Real Time Process Optimization Using a Recursive

Least Squares Impulse Weight Gradient Method

(Spine Title: Real Time Process Optimization)

(Thesis format: Monograph)

by

Yuanyuan Liu

Graduate Program in Engineering Science

Department of Chemical and Biochemical Engineering

Submitted in partial fulfillment

of the requirements for the degree of

Master of Engineering Science

The School of Graduate and Postdoctoral Studies

The University of Western Ontario

London, Ontario, Canada

August, 2008

© Yuanyuan Liu 2008

ii

THE UNIVERSITY OF WESTERN ONTARIO SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES

CERTIFICATE OF EXAMINATION Supervisor ______________________________ Dr. Arthur Jutan

Examiners ______________________________ Dr. Amarjeet Bassi _____________________________ Dr. Dimitre Karamanev ______________________________ Dr. Serguei Primak

The thesis by

Yuanyuan Liu

entitled:

Real Time Process Optimization Using a Recursive Least Squares Impulse Weight Gradient Method

is accepted in partial fulfillment of the

requirements for the degree of Master of Engineering Science

Date__________________________ _______________________________

Chair of the Thesis Examination Board

iii

Abstract

The chemical industry is increasingly compelled to operate profitably

in a very dynamic and global market. Real-time optimization plays an

important role in plant operation because it is at the level of the control

hierarchy at which business decisions are integrated into the operation.

Model-based optimization schemes require accurate models which are

sometimes hard to obtain in practice. The advantage of direct search

optimization becomes obvious when an explicit model is unavailable. In this

work, a new recursive least squares impulse weight, gradient based,

feedback optimization method is developed and tested in both simulation

and experimental applications. For a linearized dynamic system, the

gradient of the system output with respect to the system input can be

interpreted as the first non-zero impulse weight of the system. This

relationship was confirmed by open-loop and closed-loop tests on linear and

non-linear static and dynamic systems. This gradient information can thus

be used for extremum seeking. Closed-loop control real time case studies

demonstrated the ability of the new gradient based optimization method to

follow different time-varying optimal policies using a wide range of input

signals. Simulation studies also demonstrate that a strong advantage of this

new method is its ability to handle processes with changing dead time and

model structure.

iv

Keywords: Real-time optimization, Direct Search Method, Gradient Based

Optimization, Extremum seeking, Dead Time

v

Dedication

To my wonderful mother Guizhen Li

To my beloved grandmother Ping Cui

and

To my uncle Ming Li who has supported me all the way since the

beginning of my studies.

vi

Acknowledgement

I am deeply indebted to my supervisor Professor A. Jutan. Thank you

for accepting me to your research group and your kind help, excellent

guidance, continuous support and encouragement throughout my research

and the writing of this thesis. Words cannot express how grateful I am. I

feel extremely fortunate and honoured to be your last student. I would like

to thank Cheryll Rose Jutan, for her encouragement and kind support

during my study.

I would like to extend my thanks to the Department of Chemical and

Biochemical Engineering and its faculty and staff for the financial, academic

and technical support.

I would also like to thank all who have wished me good and made my

stay at London all the more pleasant.

Last but not least, I sincerely thank my parents Zhenwei Liu and

Guizhen Li for their love and constant support over all these years.

vii

Table of Contents

Title Page········································································································ i Certificate of Examination·············································································ii Abstract·········································································································iii Dedication······································································································v Acknowledgements·······················································································vi Table of Contents·························································································vii List of Figures·······························································································xi List of Acronyms ························································································· xv Nomenclature······························································································xvi 1 Introduction ··················································································1

1.1 General ······················································································2

1.2 Thesis Objective ··········································································4

1.3 Thesis Overview ··········································································4

1.3.1 Comparison of Some Recent Real-Time Optimization Methods ······5

1.3.2 A New Recursive Least Squares Impulse Weight Gradient Optimization Method······························································5

1.3.3 Simulation Examples······························································5

1.3.4 Real-Time Optimization ··························································6

1.3.5 Summary and Future Work ·····················································6

2 Comparison of Some Recent Real-Time Optimization Methods ······8

2.1 Introduction················································································9

2.2 Problem Formulation ································································· 10

2.3 Direct Search Methods ······························································· 11

2.3.1 Simplex Methods·································································· 11

2.3.1.1 Nelder-Mead Simplex Method··········································· 11

viii

2.3.1.2 Dynamic Simplex Method ················································ 16

2.3.2 Sign-Sign Optimization Method·············································· 18

2.3.3 Perturbation Method ···························································· 25

2.3.4 A Fuzzy Logic Based Approach··············································· 29

3 A New Recursive Least Squares Impulse Weight Gradient Optimization Method ··································································· 38

3.1 Introduction·············································································· 39

3.2 Impulse Weight Gradient Method ················································ 39

3.3 Theory of Recursive Least Squares Algorithms······························· 43

3.3.1 Least Squares······································································ 43

3.3.2 Recursive Least Squares ······················································· 44

3.3.3 Recursive Least Squares with Exponential Forgetting ··············· 46

3.4 Gradient Optimization ······························································· 49

3.5 Optimization Examples ······························································ 50

4 Simulation Examples ··································································· 53

4.1 Estimate Gradient using RLS······················································ 54

4.2 Gradient Estimate Simulation····················································· 55

4.2.1 A Linear Static System Example ············································ 55

4.2.2 A Second Order Non-Linear Static System ······························· 57

4.3 Combining the Perturbation Method with the Proposed Gradient

Method ···················································································· 61

4.3.1 Static Example ···································································· 62

4.3.1.1 Effect of random different test signals································ 67

4.3.2 Nonlinear Dynamic Example ················································· 68

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4.3.2.1 Simulation 1 ·································································· 71

4.3.2.2 Simulation 2 ·································································· 72

4.3.3 Comparison of proposed RLS Impulse Weight Gradient Method with Sign-Sign Method·························································· 73

4.4 Processes with Dead Time··························································· 74

4.5 Processes with Variable Dead Time ·············································· 79

4.6 Processes with Variable Dead Time and Changing Model Structure ·· 81

5 Real-Time Optimization ······························································· 84

5.1 Real-Time Application to CST······················································ 85

5.1.1 Experimental Setup······························································ 86

5.1.2 Experimental Runs ······························································ 89

5.1.3 Discussion··········································································· 93

5.2 Real-Time Application to Pressure Tank ······································· 94

5.2.1 Experimental Setup······························································ 95

5.2.2 Experimental Runs ······························································ 97

5.2.3 Discussion··········································································101 6 Summary and Future Work ······················································· 103 Appendix A –Experimental Setup ················································· 107

A.1 Data acquisition board ······························································107

A.2 MATLAB applications·······························································116

Appendix B – MATLAB Codes ······················································· 122

B.1 Generate simulated data ···························································122

B.2 S-Function of estimating first non-zero impulse weight for a linear

static problem·········································································122

x

B.3 S-Function of estimating first non-zero impulse weight for a second

order nonlinear static problem ···················································125

B.4 Generate data for a process with variable dead time and model·······128

B.5 S-Function for calculating delay ·················································129

B.6 S-Function of Gradient method on processes with dead time···········132

B.7 S-Function of Gradient method on CST and Pressure Tank ············135

Appendix C - Conversion Tables ··················································· 139

C.1 CST temperature measurement ·················································139

C.2 Pressure tank ··········································································140

Curriculum Vitae·········································································· 141

List of Figures

1.1 General structure of model-based RTO···········································3

2.1 Operations used in Nelder-Mead simplex algorithm ······················· 15

2.2 Successive reflection in 2-dimensional space ································· 17

2.3 Feedback optimizer ··································································· 18

2.4 The results from One-variable-at-a-time strategy ·························· 22

2.5 Contour plot of the Eq. (2.3)························································ 23

2.6 Optimization results of Eq. (2.3)·················································· 24

2.7 3-D optimization using Sign-Sign method ····································· 25

2.8 The principle of the perturbation method ······································ 26

2.9 The perturbation extremum seeking scheme ································· 27

3.1 Basic input-output configuration ················································· 40

3.2 Feedback optimization loop ························································ 49

4.1 Estimation of first impulse weight using RLS with a forgetting factor

0.99 and correlation method. Sampling time is 1s ··························· 54

4.2 Gradient estimation for a linear static system using Simulink

Toolbox ··················································································· 56

4.3 Gradient information for a linear static system ····························· 57

4.4 Gradient estimation using Simulink Toolbox for a second order

nonlinear static system······························································ 59

4.5 Gradient information for a second order nonlinear static system with

xi

se ······································································ 60 ( )= +sinx t noi

xii

4.6 Feedback optimization loop for a static system ······························ 61

4.7 Optimization feedback loop for Eq. (4.5) ······································· 63

4.8 Block diagrams in Simulink Toolbox for a static system·················· 64

4.9 Output for Example 4.5 ····························································· 65

4.10 Gradient and input information for Example 4.5·························· 66

4.11 Gradient comparison between RLS and Correlation method ·········· 66

4.12 Actual and optimal process input and output values for example 4.6

with random number as input signal··········································· 67

4.13 Feedback optimization loop for a nonlinear dynamic system ·········· 68

4.14 Simulation results for equation 4.8 ············································ 71

4.15 Simulation results for equation 4.8 with ramp optimal output········ 72

4.16 Minimum seeking for quadratic function 4.9 ······························· 73

4.17 Feedback optimization loop for processes with dead time··············· 75

4.18 Block diagrams in Simulink Toolbox for a process with dead·········· 77

4.19 Simulation results for equation 4.9 with dead time=3 ··················· 78

4.20 Dead time and gradient information for equation 4.10 ·················· 80

4.21 Dead time and gradient information for equation 4.11 ·················· 82

5.1 Continuous Stirred Tank ··························································· 85

5.2 Experimental setup for CST ······················································· 87

5.3 Picture of the CST used for experimental runs ······························ 88

5.4 Variation in CST temperature with both water valve and steam valve

25% open·················································································· 89

5.5 CST Real –Time optimization diagram········································· 90

xiii

5.6 Simulink© block diagrams for Real time Optimization of a CST······· 91

5.7 Real-Time Optimization-output of objective function Eq.(5.1) ·········· 92

5.8 Real-Time Optimization-temperature condition····························· 93

5.9 Pressure Tank·········································································· 94

5.10 Experiment Setup for Pressure Tank System ······························ 95

5.11 Picture of Pressure Tank ·························································· 96

5.12 Pressure Tank Real –Time optimization diagram························· 97

5.13 Simulink block diagrams for Real time Optimization of a Pressure

Tank ····················································································· 98

5.14 Optimization results for the pressure tank system with objective

function given as Eq. (5.2): objective output································· 99

5.15 Optimization results for the pressure tank system with objective

function given as Eq. (5.2): Pressure condition inside the tank ······100

A.1 Digital input ports of the data acquisition board···························108

A.2 Analog input ports of the data acquisition board ··························109

A.3 Thermocouple ·········································································109

A.4 Experiment setup for voltage measurement·································111

A.5 Experiment setup for valve control·············································111

A.6 Advantech Device Manager·······················································113

A.7 Device Setting·········································································114

A.8 Advantech Device Test ·····························································115

A.9 Real time toolbox library ··························································116

A.10 Simulink model for temperature measurement ··························117

xiv

A.11 Select ‘pci1710.rtd’ as the hardware driver ································118

A.12 Setup the connection mode and the input/output signal range ······119

A.13 Choose input range 0V ~ 10V ··················································119

A.14 Choose output range 0V ~ 10V·················································120

A.15 Simulink model for valve control ··············································121

xv

List of Acronyms

RTO Real Time Optimization

RLS Recursive Least Squares

CST Continuous Stirred Tank

DSM Dynamic Simplex Method

PID Proportional Integral Derivative

DT Dead Time

ARX Auto Regressive and eXogenous

MSE Mean Squared Error

SISO Single Input Signal Output

MPC Model Predictive Control

LMS Least Mean Squares

NLMS Normalized Least Mean Squares

Nomenclature

Chapter 2. Comparison of some Recent Real-Time

Optimization Methods

Symbols

f Function value

f Average function value

x Process inputs

y Process outputs

J Objective function

N Number of dimensions

S A Simplex which has vertices 1N +

0S Initial simplex

Greek letters

θ Time varying parameters

φ Measurement noise

ρ Scalar parameter of reflection

κ Scalar parameter of expansion

γ Scalar parameter of contraction

σ Scalar parameter of shrinkage

μ Step size of the Sign-Sign method

xvi

Subscripts

c Centroid

e Expansion

i Inside contraction

o Outside contraction

r Reflection

t Time, s

Chapter 3. A new Recursive Least Squares Impulse

Weight Gradient Optimization Method

Symbols

G Transfer function

q Shift operator

n Noise model

v Impulse response

r Covariance

P Covariance matrix

h Sampling interval

Greek letters

β Correlation

λ Forgetting factor

xvii

Chapter 4. Simulation Examples

Symbols

u Inputs

iF Input linear block

oF Output linear block

L Laplace transforms

Greek letters

θ Time varying parameters

δ Gain

Subscripts

nn Parameters of ARX model

nk Delay

Chapter 5. Real-Time Optimization

T Temperature, °C

P Pressure, psi

xviii

1

Chapter 1

Introduction

2

1.1 General

For the chemical process industry to remain competitive in the global

market, the determination of optimal control strategies which maximize

profit, minimize operating costs is becoming one of the main goals in the

design and operation of chemical processes [2]. Hence, real-time optimization

(RTO) is becoming an important concept because it is at the level of the

control hierarchy at which business decisions are integrated into the

operation. The benefits from RTO include [3]:

1. Improved product yields and/or quality;

2. Reduced energy consumption and operating costs;

3. Increased capacity of equipment, stream factors;

4. Less maintenance cost and better maintenance of instrumentation;

5. More efficient engineers and operators as a tool for process trouble

shooting and operation;

6. Tighter, lower cost process designs if new plant designs include a real

time optimizer.

Real-time optimization is most beneficial for processes with a wide range

of operating conditions due to variability in product prices and material costs,

large variations in ambient conditions or economic trade-offs and constraints.

To implement model-based RTO, an accurate process model is necessary.

A general structure of model-based RTO is shown in Fig. 1.1.

3

Fig.1.1 General structure of model-based RTO

Firstly, in the data validation step, the validity of data is checked before

using it for parameter estimation. In the model update step, the reconciled

data is then used to compute a new set of model parameters such that the

plant model represents the plant as accurately as possible at the current

operating point. The updated model is optimized in the optimization step.

After that, the optimization results are analyzed and transferred to the

computer control system for implementation.

Many applications in chemical engineering deal with nonlinear models of

complex physical phenomena. Therefore, the model used for plant design will

usually not represent the real process exactly. Under these circumstances,

the model free RTO is more attractive since it does not require an explicit

model, and based entirely on the process measurements.

4

The objective of this research is to develop a new online, non-model based,

recursive least squares, optimization scheme, to minimize operational cost

and maximize plant profitability while maintaining the desired product

quality.

1.2 Thesis Objective

Most chemical processes are operated under changing operating

conditions which result in changing optimal operating conditions. Good

tracking of optimal operating conditions is important for real-time

optimization. Model-based optimization requires an accurate model to be

applied in practice. However, some process models are either too difficult or

too costly to obtain or update due to process nonlinearity, complicated

physical phenomena, insufficient and noisy measurements or time-varying

parameters. Under these circumstances, the ‘direct search’ (non-model based)

optimization approach is more attractive since an explicit model is

unnecessary. However, most direct optimization methods deal with a static

(i.e. non-dynamic) optimum. In this work, we present a new Impulse Weight

Gradient-based Optimization approach with the ability of handling a

dynamic or a moving optimum.

1.3 Thesis Overview

This thesis is divided into the five chapters outlined below.

5

1.3.1 Chapter 2. Comparison of Some Recent Real-

Time Optimization Methods

In this chapter, some recent successfully non-model based optimization

methods are introduced and compared. They include the Nelder-Mead

Simplex Method and its extension-the Dynamic Simplex Method, the Sign-

Sign Optimization Method, the Perturbation Method, and the Fuzzy Logic

based Method.

1.3.2 Chapter 3. A New Recursive Least Squares

Impulse Weight Gradient Optimization

Method

In this chapter, the Recursive Least Squares (RLS) algorithm is

formulated and compared to the Correlation based impulse method [5]. The

principle of gradient based optimization is then formulated based on the RLS

method.

1.3.3 Chapter 4. Simulation Examples

The Recursive Least Squares (RLS) algorithm is used to estimate the

impulse weights for linear and nonlinear functions. The new Impulse Weight

Gradient Optimization Method is applied to several mathematical functions.

The optimization results are compared with an earlier dynamic optimization

method, the Sign-Sign Method. The new Gradient Optimization Method is

6

also tested on processes with dead time, processes with variable dead time

and processes with variable dead time and a changing model structure. The

simulation results are compared to the Sign-Sign Method.

1.3.4 Chapter 5. Real-Time Optimization

In order to fully test the effectiveness in an experimental setup, the new

RLS Impulse Weight Gradient Optimization method is applied to Real-Time

processes, a Continuous Stirred Tank (CST) and Pressure Tank.

1.3.5 Chapter 6. Summary and Future Work

A summary of major contributions of this thesis is given in this chapter

and some possible applications for future studies are discussed.

7

Bibliography

[1] Xiong, Qiang (2000). Optimization, Control and Identification of

Chemical Processes. M.E.Sc Thesis, University of Western Ontario.

[2] M. Hanke and P. Li (1998). Simulated Annealing For the

Optimization of Chemical Batch Production Processes.

[3] S. Engell (2006). Feedback Control for Optimal Process Operation.

Process Control Laboratory, Department of Biochemical and

Chemical Engineering, D-44221 Dortmund, Germany.

[4] A Toumi, M. Diehl, S. Engell, H.G. Bock, J.P. Schloder, Finite

horizon optimizing of control advanced SME chromatographic

processes, in: 16th IFAC World Congress, Fr-M06-TO/2,2005.

[5] M. Sheikhzadeh, S. Rohani and A. Jutan (2006). A New Non-Model

Based Extremum Seeking Controller. Optimal Control, Applications

and Methods Journal.

8

Chapter 2

Comparison of some

Recent Real-Time

Optimization Methods

9

2.1 Introduction

Real-Time optimization (RTO), or on-line operations optimization has

received a growing appeal in the process industries because of its promise for

reducing operation cost and improving process profitability [2]. Most chemical

processes are characterized by severe nonlinearity, use of inaccurate models,

presence of operational constrains, insufficient and noisy measurements,

time-varying parameters as well as ever changing operating conditions. The

operating conditions are affected by both the physical drifting of the process

itself, such as fouling of a heat exchanger, deactivation of the catalyst in a

reactor, temperature and flow-rate fluctuation of the feed etc. and the

changes in product demand and economic conditions. Real-time optimization

is attractive because it can account for these changes and drive the process

continuously toward its optimal operating point. All kinds of real-time

optimization fall into two categories: model-based approaches and direct

search approaches (or non-model based approaches) [3]. In model-based real-

time optimization, an updated process model is used to determine the

operating conditions for the plant which will act to increase the operating

profit, therefore, the success of model-based approaches rely highly on the

availability of a good process model. However, some chemical processes are

too difficult and/ or costly to model, and even if a model is obtainable it may

be difficult to update due to its complexity and nonlinearity. Thus, the

model–free direct search optimization approaches becomes preferable when a

10

good process model is unavailable, as these methods are based entirely on the

relationship between input and output collected online.

Several successful direct search optimization approaches have been

reported in the literature, such as, Dynamic Simplex Method (DSM) [4], Sign-

Sign Method [5], Fuzzy Logic Based Method [6], Perturbation Method [12],

etc. The algorithms and performance of these methods are reviewed in this

chapter.

2.2 Problem Formulation

A general nonlinear time-varying system to be optimised can be stated

as:

( , , ) ,ty f x tθ φ= + (2.1)

(2.2) ( , )J g y x=

In this formulation, f is a nonlinear function that defines a relationship

between the process input x and process output y , φ is the noise, J is the

objective function to be optimized. The process has a moving optimum due to

the time-varying parameters tθ and .The value of y is calculated by

evaluating the function

t

f in simulation or measurement in a real-time

application.

11

2.3 Direct Search Methods

The direct search methods do not need a process model in order to follow

the optimum operating conditions. This implies that the data required for

building and updating process models become unnecessary in direct search

methods, which greatly saves the time and cost associated with

measurements. Direct search methods are reasonably straightforward to

implement and can be applied almost immediately to many nonlinear

optimization problems [8].

2.3.1 Simplex Methods

After it was first proposed by Nelder and Mead in 1965 [9], the Simplex

search method has enjoyed enduring popularity. Based on the basic simplex

search method, a dynamic simplex method (DSM) [4] was developed in order

to track the moving optimum.

2.3.1.1 Nelder-Mead Simplex Method

A simplex S is a convex hull with N+1vertices in an N dimensional

space . When optimizing experimental systems using a simplex, each

vertex corresponds to a set of experimental conditions. For example, there is

a process with N input variables

Nℜ

{ }1

N

j jx

= , N+1 experiments are carried out in

12

a set of N dimensional experimental conditions. The corresponding

experimental results are labeled in a descending order:

1 2 ...... N 1f f f +> > >

Because the intention is to maximize the objective function f , 1f is referred

to as the best point, 1Nf + the worse point, and Nf the next to the worse point.

In the optimization algorithm the worst points are discarded and several

other movements are carried out for finding new experiment conditions

favorable for optimization. The possible movements include reflection,

expansion, contraction and shrinkage. The rules for the simplex algorithm

are presented as:

A) Rank

Rank the N+1vertices from best to worst as:

1 2 ...... N 1f f f +> > >

B) Reflection

By performing reflection the worst vertex is flipped through the centroid of

the opposite face and a new vertex rx is obtained. The new vertex is:

( )1r c c Nx x x xρ += + −

Where 1ρ = , 1

1 N

cj

jx xN =

= ∑ is the centroid of vertices excepting the worst vertex.

The new vertex ( )r rf f x= is then calculated and evaluated. If 1 r Nf f f≥ ≥ ,

which means that the response at the reflection vertex is not better than the

response at the previously best vertex neither worse than the response at the

13

next to the worst vertex. There is no clear evidence that the simplex is

moving in a particularly desirable or undesirable direction. Hence, the

reflected point is accepted as the vertex of the new simplex and the old worst

vertex is discarded.

C) Expansion

The expansion step allows for a more aggressive move by doubling the

length of the step from the centroid to the reflection point. If the reflected

point satisfies 1rf f≥ , which suggests the simplex is moving toward a

desirable direction, then an expansion movement is operated as:

( )1e c c Nx x x xκ += + −

where . If2κ = ( )e e rf f x f= > , accept ef and terminate the iteration, otherwise,

accept rf and terminate the iteration.

D) Contraction

If r Nf f≤ , which means the reflected point is worse than the next to the

worst point, it makes sense to perform a contraction move allowing for more

conservative moves by halving the length of the step from the centroid to

either the reflection point (outside contraction )or the worst point (inside

contraction) [8].

(1) Outside contraction.

If 1N r Nf f f+ ≤ ≤ , an outside contraction is conducted,

( )o c r cx x x xγ= + −

14

where 12

γ = , if o rf f> , accept of and terminate the iteration; otherwise

perform a shrink move.

(2) Inside contraction

If 1r Nf f +≤ , perform an inside contration

( )i c c rx x x xγ= − −

If i rf f> , accept if and terminate the iteration; other wise perform a shrink

move.

E) Shrinkage

In the shrinkage stage, a new simplex with vertices { } 1~

1

N

j

x+

=

is formed by

reducing the lengths of the edges adjacent to the current best vertex by half.

( )~ 1

1 1

1

, 2,3,..., 1jj

x jx

x x x j Nσ

=⎧⎪= ⎨ + − = +⎪⎩

Where 12

σ = .

The illustration of above operations for a two dimensional process is visually

demonstrated in Fig.2.1.

15

Fig.2.1: Operations used in Nelder-Mead simplex algorithm

16

2.3.1.2 Dynamic Simplex Method

Although the Nelder-Mead simplex method has been extensively used

in industries and research, especially in chemical engineering and chemistry,

it still has some limitations. It works well for processes with static optima;

nevertheless, it cannot deal with processes with moving optima because it

locks itself when the measurement is contaminated by noise or the process

optimum moves with time. Also it is not sensible to apply the simplex method

to a process with too many control variables, since the number of function

evaluations soars when the dimension of the problem increases. Based on

Nelder-Mead simplex method, the Dynamic Simplex Method (DSM) [4] was

developed to handle these limitations.

The Dynamic Simplex algorithm is presented as follows,

The initial simplex is { } 1

0 1

N

j jS x

+

== .

(1) Sort the vertices of the simplex from best to worst:

1 2 ...... N 1f f f +> > >

(2) Successive reflection

Reflect the worst point 1Nf + through the centroid of the opposite side and

obtain a new vertex 2Nx + , thus we have a new simplex { } 2

1 2

N

j jS x

+

==

1

2 12

2 N

N jj

x x xN

+

+=

= −∑

17

And then reflect the first point in the new simplex to obtain a new vertex 1S

3Nx + and a new simplex { } 3

2 3

N

j jS x

+

== . Keep repeating the reflection for M times

and we have { } 1

1

N M

M j j MS x

+ +

= += and a series of new simplices { } 1

Mi i

S=

(3) Select the initial simplex for the next iteration

In this step the average function value of the simplices { } 1

Mi i

S=

are calculated

as:

1

1

1 , 1,2,...,1

N i

i jj i

f f iN

+ +

= +

= =+ ∑ M

Select the simplex whose function value is the maximum and set as

the initial simplex for the next iteration. The response at the best point

maxS maxS

1x is

then re-measured.

The visual description is shown in Figure 2.2.

3x

6x

1x2x

4x

5x

0S

1S

2S

3S

Fig.2.2 Successive reflection in 2-dimensional space

18

The initial simplex is , the worst point 0S 1x is reflected to point 4x and a

new simplex is formed, and then new simplices and are obtained by

reflecting

1S 2S 3S

2x , 4x to 5x , 6x .

Two parameters should be chosen carefully, they are the simplex size and

the successive reflection M. The simplex size should be larger than the

distance that the optimum moves during one iteration in order to track the

moving optimum. A reasonably large simplex size favors reducing the effect

of noises.

2.3.2 Sign-Sign Optimization Method

In direct search methods, gradient information is calculated directly

based on the process measurements which can sometimes corrupted by a high

noise level. The sign-sign method was developed to solve the high noise

sensitivity which can cause suboptimal performance and even instability [5].

Consider a general optimizing feedback structure:

Fig.2.3 Feedback optimizer

Input(s) x

Process

Optimizer

Output y Objective function

( , )J g y x =

19

The system has n input variables: { }1 2, ,..., nx x x x= , the rules used for

driving the objective function to the optimum is:

1n ndJx xdx

μ+ = +

Where the gradient dJdx

can be calculated approximately as:

( 1) ( )( 1) ( ) ( ( 1) ( ))

( 1) ( ) ( ( 1) ( )) ( 1) ( )J n J ndJ J n J n sign J n J n

dx x n x n sign x n x n x n x n+ −+ − + −

≈ = ×+ − + − + −

Where 1 ( 0)

( )1 ( 0)

xsign x

x+ >⎧

= ⎨− <⎩

( ( 1) ( ))( ( 1) ( ))

sign J n J nsign x n x n

+ −+ −

is the sign information of the gradient, and ( 1) ( )( 1) ( )

J n J nx n x n

+ −+ −

is the magnitude information of the gradient. Consider that while process

noise can affect the magnitude of the gradient, it is less likely to corrupt the

sign of the gradient [5]. The sign information is used as the gradient search

direction and the magnitude information is incorporated into a step size μ as:

( 1) ( )( 1) ( )

J n J nx n x n

μ α+ −

= ×+ −

Where α is a scaling factor. One way is to keep step size μ fixed is with the

Basic Sign-Sign Method, another way is to use a changing step size, which is

the Variable Step Size Sign-Sign Method.

1. Basic Sign-Sign Method

In the Basic Sign-Sign Method, the gradient search direction is updated as

follows:

20

(1) μ>0, the process input is increased if

( ( 1) ( )) / ( ( 1) ( )) 0sign J n J n sign x n x n+ − + − >

(2) μ<0, the process input is decreased if

( ( 1) ( )) / ( ( 1) ( )) 0sign J n J n sign x n x n+ − + − <

2. Variable Step Size Sign-Sign Method

The Basic Sign-Sign method is not suitable in high noise level situations

where the measurements become corrupted. Hence, a Variable Step Size

Sign-Sign method was developed in order to restrain the effect of high noise

levels.

Based on Basic Sign-Sign method, the following improvements are made

in Variable Step Size Sign-Sign method:

(a) Take more than one measurements at each iteration and use the

average value of measurements.

The algorithm is written as:

(1) μ>0, the process input is increased if

( ( 1) ( ))/ ( ( 1) ( )) 0m msign J n J n sign x n x n+ − + − >

(2) μ<0, the process input is decreased if

( ( 1) ( )) / ( ( 1) ( )) 0m msign J n J n sign x n x n+ − + − <

21

where m is the number of measures taken at each iteration, the

measurements should be taken with a short period of time. 1

1( ) ( )m

m ii

J n J nm =

= ∑

is the average value of the m measures taken at the nth iteration.

(b) Adopt different step size value in different noise level. The step size

changes as follows:

1.2 var( ( )) :Small

var( ( )) : Medium/1.5 var( ( )) : Large

m

v m

m

J nJ nJ n

μμ μ

μ

⎧⎪= ⎨⎪⎩

Where vμ is the variable step size and the numerical factors are selected by

trail and error. The variance of measures var( ( ))mJ n is used for estimating the

nose level. When noise level is small, an enlarged step size accelerates the

move. When noise level is medium, a normal step size allows neutral moves.

When noise level is large, a reduced step size keeps the move conservative.

‘Large’, ‘Medium’, ‘Small’ are process dependent terms chosen by a process

engineer and are subjective.

3. Extension to multi-input systems

The Sign-Sign method will suffer from the suboptimal solution problem if

the “one-variable-at-a-time” strategy is used [13], in which only one input

variable is adjusted at one time.

Here a hypothetical example is used to illustrate the problem occurs with

the “one-variable-at-a-time” strategy. Consider a process with two inputs:

22

(2.3) 21 2 1 283.57 9.39 7.12 7.44 3.71 5.88y v v v v= + + − − − 1 2v v

Firstly, is fixed at -8, and is adjusted between -10 and 10, which is

shown in Fig. 2.4. From the plot, it is clear that the best result of is 4. Then

is fixed at 4, and is adjusted as shown in Fig.2.4. The best value of is

-2. It is reasonable to draw the conclusion that the best point is .

However, the true best point shown in the contour plot Fig.

2.5.is . The suboptimal solution problem is caused by the

interacting term .

2v 1v

1v

1v 2v 2v

1 24, 2v v= = −

1 20.4, 0.6v v= =

1 25.88v v

In order to remove the problem, sign-sign method dynamically adjusts the

input variables, and which input variable should be adjusted is selected

randomly at each sampling point [5].

-10 -8 -6 -4 -2 0 2 4 6 8 10-2000

-1500

-1000

-500

0v2=-8

v1

y

-10 -8 -6 -4 -2 0 2 4 6 8 10-600

-400

-200

0

200v1=4

v2

y

Fig.2.4: The results from One-variable-at-a-time strategy

23

X= 4Y= -2Level= 20.05

v1

v 2

X= 0.4Y= 0.6Level= 87.6608

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Optimization locked up here

Fig.2.5: Contour plot of the Eq. (2.3)

There are three parameters which should be chosen carefully in the Sign-

Sign method, they are: sampling time , step size T μ and the window

parameter , which is the number of measurements collected for calculating

a moving direction.

m

The Sign-Sign method is used to find the optimum of equation (2.3). The

result is shown in Fig. 2.6.

24

x1

x 2

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Optimization starts here

Fig.2.6: Optimization results of Eq. (2.3)

The results in Fig.2.6 indicate that the Sign-Sign method is able to find

the optimal solution regardless of the presence of the interacting term.

The Sign-Sign method is also applied to the following tree dimensional

function with moving optima:

2 21 2 3( 10 5cos(0.02 )) ( 10 5sin(0.02)) ( 0.02 ) 5 ( )y x t x x t t2 ξ= − − + − − + − + + (2.4)

where ( )tξ is the disturbance term

25

05

1015

20

05

1015

200

5

10

15

Variable x1Variable x2

Var

iabl

e x 3

Tracked inputTrue optimum

Start point

Fig.2.7: 3-D optimization using Sign-Sign method

The Sign-Sign method is applied to some simulation and experimental

applications as well.

2.3.3 Perturbation Method

This method is one of the oldest and most popular methods to estimate

the gradient for extremum control. The gradient of the process is indirectly

measured by analyzing the system response to a sine wave input. This

gradient is then passed to a feedback control structure which drives the

gradient to zero. As shown in Figure 2.8, the signal is slightly distorted by a

since wave and the correlation between the output and the input indicates at

which side of the maximum the operating point is located [14].

26

Fig.2.8: The principle of the perturbation method

The simplest nonlinear approximation of a nonlinear function

( )f u expanded around the optimal ( '( ) 0f u = ) as follows:

''*( ) ( )

2f * 2f u f u u= + − (2.5)

Any quadratic function ( )f u can be approximated locally by Eq. (2.5) using a

standard Taylor Series. The algorithm is to minimize *u u− so that the output

( )f u is driven to its optimum *f . For a maximizing problem, , for a

minimizing problem .

'' 0f <

'' 0f >

The basic extremum seeking loop is shown in Fig. 2.9.

27

( )f u

+ks

− × ss h+

sina tω sin tω

u

*u *f

ξ

y

Plant

u

Fig. 2.9: The perturbation extremum seeking scheme

The perturbation signal sina tω is injected into the plant in order to

measure the gradient information of ( )f u . denotes the estimate of the

unknown optimal input and

u

*u * ˆu u u= − denotes the estimation error. Thus,

(2.6) * sinu u a t uω− = −

Substituting the above equation into Eqn. (2.5),

* 2'' ( sin2fy f u a t)ω= + − (2.7)

By expanding this expression further, gives

2

* 2 2'' '''' sin sin2 2f a fy f u af u t tω ω= + − + (2.8)

By applying the basic trigonometric identity 22sin 1 cos 2t tω ω= − , the

expression can be organized as:

28

2 2

* 2'' '' '''' sin cos 24 2 4

a f f a fy f u af u t tω ω= + + − − (2.9)

The high pass filter ss h+

serves to remove *f and 2 ''4

a f , hence.

2

2'' ''[ ] '' sin cos 22 4

s f a fy u af u ts h

tω ω≈ − −+

(2.10)

Then this signal is multiplied bysin tω , giving

2

2 2'' ''sin '' sin cos 2 sin2 4f a fu t af u t t tξ ω ω ω≈ − − ω (2.11)

Again applying 22sin 1 cos 2t tω ω= − and 2cos 2 sin sin 3 sint t t tω ω ω ω= − , we

obtain

2

2'' '' '' ''cos 2 (sin sin 3 ) sin2 2 8 2

af af a f fu u t t t u tξ ω ω ω≈ − + + − + ω

u

(2.12)

Recall , since is constant, then we have , namely,

* ˆu u u= − *u ˆu = −

ˆ( ) ( )su s su s= −

We get

2

2

ˆ( ) ( )

'' '' '' ''[ cos 2 (sin sin 3 ) sin ]2 2 8 2

k ku s u ss s

k af af a f fu u t t t us

ξ ξ

tω ω ω

⎛ ⎞= − = − − =⎜ ⎟⎝ ⎠

≈ − + + − + ω (2.13)

Because we are interested in local analysis, the last term which is

quadratic in can be neglected. u

( )2~ '' '' ''[ cos 2 (sin si

2 2 8k af af a fu s u u t t ts

n 3 )]ω ω ω≈ − + + − (2.14)

The last two terms are high frequency signals which can be attenuated by

passing through an integrator, therefore, they can be neglected, which yields

29

''2

k afus

u⎡ ⎤≈ −⎢ ⎥⎣ ⎦ (2.15)

or

''2

kafu ≈ − u (2.16)

Since , we conclude that the system is stable and , i.e.,

converges to within a small distance of [15].

'' 0kf > 0u → ( )^u t

*u

A number of modifications of the perturbation principles have been made

to handle dynamics, several inputs, etc. The perturbation method has been

shown as an effective gradient search scheme. However, it is limited to

specific sinusoidal signals. Also when it comes to compensation of dynamics

the system can easily become unstable when the control parameters are

chosen poorly [14].

2.3.4 A Fuzzy Logic Based Approach

Fuzzy logic theory has been extended and extensively used in industry

since the basic theory of fuzzy sets was introduced by Zadeh (1965). A fuzzy

set is a set without a crisp, clearly defined boundary. It can contain elements

with only a partial degree of membership [22]. Fuzzy logic based approach is

one of the popular model-free approaches to system identification and control

due to the following advantages [6]:

(1) Explicit mathematical models are unnecessary

30

(2) Imprecise and uncertain inputs can be tolerated

(3) Nonlinearity can be handled effectively

(4) They can be blended with conventional control techniques.

(5) The approach is robust against variations in process parameters.

The fuzzy logic based optimizer receives crisp inputs from online

measurements and determines the degree to which these inputs belong to

each of the appropriate fuzzy sets via membership functions. Once the inputs

have been fuzzified, a set of fuzzy rules are applied to determine the fuzzy

output variable which is then defuzzified to crisp control actions.

Recall the general nonlinear time-varying system and its objective

function:

( , , ) ,ty f x tθ φ= + (2.17)

( , )J g y x= (2.18)

The fuzzy logic based optimization algorithm drives the process toward

optimal operating conditions by performing the following rules: if the last

change in the manipulated variable x has caused the objective function to

improve, keep moving the

J

x in the same direction; otherwise, if the last

change in the manipulated variable xhas cause the objective function to

drop, then move the

J

x in the opposite direction.

There are two input variables for the fuzzy logic based optimizer, one is

the last change in the manipulated variable which is L xΔ , the other one is the

change in the objective function JΔ . The output variable is the incremental

31

change in the manipulated variable xΔ . Then the fuzzy logic based optimizer

maps the input and output space of the process into fuzzy regions and defines

the membership functions for the input and output fuzzy variables.

Input variable L is subdivided into three fuzzy sets: Positive (P), Negative

(N), and Zero (Z).

xΔ

Input variable is subdivided into five fuzzy sets: Positive Small (PS),

Positive Big (PB), Negative Small (NS), Negative Big (NB), Zero (ZZ).

JΔ

The output variable xΔ is divided into seven fuzzy sets: Positive Small (PS),

Positive Medium (PM), Positive Big (PB), Negative Small (NS), Negative

Medium (NM), Negative Big (NB), Zero (ZZ).

The search algorithm for the iteration at k is described as follows:

Rule 1: when the system is operated under normal conditions

If kxΔ is P and kJΔ is PB then 1kx +Δ is PB

If kxΔ is P and kJΔ is PS then 1kx +Δ is PM

If kxΔ is N and kJΔ is PB then 1kx +Δ is NB

If kxΔ is N and kJΔ is PS then 1kx +Δ is NM

If kxΔ is P and kJΔ is NB then 1kx +Δ is PB

If kxΔ is P and kJΔ is NS then 1kx +Δ is NM

If kxΔ is N and kJΔ is NB then 1kx +Δ is PB

If kxΔ is N and kJΔ is NS then 1kx +Δ is PM

32

Rule 2: For processes operated under changing environmental conditions,

the objective function may change due to any sudden change in the

environment even if the manipulated variable x is kept the same. In order to

handle this situation, the following rules are required:

If kxΔ is Z and kJΔ is PB then 1kx +Δ is PM

If kxΔ is Z and kJΔ is PS then 1kx +Δ is PS

If kxΔ is Z and kJΔ is NB then 1kx +Δ is NM

If kxΔ is Z and kJΔ is NS then 1kx +Δ is NS

Rule 3: In situations where the objective function becomes flat, the search

may stop even if the manipulated variable changes. The flowing rules are

developed for avoiding a suboptimal solution.

If kxΔ is P and kJΔ is ZZ then 1kx +Δ is PS

If kxΔ is N and kJΔ is ZZ then 1kx +Δ is NS

Rule 4: Finally, it is necessary to provide the system with a rule that keeps

the operating condition optimal

If kxΔ is Z and kJΔ is ZZ then 1kx +Δ is ZZ

The above rules which describe different conditions shall be provided with

different rule weights, since the rules only become active when their

conditions apply. Rule 1 should be the strongest condition because it

describes the normal system operation. Rule 2 and Rule 3 have lower

strength, because they deal with special conditions. Rule 4 is only activated

33

when the system become close to the optimum point, thus it needs to be

weighted lightly.

First, the crisp inputs are fuzzified using appropriate membership

function. A membership function is a curve that defines how each point in the

input space is mapped to a membership value (or degree of membership)

between 0 and 1 [22]. After above fuzzy rules are evaluated, the fuzzy output

variable is transformed to a crisp number in defuzzification step. The

defuzzification is a mapping from a space of fuzzy control actions defined over

an output universe of discourse into a space of nonfuzzy control actions.

Many defuzzification strategies are used for performing defuzzification, for

example, Center of Gravity, Center of Sums, First of Maxima, Last of

Maxima, Middle of Maxima [23].

34

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Applications, vol.146, no.5, pp.552-558, 1999.

[14] R. Hooke & T.A. Jeeves (1961). Direct Search, Solution of Numerical

and Statistical Problems. Journal of the ACM, Vol. 8, Issue 2, 212-229.

[15] Y. Tan, D. Nesic & I. Mareels (2006). On non-local stability properties

of extremum seeking control, Automatica 42, 889-903.

[16] M. Krstic & H.H. Wang (2000). Stability of extremum seeking feedback

for general nonlinear dynamic systems, Automatica, 36, 595–601.

[17] W.L. Brogan (1974). Modern Control Theory, Quantum Publishers, Inc.

[18] B.D. Prater, S.C. Tuller & L.J. Wilson (1999). Simplex optimization of

protein crystallization conditions, Journal of Crystal Growth 196, 674-

684.

36

[19] F.H. Walters, L.R. Parker, S.L. Morgan Jr. & S.N. Deming (1991).

Sequential Simplex Optimization. CRC Press, Boca Raton, Florida.

[20] P. D. Roberts (1979). An algorithm for steady-state optimization and

parameter estimation. International Journal of System Science, 10,

719–734.

[21] K.S. Peterson & A.G. Stefanopoulou (2004). Extremum seeking control

for soft landing of an electromechanical valve actuator, Automatica 40,

1063-1069.

[22] Fuzzy Logic Toolbox User’s Guide, Mathworks Inc., (2000).

[23] D. H. Rao, S. S. Saraf (1995). Study of Defuzzification Methods of Fuzzy

Logic Controller for Speed Control of a DC Motor, IEEE Transactions,

pp.782-787

[24] S. J. Citron. (1969). Elements of Optimal Control. Holt, Rinehart and

Winston, Inc.

[25] Frederick H. Walters, Lioyd R. Parker, Jr, Stephen L. Morgan and

Stanley N. Deming. (1991). Sequential Simplex Optimization: A

Technique for Improving Quality and Productivity in Research,

Development, and Manufacturing. CRC Press, Inc. Boca Raton, Florida.

[26] G. E. P. Box, G. M. Jenkins and G. C. Reinsel (1994). Time Series

Analysis: Forecasting and Control. Third Edition. Prentice Hall,

Englewood Cliffs, New Jersey 07632.

37

[27] C. T. Kelley. (1999). Iterative Methods for Optimization. North Carolina

State University, Raleigh, North Carolina.

[28] M. Sheikhzadeh (2006). Optimization and Control of Crystallization

Processes. Ph.D Thesis, University of Western Ontario.

38

Chapter 3

A New Recursive Least

Squares Impulse Weight

Gradient Optimization

Method

39

3.1 Introduction

Chemical processes often involve strong nonlinear behavior and time-

varying dynamics. The model based optimization algorithms need process

models with high degree of accuracy to be applied in practice [1]. However,

due to many reasons (expenses, time), it is sometimes hard to obtain a

reliable model. In order to be able to track the optimum value in the absence

of a process model, all kinds of direct optimization methods arise. Some of

them were introduced in Chapter 2. Based on these methods, the search

direction and gradient information is calculated directly using the output

measurements. A new impulse weight gradient approach is presented here

for real-time optimization of nonlinear dynamic chemical processes. This

method is tested in both simulation and real time studies in chapter 4 and

chapter 5.

3.2 Impulse Weight Gradient Method

Consider a system with input ( )x t and output ( )y t as depicted in

Fig.3.1, the input and output data are available at each interval.

40

Fig.3.1: Basic input-output configuration.

Assuming the signals are related by a linear system, the relationship

between input and output can be described by a transfer function:

( ) ( ) ( ) ( )y t G q x t n t= + (3.1)

Where is the shift operator and q ( )G q is transfer function, is the noise

model.

( )n t

( ) ( ) ( ) ( )1k

G q x t v k x t k∞

=

= −∑ (3.2)

where ( ){ }v k are called the impulse response of the system.

In order to directly estimate the impulse response coefficients, we define

an impulse response model:

( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 1 2 2 ...y t v x t v x t v x t n t= + − + − + + ( ) (3.3)

There are different approaches to estimation the impulse response

coefficients. One is correlation method [2] which uses pre-whitened input

sequence to estimate the impulse response.

41

Correlation Method:

In correlation method, we multiply equation (3.3) by ( )x t k− ( )0k ≥ ,

gives:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0 1 1 ... ( ) )x t k y t v x t k x t v x t k x t n t x t k− = − + − − + + − (3.4)

If we take expectation on both side of equation (3.4) and assume that the

input sequence is uncorrelated with the process noise, that is,

we have ( )( ) 0E n t x t k⎡ − =⎣ ⎤⎦

( ) ( ) ( ) ( ) ( )0 1 1 ... 0,1,2,...xy xx xxk v k v k kγ γ γ= + − + =

where ( ) are cross covariance between input and output,

( ) are auto-covariance of input sequence.

( )xy kγ 0,1, 2,...k =

( )xx kγ 0,1, 2,...k =

Assuming that the impulse weights are effectively zero after sufficiently

large lag(k)=M.

( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 1 ... 0,1,2,...xy xx xx xxk v k v k v M k M kγ γ γ γ= + − + − = (3.5)

Writing above equation in a matrix form, gives

( )( )

( )

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

( ) ( ) ( )

( )( )

( )

0 0 11 1 0 1

1 0

xy xx xx xx

xy xx xx xx

xx xx xxxy

M vM v

01

M M vM

γ γ γ γγ γ γ γ

γ γ γγ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

−⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦M

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.6)

After substituting estimated auto-covariance and cross covariance, the

generalized least square method can be used to solve above equation.

42

In order to simplify the identification processes, correlation method pre-

whitens the input sequence first to make the input signal as uncorrelated as

possible, then the equation (3.6) simplifies to:

( )( )

( )

( )( )

( )

( )( )

( )

0 0 0 0 01 0 0 0 1

0 0 0

xy xx

xy xx

xxxy

vv

v MM

γ γγ γ

γγ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

(3.7)

The impulse weights can then be calculated easily from above equations:

( ) ( )( )

( )20

xy xy

xx x

k kv k

γ γγ σ

= = 0,1, 2,k = …

In terms of the cross correlation:

( ) ( )( )

( )2 0,1,2,

0xy xy y

xx x

k kv k k

γ β σγ σ

= = = … (3.8)

Thus, after pre-whitening the input sequence, the impulse weight is

proportional to the cross correlation between the pre-whitened input and

corresponding transformed output.

The correlation method takes advantage of the fact that pre-whitened

input simplifies the identification process, therefore the calculation is easy.

The disadvantages of correlation method are lack of accuracy and sensitivity

to noise. Also we need to carefully choose the window size which is how many

measurements should be collected before a new move.

We presented here an alternative method, the so called Recursive Least

Squares (RLS) method to estimate the impulse weights.

43

3.3 Theory of Recursive Least Squares

Algorithm

3.3.1 Least Squares

In least squares estimation unknown parameters of a linear model are

chosen in such a way that the sum of the squares of the difference between

the actually observed and the computed values, is a minimum [8].

The mathematical model of the least squares can be written in the form:

( ) ( ) ( ) ( ) ( )1 1 2 2T

n ny i v x i v x i v x i x i v= + + + =… (3.9)

Where y is the observed output, are the impulse weights

parameters to be determined, and

1 2, , , nv v v…

1 2, , , nx x … x are input sequence, and the two

vectors are

( ) ( ) ( ) ( )1 2[ ]Tnx i x i x i x i=

1 2[ ]Tnv v v v=

pairs of ( ) ( ){ }, , 1, 2, ,y i x i i t= … are obtained from an experiment. The least

squares method is used to estimate the parameters in such a way that the

outputs computed from the model in equation (3.9) are as close as possible

with the measured outputs ( )y i in the sense of least squares. That is, the

impulse weight parameters are chosen to minimize the least squares

criterion

44

( ) ( ) ( )( ) ( )2

2

1 1

1 1,2 2

t tT

i iL v t y i x i v i E Eε

= =

= − = =∑ ∑ 12

T

v

(3.10)

Where ( ) ( ) ( ) ( ) ( )Ti y i y i y i x iε∧

= − = −

( ) ( ) ( ) ( )1 2E t tε ε ε⎡ ⎤= ⎣ ⎦

By letting ( ),L v tv∂

=∂

0

⎞⎟⎠

, we have [4]

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )1

1

1 1

t tT T T

i iv t X t X t X t Y t x i x i x i y i

−∧ −

= =

⎛ ⎞ ⎛= = ⎜ ⎟ ⎜

⎝ ⎠ ⎝∑ ∑ (3.11)

Where ( )

( )( )

( )

12

T

T

T

xx

X t

x t

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, ( ) ( ) ( ) ( )1 2T

Y t y y y t⎡ ⎤= ⎣ ⎦ and ( ) ( )TX t X t is

nonsingular.

3.3.2 Recursive Least Squares

In real-time processes, the observations are obtained sequentially.

Therefore, it is desirable to estimate recursively in time, the impulse weight

parameters. In the recursive algorithm, the results obtained at time 1t − is

used to find estimates at time t .

We introduce the definition of the covariance matrix P

( ) ( ) ( ) ( )( ) ( ) ( )1

11

1

tT

iP t R t X t X t x i x iT

−−−

=

⎛= = = ⎜

⎝ ⎠∑ ⎞

⎟ (3.12)

45

Assume that the matrix ( )X t is nonsingular for all t , then

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

( ) ( ) ( )

11

1 11 1

t tT T

i iT

P t x i x i x i x i x t x t

P t x t x t

−−

= =

−

= = +

= − +

∑ ∑ T

⎞⎟⎠

1

(3.13)

Equation (3.11) may be written as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

1 1

t t

i i

v t P t x i y i P t x i y i x t y t−∧

= =

⎛ ⎞ ⎛= = +⎜ ⎟ ⎜

⎝ ⎠ ⎝∑ ∑ (3.14)

From equation (3.11), we have

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

1 1

11 1 [ ]

tT

ix i y i P t v t P t x t x t v t

− ∧ ∧− −

=

= − − = − −∑ (3.15)

Substituting in equation (3.14)

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

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

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

( ) ( ) ( )

1[ ] 1

1 1

1 1

1

T

T

T

v t P t P t x t x t v t x t y t

v t P t x t x t v t P t x t y t

v t P t x t y t x t v t

v t K t tε

∧ ∧−

∧ ∧

∧ ∧

∧

⎡ ⎤= − − +⎢ ⎥⎣ ⎦

= − − − +

⎡ ⎤= − + − −⎢ ⎥⎣ ⎦

= − +

(3.16)

where and ( ) ( ) ( )K t P t x t= ( ) ( ) ( ) ( )1Tt y t x t v tε∧

= − −

From equation (3.13) we have

( ) ( ) ( ) ( ) ( )( 11 1 1 TP t R t P t x t x t )−− −= = − + (3.17)

To avoid inverting ( )R t at each step, apply the Matrix Inversion Lemma

( ) ( ) 11 1 1 1 1 1A BCD A A B C DA B DA−− − − − −+ = − + − (3.18)

to the equation (3.17), with ( ) ( )1 1 , ,A P t B x t C I−= − = = and ( )TD x t= ,

46

(3.19) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

1 1 1T TP t P t P t x t I x t P t x t x t P t−

⎡ ⎤= − − − + − −⎣ ⎦ 1

After substituting above expression in the equation for ( )K t and simplifying,

we have

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 T 1K t P t x t P t x t I x t P t x t

−⎡ ⎤= = − + −⎣ ⎦ (3.20)

We can now summarize RLS algorithm as:

( ) ( ) ( ) ( )1v t v t K t tε∧ ∧

= − + (3.21)

( ) ( ) ( ) ( )1Tt y t x t v tε∧

= − − (3.22)

( ) ( ) ( ) ( ) ( ) ( ) 11 T 1K t P t x t I x t P t x t

−⎡= − + −⎣ ⎤⎦

1

(3.23)

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

11 1 1

1

T T

T

P t P t P t x t I x t P t x t x t P t

I K t x t P t

−⎡ ⎤= − − − + − −⎣ ⎦

⎡ ⎤= − −⎣ ⎦ (3.24)

with initial values ( ) ( )00 , 0v v P Iα= = , where α should be large.

3.3.3 Recursive Least Squares with Exponential

Forgetting

In the least squares model described by equation (3.9) the parameters

are assumed to be constant. In case of systems with varying parameters it is

necessary to extend the least squares method to account for time changing

parameters. There are two cases to consider in terms of varying parameters.

In one case, parameters are assumed to change abruptly but infrequently; in

47

the other case the parameters are assumed to change continuously but slowly.

In the case of systems with abrupt parameter change we use a technique

called covariance resetting. The covariance matrix P is then periodically reset

toα I , where α is a large number. In the case of systems with slowly varying

parameters it is useful to give more emphasis to recently observed data than

to older data. One approach to do this is to replace the least squares criterion

with

( ) ( ) ( )(2

1

1,2

tt i T

iL v t y i x i vλ −

=

= −∑ ) (3.25)

Where λ (0 1λ< ≤ ) is called forgetting factor. The least squares criterion of

Eq. (3.25) implies that a time-varying weighting of the data is introduced.

The recursive least squares estimate then becomes:

( ) ( ) ( ) ( )1v t v t K t tε∧ ∧

= − + (3.26)

( ) ( ) ( ) ( )1Tt y t x t v tε∧

= − − (3.27)

( ) ( ) ( ) ( ) ( ) ( )1

1 T 1K t P t x t I x t P t x tλ−

⎡= − + −⎣ ⎤⎦ (3.28)

( ) ( ) ( ) ( )1 /TP t I K t x t P t λ⎡ ⎤= − −⎣ ⎦ (3.29)

The memory of the estimator is given by 02

1hTλ

=−

, where h is the sampling

interval.

In the classical RLS, the covariance shrinks with time, losing its capability

to keep track of changes in the parameters. However, it can be seen that in

Eq. (3.29), the covariance matrix is divided by 1λ ≤ at each update. This

48

prevents the covariance matrix from becoming too small, and therefore

maintains the sensitivity of the algorithm and allows new data to contribute

to the parameter estimates.

Based on a well established experience, the RLS method is useful for real

time applications because each new data pair can be used to update the value

of impulse weights. The system must also be sufficiently excited in order to

keep the RLS algorithm active. During poor excitations old information is

gradually forgotten while there is only limited new information coming in.

This might cause the exponential growth of the covariance matrix and as a

result the estimator becomes extremely sensitive and susceptible to

numerical and computational errors [11].

Interpretation of impulse weights jv

Recall Eq. (3.3) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 1 2 2 ... ( )y t v x t v x t v x t n t= + − + − + +

For a static system relating y to x, ( )0v can be interpreted as the

gradient of y respect to x. For a dynamic system, ( )0v thus represents the

current gradient. In case of systems with dead time (DT), the first non-zero

impulse coefficient ( ) ( )y v j

x j∂

=∂

represents the current gradient. This

gradient value is determined by the current input ( ) ( )x j . 0v is the gain of

the effect of the current input ( )x ( )y t t on the output

For example,

49

When DT=1, ( )0v = 0 , the current gradient is ( )1v

When DT=2,v v , the current gradient is ( ) ( )0 1 0= = ( )2v .

3.4 Gradient Optimization

The basic principle behind gradient based optimization is the fact that

we push a process in the negative gradient direction until the extremum is

reached and has a gradient with magnitude of zero [7]. The new algorithm

proposed here is to use the RLS method to sequentially estimate the gradient

direction based on input/output measurements. This gradient direction is

then used in a feedback loop as a pseudo gradient measurement . The

controller (PID) then drives v to zero and when v is zero the process

optimum is achieved. These concepts are combined into a feedback

optimization loop as shown in Figure 3.2.

v

Process

Test Signal

PID Controller

+

-

y

v (Pseudo gradient measurement)

Gradient Estimator (RLS)

x

Set point=0

Figure 3.2 Feedback optimization loop

50

3.5 Optimization Examples

In order to evaluate the effectiveness of the impulse weight gradient

optimization method, it is tested on both simulated models and real-time

processes. Simulation results using the RLS method to estimate impulse

weights are also compared to that of the simpler correlation method. The

simulation results are also compared with other successful direct search

methods in chapter 4.

51

Bibliography

[1] Xiong, Qiang (2000). Optimization, Control and Identification of

Chemical Processes. M.E.Sc Thesis, University of Western Ontario.

[2] M. Sheikhzadeh (2006). Optimization and Control of Crystallization

Processes. Ph.D Thesis, University of Western Ontario.

[3] System Identification Toolbox User’s Guide, Mathworks Inc., (2000).

[4] Ljung, Lennart (1989). System Identification: Theory for the User,

Prentice Hall.

[5] S. Haykin (2001), Adaptive Filter Theory, 4th edition, Prentice Hall, NJ,

ISBN-0-13-090126-1.

[6] Box G. P., Jenkins G. M., Reisel G. C. (1994). Time Series Analysis,

Prentice Hall, Englewood Cliffs, New Jersey.

[7] Larsson S. (2001). Literature Study on Extremum Control, Control and

Automation Laboratory, Chalmers University of Technology.

[8] Karl J. Astrom, Bjorn Witternmark (1995). Adaptive Control, second

edition, Addison-Wesley Publication Company, Inc.

[9] A. Vahidi, A. Stefanopoulou, H. Peng, Recursive Least Squares with

Forgetting for Online Estimation of Vehicle Mass and Road Grade:

Theory and Experiments. Mechanical Engineering Dept., University of

Michigan.

52

[10] R. Krneta, S. Antic and D. Stojanovic (December 2005) Recursive Least

Square Method in Parameters Identification of DC Motors Models, SER.:

ELEC. ENERG. vol. 18, no. 3, 467-478.

[11] T. R. Fortescue, L. S. Kershenbaum and B. E. Ydstie (1981).

Implementation of self-tuning regulators with variable forgetting

factors. Automatic, 17, 6:831-835.

53

Chapter 4

Simulation Examples

54

4.1 Estimate Gradient using RLS

We use simulated data to test the effectiveness of the RLS algorithm for

estimation of the impulse weights, and compare the results to correlation

method. The simulated data was generated using the model given in Eq. (4.1).

( ) ( ) ( ) ( ) ( ) ( )1 0.5 2 3 0.3 4 0.4 5y t x t x t x t x t x t= − + − − − + − + − (4.1)

We can see from the equation the first non-zero impulse response is 1. In

simulation we used 1000 data points to estimate the first nonzero weight. In

the correlation method, we chose the windows size as 50, in the RLS a

forgetting factor 0.99 is chosen. Figure 4.1 shows estimation results using

RLS and correlation method respectively.

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

time,s

first

impu

lse

wei

ght

correlation methodRLS method

55

Figure 4.1 Estimation of first impulse weight using RLS with a forgetting

factor 0.99 and correlation method. Sampling time is 1s.

We can see from above figure that RLS method provides a smoother

estimate of the impulse weight and is also less sensitive to noise. It also

converges to a consistent estimate, which the correlation method does not.

4.2 Gradient Estimate Simulation

4.2.1 A Linear Static System Example

(4.2) = − + = =4 10, sin( )y x with x t and samplingtime 1

0 1]

The analytical derivative is -4. We use Matlab© command rarx which

estimates the impulse weights and choose the parameters such that

, where na and are orders of the respective

polynomials and nk is the number of delays from input to output. Specifically

[4],

[ ] [1 5nn na nb nk= = nb

ARX model structure:

( ) ( ) ( ) ( )A q y t B q u t nk= −

where ( )( )

11

1 11 2

1 nana

nbnb

A q a q a q

B q b b q b q

− −

− − +

= + + +

= + + +

…

…

The forgetting factor is chosen as 1 for a static model. Fig.4.2 shows the block

diagram of the linear static system using Simulink©. The input signal is a

56

sine wave. Sampling time is 1 sec. The s-function is a gradient estimator

using RLS algorithm. Figure 4.3 shows the simulation results.

Sine Wave1 Scope3

rarx1_linearstatic

S-Function

-1Z

Integer Delay1

-4 Gain

Add4

Constant3

10

Fig.4.2: Gradient estimation for a linear static system using Simulink

Toolbox

57

0 200 400 600 800 1000 1200 1400 1600 1800 2000-5

-4

-3

-2

-1

0

1

time,s

grad

ient

Fig.4.3: Gradient information for a linear static system

Figure 4.3 shows a good agreement between the analytical gradient of the

system and the first non zero impulse response weight, using a sinusoidal

signal with magnitude 1 and frequency 1rad/s as input to the static system.

4.2.2 A Second Order Non-Linear Static System

24y x x 10= − + − (4.3)

In this simulation, the input signal is a sine wave with unit amplitude and

1rad/s frequency, the sampling time is 0.01s. In order to examine the

robustness (to noise) of the proposed method, a Gaussian random number

58

with mean zero and variance 0.1 is added to the input signal. The forgetting

factor is chosen as 0.8 Figure 4.4 shows the block diagram of the nonlinear

static system using Simulink©. The simulation result in Figure 4.5 shows

that the gradient estimation is quite acceptable even under noisy conditions.

59

Scop

e

Zero

-Ord

erHo

ld1

Zero

-Ord

erHo

ld

Sine

Wav

e2

Sine

Wav

e

onlin

ear_

stati

S-Fu

nctio

n

Rand

omNu

mbe

r2

Rand

omNu

mbe

r1

u2

Mat

hFu

nctio

n

-1 Z

Inte

ger D

elay

1

-8 Gain

1

-4 Gain

1

Cons

tant

1

-10

Cons

tant

Fig.

4.4

Gra

dien

t est

imat

ion

usin

g Si

mul

ink

Tool

box

for

a se

cond

ord

er n

onlin

ear

stat

ic s

yste

m

60

150 155 160 165 170 175 180 185 190 195 200-20

-15

-10

-5

0

5

10

15

20

time,s

grad

ient

Estimated gradientAnalytical gradient

Fig.4.5: Gradient information for a second order nonlinear static system with

( )= +sinx t noise

61

4.3 Combining the Perturbation Method

with the Proposed Gradient Method

A nonlinear function ( )f θ can be approximated by expanding around the

optimum point:

( ) ( ) ( ) ( )''

2

2

ff f

θθ θ θ

∗∗ ∗ ∗= + −θ (4.4)

At the extremum points, the first derivative should be zero. We wish to

make θ θ ∗− as small as possible, so that the output ( )f θ is driven toward its

optimum f ∗ . For a minimization problem is made without loss of

generality.

'' 0f >

Figure 4.6 shows the basic feedback optimization loop using the new

impulse weight gradient search for a nonlinear algebraic function.

( )f θ

*θ *f

y θ

Gradient Estimator

(RLS)

0v PID Controller

-

θ

+

Plant

Test Signal

Set-point

Figure 4.6 Feedback optimization loop for a static system

62

The test signal must be uncorrelated with the system noise and be of

sufficient power to provide the enough excitation for the parameters

estimation [3]. The proposed RLS with forgetting factor algorithm is used to

estimate the current gradient of the system, and then the estimated gradient

is compared to the set point, which is zero at the optimum. A conventional

PID controller is used to eliminate the error between the process input

0v

θ and

optimal inputθ ∗ .

4.3.1 Static Example:

A nonlinear algebraic system given by:

* * * 2

*

*

( ) ( ) 0.1( )0.01 100 0.02 1000.01

Test Signal: , : 0.1 , : PID parameters: 1, 0.1, 0 Sampling Interval: 0.05 sec

f ff at time and at time

Sine Wave Amplitude FrequencyP I D

θ θ θ θ

θπ

⎧ = + −⎪

= < >⎪⎪⎪ =⎨⎪⎪ = = =⎪⎪⎩

(4.5)

In this example, the maximum value is changed from 0.01 to 0.02 at

. Figure 4.7 is the optimization feedback loop for this example. Figure

4.8 shows the block diagram of the static algebraic system using Simulink©

and Figure 4.9 shows the simulation results.

100t = s

63

( )( )

2

2

0.01 0.1 0.01 100( )

0.02 0.1 0.01 100

t sf

t s

θθ

θ

⎧ + − <⎪= ⎨+ − >⎪⎩

Gradient Estimator

(RLS) PID

Controller

0v

θ Test Signal

-

θ

+

Set-point=0

Fig.4.7: Optimization feedback loop for Eq. (4.5)

64

y

Zero

-Ord

erHo

ld

vo v

s erro

r

vo

u1

u

0.01

thet

a*

thet

a

erro

r

Zero

-Ord

erHo

ld2

Step

Sign

alG

ener

ator

1

rarx

1

S-Fu

nctio

n

PID

out

PID

PID

Cont

rolle

r

Add1

0.1 Gai

n

u2

Mat

hFu

nctio

nAd

d

1

Gai

n1

Mem

ory

0

Cons

tant

Fig.

4.8:

Blo

ck d

iagr

ams

in S

imul

ink

Tool

box

for

a st

atic

sys

tem

65

0 20 40 60 80 100 120 140 160 180 2000.01

0.012

0.014

0.016

0.018

0.02

0.022

time,s

f and

f*

f*

f

Fig.4.9: Output for Example 4.5

In this simulation, the parameters are and

forgetting factor is 1. The output is close to the true optimum and above the

true optimum because the process is quadratic.

[ ] [0 5na nb nk = 0 1]

Figure 4.10 shows the gradient information and comparison between

input and optimal input. The gradient is successfully driven to zero as it

would be for extremum seeking, and the input θ oscillated around its optimal

valueθ ∗ . The gradient information is also compared with the Correlation

method in Figure 4.11

66

0 20 40 60 80 100 120 140 160 180 200-0.1

-0.05

0

0.05

0.1

time,s

grad

ient

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.2

0

0.2

0.4

time,s

θ an

d θ*

θ*

θ

Fig.4.10: Gradient and input information for Example 4.5

0 20 40 60 80 100 120 140 160 180 200-0.1

-0.05

0

0.05

0.1

0.15

time,s

grad

ient

RLS algorithmCorrelation method

Fig.4.11: Gradient comparison between RLS and Correlation method

67

4.3.1.1 Effect of random different test signals

The input test signals are not limited to sinusoids, we can choose

alternate signal. This example uses random number with variance 0.05 and

zero mean as the input signal. Fig.4.12 shows the input and output.

* * * 2

*

*

( ) ( ) 0.1( - )0.01 at time<100 and 0.02 at time>1000.01

Input Signal: Random number, variance: 0.005 , mean:0 PID parameters: 1, 0.1, 0 Sampling Interval: 0.05 sec

f ff

P I D

θ θ θ θ

θ

⎧ = +⎪

=⎪⎪⎪ =⎨⎪⎪ = = =⎪⎪⎩ (4.6)

0 20 40 60 80 100 120 140 160 180 2000.01

0.015

0.02

0.025

0.03

time,s

f and

f*

0 20 40 60 80 100 120 140 160 180 200-0.4

-0.2

0

0.2

0.4

time,s

θ an

d θ*

θ

θ*

f*

f

Figure 4.12 Actual and optimal process input and output values for example

4.6 with random number as input signal

68

4.3.2 Nonlinear Dynamic Example:

In the modeling a nonlinear dynamic systems, one approach is to

separate the linear and nonlinear parts into different blocks [3].

Figure 4.13 shows the nonlinear plant with static nonlinearity ( )f θ and

input linear block ( )iF s and output linear block ( )0F s .

Fig.4.13: Feedback optimization loop for a nonlinear dynamic system

69

Again we consider the static nonlinearity ( )f θ in the form:

( ) ( ) ( ) ( )''

2

2

ff f

θθ θ θ

∗∗ ∗ ∗= + −θ

Where is constant but unknown. The output is described by the

following equation:

'' 0f >

( ) ( )0y F s f θ=

We try to make θ θ ∗− as small as possible so that the output is driven

towards its optimum ( ) ( )0F s f θ∗ ∗ . The optimal input and output, θ ∗ and f ∗ ,

are allowed to be time varying here. Let us denote their Laplace transforms

by

{ }{ }

*

*

( ) ( )

( ) ( )f f

L t s

L f t s

θ θθ δ

δ

= Γ

= Γ (4.7)

where and fθδ δ are the gain of the optimal input and output functions. In

some cases, the shape of the profile is known but the gain of system is

unknown. For example in gas turbines, optimal profile for output is ramp and

therefore is equal to ( )f sΓ 2

1s

[2].

These two time dependant variables can be chosen to have different

profiles (for example a step change). By allowing and *( )tθ *( )f t to be time-

varying, we can track a time-varying optimum. In some cases, we may

have *f (extremum point) change in a linear fashion but we can also allow for

non-linear change. We require an integrator in a proportional-integral

70

controller (PI) to obtain a zero steady-state error. We make the following

assumptions on the system in Figure 4.12 that underlie the analysis to follow:

[1]

1. , are asymptotically stable and proper. ( ) and ( )i oF s F s

2. are strictly proper rational functions and poles of

that are not asymptotically stable are not zeros of .

( ) and ( )fsθΓ Γ s

( )sθΓ ( )iF s

These two assumptions ensure the existence of an extremum profile. We

selected an example from Ariyar and Krstic work [1] where these assumption

hold. We then applied our algorithm to this and similar processes.

To illustrate the flexibility of the new method, we used a difficult case

studied by Krstic (2003) [1].

* *

2

*

*

( ) ( ) 5( ( ))1( )

11( )

3 20.01 200

( )0.09 200

( ) 0.02sin( )

i

o

2f f t t

F ss

sF ss s

tf t

t

t t

θ θ θ

θ

⎧ = + −⎪⎪

=⎪ +⎪−⎪ =⎨ + +⎪

⎪ ≤⎧=⎪ ⎨ >⎩⎪

⎪ =⎩ (4.8)

Several simulations are carried out to investigate the effect of the choice

of the input signal, PID parameters and parameters for the RLS algorithm.

71

4.3.2.1 Simulation 1

In this simulation, we use a sine wave signal with amplitude 0.02 and

frequency 0.1π rad/s as desired input and sampling time 0.02s. Figure 4.14

shows the simulation results.

0 50 100 150 200 250 300 350 4000

0.1

0.2

time,s

f and

f* (t)

300 310 320 330 340 350 360 370 380 390 400-0.1

0

0.1

time,s

θ an

d θ* (t)

0 50 100 150 200 250 300 350 400-0.02

0

0.02

time,s

v0

f*(t)f

θ*(t)

θ

Fig.4.14: Simulation results for equation 4.8

The simulation results show that the proposed method is indeed able to

find the optimum even though the optimum input varies with time. The PID

parameters for this example are 10 8 and 0 found by trial and error. The

forgetting factor here is 1.

72

4.3.2.2 Simulation 2

We change the optimum output in such a way that it moves in a ramp

shape, say

*( ) 0.01f t t=

Again in Figure 4.15 we see a good optimal tracking ability.

0 50 100 150 200 250 300 350 4000

1

2

time,s

f and

f*

300 310 320 330 340 350 360 370 380 390 400-0.1

0

0.1

time,s

θ an

d θ* (t)

0 50 100 150 200 250 300 350 400-0.1

0

0.1

times,s

v0

f

f*

θ*(t)

θ

Fig.4.15: Simulation results for equation 4.8 with ramp optimal output

73

4.3.3 Comparison of proposed RLS Impulse Weight

Gradient Method with Sign-Sign Method

The new gradient approach is tested on the following quadratic function:

(4.9) ( )25 1y x= − + 0

and the optimization performance is compared in Figure 4.16 with the Sign-

Sign method. The compare plots show that the sign-sign method needs less

iteration to converge to the minimum point. However, the proposed gradient

method is more accurate than Sign-Sign method after they both reach the

steady state, which is shown in table 4.1 by averaging the Mean Squared

Error (MSE) of output over the last 1000 samples.

0 50 10010

15

20

25Sign-Sign Method

time,s

outp

ut

50 60 70 80 90 10010

15

20

25RLS method

time,s

outp

ut

50 60 70 80 90 1000

2

4

6

time,s

inpu

t

0 50 1000

2

4

6

time,s

inpu

t

74

Fig.4.16: Minimum seeking for quadratic function 4.9

By averaging the MSE of output over the last 1000 samples, we compare

the numerical error between proposed impulse weight gradient method and

sign-sign method as shown in Table 4.1

Table 4.1

Approach MSE over last 1000 samples

Gradient Method 3.3059e-004

Sign-Sign Method 0.0364

75

4.4 Processes with Dead Time

One of the major advantages of the new method is that it is able to track

more complicated processes with variable dead time. This is the only known

algorithm that is able to accomplish this. The following rules are followed:

If dead time =1, use as the current gradient ( )1v

If dead time=2, use ( )2v as the current gradient, etc.

In order to demonstrate the effectiveness of the approach, the method is

tested on processes with dead time and the performances are compared with

other direct methods. Figure 4.17 shows the feedback optimization loop for

processes with dead time.

Process

Test Signal

PID Controller

+

-

yDead time Estimator

Gradient Estimator (RLS)

x

Set point=0

Fig.4.17: feedback optimization loop for processes with dead time.

76

A quadratic model with dead time given by:

, dead time=3 ( )25 1y x= − + 0

The Simulink block diagram is shown in Figure 4.18. The performance of

the proposed method is shown in Figure 4.19 and compared to the Sign-Sign

Method.

The input and output data are injected to a S-function called ‘dtime’ which

estimates the dead time for the process using cross correlation method. And

then the process input, output data and the output from the S-function

‘dtime’ are provided to the S-function ‘rarxd1’ which estimates the gradient

information.

77

Sco

pe3

Zero

-Ord

erH

old4

Zero

-Ord

erH

old2

Zero

-Ord

erH

old1

In1

Out

1

Sub

syst

em

Sin

e W

ave1

Sin

e W

ave

Sco

pe2

Sco

pe1

Sco

pe

rarx

d1

S-F

unct

ion2

dtim

e

S-F

unct

ion1

Ran

dom

Num

ber3

Ran

dom

Num

ber1

2 Gai

n

-10

Con

stan

t3

Fig.

4.18

: Blo

ck d

iagr

ams

in S

imul

ink

Tool

box

for

a pr

oces

s w

ith d

ead

time

78

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

30

time,s

outp

ut

0 10 20 30 40 50 60 70 80 90 1000

5

10

time,s

inpu

t

sign-sign methodgradient method

gradient methodsign-sign method

Fig.4.19: Simulation results for equation 4.9 with dead time=3

The Sign-Sign method and impulse weight gradient method are applied

respectively and it can be seen from the result shown in Figure 4.19 that the

Sign-Sign method cannot accommodate the changing process dead time,

however, the new gradient method is able to find the optimum and rapidly

finds the minimum value for the output 10, which corresponds to input 5x = .

79

4.5 Processes with Variable Dead Time

An even greater challenge is presented by processes that have

deadtimes that switch to new values at various times.

The data are generated by the model:

( ) ( ) ( ) ( )0.2 1 0.4 2y t u t nk u t nk u t nk= − + − − − − − (4.10)

Where sampling time=1s 1 0 10004 1000 30002 3000 5000

tnk t

t

< ≤⎧⎪= < <⎨⎪ ≤ ≤⎩

The current gradient of this model is given by the coefficient of the first input

which is 1.0. (u t nk− )

Figure 4.20 shows the simulation results of dead time and current

gradient information. It can be seen from the figure that the gradient method

is able to find the first non-zero impulse response even though the process

has a variable dead time.

This new algorithm is thus able to achieve solution to a very difficult

problem. The solution to this problem is not reported in the literature as far

as the author knows.

80

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001

2

3

4

time,s

dead

tim

e

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.5

0

0.5

1

1.5

time,s

curre

nt g

radi

ent

Fig.4.20: Dead time and gradient information for equation 4.10

81

4.6 Processes with Variable Dead Time

and Changing Model Structure

Some processes have a combination of difficulties and, for example, both

model structure and dead time can vary with time.

Consider the following process

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

( ) ( )

1 0.2 2 0.4 3 0 10004 0.2 5 0.4 6 1000 30000.8 2 0.15 3 3000 5000

u t u t u t ty t u t u t u t t

u t u t t

− + − − − < ≤⎧⎪= − + − − − < ≤⎨⎪ − − + − < ≤⎩

(4.11)

This example represents the ultimate test for the new algorithm and has

not been reported solved in current literature to the best of the author’s

knowledge.

It can be seen from the equation that not only the dead time is changing

but the model is changed after t=3000s.

Again dead time and current gradient information is shown in Figure

4.21

82

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001

2

3

4

time,s

dead

tim

e

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

0

1

2

time,s

curre

nt g

radi

ent

Fig.4.21: Dead time and gradient information for equation 4.11

It is clear from the results that the algorithm is able to pick off the first

coefficient (which represents the new gradient) of the models as they change

from one model to the next.

83

Bibliography

[1] K.B. Ariyur & M. Krstic (2003). Real-time optimization by

extremum-seeking control, Wiley-Interscience, NJ.

[2] M. Sheikhzadeh (2006). Optimization and Control of Crystallization

Processes. Ph.D Thesis, University of Western Ontario.

[3] L. Ljung (1989). System Identification: Theory for the User, Prentice

Hall.

[4] System Identification Toolbox User’s Guide, Mathworks Inc., (2000).

84

Chapter 5

Real-Time Optimization

85

In previous chapters, a new RLS, impulse weight gradient optimization

method was proposed and it was shown in simulations to perform well in

estimating the current gradient and keeping track of time-varying optimal

conditions. In a real time experimental scenario the situation can become

more challenging due to higher level of uncertainties and the presence of

noise, environmental changes, etc. In order to test the effectiveness of the

proposed method in real time optimization, the method is applied to an

experimental Continuous Stirred Tank (CST) and Pressure Tank,

respectively.

5.1 Real-Time Application to CST

In order to fully test the effectiveness of the proposed optimization method,

it is applied to a real-time chemical process, a Continuous Stirred Tank (CST)

which is shown in Figure.5.1

Fig.5.1 Continuous Stirred Tank

86

The laboratory scale 2L CST is filled with water and then sealed. Steam

and Water from regulated sources flow into and mix in a heating/cooling coil

inside the CST for controlling the temperature inside the tank. The fluid

enters the tank from the bottom, and exits through a float type steam trap at

the top of the tank. There are two manipulated variables: they are voltage

signals to the steam valve and water valve. The range of the steam and water

valves is normalized to lie between -1 with fully-closed position and 1 with

fully-open position. In the experimental runs, the steam valve is fixed at 25%

open, and the water valve was manipulated using the new gradient

optimization approach to search for the optimal temperature of the CST so

that an objective function, Eq. (5.1) was minimized.

2( 32)y T 5= − + (5.1)

5.1.1 Experimental Setup

The experimental setup is shown in Fig.5.2. A picture of the CST used

here is shown in Fig.5.3. The temperature measured by a thermocouple is

transmitted to the micro-computer through ADVANTEC data acquisition

board PCLD-8710. All the control actions were carried out by

Matlab/Simulink with the help of the Real-Time Toolbox [6]. The

optimization program was run in the Matlab/Simulink environment. The

signals were then transmitted to I/P transducers which control the valve

positions through the data acquisition board.

87

Fig.5.2: Experimental setup for CST

88

Fig.5.3 Picture of the CST used for experimental runs

89

5.1.2 Experimental Runs

Firstly, both the steam and water valve were fixed at 25% open, and the

changes in temperature shown in Fig.5.4 indicate that the process has severe

nonlinearity and significant measurement and process noise. Also the reactor

has a large time constant value and thus reacts to the changes in the input

very slowly. This has been confirmed in many previous studies [1]. Thus this

process poses a significant challenge to the new gradient optimization

method.

0 200 400 600 800 1000 120042

44

46

48

50

52

54

56

58

Time, s

Tem

pera

ture

, ° C

Fig.5.4 Variation in CST temperature with both water valve and steam valve

25% open

90

The feedback optimization diagram is shown in Fig.5.5. The Simulink©

diagram block is shown in Fig.5.6

CST 2( 32) 5y T= − +

Gradient Estimator

Steam flow=25% open Water flow

T (oc)

PID Controller -

0

Fig.5.5: CST Real –Time optimization diagram

91

2.5

stea

m

Qua

drat

ic fu

nctio

nTe

mpe

ratu

re re

cord

rarx

1

S-F

unct

ion

Wat

er s

uppl

y v

alve

Stea

m s

uppl

y v

alve

wate

r te

mpe

ratu

re

Rea

l CS

TR

0

Con

stan

t

Mem

ory1

PID

Con

trolle

r

PID

Fig.

5.6:

Sim

ulin

k© b

lock

dia

gram

s fo

r R

eal t

ime

Opt

imiz

atio

n of

a C

ST

92

During the experiment, the steam valve was fixed at 25% open and the

water valve was manipulated by the gradient optimization program to shift

the temperature of the CST in order to minimize the objective function Eq.

(5.1). Fig. 5.7 and 5.8 show the optimization results. The objective function,

Eq. (5.1) has a minimum value of 5.0 when the CST temperature reaches

32°C.

0 500 1000 1500 2000 25000

50

100

150

200

250

time,s

y

objective function :y=(T-32)2+5

Noisy experimental outputTrue optimum level

Fig.5.7 Real-Time Optimization-output of objective function Eq.(5.1)

93

Fig.5.8 shows that the value of the noisy temperature which approaches

the value of 32°C.

0 500 1000 1500 2000 250016

18

20

22

24

26

28

30

32

time,s

tem

pera

ture

T, °C

Fig.5.8 Real-Time Optimization-temperature condition for Eq. (5.1)

5.1.3 Discussion

The sever nonlinearity and measurement noise presents a great challenge

to the gradient optimization method. From the experimental results, we can

see that the proposed method is able to find the optimal temperature

condition in order to minimize the objective function, in spite of the excessive

level of measurement noise.

94

5.2 Real-Time Application to Pressure

Tank

The real-time application considered here is a pressure tank system

shown in Fig.5.9 through which compressed air flows from a regulated supply.

Two control valves are installed on the inlet and the outlet of the tank. The

pressure in the vessel and the outlet flow rate are measured by a pressure

meter and a flow rate meter respectively and transmitted to a micro-

computer via a data acquisition board. The pressure tank is operated with

outlet valve position fixed. The inlet valve was manipulated within the region

[-1, 1].

Figure 5.9 Pressure Tank

95

5.2.1 Experimental Setup

The real-time optimization was implemented in the Matlab/Simulink

environment to generate optimal control signals to the inlet control valve and

search for the optimal pressure in the vessel that maximizes the objective

function.

Figure 5.10 depicts a general control system setup. Figure 5.11 shows a

picture of the pressure tank used for experimental runs.

Fig.5.10: Experiment Setup for Pressure Tank System

96

Fig.5.11 Picture of Pressure Tank

97

5.2.2 Experimental Runs

Several experimental runs were conducted on the pressure tank using

Matlab/Simulink and the real time toolbox from Humusoft©. The objective

function shown in Equation (5.2) was used to evaluate the gradient

optimization method.

( )= − − + +236 1000 5sin(0.01 )y P t (5.2)

The feedback optimization diagram is shown in Fig.5.12. Simulink© block

diagram is shown in Fig.5.13. The optimization results are shown in Fig.5.14

and Fig.5.15.

Fig.5.12 Pressure Tank Real –Time optimization diagram

Notice that here the objective y is dynamic itself and presents a greater

tracking challenge for the algorithm.

Gas outlet=25% open Pressure

Tank

Gradient Estimator

Gas inlet P

PID Controller

( )=− − + +

236 1000 5sin(0.01)y P t

-

0

98

real

_tan

k_ou

tput

To

Wor

kspa

ce

0

Dis

play

Sco

pe2

flow

rat

e

pres

sure

Out

1

crea

t som

e op

timum

Sco

pe1

Sco

pe3

Sco

pe4

rarx

1

S-F

unct

ion

Gas

inle

t

Gas

out

let

flow

rat

e

pres

sure

Pre

ssur

e ta

nk

-0.5

Con

stan

t

0

Con

stan

t1

Mem

ory

PID

Con

trolle

r

PID

Fig.

5.13

: Sim

ulin

k bl

ock

diag

ram

s fo

r R

eal t

ime

Opt

imiz

atio

n of

a

Pres

sure

Tan

k

99

0 500 1000 1500 2000 2500 3000 3500 4000100

200

300

400

500

600

700

800

900

1000

1100

time,s

y

y=-(P-36)2+1000+5sin(0.01t)

Experimental outputTrue optimal output

Fig.5.14: Optimization results for the pressure tank system with objective

function given as Eq. (5.2): objective output

100

0 500 1000 1500 2000 2500 3000 3500 40005

10

15

20

25

30

35

40

time,s

pres

sure

,psi

Experiemental pressureTrue optimal pressure

Fig.5.15: Optimization results for the pressure tank system with objective

function given as Eq. (5.2): Pressure condition inside the tank

101

5.2.3 Discussion

In the Pressure Tank experiment, the optimum of the objective function

changes in a sinusoidal manner. The gas outlet valve was kept at 25% open

and the gas inlet valve was manipulated by the gradient optimization method.

The parameters of the PID controller were P=1, I=0.1, D=0. Fig.5.14 shows

that the objective function was driven toward its optimum rapidly and kept

at optimum by the gradient method.

102

Bibliography

[1] Xiong, Qiang (2000). Optimization, Control and Identification of

Chemical Processes. M.E.Sc Thesis, University of Western Ontario.

[2] Yide Ping (2006). Non-model Based Real Time Optimization of

Chemical Processes. M.E.Sc Thesis, University of Western Ontario.

[3] A. Jiang and A. Jutan (2000). Response Surface Tuning Methods in

Dynamic Matrix Control of a Pressure Tank System. Ind. Eng. Chem.

Res. 39, 3835-3843.

[4] B. Guo, A. Jiang, X. Hua, and A. Jutan (2001). Nonlinear Adaptive

Control for multivariable Chemical Processes. Chemical Engineering

Science 56: 6781-6791.

[5] Z. Ma and A. Jutan (2003). Control of a Pressure Tank System using

a Decoupling Control Algorithm with a Neural Network Adaptive

Scheme. IEE Proc.-Control Theory Appl., Vol. 150, No 4.

[6] Humusoft s.r.o. (2003). Real Time Toolbox for use with Simulink

version 3.12 user manual.

103

Chapter 6

Summary and Future Work

104

In this work, several non-model based optimization methods are reviewed

and compared. A new Recursive Least Squares, Impulse Weight, Gradient-

based, optimization strategy is developed. It is tested on various simulated

mathematical examples with different input signal and two real-time

laboratory processes. The experimental results are compared with another

non-model based optimization method, namely the Sign-Sign method and

show the advantages of the proposed Gradient-based method. The simulation

studies also show that the Gradient-based method is superior to the

perturbation method [1] in that it is not limited to specific sinusoidal signals.

A main advantage of the new Gradient-based method is that it can handle

processes with changing dead time and model structure. No other methods

can achieve this. The simulation studies on processes with changing dead

time and changing model structure demonstrate the possible application of

the Gradient-based method to a practical process with dead time and model

structure variation.

In its initial stage, the new RLS Impulse Weight Gradient-based method

has shown its possible application to practical problems. There are some

suggestions for further work:

1. A possible future application is to extend the new Gradient-based

method to a multivariable system. The RLS Impulse Weight Gradient

based method has only been tested on SISO system in this work. In the

next stage, problems with more variables can be examined.

105

2. Other controllers, such as Model Predictive Controller, can replace the

PID controller in the feedback optimization loop to test the ability of

more advanced controllers in this application.

3. Other algorithms can be used in estimating the gradient, such as,

Least Mean Squares (LMS) algorithm, Normalized Least Mean

Squares (NLMS) and a Kalman Filter based algorithm [2].

106

Bibliography

[1] Y. Tan, D. Nesic & I. Mareels (2006). On non-local stability properties of

extremum seeking control, Automatica 42, 889-903.

[2] System Identification Toolbox User’s Guide, Mathworks Inc., (2000).

107

Appendix A – Experimental

Setup

A.1 Data acquisition board

The type of the data acquisition board used for the lab experiment is

Advantech PCLD-8710 (Terminal board) and PCI-1710 (Internal PCI board).

Fig.A.1 shows a picture of the digital input ports of the data acquisition board.

Fig.A.2 shows the analog input ports and cold junction compensation

switches of the data acquisition board.

As we see from the pictures, the data acquisition board contains 16 digital

input ports (Channel 0 – Channel 15) and 16 analog input ports (Channel 0 –

Channel 15) with each channel (short for CH later) coupled with a ground

port.

Connection type:

There are two ways of instrumental connection:

(1) Single ended connection

(2) Double ended connection

A thermocouple is used throughout the instruction as a sample experiment

device.

108

A picture of the thermocouple is shown in Fig. A.3.

Fig. A.1 Digital input ports of the data acquisition board

109

Fig. A.2 Analog input ports of the data acquisition board

Fig. A.3 Thermocouple

110

Single ended mode (Thermocouple connection):

The ‘+’ wire should be connected to one of the channels (CH 0~CH 15) and the

‘-’ wire be connected to the ground ports (short for GND later on in this

instruction).

*Note: The CH0 should NOT be used when the cold junction compensation is

enabled.

Double ended mode:

In this case, two channels instead of one will be used. (The GND will NOT be

used) The ‘+’ wire is connected to the evenly numbered channels, say CH2

and the ‘-’ wire to the oddly numbered channels, say CH 3.

Note 1: When double ended mode is used, two channels next to each other

will be grouped together as one channel and the number of the oddly

numbered channel in the group will be assigned as the channel number. For

example, CH0 and CH1 are grouped as Channel 1; CH2 and CH3 are grouped

as Channel 3; CH4 and CH5 are grouped as Channel 5 and so on.

Note2: The double ended mode is less noise sensitive than single ended mode.

Test

The voltage measurement and valve control experiment will be conducted in

this chapter.

Hardware Connection

(1) Input part:

The experiment setup is shown below:

111

Fig. A.4 Experiment setup for voltage measurement

A power supply is connected to the data acquisition board in differential

mode. The channel 5 in differential mode is used (in this case, channel 4 and

5 will be grouped as channel 5). The positive pole is connected to the evenly

numbered channel 4 and the negative pole is connected to the oddly

numbered channel 5 (NO ground channel in differential connection mode).

And the voltage signal generated by the power supply is adjusted to 3.5V.

(2) Output part:

112

Fig. A.5 Experiment setup for valve control

The signal is sent by computer through the data acquisition board to the

converter which converts the 0-10V voltage signal to the 4-20mA current

signal and then to the transducer to control the valve. (Note that the output

from the board is voltage signal.)

Software setup

The Advantech testing software can be used to show the voltage signal being

sent to the data acquisition board.

To start the software:

Start All Programs Advantech Automation Device Manager

Advantech Device Manager

The device management window will pop up as shown in Fig. A.6.

113

Fig. A.6 Advantech Device Manager

All the boards installed on the computer will be listed in the ‘Installed

Devices’. Select the board which you want to test by clicking on the name.

Then click on the ‘setup’ button to start the ‘Device Setting’ window as shown

in Fig. A.7. In A/D Channel Configuration, choose the proper channels

(Channel 4 and 5 in our example) and set the proper connection mode

(Differential in our example). Then click on ‘OK’ to close the device setting

window and then on the ‘test’ button to start the following ‘Advantech Device

Test’ window (Fig. A.8).

After choosing the proper range for the connected channels, the voltage signal

measured by Channel 4 and 5 is shown in Fig. A.8 as 3.5V.

114

Fig. A.7 Device Setting

115

A.8 Advantech Device Test

116

A.2 MATLAB applications

Voltage measurement

The measured date can also be sent to MATLAB simulink for further

processing, which can be done by using the Real-time toolbox developed by

Humusoft.

To start the real-time tool box, type in the MATLAB command window the

command ‘rtlib’ to open the real time toolbox library:

Fig. A.9 Real time toolbox library

117

The library is very similar to the Simulink library and can be used in the

same way. The ‘Real-Time sources’ is equivalent to the Simulink sources and

the ‘Real-Time sinks’ is equivalent to the Simulink sinks. All the source or

sink blocks can be viewed by double clicking on the ‘Real-Time sources’ or

‘Real-Time sinks’ respectively.

The voltage measurement experiment done in the ‘Test’ chapter can also be

repeated in Simulink by constructing the following simulink model:

Fig. A.10 Simulink model for temperature measurement

118

The above Simulink model contains four blocks. They are ‘RT In’, ‘Lookup

Table’, ‘Display’ and ‘Adapter’ respectively.

‘RT In’ is a block which takes in data from the data acquisition board Before

this block can be used, which channel is to be measured should be chosen by

double clicking on the block and setting the channel number.

‘Lookup Table’ and ‘Display’ are standard MATLAB data conversion tables.

‘Adapter’ acts as the hardware driver for MATLAB which links the data

acquisition board with MATLAB.

To link MATLAB Simulink with the data acquisition board, one can double

click on the red ‘Adapter’ block and select the proper driver to start the device

setting dialogue box:

Fig. A.11 Select ‘pci1710.rtd’ as the hardware driver

119

Fig. A.12 Setup the connection mode and the input/output signal range

To properly setup the voltage measurement experiment, one should click

the ‘Differential A/D’ box to enable differential connection mode and then

click on the ‘Gains’ button to choose the proper input range.

Fig. A.13 Choose input range 0V ~ 10V

120

The detailed information is provided in the toolbox manual. One thing need

to be mentioned is that if data acquisition board works in double ended mode,

then channel 0 and 1 are grouped as port 1, channel 2 and 3 are grouped as

port 3, channel 4 and channel 5 are grouped as port 5 and so on.

Valve control

The control signal generated by MATLAB applications can also be sent out

via the data acquisition board to control the process peripheral devices. The

valve control experiment (see also Hardware connection) is repeated in

MATLAB/SIMULINK as follow:

(1) Clicking on the “D/A gains” button shown in Fig. A.12, we can select the

desired output range. Here we choose the output range to be 0-10 volt.

Fig. A.14 Choose output range 0V ~ 10V

(2) Constructing the following SIMULINK model:

121

Fig. A.15 Simulink model for valve control

Please note that the “RT Output” block is used for converting MATLAB

source signals to the analog signal and the input to this block can only range

from -1 to +1, that is, if we choose the output range to be 0-10 volt, then the

“RT Output” block will linearly convert the input signal (-1 to +1) into the

voltage signal (0 to 10 volt).

122

Appendix B – MATLAB

Codes

B.1 Generate simulated data

%generate simulated data using model %y(t)=x(t-1)+0.5*x(t-2)-x(t-3)+0.3*x(t-4)+0.4*x(t-5) %na=0,nb=6,nk=1 a=1; b=[0 1 0.5 -1 0.3 0.4]; u=randn(1000,1); % generate 1000 input data y=filter(b,a,u); % generate output data t=[0:999]';

B.2 S-Function of estimating first non-zero impulse weight for a linear static problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %This s-function is used to estimate the first impulse weight parameter for % %a linear static problem = − + = =4 10, sin( )y x with x t and samplingtime 1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [sys,x0,str,ts] =rarx1(t,x,u,flag) switch flag, %%%%%%%%%%%%%%%%%% % Initialization % %%%%%%%%%%%%%%%%%% case 0, [sys,x0,str,ts]=mdlInitializeSizes;

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%%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1, sys=[ ]; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=[ ]; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%%%%% % GetTimeOfNextVarHit % %%%%%%%%%%%%%%%%%%%%%%% case 4, sys=[]; %%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case 9, sys=[]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end % end sfuntmpl % %============================================================================= % mdlInitializeSizes % Return the sizes, initial conditions, and sample times for the S-function. %=============================================================================

124

% function [sys,x0,str,ts]=mdlInitializeSizes % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a % recommended practice as the characteristics of the block are typically % defined by the S-function parameters. sizes = simsizes; sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 1; % one output sizes.NumInputs = 2; % two input sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes); % % initialize the initial conditions % x0 = [ ]; % % str is always an empty matrix % str = [ ]; % % initialize the array of sample times % ts = [-1 0]; % end mdlInitializeSizes % %============================================================================= % mdlOutputs % Return the block outputs. %============================================================================= % function sys=mdlOutputs(t,x,u) persistent th h p phi %initialization if t==0

125

[th,yh,p,phi]=rarx([u(2) u(1)],[1 50 1],'ff',1); h=th(2); sys=h; %estimate the impulse weights recursively else [th,yh,p,phi]=rarx([u(2) u(1)],[1 50 1],'ff',1,th,p,phi); %RLS with forgetting factor 1 h=th(end,2); %find the first non-zero impulse weight end sys=h; % end mdlOutputs

B.3 S-Function of estimating first non-zero impulse weight for a second order nonlinear static problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %This s-function is used to estimate the first impulse weight parameter for % %a second order nonlinear static problem 24y x x 10= − + − % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [sys,x0,str,ts] =rarx1(t,x,u,flag) switch flag, %%%%%%%%%%%%%%%%%% % Initialization % %%%%%%%%%%%%%%%%%% case 0, [sys,x0,str,ts]=mdlInitializeSizes; %%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1, sys=[]; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=[];

126

%%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%%%%% % GetTimeOfNextVarHit % %%%%%%%%%%%%%%%%%%%%%%% case 4, sys=[]; %%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case 9, sys=[]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end % end sfuntmpl % %============================================================================= % mdlInitializeSizes % Return the sizes, initial conditions, and sample times for the S-function. %============================================================================= % function [sys,x0,str,ts]=mdlInitializeSizes % % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a

127

% recommended practice as the characteristics of the block are typically % defined by the S-function parameters. % sizes = simsizes; sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 1; sizes.NumInputs = 2; sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes); % % initialize the initial conditions % x0 = []; % % str is always an empty matrix % str = []; % % initialize the array of sample times % ts = [-1 0]; % end mdlInitializeSizes % %============================================================================= % mdlOutputs % Return the block outputs. %============================================================================= % function sys=mdlOutputs(t,x,u) persistent th h p phi if t==0

128

[th,yh,p,phi]=rarx([u(2) u(1)],[1 50 1],'ff',0.8); h=th(2); sys=h; else [th,yh,p,phi]=rarx([u(2) u(1)],[1 50 1],'ff',0.8,th,p,phi); %h=th(1); h=th(end,2); end sys=h; % end mdlOutputs

B.4 Generate data for a process with variable dead time and model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Generate simulated data for a process with variable dead time and model %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a=1; b1=[0 1 .2 -0.4]; b2=[0 0 0 0 1 .2 -0.4]; b3=[0 0 -0.8 +0.15]; u1=randn(1000,1); u2=randn(2000,1); u3=randn(2000,1); u=[u1;u2;u3]; %input data %generate output using model y(t)=u(t-1)+0.1*u(t-2)-0.4*u(t-3) y1=filter(b1,a,u1); %generate output using model y(t)=u(t-4)+0.1*u(t-5)-0.4*u(t-6) % the dead time changes, model doesn’t change y2=filter(b2,a,u2); %generate output using model y(t)=-0.8*u(t-2)+0.15*u(t-3) % both dead time and mode change y3=filter(b3,a,u3); y=[y1;y2;y3]; %output data t=[0:4999]';

129

B.5 S-Function for calculating delay function [sys,x0,str,ts] = deadtimel(t,x,u,flag) switch flag, %%%%%%%%%%%%%%%%%% % Initialization % %%%%%%%%%%%%%%%%%% case 0, [sys,x0,str,ts]=mdlInitializeSizes; %%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1, sys=[]; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=[]; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%%%%% % GetTimeOfNextVarHit % %%%%%%%%%%%%%%%%%%%%%%% case 4, sys=[]; %%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case 9, sys=[]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags %

130

%%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end % end sfuntmpl % %============================================================================= % mdlInitializeSizes % Return the sizes, initial conditions, and sample times for the S-function. %============================================================================= % function [sys,x0,str,ts]=mdlInitializeSizes % % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a % recommended practice as the characteristics of the block are typically % defined by the S-function parameters. % sizes = simsizes; sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 1; % one output sizes.NumInputs = 2; % two input sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes); % % initialize the initial conditions % x0 = []; % % str is always an empty matrix

131

% str = []; % % initialize the array of sample times % ts = [-1 0]; % end mdlInitializeSizes % %============================================================================= % mdlDerivatives % Return the derivatives for the continuous states. %============================================================================= % function sys=mdlDerivatives(t,x,u) sys = []; % end mdlDerivatives % %============================================================================= % mdlUpdate % Handle discrete state updates, sample time hits, and major time step % requirements. %============================================================================= % function sys=mdlUpdate(t,x,u) sys = []; % end mdlUpdate % %============================================================================= % mdlOutputs % Return the block outputs.

132

%============================================================================= % function sys=mdlOutputs(t,x,u) persistent atemp a btemp b nk z nktemp winsize if t==0 a=u(1); b=u(2); nktemp=1; else a=[atemp;u(1)]; b=[btemp;u(2)]; end atemp=a; btemp=b; winsize=size(atemp); if t<0.001 nk=1; end if winsize(1)>101 z=iddata(btemp,atemp); % package input-output data into the iddata object nk=delayest(z); %calculate the dead time nktemp=[nktemp;nk]; atemp(1:100,:)=[]; btemp(1:100,:)=[]; else nk=nk; end sys =nk; % end mdlOutputs

B.6 S-Function of Gradient method on processes with dead time function [sys,x0,str,ts] =rarxd1(t,x,u,flag) switch flag, %%%%%%%%%%%%%%%%%%

133

% Initialization % %%%%%%%%%%%%%%%%%% case 0, [sys,x0,str,ts]=mdlInitializeSizes; %%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1, sys=[]; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=[]; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%%%%% % GetTimeOfNextVarHit % %%%%%%%%%%%%%%%%%%%%%%% case 4, sys=[]; %%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case 9, sys=[]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end % end sfuntmpl

134

% %============================================================================= % mdlInitializeSizes % Return the sizes, initial conditions, and sample times for the S-function. %============================================================================= % function [sys,x0,str,ts]=mdlInitializeSizes % % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a % recommended practice as the characteristics of the block are typically % defined by the S-function parameters. % sizes = simsizes; sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 1; % one output sizes.NumInputs = 3; % three input sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes); % % initialize the initial conditions % x0 = []; % % str is always an empty matrix % str = []; % % initialize the array of sample times % ts = [-1 0];

135

% end mdlInitializeSizes % %============================================================================= % mdlOutputs % Return the block outputs. %============================================================================= % function sys=mdlOutputs(t,x,u) persistent th h p phi z nk if t==0 [th,yh,p,phi]=rarx([u(2) u(1)],[1 100 1],'ff',0.8); h=th(u(3)+1); sys=h; else [th,yh,p,phi]=rarx([u(2) u(1)],[1 100 1],'ff',0.8,th,p,phi); %h=th(1); h=th(end,(u(3)+1)); end sys=h; % end mdlOutputs

B.7 S-Function of Gradient method on CST and Pressure Tank function [sys,x0,str,ts] =rarx1realtime(t,x,u,flag) switch flag, %%%%%%%%%%%%%%%%%% % Initialization % %%%%%%%%%%%%%%%%%% case 0, [sys,x0,str,ts]=mdlInitializeSizes;

136

%%%%%%%%%%%%%%% % Derivatives % %%%%%%%%%%%%%%% case 1, sys=[]; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=[]; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%%%%% % GetTimeOfNextVarHit % %%%%%%%%%%%%%%%%%%%%%%% case 4, sys=[]; %%%%%%%%%%%%% % Terminate % %%%%%%%%%%%%% case 9, sys=[]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled flag = ',num2str(flag)]); end % end sfuntmpl % %============================================================================= % mdlInitializeSizes

137

% Return the sizes, initial conditions, and sample times for the S-function. %============================================================================= % function [sys,x0,str,ts]=mdlInitializeSizes % % call simsizes for a sizes structure, fill it in and convert it to a % sizes array. % % Note that in this example, the values are hard coded. This is not a % recommended practice as the characteristics of the block are typically % defined by the S-function parameters. % sizes = simsizes; sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 1; sizes.NumInputs = 2; sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 1; % at least one sample time is needed sys = simsizes(sizes); % % initialize the initial conditions % x0 = []; % % str is always an empty matrix % str = []; % % initialize the array of sample times % ts = [-1 0]; % end mdlInitializeSizes %

138

%============================================================================= % mdlOutputs % Return the block outputs. %============================================================================= % function sys=mdlOutputs(t,x,u) persistent a b htemp atemp btemp th z h winsize p phi if t==0 [th,yh,p,phi]=rarx([u(2) u(1)],[0 50 1],'ff',.99); h=th(1); sys=h; else [th,yh,p,phi]=rarx([u(2) u(1)],[0 50 1],'ff',0.998,th,p,phi); %h=th(1); h=th(end,1); end sys=h; % end mdlOutputs

139

Appendix C - Conversion

Tables

C.1 CST temperature measurement

*The input range should be chosen as: 05.0± v

RT In Temperature ( ) Co

-0.01807 -0.01465

-0.0002686 0.003906 0.006348 0.01172 0.01367 0.01953 0.02393 0.02686 0.03075 0.0376 0.04199 0.0459 0.05078 0.05225

6 9 19 30 34 40 43 49

54.5 58 62 70 75 79 84 86

140

C.2 Pressure tank

Air flow rate:

20.0016 0.009 2V F F= − +

Where

V: Voltage signal from the flow rate meter

F: Air flow rate

*The input range should be chosen as: 0 – 10 v

Pressure:

7521.11345.0 += PV

Where

V: Voltage signal from the pressure meter

P: Pressure

*The input range should be chosen as: 0 – 10 v

141

Curriculum Vitae

NAME: Yuanyuan Liu

PLACE OF BIRTH: Yantai, China

DATE OF BIRTH: 1982

DEGREES: Dalian Maritime University

Dalian, Liaoning, China

2002-2006, B.Eng.

The University of Western Ontario

London, ON, Canada

2006-2008, M.E.Sc.