# GeneralizedS(C,A,B)-Pairs for Uncertain Linear Infinite-Dimensional Systems

Authors: Naohisa Otsuka; Haruo Hinata
Journal: Journal of Applied Mathematics
 (2009)

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## Abstract

We introduce the concept of generalizedS(C,A,B)-pairs which is related to generalized S(A,B)-invariant subspaces and generalized S(C,A)-invariant subspaces for infinite-dimensional systems. As an application the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Further, an illustrative example is also examined.


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## Body

## 1. Introduction

In the framework of the so-called geometric approach, many control problems with state feedback and/or incomplete-state  feedback (e.g., controllability and observability problems, decoupling problems, and disturbance-rejection problems, etc.) have been studied for finite-dimensional systems (see, e.g., [1, 2]). Further, the concept of (C,A,B)-pairs was first introduced by Schumacher [3], and this concept has been used successfully to design dynamic compensators. After that Curtain extended the geometric concepts to infinite-dimensional systems and various control problems have been studied (see, e.g., [4–11]). On the other hand, from the practical viewpoint, Ghosh [12] and Otsuka [13] studied the concepts of simultaneous (C,A,B)-pairs and of generalized (C,A,B)-pairs, respectively, for finite-dimensional systems, and the parameter-insensitive disturbance-rejection problems for uncertain linear systems were studied. Then, Otsuka and Inaba [14–16] extended the concepts of simultaneous invariant subspaces and simultaneous (C,A,B)-pairs to infinite-dimensional systems. Further, Otsuka and Hinata [17] studied the concept of generalized invariant subspaces for infinite-dimensional systems.The objective of this paper is to investigate the concept of generalizedS(C,A,B)-pairs for infinite-dimensional systems and to study the parameter-insensitive disturbance-rejection problem with dynamic compensator.The paper is organized as follows. Section2 gives the concept of generalized S(C,A,B)-pairs and its properties. In Section 3, the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Section 4 gives an example to illustrate our results. Finally, some concluding remarks are given in Section 5.


## 2. GeneralizedS(C,A,B)-pairs

First, we give some notations used throughout this investigation. LetB(𝒳;𝒴) denote the set of all bounded linear operators from a Hilbert space 𝒳 into another Hilbert space 𝒴; for notational simplicity, we write B(𝒳) for B(𝒳;𝒳). For a linear operator A the domain, the image, the kernel, and the C0-semigroup generated by A are denoted by D(A), ImA, KerA, and {SA(t);t≥0}, respectively. Further, the dimension and the orthogonal complement of a closed subspace 𝒱 are denoted by dim(𝒱) and (𝒱)⊥, respectively.Next, consider the following linear systems defined in a Hilbert space𝒳:(2.1)S(α,β,γ):{ddtx(t)=A(α)x(t)+B(β)u(t),y(t)=C(γ)x(t),
where x(t)∈𝒳, u(t)∈𝒰:=Rm, y(t)∈𝒴:=Rℓ are the state, the input, and the measurement output, respectively. Operators A(α), B(β), and C(γ) are unknown in the sense that they are represented as the forms:
(2.2)A(α)=A0+α1A1+⋯+αpAp:=A0+ΔA(α),B(β)=B0+β1B1+⋯+βqBq:=B0+ΔB(β),C(γ)=C0+γ1C1+⋯+γrCr:=C0+ΔC(γ),
where α:=(α1,…,αp)∈Rp, β:=(β1,…,βq)∈Rq, γ:=(γ1,…,γr)∈Rr, A0 is the infinitesimal generator of a C0-semigroup {SA0(t);t≥0} on 𝒳, Ai∈B(𝒳)(i=1,…,p),Bi∈B(Rm;𝒳)(i=0,…,q), and Ci∈B(𝒳;Rℓ)(i=0,…,r). Here, in the system S(α,β,γ)(A0,B0,C0) and (ΔA(α),ΔB(β),ΔC(γ)) mean the nominal system model and a specific uncertain perturbation, respectively.SinceAi(i=1,…,p) are bounded linear operators, we remark that A(α) always generates a C0-semigroup and has the domain D(A(α))=D(A0) for all α∈Rp. Further, from the practical viewpoint it is assumed that the dimensions of input and output are finite.Now, introduce a compensator(K,L,M,N) defined in a Hilbert space 𝒲 of the form :
(2.3)Σ:{ddtw(t)=Nw(t)+My(t),u(t)=Lw(t)+Ky(t),
where N is the infinitesimal generator of a C0-semigroup {SN(t);t≥0} on a Hilbert space 𝒲 with the domain D(N)=𝒲, M∈B(Rℓ;𝒲),L∈B(𝒲;Rm), and K∈B(Rℓ;Rm).If a compensator of the formΣ is applied to the system S(α,β,γ), the resulting closed-loop system Scl(α,β,γ) with the extended state space 𝒳e:=𝒳⊕𝒲 is easily seen to be
(2.4)ddt[x(t)w(t)]=[A(α)+B(β)KC(γ)B(β)LMC(γ)N][x(t)w(t)],
where 𝒳⊕𝒲 means the direct sum of 𝒳 and 𝒲. For the closed-loop system Scl(α,β,γ), define
(2.5)xe(t):=[x(t)w(t)],Aαβγe:=[A(α)+B(β)KC(γ)B(β)LMC(γ)N]
with domain D(Aαβγe)(=D(A0)⊕𝒲).For the systemS(α,β,γ), we give the following invariant subspaces.Definition 2.1.
Let𝒱 be a closed subspace of 𝒳.
(i)𝒱 is said to be a generalized (A,B)-invariant if there exists an F∈B(𝒳;Rm) such that
(2.6)(A(α)+B(β)F)(𝒱∩D(A0))⊂𝒱,∀α,β.
Also F(𝒱):={F∈B(𝒳;Rm)∣(A(α)+B(β)F)(𝒱∩D(A0))⊂𝒱 for all α,β}.
(ii)𝒱 is said to be a generalized S(A,B)-invariant if there exists an F∈B(𝒳;Rm) such that
(2.7)SA(α)+B(β)F(t)𝒱⊂𝒱,∀t≥0and allα,β.
Also 𝒱(A,B;Λ):={𝒱∣𝒱 is a generalized S(A,B)-invariant and is contained in a given closed subspace Λ.}.Fs(𝒱):={F∈B(𝒳;Rm)∣SA(α)+B(β)F(t)𝒱⊂𝒱forallt≥0and  allα,β}.
(iii)𝒱 is said to be a generalized (C,A)-invariant if there exists a G∈B(Rℓ;𝒳) such that
(2.8)(A(α)+GC(γ))(𝒱∩D(A0))⊂𝒱,∀α,γ.
Also G(𝒱):={G∈B(Rℓ;𝒳)∣(A(α)+GC(γ))(𝒱∩D(A0))⊂𝒱 for all α,γ}.
(iv)𝒱 is said to be a generalized S(C,A)-invariant if there exists a G∈B(Rℓ;𝒳) such that
(2.9)SA(α)+GC(γ)(t)𝒱⊂𝒱,∀t≥0and  allα,γ.
Also 𝒱(ε;C,A):={𝒱∣𝒱 is a generalized S(C,A)-invariant and contains a given closed subspace ε}. Gs(𝒱):={G∈B(Rℓ;𝒳)∣SA(α)+GC(γ)(t)𝒱⊂𝒱 for all t≥0 and all α,γ}.Remark 2.2.
(i)  For the systemS(α,β,γ) a generalized S(A,B)-invariant subspace 𝒱 has the property that if an arbitrary initial state x(0)∈𝒱, then there exists a state feedback u(t)=Fx(t) which is independent of α and β such that the state trajectory x(t)∈𝒱 for all t≥0.
(ii)  IfA0 is a bounded linear operator on 𝒳 (i.e., A0∈B(𝒳)), then the statements (i), (ii) and (iii), (iv) in Definition 2.1 are equivalent, respectively. Further, in this case Fs(𝒱)=F(𝒱) and Gs(𝒱)=G(𝒱).Theorem 2.3 (see [17, 18]).
Suppose thatpi,qi(i=1,…,p),ri,si(i=1,…,q) are arbitrary fixed real numbers such that pi<qi(i=1,…,p) and ri<si(i=1,…,q). Then, the following three statements are equivalent.(i)
𝒱 is a generalized S(A,B)-invariant.(ii)
There exists anF∈B(𝒳;Rm) such that SA0+B0F(t)𝒱⊂𝒱(t≥0) and BiF𝒱⊂𝒱(i=1,…,q), and Ai𝒱⊂𝒱(i=1,…,p).(iii)
There exists anF∈B(𝒳;Rm) such that
(2.10)SA(α)+B(β)F(t)𝒱⊂𝒱,(t≥0)
for all αi∈[pi,qi](i=1,…,p) and βi∈[ri,si](i=1,…,q).(iv)
There exists anF∈B(𝒳;Rm) such that
(2.11)SA(α)+B(β)F(t)𝒱⊂𝒱,(t≥0)
for all αi∈{pi,qi}(i=1,…,p) and βi∈{ri,si}(i=1,…,q).The following theorem is the dual version of Theorem2.3.Theorem 2.4 (see [17, 18]).
Suppose thatpi,qi(i=1,…,p),ti,ui(i=1,…,r) are arbitrary fixed real numbers such that pi<qi(i=1,…,p) and ti<ui(i=1,…,r). Then, the following three statements are equivalent.(i)
𝒱 is a generalized S(C,A)-invariant.(ii)
There exists aG∈B(Rℓ;𝒳) such that SA0+GC0(t)𝒱⊂𝒱(t≥0) and GCi𝒱⊂𝒱(i=1,…,r), and Ai𝒱⊂𝒱(i=1,…,p).(iii)
There exists aG∈B(Rℓ;𝒳) such that
(2.12)SA(α)+GC(γ)(t)𝒱⊂𝒱,(t≥0)
for all αi∈[pi,qi](i=1,…,p) and γi∈[ti,ui](i=1,…,r).(iv)
There exists aG∈B(Rℓ;𝒳) such that
(2.13)SA(α)+GC(γ)(t)𝒱⊂𝒱,(t≥0)
for all αi∈{pi,qi}(i=1,…,p) and γi∈{ti,ui}(i=1,…,r).For finite-dimensional systems, Schumacher [3] first introduced the concept of (C,A,B)-pair. The following definition is a generalized and infinite-dimensional version of (C,A,B)-pair.Definition 2.5.
Let𝒱1 and 𝒱2 be closed subspaces of 𝒳. A pair (𝒱1,𝒱2) of subspaces is said to be a generalized S(C,A,B)-pair if the following three conditions hold.(i)
𝒱1 is a generalized S(C,A)-invariant.(ii)
𝒱2 is a generalized S(A,B)-invariant.(iii)
𝒱1⊂𝒱2.For closed-loop systemScl(α,β,γ), we give the following definition.Definition 2.6.
Let𝒱e be a closed subspace of 𝒳e.(i)
𝒱e is said to be a generalized Ae-invariant if Aα,β,γe(𝒱e∩D(Aα,β,γe))⊂𝒱e for all α,β,γ.(ii)
𝒱e is said to be a generalized SAe(t)-invariant if SAα,β,γe(t)𝒱e⊂𝒱e for all t≥0 and all α,β,γ.The following lemma was shown by Zwart.Lemma 2.7 (see [11]).
Let𝒱e be a closed subspace of 𝒳e and the following three subspaces are introduced:
(2.14)Sorth:={x∈𝒳|[xw]∈[𝒱e]⊥for  somew∈𝒲}=P𝒳([𝒱e]⊥),S1:=[Sorth]⊥,S2:={x∈𝒳|[xw]∈[𝒱e]for  somew∈𝒲}=P𝒳([𝒱e]),
where P𝒳 is the projection operator from 𝒳e onto 𝒳 along 𝒲. Then, the following statements hold.(i)
S1={x∈𝒳|[x0]∈𝒱e}.(ii)
S1⊂S2.(iii)
Ifdim(𝒲)<∞, then S2 is a closed subspace of 𝒳 and dim(S2∩S1⊥)<∞.Lemma 2.8 (see [4, 6, 11]).
Suppose thatA is the infinitesimal generator of a C0-semigroup {SA(t);t≥0} on 𝒳, and 𝒱 is a closed subspace of 𝒳 and Q1∈B(𝒳). Then, the following statements hold.(i)
IfSA(t)𝒱⊂𝒱 for all t≥0, then A(𝒱∩D(A))⊂𝒱.(ii)
If𝒱⊂ D(A) and A𝒱⊂𝒱, then SA(t)𝒱⊂𝒱 for all t≥0.(iii)
IfSA+Q1(t)𝒱⊂𝒱 for all t≥0, then 𝒱∩D(A)¯=𝒱.(iv)
If there exists aQ2∈B(𝒳) such that SA+Q2(t)𝒱⊂𝒱 for all t≥0 and (Q1-Q2)(𝒱∩D(A))⊂𝒱, then SA+Q1(t)𝒱⊂𝒱 for all t≥0.(v)
If there exists aQ2∈B(𝒳) such that SA+Q2(t)𝒱⊂𝒱 for all t≥0 and (Q1-Q2)(𝒱∩D(A))={0}, then SA+Q1(t)x=SA+Q2(t)x for all t≥0 and all x∈𝒱.The following two lemmas are extensions of the results of Otsuka [13] to infinite-dimensional systems.Lemma 2.9.
If a pair(𝒱1,𝒱2) of subspaces of 𝒳 is a generalized S(C,A,B)-pair such that
(2.15)∑i=1qImBi⊂𝒱1⊂𝒱2⊂⋂i=1rKerCi,Ai𝒱2⊂𝒱1,(i=1,…,p),
then there exist G∈Gs(𝒱1), G(β)∈B(Rℓ;𝒳), F(γ)∈Fs(𝒱2), F0∈B(𝒳;Rm) and K∈B(Rℓ;Rm) such that
(2.16)G=B(β)K+G(β),ImG(β)⊂𝒱2,F(γ)=KC(γ)+F0,KerF0⊃𝒱1
for all (β,γ)∈Rq×Rr.Proof.
Suppose that a pair(𝒱1,𝒱2) is a generalized S(C,A,B)-pair satisfying the stated above conditions. Since ∑i=1qImBi⊂𝒱2, we remark that 𝒱2+ImB(β)=𝒱2+ImB0.
Claim 1.
ĜC(γ)𝒱1⊂𝒱2+ImB0 for all Ĝ∈Gs(𝒱1) and γ∈Rr.
To prove Claim1, choose an arbitrary element x∈𝒱1. Then, by Lemma 2.8(iii) there exists an xn∈𝒱1∩D(Ao) such that
(2.17)limn→∞xn=x.
Now, noticing that (A(α)+ĜC(γ))(𝒱1∩D(A0))⊂𝒱1 and 𝒱1⊂𝒱2 we have
(2.18)ĜC(γ)xn=(A(α)+ĜC(γ))xn-A(α)xn∈𝒱1+𝒱2+ImB(β)=𝒱2+ImB(β),=𝒱2+ImB0.
Since dimB(β)<∞, that 𝒱2+ImB0 is a closed subspace. Further, noticing that ĜC(γ) are bounded operators,
(2.19)ĜC(γ)x=limn→∞ĜC(γ)xn∈𝒱2+ImB0,
which proves Claim 1.
Next, Claims2 and 3 hold as follows.
Claim 2.
There exists aG∈Gs(𝒱1) such that ImG⊂𝒱2+ImB0.
To prove Claim2, choose a Ĝ∈Gs(𝒱1) and x(=y+z)∈Rℓ such that y∈∑i=0rCi𝒱1 and z∈ϕ with ∑i=0rCi𝒱1⊕ϕ=Rℓ. Define a linear map G∈Rn×ℓ by Gx:=Ĝy. Then, for some xi∈𝒱1(2.20)Gx=Ĝy=∑i=0rĜCixi∈𝒱2+ImB0,(by  Claim1),
which proves Claim 2.
Claim 3.
There exists aK∈B(Rℓ;Rm) and G(β)∈B(Rℓ;𝒳) such that G=B(β)K+G(β),ImG(β)⊂𝒱2 for all β∈Rq.
To prove Claim3, let {y1,…,yℓ} be a basis of Rℓ. Then, it follows from Claim 2 that there exists an xi∈𝒱2 and ui∈Rm such that
(2.21)Gyi=xi+B0ui=xi-∑i=1qβiBiui+B0ui+∑i=1qβiBiui=(xi-∑i=1qβiBiui)+B(β)ui.
Define linear maps K∈B(Rℓ;Rm) and G(β)∈B(Rℓ;𝒳) by
(2.22)Kyi:=ui,G(β)yi:=xi-∑i=1qβiBiui,respectively.
Then,
(2.23)Gyi=G(β)yi+B(β)Kyi,G(β)yi∈𝒱2,
which proves Claim 3.
Claim 4.
There exists anF†∈B(𝒳;Rm) such that Im(GC(γ)+B(β)F†)|𝒱2⊂𝒱2.
In fact, it follows from Claim2 and hypotheses of this lemma that there exists a G∈Gs(𝒱1) such that
(2.24)ImGC0|𝒱2⊂ImG⊂𝒱2+ImB0.
Let y∈𝒱2 be an arbitrary element. Then, there exist x∈𝒱2 and u∈Rm such that GC0y=x+B0u. Define F†∈B(𝒳;Rm) such that
(2.25)F†y=-u,
Hence, GC0y=x+B0(-F†y) which implies (GC0+B0F†)y=x∈𝒱2. Then, we can easily obtain
(2.26)Im(GC(γ)+B(β)F†)|𝒱2⊂𝒱2,
which proves Claim 4.
Now, chooseF*∈Fs(𝒱2) and define F0∈B(𝒳;Rm) such that
(2.27)F0=F*+F†,on𝒱1⊥,F0=0,on𝒱1.
Then, the following claim holds.
Claim 5.
One has(A(α)+GC(γ)+B(β)F0)(𝒱2∩D(A0))⊂𝒱2 for all (α,β,γ)∈Rp×Rq×Rr  and 𝒱1⊂KerF0.
In fact, at first,𝒱1⊂KerF0 is obvious. Therefore, we prove the first one. Since F*∈Fs(𝒱2) implies SA(α)+B(β)F*(t)𝒱2⊂𝒱2, it follows from Lemma 2.8(v) that it suffices to show (B(β)F0+GC(γ)-B(β)F*)𝒱2⊂𝒱2 in order to prove SA(α)+B(β)F0+GC(γ)(t)𝒱2⊂𝒱2.
Now, letx∈𝒱2 be an arbitrary element. Then, there exists y∈𝒱1 and z∈𝒱1⊥∩𝒱2 such that x=y+z. Then, we have
(2.28)(B(β)F0+GC(γ)-B(β)F*)x=(B(β)F0+GC(γ)-B(β)F*)y+(B(β)F0+GC(γ)-B(β)F*)z=(GC(γ)-B(β)F*)y+(B(β)F†+GC(γ))z.
Now, it follows from Lemma 2.8(iii) that there exists a yn such that yn→y.
Hence,(2.29)(GC(γ)-B(β)F*)yn=(A(α)+GC(γ))yn-(A(α)+B(β)F*)yn∈𝒱2.
Noticing that (GC(γ)-B(β)F*) is bounded linear operator, it follows from (2.28) and Claim 4 that
(2.30)(B(β)F0+GC(γ)-B(β)F*)x∈𝒱2,
which proves Claim 5.
Finally, define a bounded linear operatorF(γ)∈B(𝒳;Rm) such that F(γ):=KC(γ)+F0. Then, the following claim holds.
Claim 6 (F(γ)∈F(𝒱2)).
In fact,(2.31)(A(α)+B(β)F(γ))(𝒱2∩D(A0))=(A(α)+B(β)KC(γ)+B(β)F0)(𝒱2∩D(A0))={A(α)+(B(β)K+G(β))C(γ)-G(β)C(γ)+B(β)F0}(𝒱2∩D(A0))={A(α)+GC(γ)+B(β)F0-G(β)C(γ)}(𝒱2∩D(A0)),(by  Claim3)⊂(A(α)+GC(γ)+B(β)F0)(𝒱2∩D(A0))+ImG(β)⊂𝒱2,(by  Claim3and  Claim5),
which proves Claim 6. This completes the proof of Lemma 2.9.Lemma 2.10.
If a pair(𝒱1,𝒱2) of subspaces of 𝒳 is a generalized S(C,A,B)-pair such that
(2.32)∑i=1qImBi⊂𝒱1⊂𝒱2⊂⋂i=1rKerCi,Ai𝒱2⊂𝒱1,(i=1,…,p),𝒱2⊂D(A0),
then there exist a compensator (K,L,M,N) on 𝒲:=(𝒱2∩𝒱1⊥) and a subspace 𝒱e of 𝒳e such that 𝒱1=S1, 𝒱2=S2, and 𝒱e is generalized SAe(t)-invariant, where S1 and S2 are given in Lemma 2.7.Proof.
Suppose that there exists a pair(𝒱1,𝒱2) of subspaces satisfying the stated above conditions. Since 𝒱1 and 𝒱2 are closed subspaces and 𝒱1⊂𝒱2, we have 𝒱2=𝒱1⊕(𝒱2∩𝒱1⊥). Define 𝒲:=(𝒱2∩𝒱1⊥) and 𝒳e:=𝒳⊕𝒲. Let R:𝒱2→𝒲 be a bounded linear operator such that KerR=𝒱1 and ImR=𝒲. Then, there exists a R†∈B(𝒲;𝒱2) such that RR†=I, which implies R†Rx=0⇔x∈𝒱1 (see [2, p.95]). Further, define
(2.33)𝒱e:={[xRx]∣x∈𝒱2}.
Then, it follows from Lemma 2.7 that 𝒱1 = S1 and 𝒱2 = S2. Further, it follows from Lemma 2.9 that there exist G∈Gs(𝒱1), G(β)∈B(Rℓ;𝒳), F(γ)∈Fs(𝒱2), F0∈B(𝒳;Rm), and K∈B(Rℓ;Rm) such that
(2.34)G=B(β)K+G(β),ImG(β)⊂𝒱2,F(γ)=KC(γ)+F0,KerF0⊃𝒱1
for all (β,γ)∈Rq×Rr.
DefineL∈B(𝒲;Rm) and M∈B(Rp;𝒲) such that L:=F0R†, M:=-RG(0).
Noticing that𝒱1⊂𝒱2⊂ D(A0), it is easily shown that
(2.35)(A(α)+B(β)F0+GC(γ))𝒱i⊂𝒱i,(i=1,2),∀α,β,γ.
Hence, since (A0+B0F0+GC0)𝒱2⊂𝒱2 and 𝒱1=KerR, we have KerR⊂ KerR(A0+B0F0+GC0)|𝒱2 which is equivalent to that there exists an N∈B(𝒲) such that
(2.36)NR=R(A0+B0F0+GC0)|𝒱2=R(A0+B0F(0)+G(0)C0)|𝒱2.
Then, the following two Claims hold.
Claim 1.
R(A(α)+B(β)F(γ))𝒱2=R(A0+B0F(0))𝒱2  for all α,β,γ.
In fact, letx be an arbitrary element of 𝒱2:
(2.37)R(A(α)+B(β)F(γ))x-R(A0+B0F(0))x=R(α1A1+⋯+αpAp)x+RB(β)(KC(γ)+F0)x-RB0(KC0+F0)x=RB(β)KC(γ)x-RB0KC0x+R(B(β)-B0)F0x=R{B(β)KC0-B0KC0}x=0Claim 2.
(MC(γ)+NR)𝒱2=R(A(α)+B(β)F(γ))𝒱2  for all α,β,γ.
In fact,(2.38)(MC(γ)+NR)𝒱2=(MC0+NR)𝒱2=(-RG(0)C0+R(A0+B0F(0)+G(0)C0))𝒱2=R(A0+B0F(0))𝒱2=R(A(α)+B(β)F(γ))𝒱2,(by  Claim1).
Finally, we have the following claim.Claim 3.
Ae(α,β,γ)𝒱e⊂𝒱e for all α,β,γ.
Choose an arbitrary element[xRx] of 𝒱e(x∈𝒱2).
SinceR†Rx-x∈KerR=𝒱1, we have LRx=F0x. Then,
(2.39)[A(α)+B(β)KC(γ)B(β)LMC(γ)N][xRx]=[(A(α)+B(β)KC(γ))x+B(β)LRx(MC(γ)+NR)x]=[(A(α)+B(β)(KC(γ)+F0))xR(A(α)+B(β)F(γ))x],(by  Claim2)=[(A(α)+B(β)F(γ))xR(A(α)+B(β)F(γ))x]∈𝒱e,∀(α,β,γ),
which proves that 𝒱e is generalized Ae-invariant.
Since𝒱e⊂D(Aα,β,γe), it follows from Lemma 2.8(ii) that 𝒱e is generalized SAe(t)-invariant. This completes the proof of this lemma.


## 3. Parameter-Insensitive Disturbance-Rejection by Dynamic Compensator

In this section, the infinite-dimensional version of parameter insensitive disturbance-rejection problem for uncertain linear systems which was investigated by Otsuka [13] is studied.Consider the following uncertain linear systemS(α,β,γ,δ,σ) defined in a Hilbert space 𝒳:
(3.1)ddtx(t)=A(α)x(t)+B(β)u(t)+E(σ)ξ(t),y(t)=C(γ)x(t),z(t)=D(δ)x(t),
where x(t)∈𝒳,u(t)∈𝒰:=Rm,y(t)∈𝒴:=Rℓ,z(t)∈𝒵:=Rμ, and ξ(t)∈L1loc((0,∞);𝒬) are the state, the input, the measurement output, the controlled output, and the disturbance which is a Hilbert space 𝒬 valued locally integrable function, respectively. It is assumed that coefficient operators have the following unknown parameters:
(3.2)A(α)=A0+α1A1+⋯+αpAp:=A0+ΔA(α),B(β)=B0+β1B1+⋯+βqBq:=B0+ΔB(β),C(γ)=C0+γ1C1+⋯+γrCr:=C0+ΔC(γ),D(δ)=D0+δ1D1+⋯+δsDs:=D0+ΔD(δ),E(σ)=E0+σ1E1+⋯+σtEt:=E0+ΔE(σ),
where Ai,Bi,Ci are the same as system S(α,β,γ) in Section 2, Di∈B(𝒳;Rμ),Ei∈B(𝒬;𝒳), and α:=(α1,…,αp),β:=(β1,…,βq),γ:=(γ1,…,γr),δ:=(δ1,…,δs),σ:=(σ1,…,σt). Further, from the practical viewpoint, we assume that uncertain parameters satisfy
(3.3)αi∈[pi,qi](i=1,…,p),βi∈[ri,si](i=1,…,q),γi∈[ti,ui](i=1,…,r),δi∈[hi,ji](i=1,…,s),σi∈[ki,li](i=1,…,t)
where pi,qi(i=1,…,p), ri,si(i=1,…,q)ti,ui(i=1,…,r)hi,ji(i=1,…,s), and ki,li(i=1,…,t) are given real numbers.In systemS(α,β,γ,δ,σ),(A0,B0,C0,D0,E0) and (ΔA(α),ΔB(β),ΔC(γ),ΔD(δ),ΔE(σ)) represent the nominal system model and a specific uncertain perturbation, respectively.If a compensator of the formΣ is applied to system S(α,β,γ,δ,σ), the resulting closed-loop system with the extended state space 𝒳e:=𝒳⊕𝒲 is easily obtained as(3.4)ddt[x(t)w(t)]=[A(α)+B(β)KC(γ)B(β)LMC(γ)N]·[x(t)w(t)]+[E(σ)0]ξ(t),z(t)=[D(δ)0][x(t)w(t)].
For convenience, we set
(3.5)xe(t):=[x(t)w(t)],Aα,β,γe:=[A(α)+B(β)KC(γ)B(β)LMC(γ)N],Ee(σ):=[E(σ)0],De(δ):=[D(δ)0].
Then, our disturbance-rejection problem with dynamic compensator is to find a compensator (K,L,M,N) of Σ such that
(3.6)De(δ)∫0tSAα,β,γe(t-τ)Ee(σ)ξ(τ)dτ=0
for all ξ(·)∈L1loc(0,∞;𝒬) which is a set of all locally square integrable functions on (0,1), all t≥0, and all parameters α,β,γ,δ,σ.This problem can be formulated as follows.Parameter Insensitive Disturbance-Rejection Problem with Dynamic Compensator (PIDRPDC)
GivenAi,Bi,Ci,Di,Ei, find (if possible) a compensator (K,L,M,N) of (2.3) such that
(3.7)<SAα,β,γe(·)∣ImEe(σ)>:=ℒ(⋃t≥0SAα,β,γe(t)(ImEe(σ)))¯⊂KerDe(δ)
for all parameters α,β,γ,δ,σ, where ℒ(Ω) and the over bar indicate the linear subspace generated by the set Ω and the closure in 𝒳e, respectively.The following results are extensions of the results of Otsuka [13] to infinite-dimensional systems.Theorem 3.1.
If there exists a generalizedS(C,A,B)-pair (𝒱1,𝒱2) such that
(3.8){∑i=1qImBi+∑i=0tImEi}⊂𝒱1⊂𝒱2⊂{⋂i=1rKerCi∩⋂i=0sKerDi},Ai𝒱2⊂𝒱1(i=1,…,p),𝒱2⊂D(A0),
then the PIDRPDC is solvable.Proof.
Suppose that the stated above conditions are satisfied. Then, it follows from Lemma2.10 that there exist a compensator (K,L,M,N) on 𝒲:=(𝒱2∩𝒱1⊥) and a subspace 𝒱e of 𝒳e such that 𝒱1=S1,𝒱2=S2 and 𝒱e is generalized SAe(t)-invariant. Further, it can be easily shown that ImEe(σ)⊂𝒱e⊂KerDe(δ). Then,
(3.9)<SAα,β,γe(·)∣ImEe(σ)>⊂<SAα,β,γe(·)∣𝒱e>=𝒱e⊂KerDe(δ)
for all parameters α,β,γ,δ,σ which implies the PIDRPDC is solvable.Corollary 3.2.
Assume thatV(∑i=0tImEi;C,A) and V(A,B;⋂i=0rKerDi) have the minimal element 𝒱1* and the maximal element 𝒱2*, respectively.  If ∑i=1qImBi⊂𝒱1*⊂𝒱2*⊂⋂i=1rKerCi,Ai𝒱2*⊂𝒱1*(i=1,…,p), and 𝒱2*⊂D(A0), then the PIDRPDC is solvable.


## 4. An Illustrative Example

Consider the following system with uncertain parametersα,β,γ∈R:
(4.1)∂x(t,η)∂t=∂2x(t,η)∂η2+{α∫01x(t,μ)(ϕ1(μ)+ϕ2(μ)+ϕ3(μ))dμ}(ϕ1(η)+ϕ2(η)+ϕ3(η))+(ϕ1(η)-2ϕ2(η))u(t)+β{ϕ1(η)+ϕ2(η)+ϕ3(η)}u(t)+{ϕ1(η)+ϕ2(η)+ϕ3(η)}ξ(t),y(t)=∫01x(t,μ)(2ϕ1(μ)-ϕ2(μ))dμ+γ∫01x(t,μ)(ϕ1(μ)+ϕ2(μ)-2ϕ3(μ))dμ,z(t)=∫01x(t,μ)(ϕ1(μ)+ϕ2(μ)-2ϕ3(μ))dμ,
where x(t,η) is the temperature distribution of a bar of unit length at position η∈(0,1) and time t≥0. Moreover, u(t)∈R, ξ(t)∈R, and z(t)∈R are the input, the disturbance, and the controlled output, respectively, and ϕk(η)=2sin(kπη), for η∈(0,1),k≥1.Now, let a Hilbert space𝒳:=L2(0,1) which is a set of all square integrable functions on (0,1). Then, we remark that {ϕk;k≥1} is an orthonormal basis of 𝒳=L2(0,1). Further, define the following operators as(4.2)A(α):=A0+A1,α∈R,(A0x)(η):=d2x(η)dη2,x∈D(A0):={x=∑k=1∞xkϕk∈𝒳∣∑k=1∞|λkxk|2<∞},(A1x):=〈x,ϕ1+ϕ2+ϕ3〉(ϕ1+ϕ2+ϕ3),x∈D(A1):=𝒳,B(β):=B0+βB1,β∈R,B0u:=(ϕ1-2ϕ2)u,u∈D(B0):=R,B1u:=(ϕ1+ϕ2+ϕ3)u,u∈D(B1):=R,C(γ):=C0+γC1,γ∈R,C0:=〈x,2ϕ1-ϕ2〉,x∈D(C0):=𝒳,C1:=〈x,ϕ1+ϕ2-2ϕ3〉,x∈D(C1):=𝒳,Dx:=〈x,ϕ1+ϕ2-2ϕ3〉,x∈D(D):=𝒳,Eξ:=(ϕ1+ϕ2+ϕ3)ξ,ξ∈D(E):=R,
where 〈·,·〉 means the inner product in 𝒳.Then, it is easily seen thatA0 is the infinitesimal generator of a C0-semigroup {SA0(t);t≥0} on 𝒳 and that for each k≥1ϕk is an eigenvector of A0 belonging to an eigenvalue λk=-k2π2. Moreover, the operators A1,B0,B1,C0,C1,D, and E are all bounded. Then, by using these operators we can rewrite the above system as
(4.3)ddtx(t)=A(α)x(t)+B(β)u(t)+Eξ(t),y(t)=C(γ)x(t),z(t)=Dx(t).SinceSA(α)(t)ImE⊄KerD for all t>0, one can see that the controlled output z(·) of the original system (4.3) is influenced by disturbunces ξ(·).Let us define two closed subspaces𝒱1 and 𝒱2 of 𝒳 as
(4.4)𝒱1:={a(ϕ1+ϕ2+ϕ3)∣a∈R},𝒱2:={a(ϕ1+ϕ2+ϕ3)+b(ϕ1-ϕ2)∣a,b∈R}.
If we introduce an operator G∈B(R;𝒳) as
(4.5)Gs:={(λ2+λ3)ϕ1+(λ1+λ3)ϕ2+(λ1+λ2)ϕ3}s,s∈R,
then the condition (ii) of Theorem 2.4 for the system (4.3) is satisfied, and hence it follows from Theorem 2.4 that 𝒱1 is a generalized S(C,A)-invariant. Moreover, if we introduce an operator F∈B(𝒳;R) as
(4.6)Fx:=〈x,λ1ϕ1+λ2ϕ2-2λ3ϕ3〉,x∈𝒳,
then it follows from Theorem 2.3 that 𝒱2 is a generalized S(A,B)-invariant. Thus, we can see that the (𝒱1,𝒱2) is a generalized S(C,A,B)-pair and it is easily shown that the pair (𝒱1,𝒱2) satisfies the conditions of Theorem 3.1 for the system (4.3), that is,
(4.7){ImB1+ImE}⊂𝒱1⊂𝒱2⊂{KerC1∩KerD},A1𝒱2⊂𝒱1,𝒱2⊂D(A0).Therefore, it follows from Theorem3.1 that the PIDRPDC is solvable by using a dynamic compensator which is constructed in terms of the proof of Lemma 2.10. In fact, we can obtain the dynamic compensator Σ in a Hilbert space 𝒲 described by
(4.8)Σ:{ddtw(t)=Nw(t)+My(t),u(t)=Lw(t)+Ky(t),
where 𝒲:=𝒱2∩𝒱1⊥={aψ;ψ=ϕ1-ϕ2,a∈R} and(4.9)Nw:=-(4λ1+4λ2-9λ3)w,Lw:=〈w,-(λ1+2λ2-3λ3)ψ〉,forw∈𝒲,My:=(2λ1+λ2-3λ3)yψ,Ky:=(λ1+λ2-2λ3)y,fory∈R,
which solves the PIDRPDC.


## 5. Concluding Remarks

In this paper we studied the concept of generalizedS(C,A,B)-pairs and its properties for infinite-dimensional systems. This concept is an extension of generalized (C,A,B)-pairs investigated by Otsuka [13] to infinite-dimensional systems. After that a parameter insensitive disturbance-rejection problem with dynamic compensator was formulated and its solvability conditions were given. Further, an illustrative example was also examined.In the present investigation, it should be pointed out that the sufficient conditions of Theorem3.1 is not easy to check. As future studies it is useful to investigate the existence conditions and computational algorithms of the minimal element 𝒱1* and the maximal element 𝒱2* of Corollary 3.2 in order to check easily the solvability conditions of the PIDRPDC. Further, we need to study stabilizability problems for the parameter insensitive disturbance-rejected systems.



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*Source: 169790-2009-11-08.xml*