Process Identification Based on a New Relay Feedback Experiment

Micheile Desarmo

A thesis submitted in confonnity with the requirements

for the degree of Master of Applied Science

Graduate Depart ment of Chernical Engineering & Appiied Chemistry

University of Toronto

@Copyright by Michelle Desarmo(1998)

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To rny parents

PROCESS IDENTIFICATION BASED ON A NEW RELAY FEEDBACK EXPERIMENT

Master of Applied Science, 1998, by Michelie Desarmo,

Graduate Department of Chernical Engineering and Applied Chemistry,

University of Toronto

Abstract

A novel identification experiment based on relay feedback is proposed which automaticdy

generates a nonperiodic binary process input signal by interniittently placing additional

dynamics into the relay feedback loop. The input signal is rich in frequency content, per-

rnitting identification of a multiple-point frequency or step response model from a single

experiment. The Frequency Sampling Filter (FSF) model is used to fit the models to the

relay generated data using a least squares estimator. A recursive version of this algorithm

has been incorporated into a prototype software application that combines the new relay

experirnent with on-iine FSF mode1 identification. The prototype has been applied to a

pilot-scale, stirred t d heater process to demonstrate the new method in practice.

Acknowledgement s

1 wish to express my sincere thanks to the many people who have supported me in various

ways during this project. 1 would especially like to acknowledge the foUowing people:

* My supervisor, Dr. William Cluett, for his guidance throughout my work at U of T, and also for his bancial support.

* Dr. Liuping Wang for her advice and extraordinary enthusiasm.

9 Feilow members of the Process Control Group for many interesthg conversations and

lots of advice. Special thanks to:

- Alex Kalafatis for sharing his LabVIEW knowledge.

- Joe Tseng for being on cd1 every time the cornputers did something unexpected.

- Sophie McQueen for al1 her help with my presentations.

- Female members of the group for their fiiendship, and for organizing extracur-

ricular fun.

AU of my fiiends, for their inspiration.

Penny Seymour for fond memories of my experience as a TA in the first year lab.

My family for a l l the love, support and encouragement that they have given me

throughout my education.

Contents

Abstract iii

Acknowledgements iv

List of Tables vii

List of Figures x

Nomenclature xi

1 Introduction 1 . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Thesis Objectives 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review 3 . . . . . . . . . . . . . . . . . . . . 1.2.1 The Astrorn-~a&ünd autotuner 3

. . . . . . . . . . . . . . . . . . 1.2.2 A survey of autotuning applications 5 . . . . . . . . . . . . . . . . . . . 1.2.3 Describing function approximation 6

1.2.4 Alternative methods and improvements to the DFA . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . 1.2.5 Transfer func tion identification 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Thesis 10

2 Ekequency and Step Response Identification fkom Relay Data 11 . . . . . . . . . . . . . . . . . . . . . 2.1 Frequency Sampling Filter Algorithm 11 . . . . . . . . . . . . . . . . . . . . 2.1.1 Frequency sampling filter mode1 12

. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Step response models 14 . . . . . . . . . . . . . . . 2.1.3 Formulation of the least squares problem 15 . . . . . . . . . . . . . . . 2.1.4 PRESS statistic for mode1 order selection 16

. . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 FSF/PRESS algorithm 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A New Relay Experiment 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Relaydynamics 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Proposed experiment 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Experimental design 20

2.3 Application of the FSF Algorithm to Relay Feedback Data . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The desired input spectrum 23

. . . . . . . . . . . . . . . . 2.3.2 Properties of the FSF correlation matrix 25

2.3.3 Choice of the parameter N . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Experimentalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 F'urther Design Issues and Case Studies 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Characteristics of Processes with High Criticai Frequencies . . . . . . . . . 31 3.3 The Muence of Delay on the FSF Spectrurn . . . . . . . . . . . . . . . . . 35

3.3.1 First order processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Second order processes . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.3 EffectofrelayparametersontheFSFSpectnun . . . . . . . . . . . 40

3.4 A Modification to the New Experiment . . . . . . . . . . . . . . . . . . . . 41 3.5 Guidelines for Adjusting N for the FSF Mode1 . . . . . . . . . . . . . . . . 45 3.6 Noise M o d e h g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 BacklashEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.8.1 First order models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Second order models 60

4 On-Line Identification and Experimental Results 68 4.1 Recursive Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 Formulation of the recursive least squares problern . . . . . . . . . . 70 4.1.2 Updating the covariance matrix via RUD factorkat ion . . . . . . . 70

4.2 The Recursive FSF Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.1 An alternative representation of the FSF mode1 . . . . . . . . . . . . 71 4.2.2 Recursive estimation of the FSF mode! . . . . . . . . . . . . . . . . 73 4.2.3 Initialization of the RLS algorithm . . . . . . . . . . . . . . . . . . . 76

4.3 A Prototype Software Application for On-Line Identification . . . . . . . . 77 4.3.1 Codevalidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Experimental Case Study on a Pilot Scale Stirred Tank Heater . . . . . . . 83 4.4.1 Operat ing conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.2 Results of experiment # 1 . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.3 Results of experirnents # 2 and 3 . . . . . . . . . . . . . . . . . . . . 91 4.4.4 Remarks on practical issues . . . . . . . . . . . . . . . . . . . . . . . 99

5 Conclusions and Recornmendations 105 5.1 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 107

A The Relationship Between the Batch and Recursive FSF filters 111

List of Tables

4.1 S u m m q of Parameters/Options on the Interface . . . . . . . . . . . . . . . 80 4.2 Srimmary of Experiment # 3. the Tl plus delay experiment . . . . . . . . 94

List of Figures

1.1 The Basic Relay Feedback Experiment . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Relay Block 4

. . . . 1.3 Sample Data fkom the A s t r o m - ~ a ~ ~ l ü n d Autotuner: d = f 2, E = 0.5 6

Schematic Diagram of the FSF Mode1 . . . . . . . . . . . . . . . . . . . . . 13 . . . . . . . The Influence of Additiond Dynamics on Excitation F'requency 19

Block Diagram of the Proposed Experiment . . . . . . . . . . . . . . . . . . 20 Comparison of Data Generated fkom the New and Standard Relay Experiment 21 New Relay Experiment Applied to Two Different Models with Identical Input Sequences: [O O 1 1 1 0 1 0 1 O O 11 . . . . . . . . . . . . . . . . . . . . . . . 24 A Typical FSF Input Spectrum Obtained from the Proposed Experiment: input sequence [O 0 0 1 1 1 1 0 0 11 . . . . . . . . . . . . . . . . . . . . . . 26 Experimental Resdts for the STH . . . . . . . . . . . . . . . . . . . . . . . 29 Identification Results for the STH . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 New Experiment Applied to Two First Order Models with Different Dead- times (input sequence [O O 0 1 1 1 1 0 0 1 0 1 1 1 O]) . . . . . . . . . . . . .

3.2 A Comparison of FSF Spectra for Data in Figure 3.1 . . . . . . . . . . . . . 3.3 The Effect of Deadtirne on the F'requency Response of First Order Relay

Dynamics (AU time constants are 100 sec . The FSF Erequencies are shown by o.*.xt+.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 A Comparison of the Observed and Ideal Normalized Period for First Order Modek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 A Comparison of the Observed and Ideal Normalized Period for Second Order Critically Damped Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6 The Effect of Hysteresis on the UItimate Period . . . . . . . . . . . . . . . . 3.7 Block Diagram of the Modified Relay Experiment . . . . . . . . . . . . . . . 3.8 Met hodology for the Additional Delay Experiment . . . . . . . . . . . . . . 3.9 Simulation Results for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Identification Results for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Simulation Results for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Identification Results for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Simulation Results for Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

3.14 Identification Results for Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Simulation Results for Case 3b

3.16 Identification Results for Case 3b . . . . . . . . . . . . . . . . . . . . . . . . 3.17 The Effect of Normalized Deadtirne on the Magnitude of the Process Output 3.18 Simulation Results for Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Identification Results for Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Simulation Resuits for Case 5

3.21 Identification Results for Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 Simulation Resuits for Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . 3.23 Identification Results for Case 6

4.1 The Main Control Panel of the Prototype Interface . . . . . . . . . . . . . . 4.2 The Identification Control Panel of the Prototype Intexface . . . . . . . . . 4.3 Cornparison of the LabVIEW (RLS) and MATLAB (batch LS) Fkequency

Response Estimates: 6 settling tMes of undifferenced data, N = 600. n = 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 sec samphg

4.4 Schematic Diagram of the Stirred Tank Heater Pilot Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Input/Output Data for T2

4.6 On-line Identification Results for T2 Using Three Settling Time Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( n = l l )

4.7 On-Iine FSF spectrum for T2 (N = 259, n = 11) . . . . . . . . . . . . . . . 4.8 On-line Identification Results for T2 Using Three Mode1 Orders . . . . . . . 4.9 On-line FSF spectrum for T2 (N = 259, n = 13) . . . . . . . . . . . . . . . 4.10 Convergence of the T2 Model with Time (N = 259, n = 11): 1 - 6 settling

times of data = o,x.",+,solid,dotted, respect ively . . . . . . . . . . . . . . . 4.11 Input/Output Data for Tl (no additional delay) . . . . . . . . . . . . . . . 4.12 Cornparison of Input Signal Fkequency Content for Two Processes . . . . . 4.13 Convergence of the Tl Model with Time (N = 221 . n = 11): 1 - 5 settling

times of data = o,x,",+,solid, respectively . . . . . . . . . . . . . . . . . . . 4.14 Input/Output Data for the Tl Plus Additional Delay Experiment: see Table

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 On-line Identification Results for the Tl Plus Additional Delay Experiment

(see Table 4.2; continued in Figure 4.16): solid, dotted lines are the estimated step responses fkom differenced. non-differenced data respectively . . . . .

4.16 On-line Identification Results for the Tl Plus Additional Delay Experiment (see Table 4.2; continued from Figure 4.15): solid. dotted lines are the esti- mated step responses from dinerenced. non-differenced data respectively .

4.17 On-line Identification Results for T2: automatically estimated set t ling tirne. N=190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.18 On-he Identification Results for Tl: automatically estimated settling time. N=44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.19 Biased and Unbiased Relay Data: Two runs of Experiment #1 . . . . . . .

4.20 On-iine Identifkat ion Results for T2. DBerencing can be advantageous when baddash effects are signincant. Input/Output data shown in Figure 4.19 . . 103

Nornenclat ure

e-t E

limit cycle amplitude real part of the kth fiequency response coefficient: G(gWk ) imaginary part of the kth fiequency response coefficient, G(eJWk) relay amplitude addi tional delay added to the feedback loop diagonal mat rk in the RUD factorization of P(t) kth diagonal of the FSF covariance matrix (FSF spectrum) convent ional residuals PRESS residuals sum of squared prediction errors m t h step response coefficient discrete process fiequency response at frequency wk cont inuous t ime process t ransfer func t ion discrete time process transfer funct ion it h impulse response coefficient kth FSF fiIter FSF integer hannonic number RLS update gain ul t imate gain FSF mode1 order process settling time, in units of sampling i n t e d s the DFA ideal switching period associated with FSF fkequency wi RLS covariance matrix frequency response weighting function tirne, in units of sarnples process settling time, in units of time process input at time, t unit upper triangular matrix in the RUD factorization of P( t ) disturbance at tirne, t dist urbance vector predicted process output generated using without data at time,t process output at time, t process output vector backwxds shift operator delay to time constant ratio

T d process deadtirne

DFA DFT FFT FIR FSF Model PEI Controuer PRESS RBS RLS RUD Factorization STH

c04wk) sin(wk) sampling interval relay hysteresis widt h FSF parameter vector (Bequency response parameten) FSF parameter vector estimated without using data at time,t standard deviation of the noise process t ime constant kth FSF filter output at tirne, t phase angle, in rad, at frequency W, FSF regressor vect or (6lt ered input signal) FSF filter (RLS formulation) associated with the imaginary part of the kth frequency response coefficient FSF filter (RLS formulation) associated with the real part of the kth fkequency response coefficient frequency, in rad/sec kth FSF fkequency, in rad

Describing Function Approximation Discrete Fourier Transform Fast Fourier Transform Finite Impulse Response F'requency Sampling Filter Model Proportional Integral Derivative Controuer PRediction Error Sum of Squares Random Binary Signai Recursive Least Squares Recursive UD factorization Stirred Tank Heater

Chapter 1

Introduction

1.1 Motivation and Thesis Objectives

In the early eighties, Astrom and Hagglünd (19840) proposed the use of relay feedback

combined with a describing function approximation (DFA) as a simple means to estimate

the critical fkequency response of a process. The key idea behind this work was that the

relay experiment can be used as a meam to automatically excite a process at its critical

frequency. When combined with the DFA, the relay proved to be a very efficient way to

obtain process information that could be directly applied to PID controller design. Since

1984, many variations and refmements to the original method have been developed and

used in a number of applications, including autotuning of P D controllers (Astrom and

Hagglünd, 1984h Leva, 1993; Kim, 1995; Khan, 1995), initialization of adaptive controllers

(Hagglihd and Astrom, lggl), process identification (Li et al., 1991; Sallé and Astrom, 1991)

and process monitoring (Belanger and Luyben, 1996; Chiu and Ju, 1997).

Astrom and Hagglünd's approach was particularly successhil because. although the in-

formation generated by the experiment is Limited, very Little a pn'ori knowledge is required

to conduct the experiment and the data is simple to analyse and readily applicable to fa-

miliar PID controller designs (e.g. Ziegler and Nichols, 1942). The relay experiment is

attractive not o d y because it is easy to understand and implement, but also because it is

a closed loop experiment that maintains the process near its operating point. The main

limitation of current relay methods is that the generated input/output data is dominated

by a single fkequency. This means that the input spectrum is not diverse enough to permit

more than one or two fkequency response points on the process Nyquist plot to be accurately

ident ified.

The literature on relay feedback applications relies almost exclusively upon the use of a

describing funct ion approximation, which est imates a single fkequency response point based

on the assumption that the relay data has a dominant k t harmonic component. Although

this is oRen a reasonable assumption, for cases as common as a first order process, this

assumption has b e n shown to cause significant estimation errors (Friman and Waller, 1995)-

Regardless of accuracy, the DFA is limited by the fact that it yields oniy one frequency

response point per relay experiment. A direct consequence of this limited information is

that applications have been restricted to certain control strategies (eg. PID tuning using

specified gain/phase margins). In other words, the amount of information generated leaves

Little room for flexibility in controller design.

In order to obtain a more flexible model, it is necessary to gather more process infor-

mation by performing a series of dinerent relay experiments. Those who have attempted to

constntct more generdy applicable models, such as transfer hc t ions , have identified dif-

ferent fiequency response points fkom several relay experiments, and then used these points

to fit a parametric model. Selecting a model structure complicates the modelling process

by requiring the user either to impose an a prior i model structure' or to fit several types

of models and develop a detailed procedure for deducing which is the "true" model. This

means that the predictive capability of modeis generated by these methods is often com-

prornised by mode1 structure mismatch, in addition to the accuracy limitations inherited

from the DFA.

Despite this body of work, the Literature suggests that there is still a need in industry

for a simple experiment that can produce a more accurate and more complete model of the

process. Response tests that use random binary input signals are an alterriative means of

obtaining data for process identification. The non-periodic nature of these signals implies

that there is enough multifrequency information in a single experiment to construct a more

complete hequency response model. Unfortunately, a prior i knowledge of the process is

required to design such an input signal with the correct fiequency content. Also, if a

predesigned input signal is applied in an open loop experiment, there is a risk that the

process variable will drift outside the desired operating region.

Thus, the main objectives of this thesis are:

To develop a single, relatively simple relay feedback experiment which WU provide

rich multifrequency excitation that is suitable for the purpose of identifjring a general

purpose process model (e-g. a frequency and/or step response model).

To examine the use of the Frequency Sampling Filter (FSF) model structure and a

least squares estimator as a means to improve upon the accuracy of the DFA while at

the same time developing a general purpose model.

To achieve both of the above objectives while minirnizing the amount of a priori

information required fiom the user. In other words, the objective is to automaticdy

ob t ain an appropriate mult iikequency input signal, while requiring no more process

information than that required by the basic relay experiment (i.e. the sign of the

process gain). With respect to the chosen FSF mode1 structure? only an estimate of

the process settling time is required. Work will be done to detennine whether this

information can be automaticdy determined horn the relay feedback data.

1.2 Literature Review

1.2.1 The Astrom-EIagglünd autotuner

Ziegler-Nichols (1942) controller tuning rules are based upon knowledge of the ultimate

gain and ultirnate period of the process. Such information was originally obtained by using

proportional control wit h a gradually increasing gain to induce a Limit cycle in the process

output. This method was time consuming and risked operation on the verge of instability.

Astrorn and Hagglünd (1984a) proposed relay feedback as an alternative to this trial and

error method. They identified the critical point with a describing function approximation

for the relay, and directly applied this information to a Ziegler-Nichols P D controller design.

The block diagram in Figure 1.1 shows the basic relay feedback experiment, known as the

A s t r o m - ~ a ~ ~ l ü n d Autotuning experiment.

This experiment automatically excites the process neai- its critical frequency (conditions

for the existence and stability of relay limit cycles have been presented by Astrom and

Hagglünd (1984a) and Hang and Astrom (1988)). The relay block is shown in more detail

in Figure 1.2. The principle idea behind the operation of the reiay is that it creates a

3

Figure 1.1: The Basic Relay Feedback Experiment

Feedback Erxor

Process Input

Figure 1.2: The Relay Block

periodic binary input signal, u, whose amplitude, f d: is determined by the sign of the

feedback error, e. That is, the input signal switches are triggered by the zero crossings of

the feedback error, In order to minimine correlation with high hequency noise, the relay

is t-ypicdy implemented with a hysteresis of width E , which is generally set to be about

three times the estimated standard deviation of the noise. Therefore, to trigger a switch,

the feedback error must move beyond the zero crossing by a significant amount relative to

the noise Ievel,

1.2.2 A survey of autotuning applications

Many applications have been developed to exploit the relay's ability to autornatically gen-

erate speci£ic fkequency information that is directly applicable to simple controller designs.

Autotuning of PID controllers is one such area in which variations on the autotuning exper-

iment proposed by Astrom-EIagglünd have played a signincaot role. For example, Palmor

et al. (1995) have extended the Astrom-~agglünd autotuning experiment to multivariable

processes. In contrast to sequential autotuning methods (Luyben, 1987; Shen et al.: 1996),

Palmor's method allows al1 loops to be sirnultaneously closed under relay feedback. Since

the multivariable situation gives rise to infinitely many possible critical points (depending

on the ratio of the relay amplitudes in each loop), they have concentrated on developing an

iterative method for identibing the desired critical point from a user-specified parameter

that indicates the relative importance of the loops. The methods developed by Leva (1993) :

Kim (1995) and Khan (1995) have combined the relay with additional dynamic elements

such that the process is autornatically excited a t frequencies other tban the critical fre-

quency. In the case of Kim's method, a simple autotuner was proposed which uses a delay

element to operate the process at a specified phase margin. The dtimate gain and period

are directly used as PI parameters, without the need for W h e r application of tuning rules.

A more complex iterative scheme was developed by Leva, who used a low-pass filter and a

variable delay element to design a phase margin specified PID controller. Khan used two

relay experiments (with and without an integrator) in order to obtain hequency response

information at two different frequencies. fiequency domain tiining rules were subsequently

applied to these data points to determine suitable PID controller parameters. A more

complex tuniag method, designed to improve the performance of an existing controller,

Figure 1.3: Sample Data from the Astrom-~a&ünd Autotuner: d = f 2, c = 0.5

2.5

2

1.5-

was suggested by Schei (1992; 1994). Here, the relay is applied around the wcisting closed

loop system in two separate experiments (with and without an integrator in the loop). A

discrete transfer function is identified Çom the generated data and is then combined with

specifications on the maximum amplitude of the sensi t ivi ty and complementary sensi tivi ty

functions to yield new PID controller parameters.

- - - - - - -

I 1 I I 1 1 1 I I I 1 I 1 I I I

1 1 I I l 1

1.2.3 Describing function approximation

I 1 I I 1 I - 2 - - - L - - - - l !----a 1 , - -

-2.5 t 1 1 I I O 10 20 30 40 50 60 70

Time (sec)

Some typical relay feedback data is shown in Figure 1.3. It is generaily assumed that

the relay generates a periodic square wave output of amplitude, d, with a dominant first

harmonie component which corresponds to the ultimate period of the process. The limit

cycle induced in the process output variable is assumed to be a sinusoidal signal with the

same period. With these key assumptions, and a measurement of the output lirnit cycle

amplitude, a, the DFA provides a simple way to estimate the ultimate gain of the process:

The DFA is based upon a Fourier series expamion of the relay output. Consider the

square wave relay output, which is an odd function. This signal has harmonic components

at odd multiples of the fundamental frequency. The energy associated wit h each harmonic

6

decreases rapidly, giving rise to the assumption that the fundamental fkequency component

dominantes. The Fourier series expansion of the process input signal ~ ( t ) is written as:

The harmonic terms are then assumed to be negligible:

Similady, the limit cycle of the process output, y(t), is approximated as a sinusoid which

lags the input by -n rad:

y ( t ) zz -asin(wt) (1.4)

The ratio of these two sinusoids produces an estimate of the process response at the funda-

mental frequency:

Assuming that the fundamental fkequency of the limit cycle is equal to the critical frequency,

the negative inverse of the process response provides an estimate of the ultimate gain of the

process (Equation 1.1).

1.2.4 Alternative methods and improvements to the DFA

There are limitations associated with the use of the DFA. Firstly, for cases as cornmon

as a first order plus delay process, the key assumptions of the DFA break down (notice,

for example, that the process output response in Figure 1.3 is not very sinusoidal). As a

result, a great deal of research has focused on improving the accuracy of DFA. In addition,

the DFA is limited by the fact that it can only iden te one frequency response point per

experiment. The DFA's simplicity and popularity seem, in many cases, to have encouraged

research with the aim to reduce, rather than emphasize, harmonic frequencies which could

have otherwise been used to fit more complete modek.

Li et al. (1991) have reported convergence dificulties when attempting to fit transfer

h c t ion modeis wit h DFA generated frequency response est imates. Their convergence

probiems were attributed to the inherent inaccuracy in the DFA. In fact, Friman and Waller

7

(1995) recently reported that errors as large as 30% in the ultimate gain and 20% in the

ultimate per-iod codd result from anaiysing relay feedback data wit h the DFA. These authors

have eliminated the assumption of a dominant first harmonic by pretreating the signal with

an expression which isolates the first harmonic component of the output. Estimates of

the critical bequency were &O improved with the development of an iterative experiment,

which replaces the relay with a saturathg gain element in order to minimixe the phase

Iag contribution of higher harmonies. Sung and Lee (1995) have also improved the DFA

accuracy by modifying the experiment. They propose that a dit her signal be added to the

relay signal such that unwanted harmonic fkequencies are reduced. Hang et al. (1993) have

examined the effects of disturbances on the Iimit cycle and DFA estimate, and suggested

methods for disturbance detection and self-correction.

Severai papers have taken the opposite approach to improving the DFA accuracy. Rat her

t han changiag the experiment to improve the DFA accuracy, many have instead selected an

alternative data analysis method for the basic relay experiment. For instance. Astrom and

Hagglücd (Astrorn, 1988: Hagglünd and Astriim, 1991) have modified their original method

by applying waveform analysis to the process output and directly estimating a transfer

function modeI. This method is intended primarily for f is t order systems, since they are

most likely to violate the DFA assumption that the h s t harmoaic component is dominant.

However, it has been observed (Sallé and Astrom, 1991) that this metho$ is sensitive to

noise and difficult to apply to high order systems. Others, such as Chang et al. (1992) have

replaced the DFA with analytical expressions for the limit cycle. Wang et al. (1997) have

taken an approach similar to Chang et. al., but developed the method for use with a biased

relay experiment so that steady state gain information can also be obtained. Huang et al.

(1 996) point out t hat, while simple autot uning met hods are reasonable for Ziegler- Nichols

t uning , the performance of model-based controllers and integral performance cri teria are

sensitive to inaccurate estimates of the delay to time constant ratio. In this case, 6rst order

plus delay parameters are more accurately estimated using time domain expressions for a

biased relay experiment .

The Discrete Fourier ~ a n s f o n n (DFT) is an alternative which has been applied by

Lundh and Astrom (1994) in automatic initialization of adaptive control systems. Fast

Fourier Transform (FFT) methods are suggested by Chiu and J u (1997) for fkequency do-

main based performance monitoring of the complementary sensitivity funct ion. Khan (1995)

fit pararnetric transfer function models wit h fkequency response points t hat were ident ified

by ushg Frequency Samphg Filters to isolate the k t harmonic fiequency response of the

relay experhent.

1.2.5 Transfer fimction identification

Fitting a transfer function model normally requires an estimate of severai frequency response

points. Since the Astrom-~aglünd Autotuning experirnent provides oniy one point, the

identification literature contains many papers that look at ways to generate additional

frequency informat ion. These met hods collect mdtifi-equency data via a series of modified

relay experiments, each of which alter the excitation frequency through the placement of

different, known dynamics inside the feedback loop. For example, Haggliind and Astrom

(1991) have suggested the possibility of identifying multiple points on the Nyquist curve

through the addition of an integrator into the relay feedback loop. Individuai fkequency

response points are sequentially identified through repeated application of the DFA to each

periodic data set, and a transfer function may then be fit using these points.

In a few cases (Sdé and Astrom, 1991; Shen et al., 1996; Wang et al., 1997), unsym-

metrical or biased relay data has b e n used to provide additiond information about the

gain. Sallé and Astrom (1991) presented the method of moments as a means of estimating

the steady state gain fiom unsymmetrical, periodic relay data. The Dual Input Describing

Function, which is an extension of the DFA, has also been used for this purpose by Shen

et. al. Given unsymrnetricai periodic data, two frequency response points may be identified

(i.e. the gain and critical point). Considering that deadtirne is often estimated fiom the

initial process response, there can be enough information to fit a transfer function model

from a single modified relay experiment.

Luyben (1987) extended the A s t r o m - ~ a ~ ~ l ü n d autotuner to the multivariable case for

nonlinear distillation colilmns. This work required oniy one relay experiment, but assumed

that the process gain was aiready known, or could otherwise be obtained. This method

was developed further (Li et al., 1991), such that the gain requirement was replaced by

information korn a second relay experiment where additional delay was added to the loop.

In both of these papers, a series of possible model structures were applied to the data. Each

rnodel then had to be evaluated and compared with other models to deterrnine which was

best.

The above methods produce more useful transfer models, but require the user to either

impose a model structure (Wang et al., 1997; Sailé and Astrom, i W l ) , or to fit a whole

range of modeh (Luyben, 1987; Li et al., 1991; Chang et al., 1992) and then develop some

method for evaluating which one is the %ruen model. The main disadvantage with this

approach to m o d e h g is that, in addition to errors which may have ben introduced by the

DFA, and the effort involved in selecting the model structure, there is dways further risk

of inaccuracy due to model structure mismatch.

1.3 Outline of the Thesis

The remainder of this thesis is organized as follows. Chapter 2 discusses the role of the FSF

algorithm in the development of a new multifrequency relay experiment. Key design issues

surrounding the new identifkation procedure are explained in Chapter 3, where a selection

of simulation case studies are provided. A recursive least squares (RLS) implementation of

the FSF algorithm is introduced in Chapter 4. Hem, a prototype designed with LabVIEW

software, is also presented. This program combines the new relay experiment with the RZlS

algorithm to provide a methodology for on-line step response identification. Experimental

results obtained by applying this prototype to a pilot-scale stirred tank heater are also

included. Finally, conclusions and recommendations are given in Chapter 5.

Chapter 2

Frequency and Step Response Identification from Relay Data

An automated, closed loop approach to frequency domain identification is introduced in this

chapter. Frequency domain rnodel parameters, fkom which a step response model may be

constructed, are estimated via the Frequency Sampling Filter (FSF) algorit hm developed

by Goberdhansingh et al. (1992). The FSF algorithm combines a standard least squares

estimator with the FSF model structure, and requires a multifiequency spectrum of input

excitation which, for a large class of systems, rnay be generated automatically by a modified

relay experiment . This chapter is organized such that Section 2.1.1 describes the FSF/PRESS identin-

cation algorit hm. Section 2.2 explains the deveIopment of the proposed relay experiment

which places additional dynamic elements into the feedback loop in order to create a broad

spectrurn of excitation. Section 2.3 deah with the issues surrounding application of the FSF

algorithm to data obtained fiom the new experiment.

2.1 Frequency Sampling Filter Algorithm

The FSF model, fmt introduced to the area of process ident *cation by Bit mead and Ander-

son (l98l), is obtained fiom a Linear transformation of the familiar Finite Impulse Response

(FIR) model. Therefore, any linear, time invariaot process which may be described by N

F R coefficients rnay be represented equivalently by N fkequency response coefficents. The

FSF model is comprised of a series of N pardel band-pass filters which act on the process

input signal. The filter outputs are then weighted by their corresponding fiequency re-

sponse coefficients and summed to form a predidion of the process output. The FSF model

carries two important advantages over the FIR model. Firstly, the fkequency domain repre-

sentation allows for truncation of high bequency parameters, which often have a negligible

contribution to the ove rd process step response. In most cases, the reduced FSF model

order is much smaller thaa that of the equivalent FIR model. Secondly, the remaining

FSF model parameters are often estimated more accurately because truncation of the high

fkequency parameters improves the condit ioning of the least squares estimation problem.

2.1.1 F'requency sampling filter model

Consider a linear, time invariant, stable process which may be described by an Nth or-

der FSF or FIR model. For a stable process with settling time

characterized by the impulse response coefficients of the process

N-L

where r-' is the backward shift operator and for a given sampling

order, N, is selected as N = T,,JAt. It is assumed that hi O for

Tset, the FIR model is

i n t e d , At, the model

al1 i 2 N

The FSF model is derived by substitut ing the following inverse discrete Fourier transform

(DFT) into Equation 2.1:

Interchanging the siimmation signs and noting t hat

we c m describe the FSF model as

where r ~ k = 27rklN rad, k = O, f 1 , . . . , f ( N - 1)/2- A generd process model which describes the relationship between the input signal, u(t)

and the process output, y (t) , is t herefore

Figure 2.1: Schematic Diagram of the FSF Model

The schematic diagram of the FSF model structure in Figure 2.1 illustrates how the

input signal, u(t) , passes through N parallel band pass filters and is weighted by the process

frequency response, G(eJWk ) , at each filter% centre frequency, wk. The sum of t hese weighted

frequency components forms the prediction of the process output, y (t ) .

An important advantage of using the FSF model structure over the FIR model is that

the transformation fiom the time domain (Fm) to the frequency domain (FSF) pemits truncation of the mode1 order. When dealing with the fiequency domain model; a reduction

in the number of parameters corresponds to the exclusion of high fiequency filters. Model

reduction in this sense is beneficial, since we are only considering stable processes, whose

high fiequency response is strongly attenuated and often subject to an unfavourable signal

to noise ratio. The reduced FSF model is written as foliows:

Here, we have introduced the effective model order, n, with the assurnption that ~ ( d " )

is negligible for 9 < Ikl 5 9. The issue of how to determine the best model order is

addressed later in this chapter, when the PRESS algorithm is introduced.

2.1.2 Step response models

In process control applications, it is often desirable to express the process response as a step

response model, rather than a frequency response model. A step response model is e a d y

obtained by substituting Equation 2.2 into the definition of the step response coefficients,

gm, m = O, l..., N - 1:

Interchanging the summaticns, and not ing t hat

the step response coefficients may be expressed as a weighted s u m of the fiequency response

coefficients:

where

Wang and Cluett (1997) have shown that reduction of the FSF model order is justified by

examining the contribution of the high fiequency parameters to the overall step response

model. For the zero fiequency, S(0, m) linearly increases from 1/N to unity as m goes

fkom O to (N - 1). For k > 0, the step response weighting hinction, S(k , m), can be broken into two separate terms which operate on the real and imaginary parts of G(eJ").

The magnitude of both the red and imaginary weighting functions decreases as frequency

(k) increases. Clearly then, G(elwo), is the most important parameter, and subsequent

higher kequency response parameters becorne less and less significant. The fact that the

step response weighting functions de-emphasize the high fkequency parameters allows for a

significant reduction in the FSF model order without a significant loss of accuracy.

2.1.3 Formulation of the least squares problem

Consider again Figure 2- 1, the block diagram of the FSF model structure. The set of filter

outputs at sampiing instant t form the elements of the data vector #(t) , d e h e d as

for the reduced FSF model. More specificdy,

where 4k(t) represents the output of the kth FSF filter, Hk(z):

A weighted sum of the elements in 4 provides a prediction of the process output as

foUows. A general process model may be written as

where v(t) represents the disturbance effect. The process may be expressed as a reduced

FSF model by combining Equation 2.14 with Equations 2.6, and 2.12:

If the data set spans M sampling intervals, then Equation 2.14 may be expressed in

matrix form as:

Y = @ Q + V

w here,

Therefore, a standard least squares estimator, based on the data matrix, a, and the process output signal, Y, yields a set of n parameters which correspond to each 6iter7s

contribution to the overall step response. These weights are contained in the parameter

vector, 8, and are simply the frequency response coefficients, G(e""k), which collectively

describe a kequency response model of the process.

Equation 2.16 is thus solved for 0 by minimizing the sum of squared prediction errors,

This yields the standard least squares solution of the FSF parameter vector:

2.1.4 PRESS statistic for mode1 order selection

The PRediction Error Sum of Squares ( P R E S S ) statistic is used as an indicator of model

quality. It is defined by 1 w - 1

PRESS = 1 e2, where e,t are the PRESS residuals:

The PRESS residuals, e-t7 are computed boom &, the FSF parameter vector which

waç estimated by excluding the data y ( t ) and $(t). This data is excluded from the analysis

to ensure that y ( t ) and ij(t) will be independent of each other, a property of the PRESS

which is essential for model validation. By contrast, the conventional residuals are defined

Since y(t) and &(t) in this case are dependent variables ( y ( t ) and $(t) are used to estimate

8), the conventional residuals are merely an indication of the quality of the model fit for

16

the rnodelling data set and do not imply anything about the model's overd predictive

capabiiity. The PRESS statistic is evaluated for a range of Werent FSF model orders,

and the reduced model order, n, is chosen such that the PRESS is minimiad. Wang and

CIuett (1996) have shown that computation of the PRESS residuals is simple when using

an orthogonal decomposition algorit hm.

2.1.5 FSF/PRESS algorithm

With the standard least squares algorithm in Equation 2.22, it shouid be noted that the

parameter estimates WU be consistent if the disturbanceo v ( t ) , and input. u(t ) : are uncor-

related. This is generdy not the case for closed loop experiments. However, in the case of

a relay experiment, a hysteresis is typicdy included in the relay to remove this correlation.

This technique wiil usudy work if the disturbance is stationary. However, in the presence of

drift ing disturbances, correlation may be present , causing bias in the parameter estimates.

For these reasons an extension of the original algorithm proposed by Goberdhansingh et al.

(1992) was developed by Patel et al. (1997). This algorithm is an iterative version of the

FSF/PRESS algorithm which simultaneously estimates process and disturbance modeis.

The algorithm is outlined as follows:

Step 1: Prefilter the inputfoutput data with the inverse of the curent noise mode1 (skip

this step the first time through)

Step 2: Estirnate the FSF parameters using the prefiltered data (similar to the original

algorit hm)

Step 3: Estimate a new noise model £iom the residuals

Step 4: Repeat steps 1 through 3, until the model has converged

Since this algorithm includes proper data prefiltering, the resulting model is assumed

to be unbiased. This dows statisticd confidence bounds to be calculated based on the

cowiance of the parameter estimates. Based on the assumption that the fltered residuals

are white, Cluett et al. (1997) have derived expressions for the confidence bounds around

the model estimate within which the true fkequency and step response lie.

2.2 A New Relay Experiment

In Chapter 1, it was mentioned that additional dynamic elements could be introduced into

the relay feedback loop to influence its behaviour. Relay dynarnics wil l now be reviewed,

and used to motivate the development of a new relay feedback experiment to create a

nonperiodic, multifrequency input signai that is well suited to the estimation of an FSF

model. This experiment is simple to implement, requiring almost no prior knowiedge of

the process to be identiiied. It should be noted, that with the new experiment, model

estimation is not restricted to the FSF model structure. The data generated kom the new

experiment could be used with any other model structure.

2.2.1 Relay dynamics

To try and obtain mcre information than just the critical point of the process, many varia-

tions on the Astrom-~a&ünd autot uning experiment have been proposed to provide addi-

t ional hequency informat ion. T hese met hods typicaily involve a modioied relay experiment ,

in which the excitation fiequency is altered through the introduction of a known dynamic

element into the feedback loop.

Figure 2.2 illustrates how some common dynamic elements inauence the fiequency of

the process response. The standard relay experiment excites the process at its critical point,

which is located at the intersection of the Nyquist plot and the negative real a&. This

corresponds to a phase iag of 180 degrees. In practice, a hysteresis is normally included in

the relay to avoid random switches generated by the output noise. The hysteresis produces

excitation at a slightly lower frequency than that observed with a pure relay. Similarly,

the use of additional delay (ive. in addition to the natural deadtime of the process) in the

feedback loop also shifts the frequency response to a point below the critical frequency. Both

delay and hysteresis allow the excitation fiequency to be lowered arbitrarily. according to

how much hysteresis or deiay is applied. In contrat, the addition of an integrator produces

a f ied 90 degree phase shift, such that the identified point always lies at the intersection

of the negative imaginary axis and the Nyquist plot.

Standard Relay Hysteresis rn Delay A Integrator 4

Figure 2.2: The Infiuence of Additional Dynamics on Excitation F'requency

2 - 2 2 Proposed experiment

Both the standard relay and integrator plus relay experiments are attractive because they

are able to generate fiequency information at specific points on the Nyquist plot. Therefore,

in the development of the new experiment it was decided that the standard relay and inte-

grator plus relay experiments would be combined in order to ensure a good rnultifrequency

input signal that did not require prior knowledge of the process. The fixed phase shift

obtained from an integrator ensures that the low fkequency range is targeted. While other

dynamics (e. g. delay, hysteresis) could be used to accomplish similar effects, it is difEcult

to determine in advance the quantity of delay /hys teresis required to produce the desired

effect.

In a standard relay experiment, the switches are triggered by zero-crossings of the error

signai. When an integrator is introduced to the loop, the relay is instead triggered by

zero crossings of the integrated error signal. The proposed experiment combines t hese two

individual experiments by alternating between the two different paths indicated by the

dashed lines in Figure 2.3. In other words, the new input signal is generated by alternating

Figure 2.3: Block Diagram of the Proposed Experiment

between having some input switches triggered by the feedback error, and ot hers triggered

by the integrat ed feedback error.

The overall effect of turning the integrator on and off (Le. alternating between the

integrated and non integrated paths) is to create a nonperiodic input signal and, in a sense?

"sweep" through a portion of the Nyquist plot (see, for example, Figure 2.4(a)). This input

(relay output) is seen to be a binary signal which consists of a combination of faster pure

relay switches and slower integrated relay switches, plus some transient switches of ~ s y i n g

lengths. The standard relay experiment predominantly yields information about only a

single fkequency because the data ( s e Figure 2.4(b)) is periodic. On the other hand, the

new relay experiment yields a data set that contains information at several hequencieso and

is suitable for anaiysis with the FSF algorithm.

2.2.3 Experirnental design

At fist glance, the new input appears to be much like a random binary signal (RBS).

However, in the new experiment, the relay output is automaticdy triggered according to

the process response. In fact, the experirnental design reduces to the specification of when

20

tirne(s8c)

(a) Nonperiodic Data Erom the New Experiment

(b) Periodic Data fkom the Standard ReIay Experiment

Figure 2.4: Cornparison of Data Generated fkom the New and Standard Relay Experiment

and when not to apply an integrator. Ln other words, the user is only required to specify a

sequence of boolean values which indicate whether or not the integrator is present during a

particular switching interval. Letting 1 denote the presence of the integrator, and O denote

the absence of the integrator, a sample input sequence is [O O O 1 1 1 1 1 O 1 1 O 1 O 1 O O].

For the most part, any input sequence is reasonable, so long as it contains a mixture

of zeros and ones. That having been said, there are a few guiding principles. Recall,

that the main objective behind the input sequence is to automaticdy produce a process

input signal with a bmad spectrum (Le. a nonperiodic signal), and emphasis on the lower

hequencies. Conventional relay experiments give only one fkequency point because they

discard transient data and analyse only the periodic portion of the data set. The new

relay experiment instead tries to maximize transients which create a diverse multifiequency

spectnun. The new experiment ais0 places an emphasis on obtaining good low fkequency

information, which is important for obtaining a step response mode1 with an accurate

estimate of the steady state gain.

For exampIe, a series of consecutive integrated switches can, on its own, provide a

great deal of fiequency information. The reôson for this is that, during the first three to

six integrated switches, there is a transient period where the switching interval gradually

increases. The initial switches therefore provide a range of middle and hîgh fiequency

information and when the transient dies away, a low fiequency limit cycle results. Hence,

to emphasize the lower fkequencies, it is necessaq to ieave the integrator in the loop for

at least three to six switching intewctls. Zn the absence of the integrator, the transient

period is brief (usuaily no more than about two switches) and the relay quickly settles into

a high kequency limit cycle. Series of non-integrated switches should therefore be present,

but perhaps fewer in nirmber, due to the fact that they produce only a smaU amount of

transient data, and generally provide less important high frequency information. Additional

diversity in the switching intervals can be obtained by alternating frequently between the

two possible feedback paths. The feedback path is changed only as the error or integrated

error signal crosses the hysteresis Limits. When this changeover occurs there will be a

discontinuity in the signal used to trigger the relay. This can create some extra variation

in the switching frequency and further enrich the input spectrum.

A set of input sequence guidelines are now presented to siimmarize the above ideas. Note

that these guidelines are not based on any prior knowledge of the process to be identified,

but rather, they are based on the idea that, for stable processes, it is generally desirable

both to create a wide range of spectral information, and to emphasize the low fkequency

components that are essential for a reliable estimate of step response model.

Input Sequence Guidelines:

* Empbasize the presence of the integrator

rn Avoid long series of non-integrated switches (no more than three switches in a row)

rn Use long series of integrated switches (three to six switches in a row)

0 Include a series of mixed switches (i. e. [ 1 O 1 O O 11)

2.3 Application of the FSF Algorithm to Relay Feedback Data

The conventional approach to analysing periodic relay data involves application of the DFA

to identify a single point on the Nyquist plot. As described in Section 1.2, this type of

approach to obtaining a transfer h c t i o n model may involve several relay experiments and

may be subject to various assumptions, approximations, and model fitting criteria. Since the

data collected from the new experiment contains signincant multifrequency information, the

DFA is no longer an appropriate anaiysis tool. However, through direct application of the

FSF algorithm from Section 2.1.1, a frequency response model may be accurately identilied

and used to construct a versatile step response mode1 of the process. The only information

required to construct the FSF model is an informative data set and an estimate of? N : the

process settling time. This section will verify that the modified relay experiment is able to

automatically provide the informative data set.

2.3.1 The desired input spectrum

The new experiment automatically generates the process input signal by using the re-

lay to detect appropriate switching intervals £rom the observed process response. Fig-

ure 2.5 presents simulation data fiom two ditferent processes using the same input sequence:

[O O 1 1 1 O 1 O 1 O O 11. The shape of the two data sets are very similar, t hough experiment

(b) runs about three times longer than experiment (a). Clearly, each process in conjunction

23

- O 10 20 30 40 50 60

time (sec)

(a) Process ModeI: G(s) = &

-21 I , I I I O 50 100 150 200

time (sec)

e-l. (b) Proceis Model: G(s) =

Figure 2.5: New Relay Experiment Applied to Two Different Models with Identicai Input Sequences: [O 0 1 1 1 0 1 0 1 0 0 11

with the relay experiment has created an input signal specific to the time scde of its own

dynamic response. The question rem- as to whether or not the generated input signal

fkom this generic input sequence produces a range of fkequency information that is desirable

with respect to ident-g an accurate FSF model.

R e c d from Section 2.1.1 that the key parameters of the FSF

set of harmonic fkequencies

27r 47r 67r Sn (n - l )?r wk = O? &-, &-, k-, *-?. . . *

cet Tse t Tse t Tset T'et

model correspond

radltime

to the

(2.28)

The first few FSF frequencies are most significant in terms of their contribution to the

process step response. Moreover, in terms of control applications, a good estimate of the

process frequency response at w t = O (i.e. the steady state gain) is important. The ideal

FSF input signal is therefore one which has good kequency content at each wk in Equation

2.28, and ernphasizes the spectral components which correspond to the centre frequencies

of the first few filters.

Consider the Nyquist plot of Figure 2.2, whieh previously iilustrated the fkequency

response points that are identifiable under relay feedback with and without additional dy-

namics in the loop. Notice that the point identified by the integrator-plus-reIay experiment

is always Iocated at the intersection with the negative imaginary axis, and the point identi-

fied by the pure relay is located at the intersection with the real axis. It has been observed

through a number of case studies (Chapter 3) that the centre fkequency of the first FSF

Hter, wl = 2n/TSet, often falls in the neighbourhood of the negative imaginary axis. Inte-

grated switches therefore serve to emphasize low frequency content in the frequency range

picked up by the first filter. Higher FSF frequencies, which are distributed dong the Nyquist

plot towards the negative real axis and beyond, correspond to the fkequency information

obtained fkom transients and plain relay switches.

2.3.2 Properties of the FSF correlation matrix

In order to obtain an accurate mode1 it is essential that the data set be informative at aU of

the "important" frequencies. F'ortunately, the correlation matrix, a* a, offers an indication as to the quality of the input excitation. Recd that the kth col- of the data matrix, @

(see Equation 2.19), corresponds to the output of the kth FSF filter, whose centre frequency

I k Ith Fitîer

Figure 2.6: A Typical FSF Input Spectrum Obtained fiom the Proposed Experiment: input sequence [O O O 1 1 1 1 O 0 11

is wk. The diagonal elernents of @*a are representative of the amount of input signal energy at each of the important FSF fiequencies because they are proportional to ( U ( e J W k ) ( * , the

periodiogram of the input signal at those fiequencies (Wang and Cluett, 1997) :

A plot of these diagonal elements (the "FSF spectrum") provides an approximation of

the input signal spectnun at the fkequencies Wk7k = 07kll&Z, ... ? i ~ ( n - 1)/2. The FSF

spectrum will prove to be a useful indicator which can be used to assess and, if necessq,

shape the input signal in order to obtain the desired fiequency content.

A standard relay experiment generates periodic, first harmonic dominant process input

data. In this situation, there is not enough diversity in the input spectrum to support more

than about one or two pairs of FSF filters. The input signal in this case would produce a

@*@ matrix with essentially only one dominant diagonal element.

In contrast, the new experiment7s input spectrum is rich over a wide range of fkequencies.

Figure 2.6 illustrates the shape of a typicd FSF spectnrm that results kom a mixed input

sequence. Strong spectral components in the low and middle fkequency range, corresponding

to large diagonal elements in the @*@ matrix (especially at the filters corresponding to

lkl = l? 2)? con6rm that the excitation is broadly distributed.

2.3.3 Choice of the parameter iV

The nature of the FSF mode1 applied in this work ailows the user to fit a mode1 with only

a minimum amount of prior knowledge about the process. The only required information

in this approach is an estirnate of N (the settling time in units of sampling intervals).

More importantly, this single piece of information is only required after the data has b e n

collected, allowing the user to adjust this parameter off-line to find the %estr results.

Another approach is to try and estimate N directly fiom the experiment itseif. The

reader will r e c d fiom Section 2.2.3 that the integrator plus relay switches have a long

transient period that often settles, after several switches, into a E t cycle with a period

of approximately N sampling instants (fiequency of 2x /N rad). A reasonable starting

estimate of N/2 can therefore be taken as the duration of the third or fourth consecutively

integrated relay switching intenml.

2.4 Experimental results

Thus fa., the new identifcation method has b e n motivated and discussed in a qualitative

sense. Preliminary results are presented here to summarize the above discussion and demon-

strate the method's capabilities via an experimental case study. This example is intended

simply as an illustrative result. huther simulation work and experimental results will be

presented in Chapters 3 and 4 to provide a more thorough evaluation of the method.

The new relay experiment was performed on a pilot scale stirred tank heater (STH)

apparatus (see Chapter 4 for more details on the STH process). This process consists of

a continously stirred water tank which contains steam coils. Cold water enters the tank,

is heated by the s t e m coils, and is then purnped out through a long piece of piping. A

Ievel controller maintains the tank level at a steady d u e (22cm in this experiment) by

manipulating the inlet water flowrate. The STH has approximately first order plus delay

dynarnics, where the manipulated input is the steam valve position, and the process output

is the temperature of the water exiting the tank, measured by a thermocouple placed in

the outlet pipe. Experimental results were obtained by setting the relay amplitude and

hysteresis width to be d = f 5% and t = 0.25"C, respectively. The input sequence was

[ i l 1 i O l l O O ] .

Figure 2.7 shows the input/output data fkom t his expriment. Approximately three

settling t h e s of data were coilected for analysis with the FSF/PRESS algorithm described

in Section 2.1.5. The settling tirne of the STH was initiaily estimated as N = 600 sec, based on the duration of the third consecutive integnted switching interval, which begins

at about t = 350 sec (the k t switch was so rapid that it was neglected). This estimate of

N yielded a step response estimate that seemed to settle out a Little faster than 600 sec,

and so N was adjusted to 570 sec. The diagonal elements of the correlation matrix (see

Figure 2.7(b)) were exaLzLined to evaluate the frequency content of the input signal. The

FSF spectrum seems to have a desirable peak in the Low frequency range, as well as good

information at aU fiequencies. The mode1 was restricted to a maximum of eieven filters

(n = 11). The fkequency and step response models shown in Figure 2.8 were found to be in

good agreement with expectations based upon past experience with the process. The 99%

confidence bounds on step response also indicate that the mode1 is reasonably accurate.

It is &O interesting to note that, in Figure S.B(a), the FSF fkequencies, wl and wl, are

located quite close to the negative imaginaq and negative real axes, respect ively. This shows

t hat one of the key assumptions of the experimental developrnent holds for t his process (i.e.

that the integrator plus relay and standard relay experiments generate information in these

regions). En ter- of the step response developrnent, these kequencies are two of the most

important components of the input signal. The transient data, obtained by combining

the relay and relay plus integrator experiments into one new meriment: &O provides

additional information about the gain and higher kequencies. The results show t hat , overail,

the new relay experiment provides good excitation for step response identification in this

case.

1 1 1 1 I L

O 200 400 600 800 Io00 1200 1400 1600 Time (sec)

(a) Input/Output Data

t 2 3 4 I k Ith Aller

(b) The FSF Spectrum

Figure 2.7: Experirnental Results for the STH

I 1 I L I I 1

O 0.1 0.2 0.3 0.4 0.5 0.6 Red

(a) Fkequency Response Model (Nyquist plot)

(b) Step Response Model

Figure 2.8: Identification Results for the STH

Chapter 3

Further Design Issues and Case St udies

Introduction

This chapter will address some firrther design issues for the new relay experiment, such as

the effects of deadtime, noise modelling, and choice of the FSF parameter, N. Each of these

issues will be separately motivated and discussed. The ha1 part of this chapter presents

several illustrative case studies, from which many of the key design issues were originally

ident ified.

3.2 Characteristics of Processes with High Critical F'requen-

The relay signal is triggered according to how quickly or slowly the process variable responds

to the input signal. It is this characteristic of the relay which provides us with a closed

loop experiment, and produces an input signal that depends on the particular process under

examination. The new relay input experiment has been designed to generate oscillations

which occur at the crossover frequency and at the fkequency where the Nyquist curve in-

tersects the negative imaginary axis. For some processes, the new experiment will generate

very fast oscillations t hat may not be useful for obtaining an accurate step response model.

One class of systems that produce this kind of behaviour are fist order plus delay processes

where the deadtime is s m d relative to the time constant.

Consider the simulation results presented in Figure 3.1. These data sets were obtained

using the same input sequence on two s i d a r first order models. Note that the time con-

-2 1 I I I 1 1 O 200 400 600 800 1ûûû 1200 1400

Time (sec)

-21 t I I 1 O 500 1000 1500 2000 2500

Time (sec)

Figure 3.1: New Experiment Appiied to Swo Fust Order Models with Dinerent Deadtimes (input sequence [O O O 1 1 1 1 O O 1 0 1 1 1 O])

Figure 3.2: A Cornparison of FSF Spectra for Data in Figure 3.1

Figure 3.3: The

5 sec deadtirne 25 sec deadtirne 70 sec deadtime

-0.8 I 1 -0.5 O 0.5 1

Real

Effect of Deadtime on the F'requency Response of First Order Relay Dy- namics (AU time constants are 100 sec. The FSF frequencies are shown by o,*,x,+.)

stants for these two processes are identical (100 sec) and their settling times are close (525

and 570 sec). The only difference between these models is the amount of deadtime. Based

on the assumption that the relay switching intenml is related to the time scale of the process.

it might be expected that these two experiments would yield similar data and, therefore,

similar FSF spectra. Yet, a comparison of the two data sets in Figure 3.1 shows a remark-

able difference between the duration of the two experiments. Also notice the Merence in

the shape of the two graphs of Figure 3.2, which show the distribution of input energy across

the FSF filters. Figure 3.2(b) (the more sluggish process) has most of its energy at the h s t

and second pair of filters, while Figure 3.2(a) has more energy at the second and third pairs

of filters. The overall energy distribut ion is also quite different. The input energy generated

by the slow process is concentrated at the low kequencies, wMe the faster process creates a

spectrum that is richer across a wider range of fkequencies. Obviously, the only factor that

can account for this unexpected behaviour is the amount of deadtime in the process.

The reason for this is that the crossover fiequencies for these processes are very different ,

and therefore yield very dinerent relay generated excitation. This effect is studied more

closely in Figure 3.3. These plots represent the frequency response of a first order process

with w i n g amounts of deadtime. As a reference, the location of the first several FSF

frequencies have been indicated on each plot. As deadtime decreases fkom 70 to 5 sec,

the crossover frequency moves kom the neighbourhood of w:! to the neighbourhood of wzs-

The case with no deadtime at ail, WU not oscillate under relay feedback because, even at

very high fkequencies, a pure k t order process never attains a phase lag of 180 degrees.

Since deadtime contributes to the phase lag, it is instrumental in determinhg the crossover

frequency. As deadtime increases, the crossover is pushed to a lower frequency range, and

the new relay input signal behaves more idedy with respect to the frequency content that

is desired for step response identiiication. In addition, the magnitudes of the Nyquist plots

in Figure 3.3 also predict that processes with high critical fiequencies will tend to respond

to a relay experirnent with s m d output amplitudes.

Clearly then, there is a concern that a fust order process with srnail deadtime might

generate an input signal that is deficient in low Erequency information. It is therefore

necessary to consider these processes in more detail, with the idea of modifjring the relay

experiment so that it wiU provide good low fkequency excitation for this special case.

3.3 The Influence of Delay on the FSF Spectrum

3.3.1 First order processes

The previous example has shown that process deadtime is an important factor in deter-

mining the type of input excitation obtained from the new reIay experiment. This section

analyses the iduence of delay in more detail. More specifically, the following results will

demonstrate that it is ac tudy the nonnalizeà deadtirne, R, defined as the ratio of the

deadtime and the time constant, which determines the fkequency content of the excitation

obtained fkom the new experiment . Consider a first order plus delay process described by the transfer function

The settling time of this process is estimated as

At this stage, we select an arbitrary frequency, W i , definecl by

where i , referred to as the frequency index, can take on the vdue of any red number.

Combinîng Equations 3.3 and 3.2, fiequency may now be given in terms of the norrnalized

deadtime, R = T&:

The phase shift of the process at this arbitrary frequency is therefore

Clearly, phase shift in this case is a function of two variables: frequency (determined by

i) and normalized deadtime (R). Since a process under relay feedback oscillates with a h e d

phase shift, R is the o d y variable that determines the oscillation frequency index number,

i. Therefore, by taking the phase shift as -r rad, and given R as a property of the process,

Equation 3.9 can be solved for the crosswer fiequency index number. This explains why

the relay swi t ching fiequency is de termined soley by normalized deadt ime. S imilarly, the

frequency index number for the integrator plus relay experiment can be solved from the

following equat ion:

In Sections 2.3.2 and 2.3.1, it was establiçhed that a desirable input signal shodd con-

tain strong information around the first two FSF filters. For the new experiment, i t is

therefore desirable that the fundamental fiequency of the integrator plus relay experiment

will be approximately equd to the fbst FSF frequency. With respect to Equation 3.10, this

means that if the index number, i, for the integrator plus relay experirnent is less than or

approximately equal to one, then the input signal generated by the new experiment wiil

contain sufEcient low fiequency excitation. A range of R d u e s for which this assumption

holds can be established by combining the phase shift of the first FSF frequency,

relay+integrator

- - ideal (FSF) period

Figure 3.4: A Cornparison of the Observed and Ideal Normalized Period for First Order Models

with the phase shift of an integrator plus relay experiment (Equation 3.10). That is, an

"ideal" R value is one which satisfies

Although an approximate range of ided R values could be established by solving Equa-

tion 3.12 numerically, we choose here to graphicdiy dernonstrate the range of ideal R values,

for which the actual and desired oscillation periods are similar. Let Pl be the desired oscil-

lation period associated with the fundamental FSF frequency, wl = 27~/T,,~:

A general relationship between R and the desired normalized oscillation period may be

In order to identify the range of R values for which a relay feedback system produces

desirable excitation from the perspective of an FSF model, a number of simulations were

performed using first order models with R values raaging kom O to 2. These simulation

results are shown in Figure 3.4, dong with the dashed curve representing the desired nor-

malized oscillation period that is d e h e d by Equation 3.14. Consider the integrator plus

relay curve, compared with that of the desired normalized oscillation period. The curves

intersect at R = 0.6, indicating that for R < 0.6 the relay pius integrator experiment wiil generate an oscillation with a period that is less than the desired period (Le. an oscilla-

tion that is faster than the ided FSF &equency). For R > 0.6, the observed oscillation is slightly slower than the desired FSF fiequency, and thus gives adequate infurmation about

the low frequencies. The fact that the standard relay curve lies beneath the desired curve

for all R dues, confirms that the integrator is essentiai for obtaining good low frequency

informat ion.

Now that R has been isolated as the determining factor for the fiequency content of

the relay signal, it is clear why the FSF spectra varied disproportionately between the two

examples presented at the beginning of this chapter (Figure 3.1). The R values of these

examples were 0.25 and 0.70. Referring back to the spectral distributions in Figure 3.2, it is

ciear that the one with the smaller R value of 0.25 oscillated too quickly and was therefore

not rich in kequency content at the fkst filter. In contrast, the process with the Iarger R

value of 0.70 produced more low kequency information near the centre frequency of the

fkst filter.

3.3.2 Second order processes

Up until now, this chapter has dealt exclusively with 6rst order processes. If desired,

the andysis could be extended to include other types of processes such as second order

systems. This is not necessary though because higher order processes typically react to

change with a more sluggish initial response than fnst order systems. For instancet before

delay effects have even been included, the phase versus fkequency profile of a pure second

order model drops off much faster than it does for a first order model. This means that the

critical fkequency of a high order process is naturdy located in a lower fkequency range-

Therefore, it is not surprising that most processes are not nearly as sensitive to deadtirne

issues as in the h s t order case.

To c o b this statement, consider the observed and ideal periods for the criticdy

damped second order case, which can be derived similady to the first order case above.

By redefining the process settiing tirne as Tset x 77 + Td, Equations 3.14 and 3.12 can be

relay + integrator

- - ideal (FSF) period

01 I 0 Y 1 I I I 1 O 0.2 0.4 Ob 0.8 1 1.2 1.4 1.6 1.8 2

Nomaiiied Deadtirne, R

Figure 3.5: A Cornparison of the Observed and Ideal Normalized Period for Second Order Critically Damped Models

redefined for a second order criticaily damped process:

The results in Figure 3.5 show that, for this case. the h t FSF fkequency fi& between the

standard relay and relay plus integrator curves for a.li values of R examined. Except for

very s m d R values, the integrator plus relay oscillations will occur at a kequency t hat is

slightly lower than desired. Thus, for ahost all cases, the critically damped second order

model wiil generate good data for identification of an FSF model.

While it is important to recognize the issues presented in this chapter, it should be

emphasized that a large class of processes will satisS. the spectral guidelines as they were

presented in Section 2.3.2. However, for special cases where the critical frequency is high.

there is sometimes a need to consider a modified experiment which can enhance the low

frequency region of the input spectrum.

3.3.3 Effect of relay parameters on the FSF Spectrum

In practice, it is desirable to avoid any correlation between the process input and the noise.

Therefore, the hysteresis widt h is normdy selected to be about three times the standard

deviation of the noise. In addition, the relay amplitude is usually rninimixed in order to avoid

disniption in the plant. However, it is important to consider that, if the amplitude is too

s m d , there wiil be a poor signal to noise ratio and the presence of valve nodinearities may

becorne significant. Therefore, the relay amplitude should be set as srnail as is reasonably

possible given the constraints.

Fortunately, a conservative choice of relay parameters can often prove beneficial in terms

of providing sdiicient low fkequency e x i t at ion. The combinat ion of small relay arnplit ude

and large hysteresis width pushes the limit cycle to a lower fkequency. This effect is most

pronounced in processes that actually require additionai low fkequency excitation. As an

illustration of this effect, the simulations used to generate Figure 3.4 have been rerun with

an increased hysteresis width and are presented along with the original results in Figure

3.6. It is clear fiom the figure that an increased hysteresis width produces a noticeably

slower relay limit cycle for processes with small R values. As R increases, or if an integrator

is added, the effect of hysteresis becomes almost negligible. Consequently, it is generally

recommended that the relay parameters be chosen conservatively, so that for process with

srnail R values, the low frequency information wiil be enhanced.

These recommendations are confirmed in theory by the DFA of a relay of amplitude, d?

and hysteresis width, é:

The negative reciprocal of the DFA is a straight line parallel to the real axis:

The operating point of the limit cycle is located at the intersection of the process Nyquist

curve and t his line. The imaginazy part of Equation 3.18 t herefore determines the phase shift

created by the combined effect of the hysteresis and relay amplitude. Therefore, decreasing

the relay amplitude and/or increasing the hysteresis width is a simple way to enhance the

low frequency region of the spectrurn.

Figure 3.6: The Effect of Hysteresis on the Ultimate Period

3.4 A Modification to the New Experiment

As illustrated above, a conservative choice of relay parameters can help a process with a

small R value with respect to obtaining sufficient low frequency excitation. However, since

there are practical limits on how much these parameters can be adjusted, it is also worth

considering a modScation to the new relay experiment. A modified experiment is now

proposed which d o w s the excitation frequency to be manipulated via the addition of a

user-specified delay block into the feedback loop (see the block diagram in Figure 3.7). The

delay block is placed after the relay, in series with the process, so that the relay sees a

modified "process" (indicated by the dashed box in the block diagram) with an R value

that is larger than that of the real process. Fkom an identification perspective, the data

used to construct the FSF mode1 is still the actud process input and output signals.

A general approadi to implementing the modified experiment is given in the flowchart

of Figure 3.8. It is proposed that a few settling times of data be initially collected according

to the original proposal for the new relay experiment (i.e. no delay block is present). At

this point, the data may indicate whether or not additional delay is required. If the data

seems rich enough in low frequency information, the experiment may continue as usual.

Otherwise, the initial part of the data set may provide good medium and high fiequency

information, and the remaining part of the experiment can focus on using the delay block

Figure 3.7: Block Diagram of the Modified Relay Expriment

to obtain more low fkequency information.

There are several qualitative characteristics that are indicative of a process which might

benefit Çom the modified experiment. Rapid switching of the relay output is the most

obvious indicator. In this case, a s m d process output amplitude would indicate that,

either the process gain is quite small, or the switching frequency is too fast. For example, in

the case of a first order process, a switching frequency that is too fast c m often be identihed

by the sawtooth-like shape of the process response. If a good prior estimate of the settling

time is available, then the FSF spectrum itself will indicate any weak frequency components.

Finally, if the tail end of the estimated step response mode1 seems to wander or appears

"droopy" , then the spectrum may not have enough low frequency information. A drooping step response may &O be an indicator that the settling time estimate has been chosen too

large; but if it is reduced, and the step response still does not settle out properly, then the

relay is likely switching too fast and not generating sufficient low frequency information.

If needed, the delay block should be added such that the delayed integrator plus relay

switching in tends are approximately one half to one quarter of the settling t ime. Since this

I Start the new relay expriment with the basic mixed input sequence (Le. something like [ 1 1 1 1 0 0 1 0 1 ..-] ) 1 *

After 1 or 2 settiing times of data are coltected check to see if more Iow frequency information is necessary: - is the input switching npidly? does the p m e s s response have a srnail amplitude and resernble a choppy sawtooth signal? - is it known chat the process has a smdI amount of delay relative to the time constant?

1

Decrease amplitude andfor increase hysteresis. Cs Iow frequency information still needed?

YES

Continue with the mixed input sequence until enough data has k e n collected.

NO

With the inregraior acrivared. invoduce additional defay for 3 or 4 switches. Adjust delay until: - the switching interval is mughly half the settling time. - the pmcess output amplitude increases. and appears Iess choppy or sawtooth-Iike

If the delay is satisfactory. but more data is still required. then hold the additional delay constant. while returning to the mixed input sequen