Frequency response matching methods for the design of digital control systems

FI¿DQUDNCY IùESPONSE MATCHING METHODS

FOR,

THE DESIGN OF DIGITAL CONTROL SYSTEMS

Jianfei Shi, B.E.

Thesis submitted to

Department of Electrical and Electronic Engineering for the

degree

of

Master of Engineering

^Science

The UniversitY of Adelaide

July,

1984

0''ttla-.'t.¡{e

¡/

^: _'; ^- f

i'-

^,!i

'l This thesis

^embodies

the results

^of

supervised

project

work makìng ^uP two

thirds of the

work

for

^the

degree.

J.

SHr,

'\1 'r' r..\:r-:_ ^-

':/'

I\"

.-

ll, \, l:

' .{,. ,/tÎ

EXAMINERIS REPORT OF M.E[íG.SC. THESIS;

ItFrequency reaponse matchtng methods for t,he deeign of digital control systemsrr

ERRATA OR OMISSIONS

the relatíon for Èhe zero Zy ln terma of other parameters fs omitted. It is

zL- I *

tan(q+9oo)(r2+n2-n¡

I-Èan(q+90 ) ( r-R) p.3-8 :

p.3-10:

Fourth

line, (correction)

U

-

(C-A)(D+1)-Bc

p3-12:

The term c2

+

^(n-L)Z

pA-lO:

pA-15 to

pA-28

pA-16:

4D

under

the

square rooÈ

sign in

both expressions should be

posftive,

not. negative.

Eqn D-l6.

The numerator Èerm should be

A-8, not

A*8.

The omfssion

of

values

for the

normalised parametere

E./T

^(C/L

sEep t,ime-response) and

ø"T

^(O/L frequency-response) render Èhe

sets of speciffcat,lons fn

each

of

^Èhese domaíns ÍncompleEe.

I

recommend Èhat

the tables for

damping

ratio

€=0.7 ^be recalculat,ed

to lnclude

these

values,

and

the results

be

fncluded

with

an

errata

sheet

ín

Èhe copfes

of the thesis.

The varlablee E" and ø" should be deffned under Ehe appropriate headfng.

I I

I

l

I

I

'

I

i f

\ ,t

,II

T

: Ii

¡

DEDICATED TO

MY PAR,ENTS

TABLE OF CONTENTS

Abstract Declaration

Page

vii ix

x xi xiv xv Acknowledgement

List of figures List of tables List of symbols

Chapter

l.

INTRODUCTION

l.l

Digital control systems

1.2

Advantages of frequency response methods

1.3

Classification of frequeucy ræpotrse methods f

.4

Frequency response matching design methods

1.5

^Summary of the current frequency matching ^design methods

1.6

Objectives, approaches and achievements

L.7

^{The contents}

2.

^{TREQUENCY RESPONSE} MATCHING DESIGN METHODS

2.1

Formulation of the desigu problem

2.2

Dominant Data Matching design method (DDM)

2.2.L

Description of the design method

2.2.2

Deficiencies in and modifications to the DDM method

2.3

Complex-Curve Fitting design method (CCF)

2.3.1

^{Rattan's design method}

2.3.2

Numerical integration

2.3.3

Detrimental effect of the weighting factor

l-l

1-l

1-1 L-2 1-3

t-4 l-6 l-8

2-l 2-t

2-4 2-4 2-7 2-9 2-9 2-12 2-13

Chapter ^Page 2-16

2-t6

2-20 2-21 2-2t 2-22 2-25 2-30 2-33 2-37 2.4

2.4.1 2.4.2 2.5

2.5.1 2.5.2 2.5.3 2.5.4 2.6 2.7

Iterative Complex-Curve

Fitting

^design method (ICCF) Description of the ICCF method

Numerical ^examples

SIMplex optimization-based design method (SIM) Control system design via non-linear programming Formulatiou of the non-linear programming ^problem Simplex method for function minimization

Application to the design of digital controllers

Random Searching Optimization-based ^design method (RSO) Sumrnary

3.

DETERMINAÎION

OF FREQUENCY RESPONSE MODELS

3.1

Flequency response models

3.1.1

^General description of defining a frequency response model

3.L.2

Limitations on using the frequency response of an existing system as a model

3.2

Second-order z-transfer function :rs a frequency response model

3.2.1

^{Use of the digcrete transfer as the model}

3.2.2

Second-order a-transfer function model

3.2.3

Conversion of specifications ^between the time and complex z- domains

3.2.4

Fbequency response oI the open-loop system

3.2.5

trÌequency response of the closed-loop system

3.2.6

Numerical studies on relationships of various specifications for discrete systems

3.2.7

Procedures to define a z-transfer function ^{as a frequency}

3-1 3-2 3-2

3-4

3-5 3-5 3-7

3-8 3-9 3-12

3-13

response model

tv

3- 18

Chapter

4.

Page

DESIGN OF DIGITAT CONTROLLERS BY MEANS OF

THE FREQUENCY RESPONSE MATCmNG

METHODS

^4-r

4.1

Introduction

Objectives of the design studies Dynamic characteristics of plants Design specifications

Selection of the sampling period Flequency response models

4.f

.6

Order of the discrete transfer function of a digital

controller

4-10 4.1.1

4.t.2 4.1.3 4.t.4 4.1.5

.

^4-10

4-1 4-2 4-5 4-5 4-6 4-9

4-15

4-t6

4-23 4-28 4-32

4-34 4-36

4-37

4-39

4.1.7

Simulation and assessment

4.2

^{Design studieg of Group}

I -

Comparisou of the various design methods

4.2.1

^Desigus for Plant

I 4.2.2

^{Desigus for Plant}

II

4.2.3

Discussiong of the simulation results

4.2.4

Comparison of the design methods

4.2.5

Conclusions

4.3

^{Desigu studies of Group 2}

-

Effect of the discrepancy ^between the primary frequency range of the model and that of the closed-loop system on the frequency matching

4.3.1

^{Design studies based on Plant}

I 4.3.2

^{Design studies based on Plant}

II

4.3.3

Design studies based on matching a continuous frequency response for Plant

I

4.3.4

Discussions of the simulation results

.

^4-32

4-42

4.3.5

Conclusiong 4-44

Chapter ^Page

4-46 4-46 4-5r

5-l 5-l

5-2 5-5

5-r0

6-l

A-l

A-3 A-5 A-8 A-13 4.4 Deaigu studiee of Group 3

Evaluation of the optimization-based design methods Designs with different

initial

^estimates

4.4.t

4.4.2 Conclusions

5.

^{EYBRID FREQUENCY RESPONSE ANALYSIS}

5.1

Deûciencies of discrete frequency response analysis

5.2

^{Hybrid frequency} response analysis

5.3

Numerical ercample

5.4

Conclusions

6.

^{SUMMARY AND} CONCLUSIONS

Appendix

A

^Objective function E as fuuction of ^À

B

^{Variation range of the parameter}

c C

^{Time response analysis of,}

hþ)

D

Frequency response analysis ^of, g(z) and h(z)

E

Stability of a second-order discrete system

F

^Table

A-r -

Bibliography

Relationships between the design specifications in the time, frequency

and compLex z- domaine for second-order discrete control

systems

^{A-f 5}

B-r

vl

ABSTRACT

The frequency response matching technique for the synthesis of digital control systems has been investigated. The basic philosophy

of this

technique

is to

design a digital controller so

that

^{the frequency response of the designed} closed-loop system matches a specified frequency response model. The approaches described in the current literature include Rattan's complex-curve

fitting

method and Shieh's dominant data matching method.

Two new design methods are proposed

in

this thesis. The first one is the rúer- atiae comples-curue fitting (ICCF) ^design method ^based on Rattan's algorithm. With

its

iterative calculations, the new method improves the frequency matching accuracy significantly by eliminating

in

Rattan's algorithm the frequency dependent weighting factor which severely degrades the matching accuracy

at

high sampling frequencies.

The ^{second one is} the simplæ optimization-bøsed (SIM) design method.

With

the aid of non-linear constraints on the controller parameters, this method provides

a

^good

compromise between the desired system frequency response and the required controller characteristics to avoid problems such as an excessively high controller ^gain or an oscil- latory control signal.

The non-linearity of the Shieh's method in the case of the design of a controller with an integrator is removed by choosing the appropriate controller form and dominant frequency

points.

As a result, the relevant computational algorithm is considerably simplified.

The determination of a frequency response model from design specifications given in the time, frequency and a- domains is discussed.

It

is shown

that

^{the choice of the} model may be critical

to

the succcess of the frequency matching,

in

particular when there is discrepancy between the primary frequency range of the system under design and that of the model. To help select an appropriate z-transfer function as a model, an

e¿rsy-to-use approach is developed which is based on a comprehensive investigation ^on the dynamic performances of second-order discrete systems.

A number of design studies is conducted in order to ^{assess the} frequency matching design methods. The frequency matching accuracies and time responses of the designed systems form the basis for the comparative evaluation. The resuits show that ICCF and SIM methods proposed in this thesis are superior to current methods. The effect of the discrepancy between the primary frequency ranges on the matching accuracy and the convergency of optimization algorithms are illustrated as well.

The hybrid frequency ^{response is} defined for the system containing both discrete- and continuous- time components. Unlike the commonly-used discrete frequency ^re- sponse, which is derived from the system z-transfer function and provides no informa- tion about the time response between sampling instants, the hybrid frequency response includes the characteristics of the inter-sampling time response.

vul

DECLARATION

I

herc,by declo,re tlt'øt, (ø) the thesis contøins no møterial which høs been øccepted

for

^{the awø¡d}

of

^{any other d'egree or diplomø}

in

^øny

uniaersity and thøt, to the best of my knowledge ønd belief, the thesis contains no materiol preúously ^published or written by ønother Peîson' encept wherc d,ue refercnce is mød'e

in

thc tert of the thesis; and (b)

I

consent to the thesis being mød'e øaailøble for photocopying and loøn

if

applicable.

Jiønfei ^Shi

ACKNOWLEDGEMENTS

It

is good fortune

to

have

Dr. M.J.

Gibbard as my supervisor.

I

^{am deeply}

indebted for his encouragement throughout the entire postgraduate studies and for his guidance and criticism given in the prepa.ratiou of this thesis.

I

would like to thank

Dr.

F.J.M. Salzbom, of the Department of Applied Math- ematics, for his etimulating lecture in the freld of optimization and for his helpful sug- gestions.

I

would also like to thank

Mr.

P. Leppard, of the Department of Statistics, for permission to use his computing program at the early ^{stage of} this research project.

I

wish

to

express my ^gratitude

to Dr.

B.R. Davis

for

his help given

in

many interesting discussions.

The assisstance of the staff of the department is gratefully acknowledged. Special thanks are due to

Mr.

R.C. Nash for his unsparing support.

As a student from a non-English-speaking country,

I

^am very grateful to Mrs. ^J.

Laurie, of the student counselling service, for her two-year English tutorial work which has helped me to complete this thesis.

Finally,

I

wish to express my sincere appreciation to the University of Adelaide for the Scholarship for Postgraduate Research which enabled me to pursue my studies towards the degree of Master of Engineering Science in electrical engineering.

x

Figure

2-l

2-2 2-3

2-t 2-6 2-T

LIST OF FIGURES

Location of the closed-loop poles for the numerical studies

Page

Configuration of the closed-loop system incorporating a digital controller

Discrete control system with unity feedback

Effect of the weighting factor lPn(ir)12 in Rattan's CCF method on the accuracy of frequency matching

2-4

^{[\equency responses of the} closed-loop systems designed by the ICCF method

Flow chart for the simplex optimization method

Illustration of the simplex method

in

the 2-dimensional case

Flow chart for the random searching optimization method

Flow chart for determining the frequency response model Location of the zero 21 of. h(z) in terms of the parameter

c

2-2 2-3

2-t5

2-20 2-27 2-28 2-35

3-3 3-6 3-14

3- 15

3- 16

3-16

3-t7

3-17 3-20 3-1

3-2 3-3 3-4

3-7 3-6

Relationship between the maximum overshoot Mo and the phase

margin PM at uoT

:0.7

3-5

Relationship between the maximum overshoot Mo and the resonance peak value

M,

Relationship between the peak time ratio

tfi

and the normalized closed-loop bandwidth w6T as a function of

a

and

f

Relationship between the normalized peak time ørúo and

a

as a

function of

{

3-8

Relationship between the malcimum overshoot Mo and

a

as a

function of

{

Unit-step responses of. My(z) and M2(z)

3-9

Figure

4-l

4-2 4-3 4-4 4-5 4-6

Closed-loop digital control system with unity feedback Unit-step response of Plant

I

Unit-step response of Plant

II

Open-loop frequency response of Plant

I

Open-loop frequency response of Plant

II

Root loci of the control systems designed by the SIM method with

the desired digital controller D(z) with

1:0.3

^s

Frequency responses of the closed-loop systems in the design

trÌequency and time responses of the closed-loop systems in the design studies of Group 2 for Plant

I

, designed by the ICCF method

Values of WIAE of the closed-loop systems in the design studies of Group 2 for Plant

I

_, ^{designed by the ICCF method}

xii

Page 4-2

4-6 4-6 4-7 4-7

and without constraints

4-7

^{Flequency responses of the} closed-loop systems in the design studies of Group 1 for Plant

I

4-8

^{Time responses of the} closed-Ioop systems in the design studies of Group

I

^{for Plant}

I

4-9

^Values of WIAE of the closed-loop systems in the design studies of Group

I

for Plant

I

4-10

Magnitude of frequency responses of Plant

II

_, the model

Mq¡

and

4-18

4-21

4-22

4-23

4-24

4-26

4-27

4-28

4-35

4-38 studies of Group

I

for Plant

II

4-12

^{Time responses} of the closed-loop systems in the design studies of Group 1 for Plant

II

4-13

^{Values of WIAE of the} closed-loop systems in the design studies of Group

I

for Plant

II

4-t4 Magnitudes of the frequency responses of arbitra¡ily-selected discrete systems with different primary frequency ranges due to the selection of different sampling periods .

4-ll

4-15

4-16

4-39

Figure 4-17

Page trÌequency and time responses of the closed-loop systems in the design

studies of Group 2 for Plant

II

_, ^designed by the ICCF

method

^4-4O

Values of WIAE of the closed-loop systems in the design studies of Group 2 for Plant

II

_, ^designed by the ICCF method

4-19

Unit-step responses of the continuous model

M"(t)

and the discrete model Mz(z) with

Î--Q.g

^s

4-20 Flequency and time responses of the closed-loop systems in the design studies of Group 2 for Plant

I

, designed by the ICCF method,

matching the continuous frequency response model

M.(ir) 4-21

^Values of WIAE of the closed-loop systems in the design studies

of Group 2 for Plant

I

_, ^designed by the ICCF method, matching the continuous frequency response model

M"(ir)

4-22 4-r8

4-23

4-41

4-42

4-43

4-44

4-49

4-50

5-3

5-6

o-t

5-9 Results of the design studies set A

in

Group 3, based on the SIM

and RSO methods with the arbitrarily-assigned

initial

^values

Results of the design studies set B in Group 3, based on the SIM and RSO methods with the initial values obtained from the design based on the DDM method

Block diagram of a closed-loop hybrid digital control system Hybrid and discrete frequency responses of the closed-loop

systems for Plant

I

_, ^designed by the ICCF method

5-3

Unit-step time responses of the model and the closed-loop systems for Plant

I

_, ^designed by the ICCF method

Magnitudes of the frequency responses

H(jr), ltj"), M(jr),

D.nU"), G¡G.(jw\

and

G¡(iø)C(ir)

for the hybrid control

5-1 5-2

5-4

system of Plant

I

_, ^designed by ICCF with

1:0.5s

A-1

Position of the zero

Zl

relative to the poles

P

and

P

in terms of

a

^A-3

Table 4-1 4-2 4-3

4-8 4-9 4-10

4-LL

LIST OF TABLES

Objectives of the design studies

Sampling period

I

^selected for the design studies

Comparison of the frequency matching design methods

Sampling period

?

selected for the design studies of Group

2

^.

Coeffi.cients of the controller transfer functinons in the design studies of Group ²

Page 4-4 4-9

4-4

trYequency response models and their time domain performance parameters for comparison with the control specifications

Dominant data points w; (radfs) used in the design studies based on the DDM method

4-5

Constraints on the gain and pole-zero locations of

D(z)

in the designs

for Plant

I

^{based on the SIM method}

4-6

Coefficients of the controller transfer functions in the design studies of Group

I

4-7

^{CPU time} consumption of the various design methods in the design studies for Plant

I

with

? :

0.5 s, of Group

I

4-10

4-t7

4- 18

4-20

4-31 4-33 4-36

4-37

4-45

4-47

4-48 Effect of the discrepancy between the primary frequency range

of a model and that of a closed-loop system on the design

of a digital controller

4-12

^Values of the frequency points

(radls)

used in the calculation of the error .E

4-13

Parameters of the z-transfer functions of the digital controllers for the design studies of Group 3

Relationships between the design specifications in the time, frequency and complex z- domains for second-order discrete control

systems

^A-15

A-l

xlv

LIST OF SYMBOLS AND ABBREVIATIONS

This list summarizes symbols and abbreviations commonly ^used in this

study.

The page number indicates the ^place where the symbol or abbreviation is defined.

Symbol

ø

coefficients of the z-tra¡sfer function of a plant

with

ZOH

b coefrcients of the z-transfer function of a plant

with ZOH

^2-3

e

squared-error of the frequency response matching 2-t6

el weight ed-squared-error of the frequency response matching 2-16

êe¿

steady state error ^3-8

g(z)

open-loop z-transfer function of ^{a discrete} second-order system 3-7 Page

.

^2-3

h(")

m n

p

closed-loop z-transfer function of ^{a discrete} second-order

system

^3-7

order of the numerator polynomial of the z-transfer function of ^a controller 2-1 order of the denominator polynomial of the z-transfer function of a controller 2-l order of the denominator polynomial of the a-transfer function of ^a plant with ZOH

or

^poles of a z-transfer function

q

order of the numerator polynomial of the z-transfer function of ^a plant

.

^4-6

2-3 2-23

3-2

2-L 2-L 3-8

with ZOH tp peak time

tc settling time

a

coefficients of the z-transfer function of a digital controller

y

coefficients of the z-transfer function of a digital controller

y(ú)

output signal of a closed-loop system

z

^{zeros of a} z-transfer function

.

^2-23

Symbol

A

coefrcient of a second-order discrete transfer function

B

coefficient of a second-order discrete transfer function C

D

coefficient of a second-order discrete transfer function coefficient of a second-order discrete transfer function

Page 3-1

3-l 3-l

3-1 2-2 2-LO 2-23 2-2 2-3 2-2 2-2 2-3 3-2 2-3 2-5 2-3 2-3 3-2 3-2 2-9 3-6 2-9 3-2 z-3 2-5 2-2 1-3

2-t4 .D(

)

transform or frequency response of a digital controller

E

^squ ared-matching-error function

or

^{objective function}

ø(r)

^transfer function of an error ^signal

E(")

z-transfer function of an error signal

.F'(r)

transfer function of a control element in the system feedback loop

G(r)

^transfer function of a continuous-time plant

GnG(

)

transform or frequency response of a plant with ZOH

GM

gain margin

¡f( )

transform or frequency response of a closed-loop system

I

imaginary part of a complex number

M( )

transform or frequency response of a closed-loop model

Mq( )

transform or frequency response of a open-loop model

Mp

ma>rimum overshoot

M,

^{resonance peak value}

lV(

)

numerator polynomial

P

^{open-loop or} closed-loop poles

P( )

denominator polynomial

PM

^phase margin

A( )

transform of frequency response of an open-loop system

R

^real part of a complex number Æ(

)

transform of an input signal

T

sampling period

sampling period of a frequency response model xvi

T*

Symbol

U(")

z-transfer function of the output signal of a digital controller WIAE weighted integral absolute error

f

(

)

transform of the output signal of a closed-loop system

Z

^{zero of a} control ^system

reflection coefrcient in the simplex method

angle to define the relative location of a zero to a pair of complex poles

contraction coefrcient in the simplex method expansion coefficient in the simplex method weighted-squared-error function

magnitude of a complex number in decibels auxiliary optimization variable

parameters of the constraints for a controller parameters of the constraints for a controller system damping ratio

magnitude of a complex number phase angle of a complex number frequency variable in rødf s weighted frequency variable

bandwidth

(-3

dö) of a closed-loop frequency response gain cross-over frequency

phase cross-over frequency

undamped natual frequency of a closed-loop system system oscillatory frequency

primary frequency range of a discrete system

primary frequency range of ^a frequency response urodel resonant frequency

sampling frequency

Page 2-3 4-t2 2-2 3-6

d,

or

B

"l e

f

À

l.t

v

€

î

,þ

w

u'

ub wc

ug ufl uo up

up*

ur

uo

2-26 3-5

2-29 2-28

2-to 2-23 2-24 2-24 2-24 3-7 3-10 2-23 2-3 4-12 2-8 3-9 3-10 3-7 3-7 2-4 4-36 3-2 2-3

Abbreviation

CCF -

^Rattan's complex-curve ûtting design method

DDM -

^Shieh's dominant data matching desiga method ICCF

-

^iterative complex-curve

fitting

design method

RSO -

^{random searching} optimization-based design method

SIM -

simplex optimization-based design method

ZOH -

zero-order hold

Page 2-9 2-4 2-16 2-33 2-21 2-3

xvlu