−+−=Ψ kykyNkyANkAyNkyk βαλ

(4.15)

Equation (3.3) is then expressed by

( ) ( ) ( ){ } 22

11

1min1min Ψ−Φ=Ψ−Φ= ∑=

Kn

kKkn

KJK

n

kK (4.16)

subject to the stability constraints imposed by the second-order IMC model:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

>

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

<

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−⇒

⎪⎩

⎪⎨

⎧

<>−+>−−

000001

001101

101111111

000001

001101

101111111

10101

1

2

1

1

1

1

2

1

1

1

2

21

21

βαβαβ

βαβαβ

ααααα

or

(4.17)

46


where

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΨΨΨΨΨΨΨΨΨ

=Ψnnn

333

222

111

212121

L

L

L

(4.18)

( )[ nΦΦ=Φ L1 ( )] (4.19)

Consequently, IMC model parameters are obtained by solving the constrained

least square problem as stated in Eq. (4.16). It is evident that this solution not only

depends on the database for VRFT design but also the design parameter A in the

reference model.

The control action taken by the IMC controller is calculated by:

( ) ( ) ( ) ( ) ( )[ ]2111 21 −−−−−

+−= kekekekuku ααβλλ (4.20)

where ( ) ( ) ( ) ( )kykykrke +−= and ( )kr is the set-point at k-th sampling instance.

4.3 Enhanced VRFT Design Method

Similar to Section 3.3, when the EVRFT method is used to design an adaptive

IMC controller, tuning parameter A in the reference model is updated at each

sampling instance as it has a direct impact on the IMC model parameters calculated

using the VRFT method.

The objective of adjusting reference model parameter A is to minimize the

following quadratic function:

( ) ( ) ( ) ( )[ ] ( ) ( )[{ }222 11ˆ11

21

−−++−+−= kukuwkykrwkJ ]

]

(4.21)

where is a weight parameter and [ 1,0∈w ( )1ˆ +ky is the predicted output using the

reference model at the (k+1)-th sampling instance which can be calculated by:

( ) ( ) ( ) ( ) ( )1121ˆ 2 −−+−+−−=+ NkrNkrkyAkAyky βα (4.22)

47


By the steepest descent method, the updating algorithm is derived as:

( ) ( ) ( ) ( )( )kA

kJkkAkA

∂∂

−=+ 21 η (4.23)

where ( )kη is the learning rate.

The following summarizes the computational algorithm for the proposed

EVRFT design of adaptive IMC controller:

Step 1 The process Input (u(k)) and output (y(k)) identification data that characterize

the dynamics of nonlinear system are assumed to be available and the off-line

database for EVRFT design is constructed as ( ) niix ~1= ;

Step 2 At the k-th sampling instance, based on the current database for EVRFT

design, the relevant dataset is selected according to Eq. (2.4) and then the IMC

model parameters are calculated by solving the minimization problem given

by Eq. (4.16) subjected to the constraint Eq. (4.17). The manipulated variable

is obtained by Eq. (4.20). ( )ku

Step 3 The database for VRFT design is augmented by appending the current process

data ( )ky and ; while the tuning parameter A is updated using Eq. (4.23). ( )ku

Step 4 Set and go to Step 2. 1+= kk

4.4 Examples

Example 1 Considering the continuous polymerization reaction studied in

Chapter 3 with model parameters and steady-state operating condition given in Table

3.1 and 3.2.

To apply the EVRFT method to design the adaptive IMC controller, the

identical off-line process data used in Chapter 3 are employed. The tuning parameters

specified for EVRFT design are as follows: the IMC filter parameter 7.0=λ , the

48


initial value of parameter A = 0.6, initial learning rate 045.0=η , weight parameter

used in the objective function Eq. (4.21) given by 1.0=w , the updating parameters

for learning rate and 02.1=incl 55.0=decl , and the number of data points in the

relevant dataset is set to 1200.

To evaluate the performance of the adaptive IMC controller designed using

EVRFT method, +50% and -50% set-point changes are conducted, as illustrated in

Figures 4.2 and 4.3. The updating of tuning parameter A and IMC parameters is

shown in Figures 4.4 and 4.5.

For the purpose of comparison, the four PID controllers designed using VRFT

method with both first-order and second-order reference models developed in Chapter

3 are considered. As can be seen from Figure 4.2, although VRFT design using

second-order reference model is able to match the adaptive IMC design for +50% set-

point change, it gives slower performance than the latter design for -50% set-point

change. In this case, the PID controller designed using VRFT method with first-order

reference model gives the worst control performance. On the other end, Figure 4.3

shows that, while comparable performances for -50% set-point change are observed

for both VRFT and adaptive IMC designs, the latter gives better performance than the

two VRFT designs for +50% set-point change. Table 4.1 summarizes the tracking

errors of the abovementioned servo responses.





Set- point change

A=0.95 A=0.5 A=0.77 A=0.3

EVRFT based IMC design

A=0.77 A=0.3 +50% 1836 2920 868 2479 901 -3.7% 63.7% -50% 2984 356 1465 387 339 76.9% 12.4%

49


Figure 4.2 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A=0.77);

dashed line: VRFT using 1st-order model (A=0.95)

Figure 4.3 Responses for +50% (top) and -50% (bottom) set-point changes. Solid line: EVRFT; dotted line: VRFT using 2nd-order model (A = 0.3);

dashed line: VRFT using 1st-order model (A = 0.5)

50


Figure 4.4 Updating of parameter A and IMC parameters (+50% set-point change)

Figure 4.5 Updating of parameter A and IMC parameters (-50% set-point change)

51


Next, to evaluate the robustness of the proposed control strategy, it is assumed

there exist 10% modeling error in the kinetic parameter and 20% modeling error in

the coefficient . Figure 4.6 illustrates that the proposed EVRFT design gives

satisfactory performance in the presence of modeling error.

Ik

mM

Finally, to study the sensitivity of the proposed method with respect to noise,

both process input and output data are corrupted with 1% Gaussian white noise. It can

be seen from Figure 4.7 that the proposed controller can yield reasonably good control

performance in the presence of process noise.

Figure 4.6 Responses for 50% (top) and -50% (bottom) set-point changes in the presence of modeling error

52


Figure 4.7 Responses for 50% (top) and -50% (bottom) set-point changes in the presence of process noise

Example 2 Considering a catalytic continuous stirred tank reactor (CSTR) as

shown in Figure 4.8. The dynamics of the reactor are described by the following

equations (Demuth et al., 2007):

hwwdtdh 2.021 −+= (4.24)

( ) ( )( )22

122

11 1 b

bbbbb

b

CkCk

hwCC

hwCC

dtdC

+−−+−= (4.25)

where is the liquid level, is the product concentration at the output of the

process, is the flow rate of the concentrated feed , and is the flow rate of

the diluted feed . The input concentrations are set as

h bC

1w 1bC 2w

242bC 9.1 =bC and 1.02 =bC

12

.

The kinetic constants associated with the rate of consumption are and 1=1k =k .

53


The product concentration, , is the process output and the flow rate, , is the

process input. To simplify the demonstration, the flow rate of the diluted feed is set as

. The nominal operating condition of the reactor are ,

and .

bC 1w

20=( ) 1.02 =tw

25.12=h

5839.bC

6.01 =w

Figure 4.8 The catalytic continuous stirred tank reactor

A salient feature of the above reactor is that the sign of its steady state gain

may change as the operation condition changes (see Figure 4.9). The database for

EVRFT design is generated by introducing uniformly random steps with distribution

of and the switching probability of 0.2 at every sampling time to the process

input . With a sampling time of 0.2, 1500 input/output data as shown in Figure 4.10

are collected to build the database. The process input and output are scaled by

]2.1,0[

1w

6.06.0~ −

=uu and

20.583920.5839~ −

=yy respectively.

54


Figure 4.9 Steady-state curve of catalytic CSTR

Figure 4.10 Input-output data used for constructing the database (catalytic CSTR)

55


To proceed with the EVRFT design of adaptive IMC controller, the tuning

parameters specified are as follows: the IMC filter parameter is fixed as 01.0=λ , the

initial value of parameter A = 0.4, initial learning rate , weight parameter

used in the objective function Eq. (3.18) given by

3105.2 −×=η

7.0=w , the updating parameters

for learning rate and 02.1=incl 55.0=decl , and the number of data points in the

relevant dataset is set to 1200.

To evaluate the performance of the adaptive IMC designed using EVRFT

method, two respective set-point changes from the nominal operating condition to

21.5 and to 19.5 are conducted, as illustrated in Figure 4.11. The corresponding

updating of tuning parameter A and IMC parameters in the above servo responses is

shown in Figures 4.12 and 4.13.

For the purpose of comparison, PID controllers designed using the VRFT

method with both first-order and second-order reference models are developed. The

tuning parameter A = 0.96 is selected for the first-order case and A = 0.74 is selected

for the second order case to achieve the minimum tracking errors for the two set-point

changes. It is evident from Figure 4.11 that the adaptive IMC controller designed

using EVRFT method gives faster response for both set-point changes than the two

PID controllers designed by VRFT method. Moreover, the PID controller designed

using second-order reference model shows better performance than that with a first-

order reference model. The tracking errors for all methods are summarized in Table

4.2.


Set-point change to



EVRFT based IMC design


21.5 0.1385 0.0977 0.0754 22.8% 19.5 0.2270 0.1753 0.1384 21.1%

56


Figure 4.11 Closed-loop responses for set-point changes to 21.5 (top) and to 19.5 (bottom). Solid line: EVRFT; dotted line: VRFT using 2nd-order model; dashed line:

VRFT using 1st-order model

Next, to evaluate the robustness of the proposed control strategy, 10%

modeling errors in the kinetic parameter and are considered. Figure 4.14

illustrates that the proposed EVRFT design gives satisfactory performance in the

presence of modeling error. Furthermore, to study the sensitivity of the proposed

method with respect to noise, both process input and output data are corrupted with

1% Gaussian white noise. It can be seen from Figure 4.15 that the proposed controller

can yield reasonably good control performance in the presence of process noise.

1k 2k

Finally, to further evaluate the performance of the proposed method, it is

assumed that there exists time delay in the output measurement of five sampling time.

In this case, most of tuning parameters for EVRFT design are identical to those used

for the case in the absence of time delay, except that the initial values of A and

57


learning rate η are changed learning rate η are changed as A = 0.5 and ,

while the IMC filter parameter is fixed as

4103.8 −×=η

5.0=λ .

Figure 4.16 shows the resulting performances of ERFT design for the same

set-point changes described previously. The corresponding updating of controller

parameters is shown in Figures 4.17 and 4.18. For the purpose of comparison, a PID

controller designed using VRFT method with second-order reference model is

developed. The tuning parameter A = 0.788 is selected to obtain minimum racking

error for the two set-point changes. The tracking errors for both EVRFT and VRFT

designs are summarized in Table 4.3. It can be seen from Figure 4.16 that adaptive

IMC controller designed using EVRFT method gives better overall performance than

that obtained by the PID controller designed using VRFT with second-order reference

model.

Table 4.3 Tracking errors of VRFT and EVRFT designs for time delay case

Set-point change to

VRFT EVRFT based IMC design

% improvement

21.5 0.1279 0.1062 17.0% 19.5 0.2318 0.2058 11.2%

58


Figure 4.12 Updating of parameter A and IMC parameters for set-point change to 21.5

Figure 4.13 Updating of parameter A and IMC parameters for set-point change to 19.5

59


Figure 4.14 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom)

in the presence of modeling error

Figure 4.15 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom)

in the presence of process noise

60


Figure 4.16 Responses for set-point changes to 21.5 (top) and to 19.5 (bottom) in the presence of time delay. Solid line: EVRFT; dotted line: VRFT

Figure 4.17 Updating of parameter A and IMC parameters set-point change

to 21.5 in the presence of time delay

61


62

Figure 4.18 Updating of parameter A and IMC parameters for set-point change

to 19.5 in the presence of time delay

4.5 Conclusion

In this chapter, an adaptive IMC design method is proposed. The proposed

method makes use of the VRFT design and achieves the adaptive nature by updating

the database, selecting a relevant dataset and most importantly, updating the reference

model parameter A at each sampling instance. Moreover, a second-order reference

model is employed for the VRFT design instead of the commonly used first-order

reference model to achieve better control performance. The simulation results

demonstrate that the proposed method gives improved performance over the

conventional VRFT design.

Chapter 5

Conclusions and Further Work

5.1 Conclusions

In this thesis, the Enhanced Virtual Reference Feedback Tuning (EVRT)

method is proposed and two adaptive controllers, namely adaptive PID controller and

adaptive IMC controller, are developed using the EVRFT method. This method make

use of information extracted from the process input and output database to obtain

controller parameters at each sampling instance. The adaptive nature of this method is

achieved by: (i) update the database by adding the current process data into the

database; (ii) using the relevant dataset selected from the current database based on k-

nearest neighborhood criterion; (iii) update the parameters in the reference model.

Moreover, a second-order reference model is used in both designs instead of the first-

order reference model normally used in the literature.

Simulation results are presented for both nominal cases and cases in the

presence of modeling error and process noise. Both adaptive PID and IMC designs by

the proposed EVRFT design are observed give faster and less oscillatory responses

63

Chapter 5 Conclusions and Further Work

64

than the PID controller designed using VRFT method. In summary, the EVRFT

method is a useful strategy for adaptive controller design for nonlinear processes.

5.2 Suggestions for Further Work

There are few open questions that need to be further studied. Some possible

topics for future research are listed below.

The present study focuses on the servo performance in the controller design. It

is thus of practical importance to extend the EVRFT design to address disturbance

rejection performance. Furthermore, the EVRFT method may be applied to more

complicated control systems such as cascade control. Finally, the application of the

EVRFT method to the multivariable systems may also be an interesting topic.

Appendix A

Derivation of the 2nd-order Reference Model

Consider the following second-order transfer function with a zero-order hold:

( )tsN

ts

ess

e Δ−Δ−

⋅+

⋅−

2111

λ (A.1)

where is the process time delay, tN Δ⋅ tΔ is the sampling period and λ is the time

constant. Applying Z-transform of Eq. (A.1) obtains

( )( )

( )

( ) ( ) ⎥⎥⎦⎤

⎢⎢⎣

⎡

−

Δ−

−−

−−=

⎭⎬⎫

⎩⎨⎧

+−

+−⋅−=

⎭⎬⎫

⎩⎨⎧

⋅+

⋅−

−

−

−−−−

−−Δ−Δ−

21

1

111

21

2

111

111

1111

111

AztAz

Azzzz

sssZzze

sseZ

N

NtsNts

λ

λλ

λλ

λ

(A.2)

where λt

eAΔ−

= and hence λtA Δ

−=ln . Equation (A.2) can thus be simplified as:

( ) ( )[ ]( )

( )221

11

21

121

211lnln1

−−

−−−

−

−−−

+−+

=−

−−++−⋅

zAAzzz

AzzAAAAAAAzz

NN βα

(A.3)

where AAA ln1 +−=α and . AAAA ln2 −−=β

65

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