Robust Adaptive Control of Interconnected Systems

Robust Adaptive Control of Interconnected Systems

Copyright © IFAC Identification and System Parameter Estimation, Beijing, PRC 1988 ROBUST ADAPTIVE CONTROL OF INTERCONNECTED SYSTEMS Yun-Qi Lei, Lian...

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

ROBUST ADAPTIVE CONTROL OF INTERCONNECTED SYSTEMS Yun-Qi Lei, Liang-Qi Zhang and Zheng-Zhi Wang Dept. of A.utomatic Control, Challgsha Institute uf Tech., ChaIlgsha , PRC

Abstract: In this paper we consider the robust adaptive control of interconnected systems whose subsystems contain parasiticses (unmodelled dynamics) and disturbances. Motivated by the idea that restrain the adaptively caculated parameters by the control signals , we give a robust adaptive control law and its generalized for., in which the control weighting has been considered. The new laws are suitable to be used in the cases of decentralized adaptive control of interconnected systems and they are different from the traditional adaptive law (Narendra and Valavani , 1978) and the famous a-modification (Ioannou and Kokotovic,1985;Ioannou and Tsakalis,1986) and el-modification (Narendr and AnnasNaay,1987). By use of the Lyapunov techniques, the robustness properties of the new laws in respect to parasiticses and disturbances are analyzed. Under the function independence condition or the norm nonzero condition, if the frequences of parasitics are high enough, the control weighting is not too heavy and the coupling between subsystems is .oderate,

then there must exist a

region of convergence Dc and a

residual

set Or such that the system variables started from Dc will keep bounded and converge to Dr. Besides , the rates of convergence are exponential. Keywords: Robustness, Theory , Stability. 1.

Adaptive

Control,

I NTRODUCT I ON

Decentralized Control,

Control

be used in the later case, but however this paper will give another kind of method to get good robustness in the adaptive control of interconnected systems . It's worthwhile to notice that the unstability of adaptive control systems are always accompanied by disconvergence of parameter vector which is caculated through a adaptive law and accompanied by the infinite growth of the control input signal u(t). This fact prompt us using control signal to prevent the parameter vector from disconverging and at the same time keep other variables in the system bounded. Follow the above idea and motivated by the results of optimal control of ARMAX model ( Astrom , 1970; Goodwin,Johnson and Sin,1982) we use e~(t ) =y(t)-ym(t)+~u(t) instead of 8 q (t)=y(t)-Ym(t) in adaptive c o ntrol of ,nterconnected system to caculate the parameter vector , this gives a new parameter adaptive law as

It's well known that there are some ideal assumptions in adaptive control systems,for example the order of plant should be known. If those assumptions are violated , the "hole adaptive control system will no longer be stable . Unfortunately those assumptions may be violated easily. The counterexa.ples given by (Rohrs and co"orkers , 1982) and (Ioannou and Kokotovic,1985) clearly showed this fact. Very possibly this is one of the important reason why the successful applications of adaptive control theory are not as many as people expected. Those counterexamples reflect the severe weak points of unmodified adaptive control and at the same time they are serious challenge to the adaptive control theory. Since the counterexamples given by (Rohrs and coworkers,1982) were published many researchers have turned to study the robustness of adaptive control systems and several methods have been proposed to enhance the robustness(please refer the References) , A satisfied solving of this robustness problem is needed in the aspect of both theory and applications .

ei ui(t i

-Tieo i"i

- >-iTiui ( t)"i ( t)

(18)

( 1b)

eT(t)wi(t)

which " i l l be used in this paper to get robustness of de c entralized adaptive control of interconected systems . For the derivation of the law , please refer (Lei , Zhang and Wang,1987) . The new law is clearly different from the traditional one (Narendra and Valavani,1978), and different from the e1-modification and the 0 modification which respectively are

adaptive As in the case of integrated control syste.s, the decentralized adaptive control of interconected system also exists the same robustness problem when there are parasitics and disturbances in the plant. The methods used in the former case such as a-modification(Ioannou and Kokotovic,1985; Ioannou and Tsakalis , 1986) , the e l modification(Narendra and Annaswamy,1987), the dead zone method (Samson,19B3) and the normolization method (Praly,19B4) etc . may

(2 )

r >0

28 1

( 3 )

282

Yun-Qi Lei, Liang-Qi Zhang and Zheng-Zhi Wang

law we aay attain to two aias. First, an iaplicit control weighting in decentralized adaptive control has been introduced which is reasonable because the control power or control energy of a real systea is liaited and should be punishied when the control signal becoae too large. Second, as we will show in the following that the new law which lIay be naaed control weighted adaptive law possesses of a good robustness property in respect to parasitics and bounded disturbances. It has been showed that under the function independence condition or the norm nonzero condition, if the frequencies of parasitics are high enough, the control weighting is not too heavy and the coupling between subsystems is 1I0derate, then there must exist a region of convergence Dc and a residual set Or such that the systea variables start froll Dc will keep bounded and converge to Dr. Besides, the rates of convergence are exponential. The contents in the following are arranged as: Section 2 presents problem statement, Section 3 gives the robustness analysis of decentralized adptive control of interconnected systems, Section 4 gives the generalized control weighted adaptive law and shows the robustness of it . Section 5 is then the concluding rellarks.

T'

Ya i

.,/1;+1.

= hili xIIi; X,.isAmiXa.i·""'irj ,X,.;/t). K( 7 ) (8 )

(i=l , ..... N) No" we aill at designing local controllers to keep the whole syste. stable by using only the local information as if there are no

interconnectons,

no

disturbances

and

no

parasiticses. If we apply the traditional adaptive control law (9),(10) , (11) in the following to the real system (4)-(6) .1 vi wi1 ' 2 vi

/\ivi + qiui

(9b)

cI(t)vi /\ i v f

+

( lOa)

qiYoi

w2i

i do(t)Yoi

ai

-Tieoiw i ;

U

aIw i

i

(9a)

+ dI(t)vf

(lOb)

Ti = TT1

> 0

(11 a) (lIb)

Where 2 1 WI(t)=[r i (t).v I(t)'Yoi(t) , v I(t)];

er< t ) = [ Ko i ( t

) . c I ( t ) . do i ( t ) ,d I ( t ) ] ,

A

i is a stable matrix of order nixn i , (/\ i.qi) is a controllable pair and

2. PROBLEM STATEMENT The interconnected deration is

system

under

consi-

N

+ I

(AsijXsj)

j ~i

Aizi

=

+

Gsix si

N

(5 )

I(GsijXsj)

+

(4)

j~i

Re

(A i) > 0; (i = 1

, 2 , ... , N)

(6 )

Where for i=1 , 2 , .. . , N, xi(t) are state vectors of order n i . Zi(t) are parasitical vectors (unmodelled dynamics) with order lIi' ui(t) and Yi ( t) are scalar control signals and output signals respectively. t · are saall parasitical factors and dOi(t~ are bounded disturbances , others are syste. structural matri~s or vactors with siutable dillensions and they are tille-invariant. The ideal systell when set E.i=O . Asij=O . Gsij=O and Doi(t)=O (i= 1 , 2 ..... N) satisfies the standard conditions same as in (Narendra and Valavani , 1978) . they are : (1) ni are kno"n . (2) every isolated subsystell (Asi . bsLhsi) (ideal) is co.pletely observable and controllable. ( 3) the transfer functions of isolated subsysteas Wi(s)=kiZi(s) / Ri(s) are assumed to be that ki > O ( without loss of generality) and Zi(s) are monic Hur,:itz polynomial and the relative degree n~ of Wi(s) is equal to 1. A reference model can be introduced in accordance with the ideal systell, which is (Narendra and Valavani , 1978 , Ioannou and Kokotovic,1985)

qi=[O ' O, .... I]T. As pointed out in lIany published papers (Ioannou and Kokotovic, 1985; Rohrs and co-workers , 1982; etc.) , the whole system (4)-(6), (7)-(11) will propably be unstable due to the existence of parasiticses and disturbances. To OVercome this difficulty, Ioannou and Kokotovic (1985) gave a a-IIodification Ilethod (see (2)) by introducing a saall gain feedback in (11) and Narendra and Annaswamy(1987 ) gave a el-modification (see (3) ). However. as We have said in section 1 that the unstability of (4)-(6) , (7)-(11) is always accompanied by the phenomenon that ~ai~ drift to infinite and ui(t) becolle too large. Motivated by the results of generalized minimum variance optillal control of ARMAX mOdel, so we lIay use control signals ui(t) in (11) to prevent ~aill from drifting to infinite, and at the same time to make ui(t) weighted, this idea is also supported by sOlle simUlation results . Replace eoi(t)=Ysi(t)-Ymi(t) in ( 1 1 ) b Y e~ i ( t ) = y s i ( t ) - y mi ( t ) - "i u i ( t ) ( "i >0 is a weighting factor ). then (11) becolles -Tieoiwi- ~iTiui(t)Ni(t)

(12a) (12b)

aIwi(t)

This is the new adaptive control law in which the control weighting has been considered. Now the systell (4)-(6), (7)(10 ) and (12 ) can be changed to be (Ioannou and Kokotovic.1985) N

Aci'i+bci
N +

r

jU

~j J i j

1j + £i f 2 i

(13 )

Robust Adaptive Co ntrol of Interconnected Systems

= -Tieoiw i - ~iTiuiwi(t) £ ~ i = A ii '1i + Et f 1i Si

A~iPci

PciAci

+

vi'O,

P ci

=

(15)

by using (13)-(18) inequality , then

( 16 )

Vs (

h ,Q i)

I:'

-qiqI -

and

N i

i=l

=

283 the

triangular

\I ~i 11 2

i [ - \; i N

viLi +

P;i 70 ,L i LI , O

Where

-

11 ~i 11

rg ij U'ijt!+II'i If

.oi

j~i

(<» I w i) 2 '>..i / 2

+

K

.*

Ko

U~i 12 *

\. / 2

+ K 1 K:\ / 2] ( 17)

Where

s~p

mo i

(

IIPcill [ JiDci(t)II

N

+jEIIACijllllxcj(t)!t ]) ( 18) Ke* Now , the problem is how about the stability of (13)-(18) , this is the main results of this paper , we present it in the next Section .

max:>.(s*s*T)

Because S >O and choose

~i

such that

Now 3. ROBUSTNESS ANALYSIS OF THENEW ADAPTIVE CONTROL LAW Define N

S=(Sij)NXN

~i S . . =(

1J In

:>'s i ; ( , i '

°)i = j ; "s i = ~ Vi min

_ ~ ( ' i g i j + fj g j i ) i

~j

Del

1>\( LD

r.

('iKe*Kl:l.i / 2+2*·;i

ff/~i)

i=l

; g i j =\1 Pc i Ac i j 11

the

following Dei(i = 1, ... , 4) denote whic~ can be ~eter.ined by the sum 0 f k ~ sup 11 d 0 i ( t ) U , k ~ Ei and a po s i t i v e constant .ultiplied by the norm of Some matrixes of the interconnected system and the reference model .

Hen c e, lI'i ll and IQ~wil are bounded. If Al is sat i s fie d , the n 11 Q i 11 ( i = 1. 2, ... , N ) are b o unded. If A2 i s true, assume

constan~s

The robustness properties of the new adaptive law are given in the following theore.s . The 0 re. 1: ( t i 0 , d 0 i ( t ) I! 0 , i = L 2. , .. . , N ) Assu.e that (function independence) : Al: I<»IIJ\iI...-_if and only if ".;11-00 or I i" ------00 or A2.: let W=lim(Wi(t», no component of W can be represented linearly by other components of it . If Al or A2. is satisfied and S ' O , then there must exist such that if O < :>'i < then v ariables ~i , qi , Qi start from any bounded initial state will keep bounded and converge to the following residual set

"i

"i

Drl

+

~,

I

i "i ( ~ w i ) 2)

),~i ' O , Th. detailed in Section (Ioannou and and Wang,1987 Bri.f Proof:

11 will 2 define

~

~

(Ql ( t» is the Jth c o mponent of Qi' then 1 ( t ) Wh1Ch select o ut the has the fastest rate when it tenq to infinite , we 1 2 n1+2] now den o te wi=[wi, ... ,w i and consider

ai

then

this

equation

(* )

where (* ) i s

However

Ki; , ... , K in are c o nstants. controdi c tory to A2.

QED.

F o r the sake o f simpl i city when ~i~o d o i""O, let bii=O

and

Theorem 2: ( f, i "O , doi"O,bii=O) If Al or A2 is satisfied and S ' O, then there exist some contants t i , "ii ' ''2i ' ...... such that i f ti-[O, fl) and "ic["ii ' ''2i], then the variables start fro. the regi o n of convergence

0e1)

'i ' O

pr oo fs ( including t hose proofs 4 ) are o mmitted here , but Kokoto v ic , 1985 and Lei , Z h ang ) are helpful to read .

Ko 11 ' i 11 2 + K 1

( K1 '

°, °)

wi l l keep bounded res i dual set Dr2 .

and

converge

N

KO >

+d

2i

IItU

( dli , d2 i ' d3i Brief Proof: Define

2 +d3i (Q T w ) 2 )

' 0)

to 2

.s.

D e2 )

the

284

Yun-Qi Lei. Liang-Qi Zhang and Zheng-Zhi Wang Theorea ;: ~~i--+O , doi~O) If IIWill lOi , 4i >0 is a constant, and S >O, then there exists ~i,such that if ~ie(O ' ~il then the variables l i ' Qi ' Wi start fro. any bounded initial state will keep bounded and converge to Dr at the rate of not slo.er than exp(-&o 2 ' t)

J

procedure s

Follow the saae Theoreal , then

of

proof

,

of

N

if IIWil1 ~o, the above c o nclusion s t i l l hold but thiS case is equivalent to the cas e t hat 4 Z i s s ma l l . [" ~ me a n s "d 0 e s n o t tend t6 zero [~O(t--)l."

°"

QED .

Brief Proof : Theorem 1, N

Remark: The Theorem 1 and Z have indicated the tolerance a b i l i t y of the control weighted adaptive law to the parasitics and disturbances . They state that if the strength of interconnections is moderate (S >O) and the control weightings are not too heavy , then the system variables wi 11 keep bounded for the parasiticses when 0 < ~ < "i can not b e t 0 0 1 a r ge 0 r the control signals may be suppressed too heavy and then the systea can not get enough power or energy to make the whole system stable .

l.1.

Intuitively , the function independence assumption Ai or AZ can be met easily in soae case , especially when the reference input r(t) contain rich frequences or Di(t) contain rich frequences .

Vs

~-t

u i ( t) .

4 . THEGENERALIZED CONTROL WEIGHTED ADAPTIVE LAW AND ITS ROBUSTNESS

Z

if IIwill T >Q

-Tie oi • i ~i > O ,

4:

For (4)-(10) theore. 4.

and

1: f~'

i f I1 101 i I ~ ~ 0 , also hold.

theorea 3 and

0;

doi'O ,

bii~O)

hi >O,

the

cl l

>O , C o>O , /)«1 / 4.

ab 0 v e

con c 1 us ion s

are

Vs

T>

have

(

Proof : Define Vs ( ~ , Q , ~ ) same as in Theorea Z,

( 19b)

(ZO) , .e

(£i' O,

i =1

trace(Ui(t))

Ti>O

{t+T. c onsider ' t Vs

~ 0,

N

if

(ZO )

/IQjIIZl+De3

If IIW · I I Z ! A Z , A · >0 and s >O , then there 1 1 1 ••• * ex i st s o me c o nstants £.i ' " l i ' hZi' fi
( 19a)

~iTiUi ( t)wi

Z

-z-~i

-

constant.

is a

Theorem

9 i ( t)wI(t)

-

~i t:>

Z

QED.

In order to use more information of ui(t), notice that Ui(t)=&i WiT that is ui(t)=trace ( Ui(t) )

Ui(t) is a generalized representation of ui(t). Now , replace ui(t) in (15) by Ui(t ), .e aay say that information of the correlation matrix of 9i ( t ) and Wi(t ) has been used , we now get the generalized form of the adaptive law of ( 15 ) :

II~ill

i =1

But s t i l l , the assumptions Ai and AZ may be furture replace~ by the norm nonzero condition IIWil1 Z24i which is weaker than Ai or AZ and the results may be further developed , the key point is that more information about ui(t) should be used in adaptive law. In the following section , a generalized fora of the control weighted adaptive law .. i 11 be given which uses Ui ( t) instead of

~ i [-Z-

11"i l l

° is

a

Z

~

(t

0,

consider tt

+T . Vs < 0 ;

constant. QED.

Re~ark :

the norm nonzero condition 11 Wf(t)1I >A aay be aet very easily say in the case ; f 1 I r ( t ) Il.4i . The condition 11 will ~ will satisified if £i~O and dl(t)~O.

°

In the above theoreas , i t requires that S >O, that is, the coupling between the subsysteas should be aoderate , this is a necessary condition .hich have to be

285

Robust Adaptive Control of Interconnected Systems satisfied even in the case of nonadaptive control of interconnected systells(Ioannou and Ko k 0 t 0 vie, 1985) f.i I!I [ 0 , ~ I ill P 1 i cat e s that if the frequenices of parastics are too low, then the parastics can not be viewed as parasitics any longer and should be added into the dOllinant party. If ~i are sllall, the rate of convergence will be s low, but wee ann 0 t use too 1 a r g e "i 0 r the control signals will be weighted too heavy and then the system can not get enough power or energy to lIake the whole adaptive system stable, this is the reason why i t requires the condition "i,d"ii''''2i]'

i

5.

CONCLUDING REMARKS

We have presented the control weighted adaptive law for the decentralized adaptive control of interconnected systems and have shown its robustness in theorem 1 and 2. By use of the correlation matrix Ui(t) instead of ui(t), we have given the generalied form of the control weighted adaptive law , the robustness are also given . But in the later case, the condition 11 loIi 11 ! or U loIi 11 ~O is much weaker than assumption A1 or A2 because the introducing of correlation matrix Ui(t).

Af

In Section 3, we only use the information of trace(Ui(t» which is ui(t) in e~l(t) (see (12) ) , so we can only guarantee the boundness of I
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K.S.

and

A.M.

Annaswasay(1987),

HA

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L.(1984},

"Robust

Model

Reference

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