Robust Adaptive Control for Systems with Unknown Time Delay

Copyright © IFAC Identification and System Parameter Estimation, Beijing, PRC 1988

ROBUST ADAPTIVE CONTROL FOR SYSTEMS WITH UNKNOWN TIME DELAY Ma Runjin* and Zhang Yongguang** *Division of Engin eeri ng, Northern China L'niversit)' of Technology, Beij'ing, PRC **/ nstitute of Systems Science, Academia Sinica, Beijing, PRC

Abstract.

Usualy a adaptive controler most in use is based on a lower - order model, due

to the simpleness o f model the unmad e led part of the true system often makes the co ntrol - 'C s be not robust. This paper considers a robust class of models that r.(s)=Ke /( 5-5 ) " , 1 (s - sn)' where

~

is time delay. For th is c lass of model the unmodeled part of the system

can be converted into a part of unknown time delay. Thus, the sampled system of G(s) has a fractional time delay time delay

p

p,

Basing on combination of esti mating the fractional

O ~p< l.

and co n struct ing a self - tun ing

controler, a rob ust adaptive co ntr ol

scheme is given in the paper . Th e simulations show that this adaptive control algorithm has indeed robustness.

Keywords .

Sampled data .ystem; Fractional time delay; minimum phase;

Self-tuK~ng

r egu -

lat or .

I NTRODUCTI ON

contrast with it,

in this paper we reduced order of

model further by using rat:onal aoproximation to For the systems having unknown time delay

~

t here

unknown time dela y and give a adaptive control by

are several adaptive control algorithms in the lit-

conbinating the estimation of fractional time delay

eratures, for example, Kurz and Godecke (1981).

and a self -tuntng

Wong & Bayoumi (1982), Mosca & Zappa (19 83 ) and De

pear good robustness also.

controler. The simulations ap-

Keys er & Van Cauwenberghe (19 85 ) . However, most of them are based on a assumption that time dela y

~

is a integer times of the sampling interval To.

A EXAMPLE WITHOUT ROBUSTNESS

Obviously, this ass umpti on is not natural. Usualy ,

1: =(d. El )To, where d is a integer and 0,. p < 1, this

P

is called fractional time delay . Zrang

&

Example 1 . Ma

Consider a third order system

G(s) = 1/(10s-1)(2.5s-1)(2s-1),

To=5s

(1 )

(19 88 ) gave much insight about fractional time delay, the analysis sh owed that the zeros of sam-

it has one main pole that 5 = - 0 . 1. According to 1

pled system are strongly effected by fractional

classical procedure it can be reduced ord er as

time de l ay

p .

Therefore, most adaptive controler

first order s ystem G' (s) =1/(105-1). The corre -

which depends on minimum phase condition would not

sponding impulse transfer function has following

be robust due to without considering fractional

form:

time delay

p H( z )

Recently, Souza, Goodwin, Mayne & Palaniswami

tor for sampled s ystem ( 2 ) . The simulation result

struct adapti ve control by using rational approxi ~.

(2)

lie designed a mini..mum vari..ance self-tuning regula -

(19 86 ) has given a interesting new thought to con -

mation to unknown time delay

b z- l / ( I.az - l )

.ppoared in Fig.l shows that the control is failed

Simulations given

to divergent. However, if we use a control model

in the paper show the adaptive control has good

having unknown time delay term

robustness. However, the cost is that the order of

€-~s as doing in

this paper, the .daptive cont rol by using the algo -

model will be increased one or two of order . In

rithm with considering fractional time delay given

287

288

Ma Runjin and Zhang Yongguang

later in this paper makes the closed loop s ystem be controled and have much better performa nce . The

Especially, for the case of n =2 zi ( p ) is inside the unit disc and z2(f )-OO ,s

p _ l . and the

result is showen in examp le 2 and Fi g . 2. relatio n hetween p and zi ( P ) for first order sys_ tern approx imately is SAMPLED SYSTEM WITH FRACTIONAL TIME DeLAY

Cousider a continuous system :

(5 ) and for second order system is ( 6)

(3)

where

q=4 or 5 .

Sampl in g the continuous system G(s) under time in terval To, thus, and 0

~p<

I,

Pis

~

=( d+ P )To, whe re d is a integer

ca lled fractional time delay.

APPROXIMATION BETWEEN TIME DELAY AND UNHODELLED PART

Th e

discussio~

abAve shows a source of frac tional

time delay due to sampling, t here are, also, other

Let

30urces for fractional time delay that: a)

System existing somewhat

D(s)=e - t s

nonlin~arlty

where

discribed as linear system .

b) The lag due t o

c~ lc u lation

(7)

is

of control signal

after samp l ing.

r

is a positive integer, th e degree of ap -

proximation . Obviously, \ D(j W )\ =1 and -w"t j

also,

c) The assumed order of system is not correct,

Dr(j <...>)\

L

D(j W )=

( 1.w2 'C. 2 /r 2;'/'~

=

and

l _r tg- ( t..lt /r) . The comparision of

a nd so on.

both L D(j <...l ) and L

Dr(j LJ ) is shown in Fig.3, it

can be seen that L. Dr( j "" ) is v e ry closed to

All of these will introduce some modelling errors those make the integrated time delay be unknown and

L

D( j w ) and difference between them is less than

7% while w~ r4:.(mofe detail ca n be found in

result in robust less of adaptive co ntr01.

Sou za, Goodwi n, et ai, (1986». Elementary c lassiFor the ca s e of O.:.p< I, the impulse tran3fer func tion Can be obtained by using the modified z- trans formation (see Jury (1964». We denote the sompled

cal contro l arguments to ld one that it will be vir tually impossible to r obustly stabili ze the plant if the phase shift arising from the delay and approximation

system with zero-order hold as follows:

exceeds something like 100

0

over

the c l osed loop bandwidth. Thus, if the objective z

-d

is to design a contro l system, one is on ly inter-

este d (4)

i~

accurately modell ing the system u p to

phase shift of about 100

0

.

For the ca SE p =O, let H(z;O)=H( z ) which is obtained by orig i nal z-tr ansformation . Let zi ( p ), . .. ,

z~ ( p )

From the anal y sis above, if system ( 3) has main

be the zeros of H( z;p ) and z l' ... , zn_ l be the

poles sI ' s2' . . . , srn' the term l /( s-sm_l)

zores of H( z ). Much more insight wa s given in Zhang

( s-sn ) can be approximated as K· e

& Ma ( 1988 ) , here some main results are listed

~

- 'f1's

,where t

•

k " are adequate positive numbers. Thus,

wi t hout pr oof . (cf . Th e Appendix ) (s-s )

(i) H( z;p ) has n zeros, but H( z ) has n - l zer os. H( z;p ) h as n poles which are the same as those of

(i1) As z.i. ( ~ )

p_ O,

H(z;P) _

bandwidth. Obvious 1~1, G(s) and G' ( s ) are in the

H( z ;O)=H( z ) , O:1e of the

tends to 0, f o r example z i (0) =0. So

zeros

zi ( ~ )

-1

H(z ) , one o f the z~ (1) =oo.

So

the sample d s y s t e m i s non-minimum pha se while the

P is

main

p~leo

nOt more than 3 . In fact, control er de-

Signed based on ( 8 ) ha s good r obus tness than that

tends to 00, for example

fra ction al time delay

same class of mode I, out G' ( s ) has lower orde r. In

practice, one consider u s ualy the system having

there i s a c an ce ~la tion between 0 zero and 0 pole . (Hi) As p--. l, H( z ; P ) _ z

(8)

is an approximate of GCs) over the clo sed l oo p

H( z ), but there is one more pole that Pn_l=O.

zer os

:n

s ufficently close to 1 .

without consid2ring time delay i f we can give a

estimate of fractional time de l ay p .

Control for Systems with Cnknown Time Delay F~ACTIONAL

ESTIMATION OF

289

STR was presented in ~strom, Wittenmark (1973) and

TIME DELAY

.

Iserman (1977) . A modified STR to de al with nonIt is very difficult to f i nd an analytical expres-

p from the z e r o

sion of calculating

n<4. Znang & Ma (988) sh oued tions between

p

f )

z~(

and

z~(

P)

unless

a ppro xi mate rela-

minimum phase sampled system was given in

As ~ ["i5m

&.

Wittenmark (1985). Her e , we try to present ano ther mo difi ~ d

STR t o deal with th e system ha v ing unknown

t ime delay.

tha t

Co nsid e r the st a ble system t o be contra led as folfo r

Z;-I

l ow s:

n =1

G(s) = k e

-1:5

/(5 +5

(9)

and

1

(s+ s )

)

(13 )

n

which has one o r two main pol es. We try to model it by th e foll oW i ng mo de l: Based On th e s e relations, we c an estimat e a ppro-

C; (s )

ximal e ly f or n = 1, 2 as f o l lo\.;s :

I ) fit f o ll owi n g

mo d ~ l:

~

k'e

- t.'

s

I(s+a )

(14 )

(a:> 0)

if G(s) hriS on l y one main po l e . or

'" y( k-I ) . "a y( k- ZI -Ab ; ui k-1 )- 1'\ y( k)=a biu (k - Z) j 2

:b 3U( k-3 1.

(15)

<- l lcl

(10) if G( s) has two main p ol es, wh e re aiO, bJO, a and b

If f i t first orde r modpj . ta kp 2) Solve [ he z e r os

f\. z;l

k)

~,~O and ~i ~ O '

., no

/\. " Z2(K,

of

may b e diffe rent p os iti ve - ea! ( aJb . a > O, b>O), the same po si tive re al (a=b. a >O, b>O) or a pair

liD ),

of co mp lex conju g ate (a= b). that i s the r oo ts of

A _ I' 71' I' R ' ( Z) = b i z -. b;z.b~

sampling i nterva ll th 0r~

n:::l)and tak e \'ii\<

on ly one ze r o zi(k) [ or

i s

\'i2\ .

in eve ry

usu .1 ly zi Rnd

z2

p

2 t he sampled syst ems given in

n~l,

(4) e i the r ar e minimum phas e O r exist on l y one zero

a re

out sid e unit disc, so wc can co nveni e ntly t a k e in to

negative r ea l nu mb er. 3) E stimate

For th e ca se of

p

a ccount the estimation o f

2 in e\'('r~' s:l l"!lpl ing i nt.E:' !"'v al by

acco rding to t he loca-

tio n of z e ro ou ts i de uni t di sc . Thus. anadaptive

co ntrol a lgorithm is s hown as fo llow s :

using the f o rmula that

A

St ep 1. Se l ec t d(O), 9 (0) and Pt O) , usu a ll y

(1 1 )

6 a CO)=O a nd P( 0)= 10 1.

Step 2 . Recursl ve l y es tima t e I'

Note that

P 2 1k )

mi ght be grea ter than I or less

e

( k) b y RLS o r

Et

o ther ide ntifi cati on a l g orithm, wher e

=(a , a , l 2

than 0 if z2 >4- or 0 < z2 <4- re spe ctinl \". I n fact, fo, 2A St e p 3. Calculate the ro ots of "B/ (z) =b;z .b z

this is po ssibl e i f dt0' is llnccrrectlv se l ected . Simulations sh ow th a t d... (K i "':",i g!lt bl? grea te r i f d (O)

is l ess th ar th e t :- 'Je

be less t han 0 i f

d~ O)

is

\ 'il ;~O

g~~A:~r

~ha n

a T'c d. ( k ) "'ight

than

: r ue

~he

valu e (cf . Fi g . :. an d 51. sh 0ul~

So the pro cedu r e

Recur~ively

4)

"3

" f\. when n=2 (o r root of 1\. B , ( z )=b ;z -bi wh en n= I) .

tet z2 has r.:.aximllm ab so lute va l ue r e lative to zi ''''hen n =2. St e p 4. Es ti mate

~ 0n:~ n~~

adj~ s~

2

-b

~j k'

3S

as

f01 !o~s:

f011 0~~n g

f ( k) and d(k) bv using the

p rocedu r e gi ven in s ection S .

role:

St e p 5. Compute the mul ti - st ep pr e dictions v ( k-i \ k) , i= l, 2, ... , d(k) as fo l lo ws:

d t k-I ' -I

A

Thus. "t.. ( k ' = CC(k 1-

p : ',{\ < c:

;:

C' !SP ~· 't< : <:·

d ( k -! I A

E' \k ·"~c.

5 ) k=k-l. then back t c

Cas e I: If ~ i n i ",um

I

z

i \< I ,

t h u 5 the

5 a mr

led s y 5 tern i

5

phase, the o rig i nal STR can s t ill be used .

To compute th e multi - step pred ic tions by usi n g the

=, _"

cc ~t;~u e

the p:- o-

following model:

ced ur e. 1\

1-a (k) z l

l!~K~O\{:-;

TI:1E DELAY

Case 2: If

\

z2\~

-I A

- a (k) z 2

-2

1, fi rst fac to rize

(16 )

~' (z) =

" -I " -2 " -3 b; ( k)z +biCk)z -b ( k ) z into Band B-, wh e re

3

290

Ma Runjin and Zhang Yongguang wh ich has a pair of complex conjegate main poles

+ " + A+ -1" 1\ -1 B = b Ck)+b Ck)z = biCk) Cl-ziz ) 1 2 1\-

l+b Ck)z 1

-1

A

= l-z z

2

t hat (17)

-I

G(s)

e-

3s

2 /(9s +3s+1)Cl . 3s+1)CS+l)(0.5S+1) (24)

the n , to computing the multi-step predictions by The modelled part of (24) is that

using the following model : ,,_

All

b Ck)-b Ck)z 2 1

- 1

(25)

z- d(k) ( 18) The co ntrol result is shown in Fig.4.

Example 4. Co nsid er the first order system with

wher e

timevarying time delay that ( 19 )

(26)

t

The time delay is changed as follows: The purpose of the transformation is to make t he

0

static gain ke e p the same.

Step 6 . Determine the control u(k)

-C Ct)=

Case 1: If!z2 l
"'-80',11'0'

2.7 To

tion that 1\

uCk)

if 0~t<80To if

80To ~

t< 180To

if 180To ~ t<. 200To

1.3 To

if

0.2 To

if

200To~

t < 260To

t ? 260To

1\

[ Yr(k+d(k)+1 )+a (k)y(k+d( k) \ k) 1 1\ (k)y(k+d(k)-1 1\ \ k )-b " ( k)u(k-l) +a 2

2

(27) (20)

-'b3(k )u( k-2)] / bi Ck)

Assuming t:

1 z2 1 ~ 1, then us e the following

Case 2: If

(t )

is unkn own, the performance of the

adaptive co ntr ol by using th e algorithm propos ed in this paper is shown in Fig.5.

equation that " \ k) u(k) = [ Yr(k+d(k)+I)+a1\ (k)y(k.d(k) 1

+~2(k)Y(k + d(k)-1 \ k) -~( k )UC k- l)]

If we don't tak e accou nt into the unknown time de-

(21)

lay in the modelled parts of example 2, 3 and 4 the

Idl

r es ults will be divergent simil ar to exa mple 1. We

1

can conclude th at model c lass (13) is indeed a ro-

I b Ck)

bust modpl class and the adaptiv e co ntro l for the

where Yr(') is the set-point of output .

class by using first order or second order model Step 7. k=k +l. then back to step 2 to continue the procedure .

(14) and (15) with unknown time delay has good ro bustness .

Remark There are several mothods to compute the multistep predictions, for example, De Keyser &

REFERENCES

Cauwenberghe (1981) and Clarke, Hodgson & Tuffs (1983) .

Astrom, K. J. and Wittenmark, B. (1973) . On self tuning regulators, Automatica, 9,

185.

Astrem, K. J. and Wittenmark, B. (1985). The selfSIMULATIONS

tuning regulat ors re vi sited,

~th

IFAC

Symp . on Identification and Syst. Parameters Example 2. Consider system which is the sa me as ( 1) that

Estimation , York, U. K"

xx v.

~strom, K. J. Hagander, P. and Sternby, J. (1984), G(s) 1/( 105 +1)( 2.55-1 )(25 -1 )

(22 )

Zeros of sampled systems, Automatica, 20, 31 - 38. Souza, C. E., Goodwin, G.C. , Mayne, D.Q. and

But the modelled p3rt of (2 2 ) is changed as

Palaniswami, M. (1986) . An adaptive control (23 )

The resu l t of adaptive control introduced in this paper is presented in Fig.2. It is obviously seen that the performance is much better than that of example 1. Example 3 . Consider the fifth order system

algirithm f or linear systems having unknown time delay, Technical report EE8626, University of Newcastle, Australia.

Zhang, Y.G. and Ma, R. J. ( 1988 ). Zero< of
Control for Systems with Unknown T ime Delay

29 1

Compari ng H(z; P ) with H(z), H(z; p ) has n zeros, but

Acke rmann, J. (1972) . Abtastregelung, Springer-

H(z) has n-l zeros on l y. In addition, H(z; p ) has

Verlag.

poles which are the same as that of H(z ) , but H(z;

De Keyser, R.M.C. and Van Cauwenberghe, A.R.A. (1981). Self - -uning mul t istep predictor appli -

P)

has a additional pole: Pn+l=O .

c"tion, Automatica, 17, 167-174. Now we consider two limit processes

De Keyser, R.M.C. and Van Cauwenberghe , A.R.A. (1985). Ext en ded predi ct ion self-adaptive con trol, Pr oc . 7th IFAC Symp . on Identification and Syst. Parameters Estimation, York, U.K . ,

K~yser,

p~ o

Rnd

p_l. First, we investigate the case of

p~ O .

Based on

the co ntinuity of modified z-transformation it is

true that

1255-126e. De

3$

R.M.C. (1986). Adaptive dead-time e sti-

lim p- O

mation , Pro c . 2nd IFAC Wo rk shop on Ada p tive Systems in Co ntrol and Signal Processing , Lund,

lim

bie r )

lim

b~+I ( P )

p..... o

Jury, L1. (1964). Theor y and Application of the

p-a

Z-transformation, John Wi l ey , New York.

Verlag.

b'(O) =b., 1

( 198 3). Th e o ffset problem and k-incr emental pr e dictors in self - tuning control, Proe.

i=1,2, . .. ,n

1

(3 1 )

Let Zl' z2' ... , zn_l be the n-I zeros of H(z) in ( 28 ) and zi (f ) ' ... ,

Clarke, D.W., Hodg son. A. J .F . and Tuffs, P.S .

(30)

This means thdt

Sweden, 209 - 213 .

Is erma nn, R. (1977) . Digitale Regclsystem, Springer

H(z)

H( z ;0)

H(z; P )

P)

z~( f )

be the n zeros of H(z ;

in (29). According to polynomial theory we have

lEE,

(32 )

130 , PtD (5), 217-225. Kurz, K. and Goedec ke, W.

(1981) . Dig ital parame-

b~+l(O)=O,

Because

so there must be one zi(O) at

ter - adaptive control of pro cess with unknown

least that it equals O. Without los s of generility,

dead-time, Automatica, 17 , 245 - 252.

we can assume zi (O)=O .

Mosca, E. and Zappa, G. ( 1983 ) . A MV adaptive con troller for plants with time-varying 1/0 transport delay. Pro c . IFAC Works shop on Adaptive

According to (29) there is a 0 pol e Pn+l' so there would be cancellation between zi (O)=O and Pn+l =O as

p_O and

Systems in Con trol and Signal Processing, SAC - 5 lim

Wong, K.Y. and Bayoumi, M.M . (1982) . A self-tuning

zi+ l ( P )= zi+l (O)=zi'

i=I,2, .. . ,n-I

(33)

/4-0

algorithm for syste ms with unknown time-delay,

For the convenience of theoritically mentioning we

proc. 6th IFAC Symp. Identification and Syst .

assume that H( z; O) and H(z) ha ve a 0 zero and a 0

Parameters Estimation, 1064- 1069.

pole constantly. Next

consider the case of

'Ne

p_ l. Also based on

the con t inuity of modified z-transformation we have

APPENDIX

lim

H(z;

f'~

When E> =O the transfer function of the sampled sys -

lim

bi+l( r )=bi ~I(1 ) =bj'

lim

bi( P )= bi(!)=O

P- l -n

n

H(z)

'-a 1 z

- 1

-a z

= H(z;l) = z-I H( z )

z-d

z -a z

Because bi ( 1 )=0, thus

1

For the case of

n

(28)

- a

O
n

z~ (!)

(36)

, ~ ,

This implies that there must be at least one zi ( l )

to obtain the transfer svste~

function of the sampled

I

r~ 1

-n

n

n- l

i=1, 2, ... n-1 (35 )

1\

n

(34)

This means that

tem (with zer o-order hold) is as follows: - b z

p)

1

that equals infinite, without loss of generility we

the modified z-trans-

assume

z~ (1)= 00 .

and

z ~ (1 )

form method is used . we have ' ( D ) - ( n-l 1 b 1' ( P) z - 1 . . .. -b n-I I .z

H( z: P ) = 1-a z

-1

1

- . .. -a z

- n

n

n- l

- ... -an ) z

i=1,2, . . . ,n-1 (37)

z' Cl) n

n

bi ( f l zn- .. . -b~(r)z-b~_I(f) ( z Ta1z

z

1

-d

Z~ ( I)= oO

z-d

is equivalent to a backward shift operator

( 29) z- l, which is absorbed by

Ma Runjin an d Zha ng Yongguang

292

Input

A

Output

Esti .. "tion ---=-=:=~==------==- . - . ,--

. 4' ".

"

K

Esti .. "tion ~)

i.

" Fig. 4

Fig.