Regulator problem for infinite-dimensional systems

Systems & Control Letters 3 (1983) 31-39 North-Holland

June 1983

Regulator problem for infinite-dimensional systems Toshihiro KOBAYASHI

Department of ControlEngineering, Kyushu Instituteof Technology, Tobata, Kitakyushu 804, Japan Received 10 February 1983 Revised 20 March 1983 In this paper a regulator problem is investigated for an infinite-dimensional system with constant disturbances. The regular problem is to determine a feedback control law which stabilizes and regulates the system. A design procedure of a regulator is proposed which can be realized in finite-dimensional theories and techniques for an infinite-dimensional system.

Keywords: Infinite-dimensional system, Stabilization, Output regulation, Constant disturbances, Finite-dimensional observer.

I. System description and problem formulation We consider the system described by an evolution equation on a reflexive Banach space X:

du(t)-Au(t)+Bf(t) dt

+w,

u(O)=uo~D(A)

0
(1.1)

where u(t) ~ X is the system state vector, f ( t ) ~ E p is the control vector and w ~ X is an u n k n o w n constant disturbance vector. The operator A : D ( A ) ~ X is a closed, linear, densely defined generator of a h o l o m o r p h i c semigroup U(t) on X. The c o n t r o l f ( t ) is assumed to be H61der continuous. The operator B is a b o u n d e d linear operator from a p-dimensional Euclidean space E p to X. Then the system (1.1) has a unique solution u(t) ~ D ( A ) for t >/0, continuous for t >/0 and continuously differentiable for t > 0, given by

u ( t ) = U ( t ) u o ---~fotu(t - s)( B f ( s ) + w) ds.

(1 .2)

The controlled output y ( t ) ~ E r is given by

y(t)=Cu(t),

O < t < t 1,

(1.3)

where the output operator C : D ( C ) c X--* E ~ is linear and defined on D(A), and hence D ( A ) c D(C). T h e operator C is assumed to be A - b o u n d e d [4]. The key to finite-dimensional regulator design is to a decomposition of the state space X based on the m o d e s of the system. The operator A satisfies the spectrum decomposition assumption [4]; then a projection P exists such that

X=PX+

(1- P)X

(1.4)

and PX, (1 - P ) X form A-invariant subspaces of X. F r o m the viewpoints of system analysis and synthesis, it is practical and interesting to take P X as a finite-dimensional space. We shall assume henceforth that P X is an N-dimensional subspace. Consequently from (1.1) and (1.3),

dPu( t ) = A e P u ( t ) + P B f ( t ) + Pw, dt

Pu(O) = Pu o,

0167-6911/83/$03.00 © 1983 Elsevier Science Publishers B.V. (North-Holland)

(1.5)

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dQu(t) dt =AeQu(t)+QBf(t)+Qw

,

Qu(O)=Quo,

y ( t ) = CpPu( t ) + CeQu( t ),

June 1983 (1.6) (1.7)

where Q = 1 - P and Ap, A e are the restrictions of A to" P X and QX, respectively. PB, QB are the restrictions of B to P X and QX, restrictively. Ce, Ce are the restrictions of C to P X and QX. Ue(t ) = P U ( t ) is generated by A e and Ue(t ) = QU(t) is generated by AQ. Actually A e is bounded on P X and Up(t) is a uniformly continuous holomorphic semigroup. We also assume that the operator A e is a generator of an exponentially stable semigroup Uo(t ) such that for constants K >1 1 and a > 0 Ilfe(t)l I ~< Ke -°t,

t > 0.

(1.8)

Now we may pose the following control problem. Problem 1. Find a linear feedback control law for the system (1.1) and (1.3) such that (i) the resulting closed-loop system without a disturbance w will be exponentially stable, and (ii) the controlled output y ( t ) will be regulated so that y ( t ) ~ y d, t ~ oo, independent of w where Yd ~ E~ represents a desired constant reference vector. To our knowledge there are a few works by Pohjolainen [6] for infinite-dimensional systems. He has treated only stable systems. In his design techniques the concepts of state feedback and observer are not included. The assumption that the disturbance vector w and the reference vector are constant in time t is not the most general. We can treat polynomial signals in time t. However, the constant vectors are most important and they allow us to develop the theory without unnecessary mathematical complexity.

2. Construction of state feedback controllers In this section we construct a feedback controller which solves Problem 1. The controller consists of two parts: the stabilizing compensator (Proportional part of the controller) and the servocompensator (Integral part of the controller). The role of the servocompensator is to change the system steady state, so that the output regulation y ( t ) ~ Ya will occur. Now if we put i/(t) = y ( t ) - Y a ,

(2.1)

we obtain from (1.1) and (1.3) the following system: °

in the extended state space Xr = E r × X, which will be a Banach space when equipped with the norm "llx, --I1" I1~, + I1" II~-Before designing a compensator, it is useful to transform the state variable as follows: I~ = ~1 + S u ,

32

u = u,

(2.4)

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June 1983

where S is a bounded linear operator from X to E C By this transformation we get from (2.2)

that is,

,AQ][,],u+Is ,+[ ;w j Sw --Yd]

Pit Qit

Ap 0

0 AQ

Qu

PB QB

(2.5)

Since the operator A o is the generator of an exponentially stable semigroup Uo(t ), the inverse A~ 1 exists and is bounded. If we take Sp = 0 and S o = - CoA ~ 1, the operator S = (0, SO) is actually a bounded linear operator from X to E p. In this case, (2.5) becomes

SQQB] Pit

=

Qi~

0 0

Ae 0

0 Ao

Pu

+

Qu

;', ]:+

"SQQw- y~] Pw

(2.6)

Qw

Thus (2.2) and (2.3) become

(2.7)

,=t0 JI'u]

(2.8)

where C = (Cp, 0), SB = SoQB = - C Q A ~ '0 8 and Sw = SQQW = - C o A ~ 'Qw. For the system (2.7) we consider a linear feedback control law

f ( t ) = D ~ ( t ) + F u ( t ) = ( DS + F ) u ( t ) + D fot(y(s ) - Y a ) ds

(2.9)

where D ~ L ( E r, E p) and F ~ L ( X, EP). Then we get the closed-loop system (2.10) We can prove the following theorem in a similar way to that in [6], Theorem 3.3. Theorem 1. Suppose that there is a stabilizing control f = D~ + Fu such that the system (2.11)

will be exponentially stable. Then regulation will occur in spite of a constant disturbance w. Theorem 1 says that if the augmented system (2.11) can be stabilized, then regulation will automatically occur. The next step is to show that there exists a stabilizing control of the form f = D~ + Fu where D ~ L ( E r, E p) and F ~ L (X, E p). The key to stabilizability for the system (2.11) is a decomposition of the system state u on the modes of the system. Decomposing the state u by Pu and Qu, we obtain from (2.11)

Pit

Qit

=

0

Av

0

0

Pu

ao

Qu

+

PB

f.

e~] 33

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June 1983

For this system let us consider a linear feedback control law

f ( t ) = D~(t) + Foeu(t ) = DSu(t) + Foeu(t ) + D fot(Y(S ) -Ya) ds

(2.12)

where Fo ~ L(PX, EP). Then we have the closed-loop system

Pi~ = Qit

PBD QBD

Ap + PBFo QBFo

Pu • Qu

AO

Since P X is an N-dimensional subspace, if the (r + N ) dimensional system ([:

AC:I, [ S B I )

(2.13,

is controllable, there exist feedback control operators D and F0 such that all the eigenvalues of Ale,

[ SBD A/,,,= t PBD

Ce + SBF o ] Ap + PBFo l'

have negative real parts. The feedback operators D and F0 can be determined by pole allocation or optimal regulator design for the usual finite-dimensional system [5]. Now we have for constants K 2 >/1 and ~0 > 0

Ilexp( A/pt )ll ~ K~e -~'',

t >10.

This implies

[,t,

Pu(t)]

[.0 1] PUo

t>~

0

(2.14)

Let us estimate Qu(t). Since

Qu(t)=Uo(t)Quo+

U°(t-s)QB[D

F°] Pu(s)

ds,

from (1.8) and (2.14) we have

,

IIQu( t )ll <~gllQuoll e-°' + fogg=llQBII II[D Fo]ll

[Puo[~(O)]e_O,,_,, e-~"ds o - o , - o-o,

=KIIQu°Ile-°' + KK211QBIIII[D F°]II [Pu o

o-~

where we choose ~o such that ~0 ~=O. Consequently we obtain

IIQu( t )11~
t >/O, '

where

¢1-- ~ max(K, ~K-2IIPll 2 + (KK211QB-I1[ D Fo ]11/Io - ¢ol)211QII2 ). 34

(2.15)

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June 1983

Moreover, the estimates (2.14) and (2.15) give [~(t)

] = ~/ll~(t)ll 2 + iiPu(t ) + Qu(t)ll2

u(t) 111

Thus we have obtained the following estimate: IIU/(t)ll ~< c2 e -mint°''°)',

t >/0,

(2.16)

where cz = V/2(gff + c~) max(v~-, }/1 + IIPII2 ). It has been shown that the system (2.11) is exponentially stabilized by the feedback control law f = D~ + Fu where F - FoP. We have obtained the following theorem. Theorem 2. I f the ( r + N )-dimensional system (2.13) is controllable, then there exists a feedback control law

f ( t ) = D S u ( t ) + Fu(t) + D fot(Y(S ) --Yd) ds which exponentially stabilizes and regulates the system (1.1). Remark 1. It is shown that the ( r + N)-dimensional system (2.13) is controllable if the N-dimensional subsystem (Ap, PB) is controllable and dim range PB

Ap = r + N < ~ p + N .

In our design schemes it is not clear how fast the output y ( t ) will converge to the reference vector Yd" If for the system (2.7) we apply the feedback control law (2.12), we get the closed-loop system

it~ = A/u~( t) + w,

(2.17)

where '

, [SWwy ]

'

BD

.4 +

B F ]"

The solution is given by

u~(t) = ~ ( t ) u~(0) + £ ' ~ ( t - s)w, ds. Differentiating this in t, we obtain

it~(t) = v~(t)(A:u~(0) + w,). From (2.16) and u o E D( A) Ilite(/)ll ~< c2 e - ' i " t ° ' ~')' IIAIu~(0) + wsll, t/> 0. Since (2.4) implies ~(t) = i / ( t ) + Sit(t) = (y(t) --Yd) + SQQit(t), we get the estimate IlY(/) --Ydll <~const-e-mi"ta'~')tllAIu~(0) + wsll, t >/0.

(2.18) 35

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This estimate says that the controlled output y(t) will converge to the constant reference vector Ya with an arbitrarily assignable exponential decay rate by the feedback controller (2.12).

3. Construction of output feedback controllers Theorem 1 and Theorem 2 give the basic solution for Problem 1. However we assume the knowledge of

Su and Pu in the feedback control law (2.12). In this section we shall show that even if we use the state v of an observer in place of u, the feedback control law (2.12) still gives the solution of Problem 1. Consider the measurement output z(t) given by

z(t) = M u ( t ) ,

t >t O,

(3.1)

where the measurement operator M : D ( M ) ~ E q is linear and defined on D(A), and hence D ( A ) c D ( M ) . The operator M is assumed to be A-bounded. Now we construct an identity observer

i~(t)=Av(t)+Bf(t)-G(Mv(t)-z(t)),

0
v(0)=0.

(3.2)

Here G is a compact operator from E q tO X. Then A G = A - GM generates a holomorphic semigroup T(t) on X [7] and the solution of (3.2) is given by

v ( t ) = fotT(t - s)( B f ( s ) + Gz(s)) ds. The solution v ~ C(0, t~; X). We can prove the following lemma. Lemma 1. 1./"the N-dimensional system ( A p, Mp ) is observable and MQUo(. ) ~ L2(0, tl; E q ) then there exists an operator G such that the semigroup T( t ) will be exponentially stable.

Proof. Consider the system

~=Av(t)-GM~(t),

g(0)=~0~D(A

).

We choose G such that QG = 0, that is,

Gz = { G°zO

on°nPXf°rz~El"QX,

where GO~ L ( E q, PX). Decomposing g by P g and Qg, we have

P~=ApPv-Go(MeP~+MoQ~

),

? g ( 0 ) = P ~ 0,

Qg=AQQO,

Q o ( O ) = Q g 0.

(3.3)

Since P X is an N-dimensional subspace of V, from finite-dimensional theory [5] we can find a G O L ( E q, PX) such that all the eigenvalues of Ace = A e - G o M e have negative real parts. Thus there are constants K c >/1 and 3' > 0 such that Ilexp(acpt)ll ~< K G e -vt,

t >/0.

(3.4)

In this case we can obtain the estimate [3, Theorem 3.1] IIT(t)ll ~< c3 e -min(°/2"~)',

t >/0,

where o * 2"y and c~ = { KGIIPI] + K IIQII(1 + KGIIGolIIIMQUQ(.)ll/2TI/~-o')) • 36

(3.5)

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From (1.1), (3.1) and (3.3) the estimated error vector e = v - u satisfies

b(t)=Ace(t)-w,

e(0)=-u

0.

(3.6)

Even if the operator A 6 generates an exponentially stable semigroup T(t), there remains an estimated error in the steady state, since w is a constant vector in time t. However we can show that the feedback control law (3.7)

f ( t ) = D S v ( t ) + Fv + Dfo'(U(t ) --Yd) dt gives the solution for Problem 1. From (2.7), (3.6) and (3.7) we get the closed-loop system

= SBDS + S B F BDS + B F y = [ O 0 C]

0 0 ]fell w]

SBD BD

C + S B F ~ + Sw A + BF J L u

(3.8)

Ya '

(3.9)

,

in the extended state space Xq = X x E q X X, which will be a Banach space when equipped with the norm

II • 112x,,= II-I1~ + I1" I1~,, + II-I1~. Corresponding to Theorem 1 the following theorem holds. Theorem 2. I f the system

= SBDS + S B F BDS + B F

0 0ire ] [il

SBD BD

(3.10)

C + SBF ~ = AG/ A + BF J Lu

is exponentially stable, then regulation will occur in spite of a constant disturbance w. Next we can show that the system (3.10) will be exponentially stable, if both the semigroup T(t) generated by A c and the semigroup U/(t) generated by A/are exponentially stable. In a similar way to that for the estimate (2.16), we can obtain the following estimate for the semigroup Uc(t ) generated by Acf:

iiUc(t)ll
t>~O,

(3.11)

where min(o, to) * rain(y, 0/2), c 5 = V~3 2 + c42 and

iw:ll c4=min(c2+c2c3b/,~/2 max(c2, c2c3b/)),

b/=lmin(o, o o ) _ m i n ( o / 2 , ~ ) I "

We have the following theorem from Theorem 3 and Lemma 1. Theorem 4. I f the ( r + N )-dimensional system (2.13) is controllable, the N-dimensional system (Ap, Alp) is observable, and MoUo(. ) ~ L2(0, tl; Eq), then there exists a feedback control law (3.7) which exponentially stabilizes and regulates the system (1.1). Moreover if for the system (2.7) we apply the feedback control law (3.7), we can obtain the estimate

f
ily( t)_Yall < const, e-mi,tv . . . . /2), A ~ / / ~ ( 0 )

Lu(O)

=

-w 1

Sw

w-Y

'

t >~O.

(3.12)

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This estimate says that the controlled output y(t) will converge to the constant reference vector Yd with an arbitrarily assignable exponential decay rate by the feedback controller (3.7). However, the infinite-dimensional observer (3.2) is not so easy to realize. We can show that the system (1.1) and (1.3) is stabilized and regulated by output feedback through a finite-dimensional observer. Define the other projections PL and Qt- such that

X = Pt- X + Qt- X and Pt- X is an L-dimensional subspace where L >1 N. We construct an observer

ig(t) = A v ( t ) + P L B f ( t ) -

G(Mv(t)-z(t)),

0 < t < t,,

v(0) = 0.

(3.13)

Then we get the following system corresponding to (3.10):

[i]

=

SBDS + SBF BDS + BF

:lle] [;]

SBD BD

C + S B F / / ~ = A-a! " A + BF ] [ u u

(3.14)

This system operator A-c/is a bounded perturbation

Bt-=

-Qt-B(DS+F) o

-QtBD o

0

added to

AGf. The

operator

0

-Qt-BF] o I' o J

AGI=AGf+ JffLgenerates

a strongly continuous semigroup

UG(t), defined

by

UG(I)Uq= UG(t)uq + f o t U G ( t - s ) n L u G ( s ) u q d s , uq ~ gq. Moreover from (3.11) we obtain the estimate [2], [4]

IIUoll ~ c5 e (-min(v . . . . /2)+csllidl)t, t >1 0.

(3.15)

If we choose L such that - m i n ( y , w, 0 / 2 ) + c 5II/~t-II ~< - 8 < 0, the system (3.14) is exponentially stable. In this case the control law (3.7) still stabilizes and regulates the system (1.1) using the observer (3.13) in place of the observer (3.2). On the other hand the observer (3.13) is decomposed as follows:

Pj)(t)=(AvL--GoMpL)PLv(t)+PzBf(t)--GoMQzQLv(t)+Goz(t),

QLi~(t)=AQLQLv(t).

where Art " and A o t, are the restrictions of A to Pt-X and Qt X, respectively. Mvt " and Mot - are the restrictions of M to Pt-X and Qt-X. Moreover the restriction of G to PI_X is GO and the restriction of G to Qt-X is 0, since L >~N. We are free to choose Qt-v(0)= 0 and this implies that Qt-v(t)= O, t >10. Thus an L-dimensional compensator is given by

Pt-i)( t ) = ( A r t, - GoMvt-) Vt-v( t ) + Pt- B] ( t ) + Goz( t ),

f (t) = ( o s + F)et-v(t)

+ Ofot(y(s ) --Yd) ds.

(3.16)

If we choose L such that -min(~,, w, o / 2 ) + c s I I B t - I I < - 8 < 0 , Theorem 3 and 4 still hold for the L-dimensional compensator (3.16). In this case y ( t ) will converge to Ya with the exponential decay rate e-st.

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References [I] M.J. Balas, Feedback control of linear diffusion processes, Internat. J. Control 29 (1979) 523-533. [2] R.F.Curtain and A.J. Pritchard, Infinite Dimensional Linear Systems Theory (Springer, Berlin, 1978). [3] R.F. Curtain, Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input, IEEE Trans. Automat. Control 23 (1982) 98-104. [41 T. Kato, Perturbation Theory of Linear Operators (Springer, New York, 1966). [5] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscienee, New York, 1972). [6] S.A. Pohjolainen, Robust multivariable Pl-controller for infinite dimensional systems, IEEE Trans. Automat. Control 23 (1982) 17-30.

[7] J. Zabczyk, On decomposition of generators, S l A M J . Control. Optim. 16 (1978) 523-534. [8] T. Nambu, Feedback stabilization of diffusion equations, 1982 Japan JACC (in Japanese).

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