STUDY OF ESTIMATION AND OPTIMIZATION ...

STUDY OF ESTIMATION AND OPTIMIZATION TECHNIQUES

S U I T A B L E FOR MICROPROCESSOR ADAPTIVE CONTROLLERS

by

PETER SPASOV B.A.Sc.

(E.E.),

U n i v e r s i t y of Toronto,

1977

A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D

SCIENCE

in THE FACULTY OF GRADUATE STUDIES (Department

We a c c e p t

of E l e c t r i c a l

this thesis required

Engineering)

as c o n f o r m i n g t o standard

THE U N I V E R S I T Y OF B R I T I S H COLUMBIA May,

©

Peter

1979

Spasov,

1979

the

5E-6

In p r e s e n t i n g

t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r

an advanced d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s .

for extensive

study.

copying of t h i s thesis

be g r a n t e d by the Head o f my Department o r It i s understood that copying or p u b l i c a t i o n

o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written

permission.

Department o f

Electrical

Engineering

The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date May 17.

B P 75-51

1 E

1979

ABSTRACT

Adaptive controllers

are

controllers

unknown o r c h a n g i n g e n v i r o m e n t s . conventional controllers the

plant

that

that perform o p t i m a l l y i n

One c l a s s

of adaptive

tune themselves.

s y s t e m p a r a m e t e r s and o p t i m i z i n g t h e

estimates.

It

is desired

to have a l g o r i t h m s

This Is

controllers done b y

a microprocessor

is

that

are

short

PID c o n t r o l o f a second o r d e r

tions.

system.

i n both

with

cubic splines

The o t h e r

to

technique,

find

of the

In general, more t h a n

plant

are

estimated

the

coefficients

o f the

input

and f i n d s

is

response equation.

the phase tangents by

w i t h Walsh F u n c t i o n s .

each o f these a l g o r i t h m s

data

b y two

Identification,

is

estimated

to

1000 l i n e s o f c o d e w i t h e x e c u t i o n t i m e s o f l e s s

once the measured

a

itera-

characteristic

c a l l e d Walsh F u n c t i o n Parameter

output

i n a few

involves curve f i t t i n g of a step

(WFPI) u s e s a s q u a r e wave t e s t correlation

to o p t i m i z e b o t h

These a l g o r i t h m s n o r m a l l y converge

One t e c h n i q u e

device

s y s t e m and l e a d - l a g compensation o f

The p a r a m e t e r s o f a s e c o n d o r d e r

techniques.

these

possible.

G e n e r a l i z e d G e o m e t r i c P r o g r a m m i n g (GGP) i s u s e d

servomotor

estimating

c o n t r o l l e r b a s e d on

p r o g r a m l e n g t h and e x e c u t i o n t i m e s o . t h a t i m p l e m e n t a t i o n i n a s u c h as

are

available.

require than

no

1 second,

TABLE OF CONTENTS Page ABSTRACT

i i

TABLE OF CONTENTS

i i i

SYMBOLS

v

ABBREVIATIONS

vi

L I S T OF ILLUSTRATIONS AND TABLES

v i i

ACKNOWLEDGEMENTS

viii

I.

INTRODUCTION - ADAPTIVE CONTROL

1

II.

OPTIMIZATION USING GENERALIZED GEOMETRIC PROGRAMMING (GGP) . . .

4

III.

IV.

V.

2.1

Introduction

to

GGP

2.2

Condensation

of PID C o n t r o l Problem

2.3

O p t i m i z a t i o n o f the

2.4

Servomotor

4

D u a l Problem - PID C o n t r o l

Lead-Lag Network Compensation

7 15 25

PARAMETER I D E N T I F I C A T I O N USING CUBIC S P L I N E S

35

3.1

The P r o b l e m o f P a r a m e t e r I d e n t i f i c a t i o n

35

3.2

Cubic Splines

36

3.3

Identifying

the

Second O r d e r

System

PARAMETER I D E N T I F I C A T I O N USING WALSH FUNCTIONS 4.1

Development

o f the

4.2

Testing

Method

4.3

Conclusion for

the

the

Method

39

51 51 54

WFPI T e c h n i q u e

IMPLEMENTATION STUDIES

61

63

5.1

F e a s i b i l i t y of Implementation

63

5.2

Test

67

o f yP Programs

iv

Page CONCLUSIONS REFERENCES

70 .

72

APPENDIX A - GGP A l g o r i t h m s

75

APPENDIX B - P a r a m e t e r E s t i m a t i o n APPENDIX C - W a l s h F u n c t i o n

Algorithms

Parameter

Identification

APPENDIX D - P I D C o n t r o l and L e a d - L a g C o m p e n s a t i o n

82 85 87

SYMBOLS

Chapter

2

g

constraint

G

i n g , f u n c t i o n f o r z = In X space

X

primal variable

6, t

dual

a

and c o s t

variables

time s c a l i n g

s

f u n c t i o n f o r GGP

factor

Y , u , C n' n' n

parameter ^

f o r 2nd o r d e r

K K

system J

c

, T . , T, 1 d

parameters

f o r PID c o n t r o l

P

, T P

parameters

for

parameters

for lead-lag

K^,

, a

Chapter C.

servomotor network

3

.

cubic spline

coefficient

S(t)

spline

function

x,

system

output

x

y

first

z

second d e r i v a t i v e of system

a

l > 2 » Yp»

a

k'

a

^k' ^k'

T

Q r

k'

d e r i v a t i v e of system

parameters for ^k'

S

k

variable for

2nd o r d e r

output output

system

identification algorithm

Chapter 4 x

sine

y

cosine type Walsh

xc,

yc

type Walsh

correlation

function

function

1^

integral

s^

sign of i n t e g r a l

Y , 12

X

where

1_ 'ii

6 -

X

1 2

Ul

ui

ui2

1

5

2

X

ui

The b a r o v e r t h e u r e f e r s

to

the

condensed v e r s i o n o f the

function.

The c o n d e n s e d f u n c t i o n a p p r o x i m a t e s

scribed

[12].

in

z=£nX s p a c e . by Bohn

It

the o r i g i n a l

f u n c t i o n as

i s a tangent hyperplane to the o r i g i n a l

de-

function i n

N o t e t h a t c o n d e n s a t i o n i s p e r f o r m e d i n t h e manner

b11

-

where

= CnXj

l

X

X

1

b i ?

b i q

2

3

X

'll

Cn

6

5

'11

described

u

= 1 +

2

x

12

_ C

2 1

c

2 1

x

l

I 1 =| —

condensation,

1

2

= 6 ,

b

1 2

1

6

3

1 2

set

x

obtain b

2

=

the

= b

n

(2.28)

b i 2

b

2

Then s i m i l a r l y

u

(2.27)

12

C o n t i n u i n g on w i t h

2

2 x

b

2

2

3

(2.29)

3

6

6 i 2

2 2

b

2 1

-b

-0,

2 3

b

2 2

'22

21

—6

1 »

=

^

t h e s e two p a r t s o f t h e 2

u

2 2

b3

&21

with

original

[12].

U

where

12

-6 12

'ii

2

f u n c t i o n condensed,

then

^22

"

=

x u 2

2

The

condensed c o s t

l1

a

g

0

= C X

l

=

L i

1

1

l?

a

X

a

%

X 2

function finally

13 i d

9 1

A

+C X 2

2

X

i

a

is

9 9

A

2

2

2

X

91

a

(2.31)

3

where c

TJ^Y

'

a

b = 7;— ,

i l

"

=

1 _ a

13»

1 2

a

=

b

22

_

b

a

12>

13 = " 13 b

(2.32A)

2

C

2

2

a

Ml

Parts

2 1

=

-bn,

o f (2.32A)

a

2

= -l-b

1

1

,

2

a 3 = -l-bi 2

(2.32B)

3

and (2.32B) c a n be r e w r i t t e n u s i n g (2.27) and

(2.29) a

= -l- 0

3

(2.80)

satisfied

positive.

then

Therefore

(2.80)

is

i n summary

also the

satisfied, conditions

thus can

insuring

be

as 3 t


0,

< b

a

> 0

2

(2.81B)

- I —J

1

equations of

for

(2.14)

C

= 0.5

g

and

(2.81C)

(2.15),

the

controller

parameters

by 1 K

C

=

(X, - b ) a t

s

33

t T, =

(Xi (X

t

In order numerical

to

b ) x

S

d

T. =

(2.82A)

2

9z

s

(

X

2

,

-

(2.82B)

2

2 " 2 ——— b

z

. b )

)

(2.82C)

d e m o n s t r a t e how r a p i d l y

example w i l l

be

the

algorithm converges,

given.

Example Data

:

Choosing

a_ = 0 . 2 5

a

C =0.5 s

t

2

s

= 0.01 =2.5

y = 1

a

X

(

0

)

= 1 .3125 = 0 .59

X x

Table 2.1 the

are

to

optimum.

close

Summaries o f t h e If

the

)

= 0 .0624

o f the

the n u m e r i c a l r e s u l t s optimum s y s t e m .

occurs very r a p i d l y .

points the

0

summarizes

step response

convergence

(

3

to

the

Note that

Other d a t a were t r i e d w i t h are

open l o o p s y s t e m has

controller settings

equations.

The d e t a i l s

the

from Table

2.1,

condensation

a step response

similar

a time d e l a y ,

convergence

the

similar rates.

same a l g o r i t h m c a n

Once X i , X 2 a n d X 3 h a v e been

may b e d e t e r m i n e d of this

initial

shows

l i s t e d i n Appendix A .

be a p p l i e d a s s u m i n g no t i m e d e l a y . the

As c a n be seen

optimum and w o u l d y i e l d

algorithms

and F i g u r e 2 . 1

method a r e

by a p a i r given i n

of

found,

non-linear

[12].

GGP Algorithm Results For PID Control Of A 2nd Order System

Iteration

D X

l

X

2

X

3

S

o

g

o

b

8

1

g

D

8„

2 2

P

1

1.217

0.710

0.0602

1.306

1.305

1.010

1.327

1..312

1.237

2

1.195

0.746

0.0597

1.309

1.309

1.000

1.310

1.310

1.310

3

1.192

0.753

0.0596

1.310

1.310

1.000

1.310

1.310

1.310

4

1.191

0.754

0.0596

1.310

1.310

1.000

1.310

1.310

1.310

Table 2.1

Output

Fig.

2.1

Step Response f o r O p t i m a l PID C o n t r o l

2.4

Servomotor Lead-Lag Network Compensation Often,

networks.

servomotor

systems

are

c o n t r o l l e d by l e a d - l a g

T h e r e a r e many c l a s s i c a l t e c h n i q u e s

such systems

based

compensation

available for

on c e r t a i n s p e c i f i c a t i o n s . . .

, It

is also

designing possible

[Zb J to d e s i g n such systems A servomotor

u s i n g the

GGP t e c h n i q u e s

i s often modelled

d e s c r i b e d i n S e c t i o n 2.

as

K 2

G ( ) S

(2.84)

S ( l +

P

The p r o b l e m i s

T

P

S )

to design a compensation network o f the

G (s)

K (1 = -£ 1

+

a x

s) ^—

+

X

S

form

(2.85)

c The f o l l o w i n g K (x X

+

x

A _ _ c

i

definitions

are

made

- K K P c

(2.86)

)t p__s

(1 (2.87A),

X

+ X

c p

X

3

3

(2.87C)

,

X

h

t & —

'(2.87D)

c p

2

—

X

(2.87B)

c p t a ^ —

,:.

X

X X

K)t

^

2

X X

Kt ^ —

a x

(2.87E)

X

c

"

P

2(ISE) g

0

^

(2.88)

t s

From e q u a t i o n s step

(2.84)

to

(2.88)

it

can be found t h a t ,

for a

unit

input, g

0

1 Xl = X X - + X

2

X2aXX X

k

1 — X /XjX 3

Xa XX 2

2

+ 2

2

3

2 4

(2.89)

Again,

a time c o n s t r a i n t

an a p p r o x i m a t i o n can be

i s necessary.

For the

transient

response,

made.

G * G G c p

(2.90)

G =

(2.91) s(l

Then t h e

+

T

s)

P

closed loop error

for

a step

input

is

s + 2?«., •E(s) =

(2.92) s

aK co = —

where

The p a r a m e t e r the

response.

m a i n l y by K a .

2

1 ,

2

T

of

+ 2cws+ a)

2

2£w = — T

P has v i r t u a l l y

The i n i t i a l

(2.93) P

no e f f e c t

p o r t i o n of the

A time c o n s t r a i n t

can be s e t

on t h e

transient

transient i n the

is

portion

determined

following

manner:

Define Yi - 2 e w t Y

2

-

(2.94)

(wt )

=

2

s

(2.95) T

P The e r r o r transform of This

at

time t

is

approximated by t a k i n g the

(2.92) a n d u s i n g t h e

d e f i n i t i o n s of

inverse Laplace

(2.94) a n d

(2.95).

yields

-? e(t

) = e

[cos6 + — s i n e ] 26

2

S

/ Y O

The a p p r o x i m a t i o n time of the

response.

-/YxV

is

(2.96B)

2

c l o s e to

Using

(2.96A)

the

(2.87) a n d

actual error

d u r i n g the

(2.95) , we h a v e

rise

Y

= X

2

2

- aXit = X

Now a t i m e c o n s t r a i n t

- aX

2

x

+

a

(2.97)

z

can be s p e c i f i e d .

= C X! + c x = 1

g l

3

where

h

(2.98)

2

-a

(2.99)

Y,

-

a

Y

-

a^

2

(2.100)

After

2

s p e c i f y i n g an e ( t ) , g

Y

2

can be found by s o l v i n g t h e

following

equation. -20 Y,

a +

=

/ Y

2

e (t 2

Y

- a

2

2

cos/ Y

2

- a

2

+ a sin/ Y

-

2

a

5

)

(2.101)

a

X

(2.102)

a =

( 2 . 1 0 1 ) was s o l v e d f o r v a r i o u s v a l u e s o f a a n d e ( t

) = 0.5.

It

s was f o u n d

that Y

2

= 1 + a/2

(2.103)

T h i s a p p r o x i m a t i o n was f o u n d t o b e w i t h i n of Y

2

i n the

r a n g e o f 1.0

Hence, s o l v i n g starting point. the

following

It

for Y

2% o f t h e a c t u a l

value

< a < 2.0. 2

c a n be s i m p l i f i e d

s h o u l d be n o t e d

that

from

by u s i n g

(2.103)

(2.101), Y

2

must

as

a

satisfy

condition.

(2.104)

After

condensation,

the

cost

becomes

/c X X X3\ l/c X X X \ 6

1

x

8

1

2

2

2

3

6 2

/c3Xi\

A a i

/c4X2V

a 2

(2.105)

= S0S1 1

;

where d i r e c t

6

/

u s e was made o f t h e X

a manner s i m i l a r to

Appendix A . 3 .

be e a s i l y

constraint

I

a

(see

\ ®2

(2.87)) (2.106)

and s o l v i n g t h e

a few d i f f e r e n c e s ,

dual variables

dual problem i s

The a l g o r i t h m i s t h a t are

have been found,

done

listed

discussed

in

below.

the p r i m a l v a r i a b l e s

can

calculated. CT

X

X

1

= —

x

C

3

o

2

(2.107)

= —

2

(2.108)

X3 c a n b e f o u n d b y t a k i n g t h e There are criterial

\ \

t h a t of PID c o n t r o l .

There are

Once t h e

2

I

= a - X±

k

The p r o c e s s o f c o n d e n s a t i o n in

1

l i m i t s for

requires X

the

derivative

of the

a l g o r i t h m to work.

uncondensed

Again,

the

function.

stability

that

3

X i1X 2

< 1

(2.109)

A

Another s t a b i l i t y conditions.

These

g

=0

0 X

3

x

from the

Kuhn T u c k e r

are

§0„

80

c o n d i t i o n c a n be d e r i v e d

= 0

(2.110A)

+ V81, = 0 x

(2.110B)

+

Y

2

u

8 l

Y

2

(2.HOC) ,

81 = 1

(2.110D)

We h a v e -X X g 2

g

0

x

l

3

=

+ (2X]X

0

*2

aX ) 2

2

k

(2.111)

lX go + x

3

(2.112) h - X!X )go + ( X : h

3

2

= X

+

l +

=

(2X g

1

h 3

0

+ 2a X X 2

3

:

- X

g

- 2aX

3

3

where

h = X^XQ, - X

and

g! gi

-

2aX ) 4

(2.113)

(2.114)

2 3

= -aC

(2.115)

= C

(2.116)

k

x S o l v i n g the

2 x

h

l

x

+ X

2

2

first

two K u h n T u c k e r e q u a t i o n s

of

(2.110)

for

u,

yields g

o

X l

+ g

a

= 0

v

X

Substituting i n gg a t

0

(2.117)

2

(2.117) by

(2.111)

the optimum, y i e l d s ^ 2X X - 2aX + 2a X Xi + a ^ g = X X + aX!X 2

X

3

3

1

and

2

+

(2.112)

+ aX

and by s o l v i n g

3

(2.118)

0

2

Assuming t h a t X X Using

2

> 0,

3

+ aX

l

(2.97) w i t h Y

2

Since X i > a, Y Though t h e

2

3

for

3

i m p l i e s a new s t a b i l i t y c o n d i t i o n .

> 0

(2.119)

(2.119), a condition for Y

is,

2

< . a ( 2 X i - a)

(2.120)

then a d e f i n i t e c o n d i t i o n i s 2X

Then from ( 2 . 1 0 4 )

_

2 X

3

stability condition. (2.123)

3

and

(2.123),

the

following

limit

for Y

2

is

obtained. a 4 This

It

2

< Y

< a

2

(2.124)

2

implies 1

1

—

< Ka < —

4x

T

P

s h o u l d be n o t e d

purposes.

It

(2.120)

satisfied.

is

(2.125) P

that these l i m i t s

i s p o s s i b l e to

exceed the

are u s e f u l

limits with Y

T h e a l g o r i t h m was t e s t e d u s i n g s e v e r a l e x a m p l e s . an example t a k e n system

from r e f e r e n c e

x t

P P s

The r e s u l t s 2

= 100 = 0.04 = 0.05625 of the

= 1.71 X]/ ) =

Obtain

X

0

ISE

C

s

1.70

= 1.6066 = 0.0430

,

= 0.5000

optimization algorithm a = 1.40625

Using

x

for

X

2

= 1.9917

2

computational

> a

2

as

are

long

as

One o f t h e m was

The p a r a m e t e r s o f t h i s

are K

Y

[26].

for

servo-

31

K

= 0.2436

c

,

x

The t i m e r e s p o n s e

is

c

= 0.2808,

a = 0.8875

'

shown i n F i g u r e 2 . 2 .

Verification

was d e t e r m i n e d b y s u b s t i t u t i n g t h e o p t i m u m s e t t i n g s equations. ventional

F i g u r e 2 . 3 shows t h e technique

543 o f r e f . control.

[26].

time response

of l a g network d e s i g n . A comparison w i l l

The r e s p o n s e

obtained

from the

The g r a p h i s

taken

a higher overshoot

Better

conventional design but

more r e a d i l y w i t h o u t g o i n g t h r o u g h a t r i a l

Note t h a t

the

same c o n s t r a i n t

i s s a t i s f i e d by b o t h

A l e a d d e s i g n was o b t a i n e d w i t h Y its

response

shown i n F i g u r e

2.4.

2

of 0.46

and e r r o r

= 1.3125,

from

and a

time constant

it

con-

slowly

seconds. GGP o f f e r s

procedure.

compensators. and t

p.

improved

decaying term w i t h a r e l a t i v e l y large c o n t r o l c o u l d be o b t a i n e d w i t h

optimum

i n t o t h e Kuhn T u c k e r

show t h a t GGP o f f e r s

i n F i g u r e 2 . 3 has

o f the

= 0.05

with

G>)=K^l+octs)

O.S

Fig.

2.4

Lead

C o m p e n s a t i o n U s i n g GGP

35 PARAMETER I D E N T I F I C A T I O N USING CUBIC S P L I N E S

3.

The

3.1

Problem of Parameter

In

the p r e v i o u s

Identification

chapters

i t h a s b e e n shown t h a t

s y s t e m p a r a m e t e r s and an a p p r o p r i a t e settings

f o r a c o n t r o l l e r can be o b t a i n e d .

procedures

requires

overview of the

a system model.

an a p p r o p r i a t e

s y s t e m b a s e d on a p r i o r i may b e c h o s e n .

to

use

done,

the

is

often

as

a parameter.

a

Identification:

equation

0

given i n reference

can be used

to

describe

plant

c o u l d be m o d e l l e d

c o u l d be used.

to

the

order

determine

the

of the

Since i t

system i s .

coefficients

as

is the

Onee

of the

+ a

l Y

as

d e s c r i b i n g s t a t e and p a r a m e t e r to what

purposes

is

of

d e f i n e d as

that

linear

this

thesis,

estimation,

a s t a t e and what the

following

T h i s i n v o l v e s d e t e r m i n i n g the

d e s c r i b i n g ( m o d e l l i n g ) an i n p u t / o u t p u t

9y + 32^- +

a

model

is

there

defined

definition

chosen:

Parameter linear

For the

An e x c e l l e n t

parameters.

literature

some d i f f e r e n c e s

be

a chemical process

choose what

i d e n t i f y the

In

design

d e s c r i b e d by l i n e a r d i f f e r e n t i a l e q u a t i o n s ,

i t w i l l be n e c e s s a r y

equations,i.e.

will

to

suitable

d y n a m i c s , i . e . an a p p r o p r i a t e

l i n e a r approximations

systems

remaining problem i s

of equation

knowledge of i t s

For example,

a n o n - l i n e a r system or convenient

To a p p l y s u c h c o n t r o l l e r

T h i s m o d e l must be i d e n t i f i e d .

'type'

plant

(specification),

p r o b l e m s i n i d e n t i f i c a t i o n and c o n t r o l a r e

Often

is

time c o n s t r a i n t

g i v e n the

y ..ang^r d

b

0

+ biu + b

2

+

coefficients

system of

b

the

of form

(3.1)

a

[2].

For

Adaptive Control,

m e t e r i d e n t i f i c a t i o n scheme algorithms of

that

have been proposed,

extended

i t w o u l d be

Kalman f i l t e r i n g

desirable

can be c a r r i e d out

the

most

techniques.

in

chapter.

this

This i s

Parameter

In identifying

equation

this

the

noisy,

t y p e o f scheme

are

the

cubic

splines.

3.2

Cubic

Many form

that w i l l

the

be

investigated

some known i n p u t i s m e a s u r e d .

f i t t e d w i t h a smooth c u r v e u s i n g

give the

technique

best

fit

to

curve

These

splines.

coefficients

of

the

curve.

smoothed

t o be s t u d i e d w i l l be t h a t

parameters o f a second o r d e r model w i t h time delay.,

a

of using

Splines

In a quotation

that w i l l

chapter,

i n real time.

t h a t make u s e o f s p l i n e

i d e n t i f i c a t i o n i s performed by f i n d i n g the

differential

para-

[27].

An output response to

measurements,typically

some

t y p i c a l o f t h e s e b e i n g some

A l t e r n a t e schemes have b e e n p r o p o s e d fitting

to have

order

to

from r e f .

obtain i n t u i t i v e i n s i g h t i n t o the [18] w i l l be

nature of

splines,

given.

"Draftsmen have l o n g used m e c h a n i c a l s p l i n e s , w h i c h a r e f l e x i b l e s t r i p s o f a n e l a s t i c m a t e r i a l . The m e c h a n i c a l s p l i n e i s s e c u r e d b y means o f w e i g h t s a t the p o i n t s of i n t e r p o l a t i o n - h i s t o r i c a l l y c a l l e d knots. The s p l i n e a s s u m e s t h a t s h a p e w h i c h m i n i m i z e s i t s p o t e n t i a l e n e r g y , a n d beam t h e o r y s t a t e s t h a t i : . . t h i s energy i s p r o p o r t i o n a l to the i n t e g r a l w i t h r e s p e c t t o a r c l e n g t h o f t h e s q u a r e o f t h e c u r v a - .:: t u r e of the s p l i n e . " Essentially, cal

curve,

function

segments

f o r each

instead

(intervals)

segment.

of f i t t i n g a l l of the of data are

d a t a w i t h one

fitted with a cubic

analyti-

spline

37 When t h e k n o t s interpolating

spline

S(t )

(x„, t~),

i=l,

i

...

L i .

a function

= x(t ),

±

such

is

(x., t.), 1 1

such

(x , t ) are n n

given,

that

n

2,

the

(3.2)

that rt

J

= J

is

( S(t)) dt

n

(3.3)

2

l

t

minimized.

The c u b i c s p l i n e h a s and s ( t )

are

continuous.

That

the

additional property

is,

at

the

knot points

that the

s(t),

s(t)

following

is

true, S._ (t ) 1

= S.(t.)

i

K ± (t

^ i - l

(

t

i

}

=

S.^a.) for

i

±

of

this

this

t

b

0 L

t.,

the

determine the

details

i

= C

x = t -

Given

order

higher

the

S (t)

with

(3

-

= S. ( t . )

higher

respective

For

will

)

4B)

(3.4C)

= 2, 3, . . . n For

their

(3.4A)

splines,

order

derivatives.

interval,

let

+ CX^.T + C , 2

t-

< t

these c o n t i n u i t y


z

(

3

1

)

N+l Using

( 3 . 1 0 B ) and

(3.17)

a,i =

— — ax+y

(3.22)

a

aa

(3.23)

2

=

1

For a v e r a g i n g ,

ai

then

= a

-

x

N+l

a

2

( d

N

=

a N +

N+l

1

l

a

+ z

i

)

(3.24)

N

(3.25)

a

N+l

N+l

where d

=

N+l

Vi -

V l V lV l +

N

I

( 3

i i+i - i+ii> = V i

(x y

x

y

+

D

N

( 3

-

-

2 6 )

2 7 )

1=1 W i t h Y> aj_, a , 2

known,

z transform techniques,

the

t i m e d e l a y T J J can be d e t e r m i n e d .

a step response w i t h

Using

discrete

a r b i t r a r y t i m e d e l a y c a n be

generated.

These generated values can be compared to measured values and

the time delay can be determined by their respective times. The c r u c i a l part of the entire estimation scheme i s the determination of a\, a

2

which depend heavily on knowing x^ and i t s derivatives y^

and z-^. In order for the equations of (3.14) to be independent, no more than two spline approximations (x,y,z) from the same spline i n t e r v a l should be used.

Because x,y,z are continuous at the knot points, then just the

knot points

can be used.

This makes the computation of x, y, z simpler. th

Then from (3.11) and (3.5) for the i x

= C

±

z

-

0 i

y. = C i 1

knot (3.28A) (3.28B)

i

= 2C .

±

(3.28C)

2j

Hence, the spline c o e f f i c i e n t s can be used d i r e c t l y for :the parameter i d e n t i f i c a t i o n equations.

Another reason for using only the i n i t i a l

knot points becomes apparent when the following i s considered. Substitute the spline approximation into

(2C

2

+ aC 1

+ a (C -Y))+(6C

l

2

0

+ (3aiC + a C 3

2

At x=0 have

2

)x

2

+ 2a C

3

x

+ a C x 2

3

3

2

(3.8)

+ a C )x 2

1

= 0

(3.29)

(note x= t-t.) x

2C = -a^i 2

- a C 2

0

+ a y 2

D i f f e r e n t i a t i n g (3.8) and substituting i n the spline approximation

(3.30)

(6C At

+ 2a1C2

3

+ a C )+(6a C 2

1

3

+

2a C 2

2

)T

+ 3a C x 2

= 0

2

3

(3.31)

T = 0 we h a v e 6C

Then t h e

= -2aiC

3

2

--3.2^1

o n l y t i m e when t h e

Similarly

(

3C

- = ~

2

a

3

to s a t i s f y

a

ai

2

(3.8)

strate solving

3C

3

t w i c e and s o l v e d f o r

that other for a

a

1 ?

x's

(3.35)

can't

into

Since i t

x's

and

is

desirable

following

euqation,

=

\

( x

*k-i

x, = s m o o t h e d k ^

= measured

+

2

x=0.

it

X

+

X m

k i

can be

for

used.

to keep the i s used to

spline

a l g o r i t h m as

smooth t h e

}

+

data data

estimation algorithm is

demon-:

a point

k The

seen

T h i s does n o t

s m o o t h i n g must be done b e f o r e

a simple averaging technique

\

(3.34),

i f x=0 i s c h o s e n a s

c a n n o t be

possible,

where

(3.33)

be u s e d , b u t

to n o i s y data,

i s used.

then (3.35)

then other

2

x=0,

34

3

c a n o n l y be d e s c r i b e d by a s p l i n e a t

Due fitting

e~

x = y ( 1 + - \ 3,

., - b t (e

z = x = ^

(ae"

a, b =

ie.

a,

b are

a t

a n a l y t i c response, b t

))

(3.37A)

£

-at.

,„ „.,„. (3.37B)

)

- be"

b t

)

(3.37C)

(3.37D)

poles.

are

to n o i s y data

i s shown i n f i g u r e

g i v e n i n T a b l e 3 . 1 . It c a n b e s e e n

differences

for

the

derivatives

at

The l i m i t a t i o n i n t h e a c c u r a c y o f s p l i n e f i t will

limit

the

accuracy of estimating

s t a t e very w e l l because i t ful

state values

U

s

o

o

n

e

that

second order determine

and

there are

transient

line.That is

could not

3.1

smaller values

The s p l i n e w i l l

a straight

because a l l s t a b l e

of Y O »

the

d u r i n g the

a i , a2-

i s almost

for e s t i m a t i o n purposes

same s t e a d y

its

ie.

2

A plot of a spline f i t

significant

function with

2 -aj±/ai-4a2

the

numerical results

o f the

on

D

Tab y = x = —^

where

- ae"

at

a

A s p l i n e a p p r o x i m a t i o n was t r i e d

a simulated system w i t h a = . 2 5 , a = . 0 1 . 1

a c u b i c s p l i n e can r e p r e s e n t

of

time.

response

fit

steady

not very systems a

some

use-

have

the

particular

system.

Increased narrower. narrowed, the

accuracy

T h i s works i f the the

can be o b t a i n e d by making t h e data

i s not n o i s y .

If

the

intervals

s p l i n e a p p r o x i m a t i o n becomes m o r e s e n s i t i v e

derivatives.

spline

to n o i s e ,

intervals are especially

T h e r e i s a r e a s o n why t h e t=0.

C o n s i d e r the

Taylor

ya t x(t)

expansion o f x about ya a t

2

2

1

= —2

fit

o n l y a v e r y s m a l l range

transient

further found

part

of the

update o c c u r s .

that

a^,

a

(

using a t h i r d order

of

4

2

+

parameter

response.

e s t i m a t i o n c a n o n l y be done

At steady

In practice,

n u m e r i c a l e r r o r when u s i n g s m a l l

state,

once s t e a d y

o c c u r when i t

i s known t h a t

estimation i s

restricted

the

the

are dependent

two o u t p u t

It

the

ISE of the

of the parameters

as

z = 0,

no it

p r o b a b l y due

was

to

estimation should

by an i n p u t .

Hence,

estimation for

ai=0.25,

S i m i l a r b e h a v i o u r was the

estimates

t h r e s h o l d and t h e number o f

other

parameters

accuracy of the

system.

i n (2.22). It

open l o o p i n t e g r a l once the

of the

for

chosen.

T i m e d e l a y c a n be e s t i m a t e d estimates

i e . x, y,

c a n be s e e n t h a t

spline interval, starting

c o u l d be done by c h e c k i n g t h e

with

the

b e i n g O . l y and 0 . 9 Y -

One c o n v e n i e n t m e t h o d o f c h e c k i n g t h e

known i n t e r m s

)

thresholds.

of parameter

overdamped s y s t e m s .

on t h e

times of smoothing

system i s a f f e c t e d

results

two t h r e s h o l d s

observed for other

8

values.

between

T a b l e 3 . 2 shows t h e 2

3

during

s t a t e i s approached,

Since there i s a time delay i n g e n e r a l ,

a =0.01 with

,

e q u a t i o n w o u l d be p o s s i b l e

k e p t on becoming i n c r e m e n t a l l y l a r g e r ,

2

3

t.

As m e n t i o n e d b e f o r e , the

2

2

approaches

t=0.

ya (a -a )t

3

2

g

An a c c u r a t e

a c c u r a c y d e c r e a s e s when o n e

estimates

The c l o s e d l o o p e r r o r c o u l d a l s o be

is

compared

time d e l a y i s known.

by s i m u l a t i n g t h e

response w i t h

a n d c o m p a r i n g them t o m e a s u r e m e n t s .

the The

a l g o r i t h m l i s t e d i n Appendix B . 2 uses the time s h i f t

of these points

delay simulation.

If

d e l a y w o u l d be f o u n d . of

a

a

1 5

2

are not

the

relative

to

the

exact values

Table 3.3

knot points appropriate

of a^,

a

2

.1 < x

m

Otherwise,

to s m a l l e r r o r s

i n the

3.2.

by t h e

For the

The c o n v e r s e for

the

is

the

time delay estimate

estimates of a

x^Ct),

Instead

,

exact

time time

estimates

hence

f o r example t

varies

the

such

that

o n l y s l i g h t l y due

2

t o s e e how t u n i n g o f a P I D c o n t r o l l e r w o u l d

the s

t,

a .

The r e s u l t s

x^(t )

t r u e f o r x^{t).

ISE i s

for

less

this than

test the

are

case based

< x - } ( t ) w h i c h i s why t h e s

Note that

the

shown i n

ISE i s

on

figure the

smaller.

same r e l a t i o n s h i p h o l d s

true

overshoot.

In general, for

the

of a zero the

for larger

to a range of o u t p u t s ,

estimates.

response

true parameters.

find

shows t i m e d e l a y e s t i m a t e s when t h e

I t w o u l d be i n t e r e s t i n g be a f f e c t e d

to

exact.

s h o u l d be r e s t r i c t e d < .6.

point

were u s e d ,

The Tp e s t i m a t e s become m o r e i n a c c u r a t e estimate

i n order

estimation but

the

cubic splines accuracy i s

offer

a considerably simpler

limited.

technique

Output

Fig.

3.1

x

Cubic Spline

Curve

Fitting

Comparison of Spline Approximation to Analytic Function (*refers to the spline approximation)

t [sec]

X

X

y

y

0

0

0

0

0.01

*

z [10-3]

' z* ' [10-3]

10

1.83 4.95

10

0.2364

0.2378

0.0314

0.0285

-0.217

20

0.5156

0.5165

0.0233

0.0242

-0.980

i

30

0.7033

0.7026

0.147

0.0143

-0.710

.' -0.615

40

0.8197

0.8175

0.009

0.0089

-0.447 - -0.456

-1.360

Table 3.1

Parameter Estimates of a 'Second Order System a^O.25,

Noise Variance

a

l

a =0.01,

TH=0.1y

2

Spline Interval [sec]

*2

# of times of smoothing

0.000

0.2397

0.00927

10

1

0.0115

0.2385

0.00970

10

1

0.0115

0.2078

0.00748

10

1

0.0115

0.2277

0.00998

5

3

0.0115

0.2320

0.00860

15

1

0.0115

0.267

0.0103

10

1

0.0115

0.341

0.0144

5

3

*1

TH=0.25y

*2

TH=0.25y,

a =0.35,

a =0.015 2

Table 3.2

Delay Time Estimation a =0.25,

a =0.01,

^=10 sec,

Knot Point

^=0.23,

X c

X .

c [sec] fc

a=o 2

Noise Variance=0.0115

0095

t [seed

D [sec] T

^=0.27, X

a =0. 0105 2

t [sec]

D [sec] T

.100

15.6

.100

5.7

9.9

.102

5.6

10.0

.407

25.6

.410

15.7

9.9

.409

15.9

9.7

.633

35.6

.634

25.1

10.5

.634

26.3

9.3

.775

45.6

.775

34.4

11.2

.775

36.7

8.9

.862

55.6

.862

43.7

11.9

.862

47.3

8.3

.914

65.6

.914

52.8

12.8

.914

57.7

7.9

Table 3 . 3

System Response

Fig.

3.2

x(t)

PID T u n i n g Based on E s t i m a t e d

P a r a m e t e r s , 2nd O r d e r

System

51

4.

PARAMETER I D E N T I F I C A T I O N USING WALSH FUNCTIONS

4.1

Development o f the Method In

the

identified

s p l i n e i d e n t i f i c a t i o n method, u s i n g the

system response

have a c h a r a c t e r i s t i c ters

frequency

spectrum,

c a l l e d Walsh f u n c t i o n s .

applications

i n f o r m a t i o n about

be i n v e s t i g a t e d h e r e .

c o r r e l a t i o n of the

hence,

due t o

the

output

c a r r i e d out

Also

tested

second order

The b a s i c system i s

on the

These

paramefunctions

it

has

use o f Walsh

involves testing

control

w i t h v a r i o u s known s q u a r e wave

the property

the

functions

functions

the

system

Identification is

estimates are

of being simple.

done

signals,

less

sensitive

T h i s method

was

systems.

t h e o r e t i c a l background can be found i n r e f .

g i v e n a s q u a r e wave i n p u t w i t h a f r e q u e n c y

correlation

the

b a s e d on c o r r e l a t i o n

Essentially i t

correlation process,

to n o i s e . for

systems

They h a v e been used i n v a r i o u s

w i t h a s q u a r e wave a n d s o i s e a s i l y i m p l e m e n t e d . by

Since

o f o r t h o n o r m a l s q u a r e wave

system i d e n t i f i c a t i o n [ T 3 ] . A procedure

will

time domain.

essentially

[15].

R e c e n t l y some w o r k h a s b e e n in

i n the

can a l s o be o b t a i n e d u s i n g a s e t

which are

the parameters were

i s performed by u s i n g v a r i o u s Walsh

o f co^.

[13]. Time

The average

functions.

are

(4.1) -N where

c(t)

=

u(t)

= known s i g n a l

T h e known s i g n a l s

output

are Walsh

functions.

Xi

- s q u a r e wave w i t h co = toi

X3 x

s q u a r e wave w i t h to = to 3 = 3toi

- s q u a r e wave w i t h to = L05 = 5toi

5

where x i i s signals

the

used

o r i g i n a l generated t e s t

input

for

the

Other

are

y\-

90° phase s h i f t e d

s q u a r e wave w i t h to = toi

y -

90° phase s h i f t e d

s q u a r e wave w i t h to = to

3

system.

Y5~ 90° phase s h i f t e d The s e c o n d o r d e r

3

= 3toi

s q u a r e wave w i t h to = 105 = 5toi

system has

the

following

transfer

function

2 (jto)

G

^Vj?

= (to

P

- to)

P

= |

|

G

+ j 2 ? to to P P

e

-

j

(4.2)

e

P

2£ to to t a n G = —2—2— to2 - to 2 P

where

z

It in

2

c a n b e shown t h a t

terms o f t h e i r

(4.3)

z

the

harmonics

correlation

of

the

xic a Aicos0i + — 3 x c 3

- A cosG 3

functions

approximated

form^^

cos6

3

+ — 5

(4.4B)

3

can be

cos0

(4.4A)

5

x^c - A c o s e 5

(4.4C)

5

and yic

y c 3

- A sin0 3

Since the

= Aisin9i

determined

signals

are

3

+ — 5

just

(4.5A)

5

5

plus

functions,

following.

sin8

y c = A sin0 o r minus l e v e l s ,

n u m e r i c a l l y from the

correlation

by the

sin0

-(4.5B^.

3

can be e a s i l y p e r f o r m e d Given the

+ — 3

5

the

(4.5C)

5

correlation

data.

then phase i n f o r m a t i o n

can

be

53

y c - y c/3 t a n G j = —— xjc - x c / 3 x

y c/5 — x c/5

3

5

3

Y3 = ——

(4.6A)

5

C

tan0

3

C

tan6

(4.6B)

x c

5

(4.6C)

x c

3

5

£^ a n d 03^ c a n t h e n b e d e t e r m i n e d b y e q u a t i n g

The p a r a m e t e r s and

Y5 = — -

(4.6)

(4.3). This y i e l d s OtanSx - 9 t a n 9 ) o j 3

03

(4.7A)

2 1

2

3tane! (45tan6

a, 2

o)

3

-

(25tan0

5

-

75tanG )o) 5

2 1 (

3tan6

4

-

7

B

)

5

5tanQ )^i

1

l

(4.7C) tanBc -

=

2

- oj • 2o)i o)

(9o3j

2

2

5tan6!

)tan9

1

(4.8A)

P

1

- OJ

=

2

)tan6

3

E

(4.8B)

6o)i0)

P

P

1

(250)!2

?

3

=

(o)!

?

-

3

5tan8

P

p

tanG

:

P

2

-

- 03

= P

2

)tan6

5

2

(4.8C)

IOOJIU)

P The g a i n y^, o f a s y s t e m c a n b e d e t e r m i n e d response

due t o a s t e p

input.

from the

steady

state

4.2

T e s t i n g the Method A typical

in

p e r i o d i c output

Figure 4.1.

the

data

response

f o r a s q u a r e wave i n p u t

The c o r r e l a t i o n f u n c t i o n s

i n lengths

The v a l u e

o f 30 e q u a l s e g m e n t s f o r

f o r each i s o b t a i n e d u~c~ = ( s j l j + s I 2

where

I

i

t

T h i s method i s correlation accuracies the

first

3 harmonics of c ( t )

the

part

3

0

)/K

(4.9)

particular

correlation

of the

too s m a l l ,

response

i e to > 0 ) 5 .

is

are

D a t a was s a m p l e d o v e r one h a l f Appendix C.

The i n a c c u r a c i e s

For low input

test

the

steady

state

is

only in

approached.

then m a i n l y determined by

i f toj i s

period. (>5%)

too l a r g e

accounted then

are

higher for,

important

shown i n T a b l e

The a l g o r i t h m u s e d

due t o

frequencies,

accuracy of the

lower harmonics.

that

neglected.

h a r m o n i c s d i m i n i s h more r a p i d l y . frequencies,

fact

Any i n -

Numerical accuracy

Since these are not

Similarly,

components

the

because

v e r y h i g h and v e r y low v a l u e s

The m e t h o d was t e s t e d a n d some r e s u l t s

the

I

are being used.

When to^ i s

components,

lower frequency

than

0

value

t a n e ' s would d i m i n i s h at

inaccuracy occurs.

circled.

3

t h a t h a d o c c u r r e d a r e due p r i m a r i l y t o

The t r a n s i e n t

then

(T/2).

found to be r e l a t i v e l y i n s e n s i t i v e to n o i s e

to a n d a t to = io .

frequency

period

i s p e r f o r m e d w i t h a d e t e r m i n i s t i c known s i g n a l .

determining the of

integrating

segment

b

s_^ = ± 1 w h i c h d e t e r m i n e s function = some c o n v e n i e n t

the h a l f

shown

by

+ ••• + s

2

= i n t e g r a l of the

K

were computed by

is

the the

frequency

higher harmonics

Inaccuracies

i s shown i n

used,

accuracy of the

Conversely, for

4.1.

the h i g h e r

are

shown

lower input

d i m i n i s h more r a p i d l y

due t o n o i s e c a n b e

minimized

by

a v e r a g i n g more d a t a

cant at

part

i n the

frequencies

functions

have the

5

=

-0.008053

y c

=

-0.001845

With a noise variance

measurement

The i n p u t noise

There i s

frequency

system i s

1

=

u

3ojg, 5 W Q .

3 times w i t h

(4.3),

This

:

case.

become the

(cotangents)

found to

tangent

property

s q u a r e waves o f

the m a t r i x i s

tan 6

1

-

2

(3co)

frequency is

sym-

be,

tan05

3

503

- 1

2

3oo

(5OJ)

2

1

-

15o)

9OJ

(4.10)

25. 2

- 1

(9oo)

- 1

2

(15o>)

(15OJ)

2

2

-

1

25o)

15a>

5OJ

5O)Q

(5(D)'

- 1

(25o))

2

-

1

A

where

0) Hence,

by

to/overcome-

accuracy o f the phase

3oo

(3o))

o)! =

of these values

the

0

3OJQ

=

errors

The m a t r i x o f p h a s e t a n g e n t s

Using equation

O J

o)!

i n the n o i s e l e s s

o f t h e WFPI t e c h n i q u e .

tested

t a n 9],

(a

function

F o r co^ = 1 ,

c o u l d be i n c r e a s e d

a way o f d e t e r m i n i n g t h e

o c c u r s when t h e

metrical.

the

signifi-

problem.

on a p r o p e r t y

= OJQ,

correlation

become v e r y s m a l l .

of 0.0115,

estimates based

oij

Noise plays a

because the

following values

x c 5

significant.

o v e r more p e r i o d s .

estimation of tanO's

lower fundamental

correlation

points

the

-

C0Q

accuracy of the

using three test

signals

phase tangent estimate

and the

By t a k i n g a d v a n t a g e o f t h e

symmetry

can be

checked

property.

p e r i o d i c waveform, n o i s e

can be

averaged

out.

Estimates

of the

shown i n T a b l e 4 . 2 .

phase tangents w i t h s i m u l a t e d n o i s y d a t a

The n o i s e was s i m u l a t e d b y g e n e r a t i n g

bers w i t h a uniform p r o b a b i l i t y d i s t r i b u t i o n . will

d i m i n i s h the noise

are

shown i n T a b l e s 4 . 3 .

dependent on e r r o r s

of the

from a t e s t

to

t h e most

phase tangent e s t i m a t e s .

inaccuracy.

(4.7B) i s

equation

improves

of w e i g h t i n g the

t h e most the

5

s i g n a l o f to^ =

A more

example of

s h o u l d be w e i g h t e d

i n f l u e n c e d by t a n 0 . 5

accuracy

equations

I n the

Those e q u a t i o n s

w h i c h a r e more d e p e n d e n t o n t a n 0 example,

averaging

Errors i n estimation

a l g o r i t h m would use w e i g h t i n g f a c t o r s . subject

random num-

errors.

The p a r a m e t e r e s t i m a t e s d e t e r m i n e d 0.903 r a d . / s

Sufficient

is

significantly.

i s beyond the

scope

are

strongly

sophisticated

shown, tan05

(4.7)

and

less.

In

is

(4.8) the

Eliminating this

A detailed of this

investigation

work.

58

Phase Tangent Actual

V a l u e s Determined from Test

System P a r a m e t e r s :

Test

y 1> P =

3

input:

u=l,

0 171

Variance=0.0115

-4.850

.4172

.2216

-4.833

.4154

.1133

-4.844

.4351

.2663

-4.810

.4297

.2688

Determing Phase Tangents by Averaging Noisy Data co =1.0, p =0.5 P P U!=0.903

* Noi. of Samples

tan 0

N/A

-4.835

.4198

.2283

0

4

-4.839

.4125

.2704

.0115

8

-4.838

.4198

.2426

.0115

12

-4.837

.4176

.2156

.0115

12

-4.838

.4154

.2021

.0231

20

-4.835

.4236

.2389

.0231

8

-4.842

.4136

.3650

.0346

15

-4.837

.4121

.2053

.0346

30

-4.834

.4227

.2246

.0346

1

tan 9

3

Table 4.2

tan 0

5

Noise Variance

Parameter Estimations Noise Variance=.0346, 30 samples

Table 4.3A Estimates Equation Used

w P

2

P * P (to =1.037) (to =.999) P P

.1.000

0.672

0.494

(4. 7A)., (4 . 8A)

1.232

0.471

0.495

(4.7B),(4.8B)

0.995

0.463

0.482

(4.7C),(4.8C)

1.075

0.553

0.490

Average

0.999

N/A

0.490

Average of A & C for U p 2

Parameter Estimations No Noise Table 4.3B Estimates - 2

Equation Used

P

p (w =1.001) P P

0.998

0.496

(4.7A),(4.8A)

1.012

0.491

(4.7B),(4.8B)

0.998

0.490

(4.7C),(4.8C)

1.003

0.492

CO

Average

4.3

C o n c l u s i o n F o r T h e WFPI T e c h n i q u e Investigation of this

been begun r e c e n t l y . promising. steady

It

type of parameter

The p r e l i m i n a r y

seems t h a t

has

only

been system before

g i v i n g enough t i m e f o r s i g n i f i c a n t

transients

adequate.

An i t e r a t i v e l e a s t fitting

i n v e s t i g a t i o n s have

any f r e q u e n c y t h a t e x c i t e s t h e

s t a t e i s reached, but

to occur i s

identification

squares

procedure for r a t i o n a l function

to determine a t r a n s f e r

function

has been p r o p o s e d [ 2 3 ] . A t r a n s f e r

from phase

function

frequency

curve response

can be w r i t t e n i n terms o f

even and odd f u n c t i o n s o f co . 2

Q(co ) + jcoP(co ) 2

H(jai)

2

=

(4.11)

— U(co ) 2

The phase

tangent

is

coP(co ) 2

tanG(co) =

Q(co )

tive

least

squares

find

P(co )/Q(co ) .

this

function.

2

control

+ Pico

q

0

+ q o

2

l t

response

+ • • • p co ^)io 2

M

2

2

data

+ • • • q co

(4.12)

N

N

t a n 6 o f a l i n e a r s y s t e m , an

itera-

r a t i o n a l a p p r o x i m a t i o n p r o c e d u r e can be used The t r a n s f e r

2

function

can t h e n be o b t a i n e d

to

from

T h i s method i s n o t v e r y a p p l i c a b l e f o r m i c r o p r o c e s s o r

systems because

and l e a s t

0

=

2

G i v e n the phase

(p

squares

curve

it

involves

extensive frequency response

tests

fitting.

A g e n e r a l i z a t i o n o f WFPI t o h i g h e r o r d e r s y s t e m s h a s b e e n p r o p o s e d . It

i s p o s s i b l e t o d e t e r m i n e more p h a s e

identify

higher order systems[13],

the output in

the

response

form o f

t a n g e n t s more a c c u r a t e l y

Instead

of using a Fourier

i s expanded i n t o a Walsh f u n c t i o n

series

to

series, expansion

c(t)

= aix! + a x 3

3

+ a x 5

+

5

••• (4.13)

+ b y

+ by 1

where x , y a r e

the

3

1

+ b y 5

5

+

•••

s i n e and c o s i n e t y p e W a l s h f u n c t i o n s

S e c t i o n 4 . 1 and a,b can be d e t e r m i n e d

are

the

expansion c o e f f i c i e n t s .

by c o r r e l a t i o n f u n c t i o n s .

t h e n be d e t e r m i n e d by u s i n g t h e a system of equations For example,

3

the

that

is

described

These

coefficients

The p h a s e t a n g e n t s

coefficients.

l i n e a r i n the

Using the

phase

p a r a m e t e r s c a n be

parameters of a t h i r d order

s y s t e m can be

in

can tangents,

derived.

found by

solving, /

-co.^tanO 1

/

-to

tan9} tan9

3

tan0

5

O J

2 3

tane

a

-coc tan9q 2

3

=

ai

3

3\ co

C O S

2

(4.14)

3 3

3

)

015

where

H(s)

Proposals

(4.15) s

3

+ a s

2

2

f o r making use

+ a^s

a l s o made

a

of magnitude

e v e n and odd p a r a m e t e r s o f a t r a n s f e r d e l a y have been

+

[13].

0

i n f o r m a t i o n to decouple

f u n c t i o n and e s t i m a t i n g

time

the

IMPLEMENTATION STUDIES

5.

The f e a s i b i l i t y o f i m p l e m e n t i n g t h e will

b e shown a n d t e s t s w i t h a l a g c o m p e n s a t i o n

t i o n program f o r

5.1

ref.

of.adaptive

Many i m p l e m e n t a t i o n s

self

tuning.

variants

have been

s u c h as

investigated

developed systems

for

the

still

or

a con-

variations

of conventional

controllers

[4,5].

A self

Digital

same q u a l i t y as a n a l o g P I D with integral

regulator-a

action

was

self-tuning

control

algorithm

discrete-time-randomly-distributed

[16,19,31]. (self-tuning)

development

having their fact

the

found

type where

E x t e n s i v e w o r k was done o n

c o n t r o l of

o f the

c o n t r o l parameter t u n i n g .

the

can be

b y many r e s e a r c h e r s .

feedback

c o n t r o l l e r c a n be i m p l e m e n t e d b a s e d

strategy o u t l i n e d i n Figure 1.1.

been

of the

i n references

on t h e minimum v a r i a n c e

setpoint

An a d a p t i v e the

attain

state vector

and i n v e s t i g a t e d [ 4 ] .

c o n t r o l l e r s based

integra-

out.

c o n t r o l systems

The e f f e c t i v e n e s s

i m p l e m e n t a t i o n o f PID does not

proposed

be c a r r i e d

have been

t u n i n g P I D c o n t r o l l e r was i n v e s t i g a t e d

and a v a r i a n t

algorithms

p r o g r a m and an

c o n t r o l l e r s u c h as P I D , l e a d - l a g c o m p e n s a t o r s

these are

and t h e i r

will

control

Implementation

of a p p l i c a t i o n s

[30].

ventional of

t h e WFPI t e c h n i q u e

F e a s i b i l i t y of A list

in

adaptive

development

techniques

in utilizing

mainly because of t h e i r

f o r parameter e s t i m a t i o n themselves

on the h i s t o r y

that d i g i t a l controllers

usefulness

The m a i n w o r k c o v e r e d s o f a r

The c o n t r o l l e r s based

the

on

c o u l d be

are

of analog

insensitivity, simplicity

and

conventional,

control.

"better" u t i l i z e d ,

conventional control

has

Despite

there

algorithms,

and f a m i l i a r i t y .

is

The e f f e c t i v e n e s s [3,5,28]

on c o n v e n t i o n a l c o n t r o l l e r s has been

Commercial success

t o d a t e , w i t h uP c o n t r o l u s i n g

a l g o r i t h m s has been demonstrated system

[ 1 1 ] and o t h e r s .

widely

used.

with

how f e a s i b l e

algorithms o n - l i n e , i n p a r t i c u l a r , The a n s w e r

been done. the it

It

algorithms

such systems

Adaptive versions

The q u e s t i o n may a r i s e ,

sor?

i s hard

to

is

(and o p t i o n s )

used

it

the

TDC-2000

them w i t h

an a c t u a l

s u c h as

a n d how t h e y

to

to implement

implement

d e p e n d s on many f a c t o r s

as

before.

conventional

o f t h e s e have y e t

to p r e d i c t before

i s p o s s i b l e to estimate

studied

be

adaptive

a microproces-

implementation

the m i c r o p r o c e s s o r are

has

used,

implemented.

Still,

e x e c u t i o n t i m e and p r o g r a m s i z e u s i n g

certain

assumptions. The t a r g e t m i c r o p r o c e s s o r i s assumed Hz c l o c k .

Table 5.1

algorithms.

These

on

times w i l l

the

this

instructions

The number o f c l o c k

register

branches

e x e c u t i o n time of the

f a c t o r was e s t i m a t e d

It

acquisition,

Self

to

s h o u l d be n o t e d operating an e n t i r e

r e q u i r e d was t o t a l l e d f r o m t h e

since other

to o b t a i n a time operations

a n d memory a c c e s s l a g compensator

extrapolated that

etc.

are

the

The flow

[32]. scaling,

required.

Based

program a c t u a l l y w r i t t e n , The l i n e s o f

e s t i m a t e s do n o t peripheral

machine

compensator

include

data

operations

system.

t u n i n g c o u l d be p e r f o r m e d

measurements have been o b t a i n e d .

s u c h as

from the w r i t t e n

i n t e r f a c i n g and o t h e r control

various

f o l l o w i n g manner.

to be a p p r o x i m a t e l y 5 . 5 .

c o d e r e q u i r e d was s i m i l a r l y

particular

estimates f o r the

c y c l e s was u s e d

n o t be p r e c i s e

tranfers,

program.

performance

These e s t i m a t e s were d e r i v e d i n the

number o f a r i t h m e t i c charts.

shows t h e

t o b e a TMS 9 9 0 0 w i t h a 3 M

i n a few s e c o n d s

after

I f d a t a were to be s t o r e d

open

loop

i n RAM,

it

Performance

Algorithm (Refer to Appendix)

A.l

A.2

A.3

B.l

B.2 C (Integral segments)

C (continued)

PID D

Compensator D

Estimates

E x e c u t i o n Time P r o b a b l e T o t a l (General) E x e c u t i o n Time [ms] [ms]

4+6.25/ iteration

28

1025

6

600

28

800

6

1.5+6.5/ iteration

4/knot +(.3/ data)N + 2/ s p l i n e inter-r val

.25+1.5/data l+30h(.15) per half period

Lines of Machine Code

358

625

1,500

175

N/A

. 50

12

920

4

Comments and Assumptions 3 iterations for convergence

One s t e p

4 . iterations for convergence N=# o f t i m e s of smoothing 10 k n o t s , 9 spline interv a l s , 1000 p t s . of d a t a , N=l 1000 d a t a

pts.

h=# o f s a m p l e s per i n t e g r a l segment test with signals

.5/data

N/A

50

done e v e r y Sample

.5/data

N/A

50

done e v e r y Sample

Table

5.1

3

c o u l d be sampled i n t i m e i n c r e m e n t s A typical so t h a t than

the

actual

short

P I D c o n t r o l l e r may u s e control part

is

assumed

that

16 b i t

attractive

a p p r o x i m a t e l y 100

is

and B . 2

integer

i n a yP.

data

i s acceptable.

a 16 b i t

Increasing

both execution time

A d m i t t a b l y u s i n g a TMS 9900 makes

because i t

ys.

o f a p r o g r a m w o u l d be s l i g h t l y l o n g e r

p r e c i s i o n would s i g n i f i c a n t l y i n c r e a s e

memory r e q u i r e m e n t .

as

algorithms A . 2 , B . l

1 K w h i c h can be e a s i l y accommodated It

the

adaptive

as

and

implementation

machine w i t h hardware

multiply

and

divide. To s e e how t h i s

has

an e f f e c t ,

a c o m p a r i s o n c a n b e made w i t h

c o n t r o l l e r using a Motorola 6800[28]. This i s complement

a r i t h m e t i c was p e r f o r m e d

were s e l e c t e d controller the in

from the

u s i n g 16 b i t

computed r e s u l t

r e q u i r e d an average

software.

very

integers.

t o be an o u t p u t

When c o m p a r i n g t h i s

TMS 9900 i t

12

2's bits

to a D / A .

This

o f 7 ms t o p e r f o r m c a l c u l a t i o n s u s e d

c a n be seen

to the that

estimated

the

in

performed

execution time

for

tuning algorithms would

be

long. When d e s i g n i n g a yP c o n t r o l l e r ,

one t o u s e .

Since c o n t r o l l e r s

Many m i c r o p r o c e s s o r s order

are

t o make a c h o i c e

c h o i c e was t h e

question

time,

is

which

t h e y must

and I / O o p e r a t i o n s

be

efficiently.

a v a i l a b l e and many t r a d e o f f s w o u l d b e

The yP c h o s e n was t h e this

one i m p o r t a n t

operate i n r e a l

capable of doing numerical algorithms

for

machine.

PID a l g o r i t h m , m a i n l y b e c a u s e m u l t i p l i c a t i o n had t o be

u s i n g the

in

an 8 b i t

a PID

required

[1,29]. Texas Instruments

fact

that

TMS 9 9 0 0 .

The m a i n

i t was r e a d i l y a v a i l a b l e a n d h a s

good m o n i t o r i n g s y s t e m

(TIBUG)

f o r d e b u g g i n g and an a s s e m b l e r

8002 y P r o c e s s o r L a b ) .

Other advantages are

its

factor a

(Tektronix

16 b i t w o r d l e n g t h

giving

greater accuracy, software

5.2

and t h e h a r d w a r e

multiply

and d i v i d e .

c o m p l e x i t y and e x e c u t i o n t i m e a s m e n t i o n e d

Test

reduces

previously.

o f uP P r o g r a m s

Two p r o g r a m s w e r e w r i t t e n a n d r u n u s i n g o f f - l i n e was t h e

This

data.

l e a d - l a g compensation network mentioned p r e v i o u s l y .

was a p r o g r a m t o

compute t h e

integral

segments

(I

One p r o g r a m The

other

of eqn.(4.9))

for

WFPI. The c o m p e n s a t o r for

implementing the

be r e q u i r e d

to

test

p r o g r a m was u s e f u l

i n estimating

other

A data

it

algorithms.

in real

time.

the

requirements

a c q u i s i t i o n system would

An o f f - l i n e

t e s t was p e r f o r m e d

r u n n i n g a s i m u l a t i o n o f a servomotor u s i n g optimum s e t t i n g s by the

GGP a l g o r i t h m .

The s e r v o m o t o r u s e d has

as

a time constant

by

determined of x

P

10 ms. A

The c o n t r o l p a r a m e t e r s u s e d x

= 0.9474

The s e r v o

,

A

2

= 0.7366

response

was

,

were A

3

=

0.3821

t h e n l o a d e d i n t o ROM.

The

compensator

program reads t h i s

data

A/D.

c o n t r o l l e r o u t p u t s w e r e l i s t e d i n RAM g i v i n g

The c o m p u t e d

results for

the

to

the

e v e r y 1 ms, i e a s i m u l a t e d s a m p l i n g from an

simulation results.

t e d by a 2nd o r d e r

tested

i n a s i m i l a r manner.

system s i m u l a t i o n w i t h n o i s e .

scaled hexadecimal integers

segments were computed by the

uP p r o g r a m .

method d e s c r i b e d data

i n Chapter 4.

since a data

It

was

D a t a was

The d a t a was

and l o a d e d i n t o ROM.

and phase t a n g e n t s were computed o f f - l i n e

cated)

responsible

differences.

The WFPI p r o g r a m was

into

T r u n c a t i o n e r r o r was

close

The

The c o r r e l a t i o n

generaconverted

integral functions

u s i n g t h e s e v a l u e s by

the

t e s t e d u s i n g 16 a n d 12 b i t

a c q u i s i t i o n s y s t e m c o u l d be t y p i c a l l y

12

(trunbits.

The

results

are

shown i n T a b l e

Truncation errors by

5.2.

due t o a p p r o x i m a t e r e p r e s e n t a t i o n

sums w e r e f o u n d t o b e s i g n i f i c a n t .

correct

for t h i s .

Given that

o b t a i n an i n t e g r a l segment, between 0 and H b e c a u s e

integrals

A s i m p l e t e c h n i q u e was u s e d

t h e r e are H samples

the

of

truncation error

each sample w i l l

t o b e summed of the

to

to

sum w i l l

h a v e a n e r r o r b e t w e e n 1.

be and

0.

S i n c e t r u n c a t i o n s h o u l d o n l y o c c u r f o r r o u n d i n g o f f d e c i m a l < O.5.,

the

average

can

b e done b y a d d i n g s H / 2 t o e a c h segment w h e r e s i s

integral One

truncation error

of the

sum w i l l

Hence, c o r r e c t i o n the

sign of

the

segment. can c o n c l u d e t h a t

it

is

feasible

and i d e n t i f i c a t i o n a l g o r i t h m s i n p r e s e n t (1979)

be H / 2 .

to

implement the

s t a t e of the

art

optimizing microprocessors.

Phase Tangents Determined w i t h the Actual

System P a r a m e t e r s :

Test input:

y =1, P

u=l,

-1,

Noise:

Case

Variance of

tan

TMS 9900

? =0.5, P 0

T , T

0.5

X,

±

2

+ b

3

(Appendix A . l )

80

A.3 GGP A l g o r i t h m F o r

Servomotor

Lead-Lag Network Compensation

For d e f i n i t i o n of v a r i a b l e s ,

g

Q

=

1 _ x

1

+

l

X

-2aX

see

+ 2a X

+ a X

2

3

2 3 X

X

Section

2

3

-

2 x

2.4

2a X_ + a + X X 3

T

l 3

x

"

1 X

Given

Y

2

, t

c

3

(Y

already

-a

=

s

Y

Choose i n i t i a l

2

2

- a

L

X

2 _ _

solved

c

2

X ^ ' ( a < X ^ < 2a)

4

for)

1

= Y

2

-

k

a

2

3

Y

X

aXi - a

+

2

+ X

2

2 x

1

2

- 2aXi+,

X

3

X

B = 2a X X 2

-B + / B = 2A

(A.3-1)

- a

l

C =

2

1

t f

-a X 2

X

2 4

2 1

X

2

- 4AC

2

(A.3-2)

X-

2a X 2

h X

= X

h

2

= -2aX

3

+

3

+ a X 2

- 2a X! + a

2

3

x

k

+ X X

Xi

1 2 X

1

'12

l

h

2aX h

2a X

a Xi 2

2

3

>22 x 2

3

'23

l 2

h

b

2a X3

X

>26

>25 »12 1 + 5

1 + 26

= 6

1

-