Generalized 𝑆(𝐶,𝐴,𝐵)-Pairs for Uncertain Linear Infinite-Dimensional Systems

Abstract

We introduce the concept of generalized 𝑆(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.

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 generalized 𝑆(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. Section 2 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. Generalized 𝑆(𝐶,𝐴,𝐵)-pairs

First, we give some notations used throughout this investigation. Let 𝐁(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 {𝑆_𝐴(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 space 𝒳:

𝑑𝑆(𝛼,𝛽,𝛾)∶𝑑𝑡𝑥(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛽)𝑢(𝑡),𝑦(𝑡)=𝐶(𝛾)𝑥(𝑡), (2.1) where 𝑥(𝑡)∈𝒳, 𝑢(𝑡)∈𝒰∶=𝐑^𝑚, 𝑦(𝑡)∈𝒴∶=𝐑^ℓ 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: 𝐴(𝛼)=𝐴₀+𝛼₁𝐴₁+⋯+𝛼_𝑝𝐴_𝑝∶=𝐴₀+Δ𝐴(𝛼),𝐵(𝛽)=𝐵₀+𝛽₁𝐵₁+⋯+𝛽_𝑞𝐵_𝑞∶=𝐵₀+Δ𝐵(𝛽),𝐶(𝛾)=𝐶₀+𝛾₁𝐶₁+⋯+𝛾_𝑟𝐶_𝑟∶=𝐶₀+Δ𝐶(𝛾), (2.2) where 𝛼:=(𝛼₁,…,𝛼_𝑝)∈𝐑^𝑝, 𝛽:=(𝛽₁,…,𝛽_𝑞)∈𝐑^𝑞, 𝛾:=(𝛾₁,…,𝛾_𝑟)∈𝐑^𝑟, 𝐴₀ is the infinitesimal generator of a 𝐶₀-semigroup {𝑆_𝐴0(𝑡);𝑡≥0} on 𝒳, 𝐴_𝑖∈𝐁(𝒳)(𝑖=1,…,𝑝),𝐵_𝑖∈𝐁(𝐑^𝑚;𝒳)(𝑖=0,…,𝑞), and 𝐶_𝑖∈𝐁(𝒳;𝐑^ℓ)(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.

Since 𝐴_𝑖(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∈𝐑^𝑝. 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 : 𝑑Σ∶𝑑𝑡𝑤(𝑡)=𝑁𝑤(𝑡)+𝑀𝑦(𝑡),𝑢(𝑡)=𝐿𝑤(𝑡)+𝐾𝑦(𝑡), (2.3) where 𝑁 is the infinitesimal generator of a 𝐶₀-semigroup {𝑆_𝑁(𝑡);𝑡≥0} on a Hilbert space 𝒲 with the domain 𝐷(𝑁)=𝒲, 𝑀∈𝐁(𝐑^ℓ;𝒲),𝐿∈𝐁(𝒲;𝐑^𝑚), and 𝐾∈𝐁(𝐑^ℓ;𝐑^𝑚).

If a compensator of the form Σ is applied to the system 𝑆(𝛼,𝛽,𝛾), the resulting closed-loop system 𝑆_cl(𝛼,𝛽,𝛾) with the extended state space 𝒳^𝑒∶=𝒳⊕𝒲 is easily seen to be 𝑑=,𝑑𝑡𝑥(𝑡)𝑤(𝑡)𝐴(𝛼)+𝐵(𝛽)𝐾𝐶(𝛾)𝐵(𝛽)𝐿𝑀𝐶(𝛾)𝑁𝑥(𝑡)𝑤(𝑡) (2.4) where 𝒳⊕𝒲 means the direct sum of 𝒳 and 𝒲. For the closed-loop system 𝑆_cl(𝛼,𝛽,𝛾), define 𝑥^𝑒(𝑡)∶=𝑥(𝑡)𝑤(𝑡),𝐴^𝑒_𝛼𝛽𝛾∶=𝐴(𝛼)+𝐵(𝛽)𝐾𝐶(𝛾)𝐵(𝛽)𝐿𝑀𝐶(𝛾)𝑁 (2.5) with domain D(𝐴^𝑒_𝛼𝛽𝛾)(=D(𝐴₀)⊕𝒲).

For the system 𝑆(𝛼,𝛽,𝛾), we give the following invariant subspaces.

Definition 2.1. Let 𝒱 be a closed subspace of 𝒳.
(i) 𝒱 is said to be a generalized (𝐴,𝐵)-invariant if there exists an 𝐹∈𝐁(𝒳;𝐑^𝑚) such that (𝐴(𝛼)+𝐵(𝛽)𝐹)(𝒱∩D(𝐴₀))⊂𝒱,∀𝛼,𝛽. (2.6) Also 𝐅(𝒱)∶={𝐹∈𝐁(𝒳;𝐑^𝑚)∣(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;𝐑^𝑚) such that 𝑆_{𝐴(𝛼)+𝐵(𝛽)𝐹}(𝑡)𝒱⊂𝒱,∀𝑡≥0andall𝛼,𝛽. (2.7) Also 𝒱(𝐴,𝐵;Λ)∶={𝒱∣𝒱 is a generalized 𝑆(𝐴,𝐵)-invariant and is contained in a given closed subspace Λ.}.𝐅_𝑠(𝒱)∶={𝐹∈𝐁(𝒳;𝐑^𝑚)∣𝑆_{𝐴(𝛼)+𝐵(𝛽)𝐹}(𝑡)𝒱⊂𝒱forall𝑡≥0andall𝛼,𝛽}.
(iii) 𝒱 is said to be a generalized (𝐶,𝐴)-invariant if there exists a 𝐺∈𝐁(𝐑^ℓ;𝒳) such that (𝐴(𝛼)+𝐺𝐶(𝛾))(𝒱∩D(𝐴₀))⊂𝒱,∀𝛼,𝛾. (2.8) Also 𝐆(𝒱)∶={𝐺∈𝐁(𝐑^ℓ;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(𝐑^ℓ;𝒳) such that 𝑆_{𝐴(𝛼)+𝐺𝐶(𝛾)}(𝑡)𝒱⊂𝒱,∀𝑡≥0andall𝛼,𝛾. (2.9) Also 𝒱(𝜀;𝐶,𝐴)∶={𝒱∣𝒱 is a generalized 𝑆(𝐶,𝐴)-invariant and contains a given closed subspace 𝜀}. 𝐆_𝑠(𝒱)∶={𝐺∈𝐁(𝐑^ℓ;𝒳)∣𝑆_{𝐴(𝛼)+𝐺𝐶(𝛾)}(𝑡)𝒱⊂𝒱 for all 𝑡≥0 and all 𝛼,𝛾}.

Remark 2.2. (i) For the system 𝑆(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) If 𝐴₀ 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 𝐅_𝑠(𝒱)=𝐅(𝒱) and 𝐆_𝑠(𝒱)=𝐆(𝒱).

Theorem 2.3 (see [17, 18]). Suppose that 𝑝_𝑖,𝑞_𝑖(𝑖=1,…,𝑝),𝑟_𝑖,𝑠_𝑖(𝑖=1,…,𝑞) are arbitrary fixed real numbers such that 𝑝_𝑖<𝑞_𝑖(𝑖=1,…,𝑝) and 𝑟_𝑖<𝑠_𝑖(𝑖=1,…,𝑞). Then, the following three statements are equivalent.(i)𝒱 is a generalized 𝑆(𝐴,𝐵)-invariant.(ii)There exists an 𝐹∈𝐁(𝒳;𝐑^𝑚) such that 𝑆_{𝐴0+𝐵0𝐹}(𝑡)𝒱⊂𝒱(𝑡≥0) and 𝐵_𝑖𝐹𝒱⊂𝒱(𝑖=1,…,𝑞), and 𝐴_𝑖𝒱⊂𝒱(𝑖=1,…,𝑝).(iii)There exists an 𝐹∈𝐁(𝒳;𝐑^𝑚) such that 𝑆_{𝐴(𝛼)+𝐵(𝛽)𝐹}(𝑡)𝒱⊂𝒱,(𝑡≥0) (2.10) for all 𝛼_𝑖∈[𝑝_𝑖,𝑞_𝑖](𝑖=1,…,𝑝) and 𝛽_𝑖∈[𝑟_𝑖,𝑠_𝑖](𝑖=1,…,𝑞).(iv)There exists an 𝐹∈𝐁(𝒳;𝐑^𝑚) such that 𝑆_{𝐴(𝛼)+𝐵(𝛽)𝐹}(𝑡)𝒱⊂𝒱,(𝑡≥0) (2.11) for all 𝛼_𝑖∈{𝑝_𝑖,𝑞_𝑖}(𝑖=1,…,𝑝) and 𝛽_𝑖∈{𝑟_𝑖,𝑠_𝑖}(𝑖=1,…,𝑞).

The following theorem is the dual version of Theorem 2.3.

Theorem 2.4 (see [17, 18]). Suppose that 𝑝_𝑖,𝑞_𝑖(𝑖=1,…,𝑝),𝑡_𝑖,𝑢_𝑖(𝑖=1,…,𝑟) are arbitrary fixed real numbers such that 𝑝_𝑖<𝑞_𝑖(𝑖=1,…,𝑝) and 𝑡_𝑖<𝑢_𝑖(𝑖=1,…,𝑟). Then, the following three statements are equivalent.(i)𝒱 is a generalized 𝑆(𝐶,𝐴)-invariant.(ii)There exists a 𝐺∈𝐁(𝐑^ℓ;𝒳) such that 𝑆_{𝐴0+𝐺𝐶0}(𝑡)𝒱⊂𝒱(𝑡≥0) and 𝐺𝐶_𝑖𝒱⊂𝒱(𝑖=1,…,𝑟), and 𝐴_𝑖𝒱⊂𝒱(𝑖=1,…,𝑝).(iii)There exists a 𝐺∈𝐁(𝐑^ℓ;𝒳) such that 𝑆_{𝐴(𝛼)+𝐺𝐶(𝛾)}(𝑡)𝒱⊂𝒱,(𝑡≥0) (2.12) for all 𝛼_𝑖∈[𝑝_𝑖,𝑞_𝑖](𝑖=1,…,𝑝) and 𝛾_𝑖∈[𝑡_𝑖,𝑢_𝑖](𝑖=1,…,𝑟).(iv)There exists a 𝐺∈𝐁(𝐑^ℓ;𝒳) such that 𝑆_{𝐴(𝛼)+𝐺𝐶(𝛾)}(𝑡)𝒱⊂𝒱,(𝑡≥0) (2.13) for all 𝛼_𝑖∈{𝑝_𝑖,𝑞_𝑖}(𝑖=1,…,𝑝) and 𝛾_𝑖∈{𝑡_𝑖,𝑢_𝑖}(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. Let 𝒱₁ 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 system 𝑆_cl(𝛼,𝛽,𝛾), we give the following definition.

Definition 2.6. Let 𝒱^𝑒 be a closed subspace of 𝒳^𝑒.(i)𝒱^𝑒 is said to be a generalized 𝐴^𝑒-invariant if 𝐴^𝑒_{𝛼,𝛽,𝛾}(𝒱^𝑒∩D(𝐴^𝑒_{𝛼,𝛽,𝛾}))⊂𝒱^𝑒 for all 𝛼,𝛽,𝛾.(ii)𝒱^𝑒 is said to be a generalized 𝑆_𝐴^𝑒(𝑡)-invariant if 𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(𝑡)𝒱^𝑒⊂𝒱^𝑒 for all 𝑡≥0 and all 𝛼,𝛽,𝛾.

The following lemma was shown by Zwart.

Lemma 2.7 (see [11]). Let 𝒱^𝑒 be a closed subspace of 𝒳^𝑒 and the following three subspaces are introduced: 𝑆_orth𝑥𝑤∶={𝑥∈𝒳|∈[𝒱^𝑒]^⟂forsome𝑤∈𝒲}=𝑃_𝒳([𝒱^𝑒]^⟂𝑆),₁∶=[𝑆_orth]^⟂,𝑆₂𝑥𝑤∶={𝑥∈𝒳|∈[𝒱^𝑒]forsome𝑤∈𝒲}=𝑃_𝒳([𝒱^𝑒]), (2.14) where 𝑃_𝒳 is the projection operator from 𝒳^𝑒 onto 𝒳 along 𝒲. Then, the following statements hold.(i)𝑆₁={𝑥∈𝒳|[^𝑥₀]∈𝒱^𝑒}.(ii)𝑆₁⊂𝑆₂.(iii)If dim(𝒲)<∞, then 𝑆₂ is a closed subspace of 𝒳 and dim(𝑆₂∩𝑆^⟂₁)<∞.

Lemma 2.8 (see [4, 6, 11]). Suppose that 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑆_𝐴(𝑡);𝑡≥0} on 𝒳, and 𝒱 is a closed subspace of 𝒳 and 𝑄₁∈𝐁(𝒳). Then, the following statements hold.(i)If 𝑆_𝐴(𝑡)𝒱⊂𝒱 for all 𝑡≥0, then 𝐴(𝒱∩D(𝐴))⊂𝒱.(ii)If 𝒱⊂ D(𝐴) and 𝐴𝒱⊂𝒱, then 𝑆_𝐴(𝑡)𝒱⊂𝒱 for all 𝑡≥0.(iii)If 𝑆_𝐴+𝑄1(𝑡)𝒱⊂𝒱 for all 𝑡≥0, then 𝒱∩D(𝐴)=𝒱.(iv)If there exists a 𝑄₂∈𝐁(𝒳) such that 𝑆_𝐴+𝑄2(𝑡)𝒱⊂𝒱 for all 𝑡≥0 and (𝑄₁−𝑄₂)(𝒱∩D(𝐴))⊂𝒱, then 𝑆_𝐴+𝑄1(𝑡)𝒱⊂𝒱 for all 𝑡≥0.(v)If there exists a 𝑄₂∈𝐁(𝒳) such that 𝑆_𝐴+𝑄2(𝑡)𝒱⊂𝒱 for all 𝑡≥0 and (𝑄₁−𝑄₂)(𝒱∩D(𝐴))={𝟎}, then 𝑆_𝐴+𝑄1(𝑡)𝑥=𝑆_𝐴+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 ^𝑞_𝑖=1Im𝐵_𝑖⊂𝒱₁⊂𝒱₂⊂^𝑟_𝑖=1Ker𝐶_𝑖,𝐴_𝑖𝒱₂⊂𝒱₁,(𝑖=1,…,𝑝), (2.15) then there exist 𝐺∈𝐆_𝑠(𝒱₁), 𝐺(𝛽)∈𝐁(𝐑^ℓ;𝒳), 𝐹(𝛾)∈𝐅_𝑠(𝒱₂), 𝐹₀∈𝐁(𝒳;𝐑^𝑚) and 𝐾∈𝐁(𝐑^ℓ;𝐑^𝑚) such that 𝐺=𝐵(𝛽)𝐾+𝐺(𝛽),Im𝐺(𝛽)⊂𝒱₂,𝐹(𝛾)=𝐾𝐶(𝛾)+𝐹₀,Ker𝐹₀⊃𝒱₁ (2.16) for all (𝛽,𝛾)∈𝐑^𝑞×𝐑^𝑟.

Proof. Suppose that a pair (𝒱₁,𝒱₂) is a generalized 𝑆(𝐶,𝐴,𝐵)-pair satisfying the stated above conditions. Since ∑^𝑞_𝑖=1Im𝐵_𝑖⊂𝒱₂, we remark that 𝒱₂+Im𝐵(𝛽)=𝒱₂+Im𝐵₀.
Claim 1. 𝐺𝐶(𝛾)𝒱₁⊂𝒱₂+Im𝐵₀ for all 𝐺∈𝐆_𝑠(𝒱₁) and 𝛾∈𝐑^𝑟.
To prove Claim 1, choose an arbitrary element 𝑥∈𝒱₁. Then, by Lemma 2.8(iii) there exists an 𝑥_𝑛∈𝒱₁∩𝐷(𝐴_𝑜) such that lim_𝑛→∞𝑥_𝑛=𝑥. (2.17) Now, noticing that (𝐴(𝛼)+𝐺𝐶(𝛾))(𝒱₁∩𝐷(𝐴₀))⊂𝒱₁ and 𝒱₁⊂𝒱₂ we have 𝐺𝐶(𝛾)𝑥_𝑛=(𝐴(𝛼)+𝐺𝐶(𝛾))𝑥_𝑛−𝐴(𝛼)𝑥_𝑛∈𝒱₁+𝒱₂+Im𝐵(𝛽)=𝒱₂+Im𝐵(𝛽),=𝒱₂+Im𝐵₀. (2.18) Since dim𝐵(𝛽)<∞, that 𝒱₂+Im𝐵₀ is a closed subspace. Further, noticing that 𝐺𝐶(𝛾) are bounded operators, 𝐺𝐶(𝛾)𝑥=lim_𝑛→∞𝐺𝐶(𝛾)𝑥_𝑛∈𝒱₂+Im𝐵₀, (2.19) which proves Claim 1.
Next, Claims 2 and 3 hold as follows.
Claim 2. There exists a 𝐺∈𝐆_𝑠(𝒱₁) such that Im𝐺⊂𝒱₂+Im𝐵₀.
To prove Claim 2, choose a 𝐺∈𝐆_𝐬(𝒱₁) and 𝑥(=𝑦+𝑧)∈𝐑^ℓ such that 𝑦∈∑^𝑟_𝑖=0𝐶_𝑖𝒱₁ and 𝑧∈𝜙 with ∑^𝑟_𝑖=0𝐶_𝑖𝒱₁⊕𝜙=𝐑^ℓ. Define a linear map 𝐺∈𝐑^𝑛×ℓ by 𝐺𝑥∶=𝐺𝑦. Then, for some 𝑥_𝑖∈𝒱₁𝐺𝑥=𝐺𝑦=^𝑟_𝑖=0𝐺𝐶_𝑖𝑥_𝑖∈𝒱₂+Im𝐵₀,(byClaim1), (2.20) which proves Claim 2.
Claim 3. There exists a 𝐾∈𝐁(𝐑^ℓ;𝐑^𝑚) and 𝐺(𝛽)∈𝐁(𝐑^ℓ;𝒳) such that 𝐺=𝐵(𝛽)𝐾+𝐺(𝛽),Im𝐺(𝛽)⊂𝒱₂ for all 𝛽∈𝐑^𝑞.
To prove Claim 3, let {𝑦₁,…,𝑦_ℓ} be a basis of 𝐑^ℓ. Then, it follows from Claim 2 that there exists an 𝑥_𝑖∈𝒱₂ and 𝑢_𝑖∈𝐑^𝑚 such that 𝐺𝑦_𝑖=𝑥_𝑖+𝐵₀𝑢_𝑖=𝑥_𝑖−^𝑞_𝑖=1𝛽_𝑖𝐵_𝑖𝑢_𝑖+𝐵₀𝑢_𝑖+^𝑞_𝑖=1𝛽_𝑖𝐵_𝑖𝑢_𝑖=(𝑥_𝑖−^𝑞_𝑖=1𝛽_𝑖𝐵_𝑖𝑢_𝑖)+𝐵(𝛽)𝑢_𝑖. (2.21) Define linear maps 𝐾∈𝐁(𝐑^ℓ;𝐑^𝑚) and 𝐺(𝛽)∈𝐁(𝐑^ℓ;𝒳) by 𝐾𝑦_𝑖∶=𝑢_𝑖,𝐺(𝛽)𝑦_𝑖∶=𝑥_𝑖−^𝑞_𝑖=1𝛽_𝑖𝐵_𝑖𝑢_𝑖,respectively. (2.22) Then, 𝐺𝑦_𝑖=𝐺(𝛽)𝑦_𝑖+𝐵(𝛽)𝐾𝑦_𝑖,𝐺(𝛽)𝑦_𝑖∈𝒱₂, (2.23) which proves Claim 3.
Claim 4. There exists an 𝐹^†∈𝐁(𝒳;𝐑^𝑚) such that Im(𝐺𝐶(𝛾)+𝐵(𝛽)𝐹^†)|_𝒱2⊂𝒱₂.
In fact, it follows from Claim 2 and hypotheses of this lemma that there exists a 𝐺∈𝐆_𝑠(𝒱₁) such that
Im𝐺𝐶₀|_𝒱2⊂Im𝐺⊂𝒱₂+Im𝐵₀. (2.24) Let 𝑦∈𝒱₂ be an arbitrary element. Then, there exist 𝑥∈𝒱₂ and 𝑢∈𝐑^𝑚 such that 𝐺𝐶₀𝑦=𝑥+𝐵₀𝑢. Define 𝐹^†∈𝐁(𝒳;𝐑^𝑚) such that 𝐹^†𝑦=−𝑢, (2.25) Hence, 𝐺𝐶₀𝑦=𝑥+𝐵₀(−𝐹^†𝑦) which implies (𝐺𝐶₀+𝐵₀𝐹^†)𝑦=𝑥∈𝒱₂. Then, we can easily obtain Im(𝐺𝐶(𝛾)+𝐵(𝛽)𝐹^†)|_𝒱2⊂𝒱₂, (2.26) which proves Claim 4.
Now, choose 𝐹^∗∈𝐅_𝑠(𝒱₂) and define 𝐹₀∈𝐁(𝒳;𝐑^𝑚) such that
𝐹₀=𝐹^∗+𝐹^†,on𝒱^⟂₁,𝐹₀=0,on𝒱₁. (2.27) 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)∈𝐑^𝑝×𝐑^𝑞×𝐑^𝑟 and 𝒱₁⊂Ker𝐹₀.
In fact, at first, 𝒱₁⊂Ker𝐹₀ is obvious. Therefore, we prove the first one. Since 𝐹^∗∈𝐅_𝑠(𝒱₂) implies 𝑆_{𝐴(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 𝑆_{𝐴(𝛼)+𝐵(𝛽)𝐹0+𝐺𝐶(𝛾)}(𝑡)𝒱₂⊂𝒱₂.
Now, let 𝑥∈𝒱₂ be an arbitrary element. Then, there exists 𝑦∈𝒱₁ and 𝑧∈𝒱^⟂₁∩𝒱₂ such that 𝑥=𝑦+𝑧. Then, we have (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. (2.28) Now, it follows from Lemma 2.8(iii) that there exists a 𝑦_𝑛 such that 𝑦_𝑛→𝑦.
Hence, (𝐺𝐶(𝛾)−𝐵(𝛽)𝐹^∗)𝑦_𝑛=(𝐴(𝛼)+𝐺𝐶(𝛾))𝑦_𝑛−(𝐴(𝛼)+𝐵(𝛽)𝐹^∗)𝑦_𝑛∈𝒱₂. (2.29) Noticing that (𝐺𝐶(𝛾)−𝐵(𝛽)𝐹^∗) is bounded linear operator, it follows from (2.28) and Claim 4 that (𝐵(𝛽)𝐹₀+𝐺𝐶(𝛾)−𝐵(𝛽)𝐹^∗)𝑥∈𝒱₂, (2.30) which proves Claim 5.
Finally, define a bounded linear operator 𝐹(𝛾)∈𝐁(𝒳;𝐑^𝑚) such that 𝐹(𝛾)∶=𝐾𝐶(𝛾)+𝐹₀. Then, the following claim holds.
Claim 6 (𝐹(𝛾)∈𝐅(𝒱₂)). In fact, (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)),(byClaim3)⊂(𝐴(𝛼)+𝐺𝐶(𝛾)+𝐵(𝛽)𝐹₀)(𝒱₂∩𝐷(𝐴₀))+Im𝐺(𝛽)⊂𝒱₂,(byClaim3andClaim5), (2.31) 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 ^𝑞_𝑖=1Im𝐵_𝑖⊂𝒱₁⊂𝒱₂⊂^𝑟_𝑖=1Ker𝐶_𝑖,𝐴_𝑖𝒱₂⊂𝒱₁,(𝑖=1,…,𝑝),𝒱₂⊂D(𝐴₀), (2.32) then there exist a compensator (𝐾,𝐿,𝑀,𝑁) on 𝒲∶=(𝒱₂∩𝒱^⟂₁) and a subspace 𝒱^𝑒 of 𝒳^𝑒 such that 𝒱₁=𝑆₁, 𝒱₂=𝑆₂, and 𝒱^𝑒 is generalized 𝑆_𝐴^𝑒(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∩𝒱^⟂₁). Define 𝒲∶=(𝒱₂∩𝒱^⟂₁) and 𝒳^𝑒∶=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 𝒱^𝑒𝑥∶={𝑅𝑥∣𝑥∈𝒱₂}. (2.33) Then, it follows from Lemma 2.7 that 𝒱₁ = 𝑆₁ and 𝒱₂ = 𝑆₂. Further, it follows from Lemma 2.9 that there exist 𝐺∈𝐆_𝑠(𝒱₁), 𝐺(𝛽)∈𝐁(𝐑^ℓ;𝒳), 𝐹(𝛾)∈𝐅_𝑠(𝒱₂), 𝐹₀∈𝐁(𝒳;𝐑^𝑚), and 𝐾∈𝐁(𝐑^ℓ;𝐑^𝑚) such that 𝐺=𝐵(𝛽)𝐾+𝐺(𝛽),Im𝐺(𝛽)⊂𝒱₂,𝐹(𝛾)=𝐾𝐶(𝛾)+𝐹₀,Ker𝐹₀⊃𝒱₁ (2.34) for all (𝛽,𝛾)∈𝐑^𝑞×𝐑^𝑟.
Define 𝐿∈𝐁(𝒲;𝐑^𝑚) and 𝑀∈𝐁(𝐑^𝑝;𝒲) such that 𝐿∶=𝐹₀𝑅^†, 𝑀∶=−𝑅𝐺(0).
Noticing that 𝒱₁⊂𝒱₂⊂ D(𝐴₀), it is easily shown that (𝐴(𝛼)+𝐵(𝛽)𝐹₀+𝐺𝐶(𝛾))𝒱_𝑖⊂𝒱_𝑖,(𝑖=1,2),∀𝛼,𝛽,𝛾. (2.35) 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 𝑁𝑅=𝑅(𝐴₀+𝐵₀𝐹₀+𝐺𝐶₀)|_𝒱2=𝑅(𝐴₀+𝐵₀𝐹(0)+𝐺(0)𝐶₀)|_𝒱2. (2.36) 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, let 𝑥 be an arbitrary element of 𝒱₂: 𝑅(𝐴(𝛼)+𝐵(𝛽)𝐹(𝛾))𝑥−𝑅(𝐴₀+𝐵₀𝐹(0))𝑥=𝑅(𝛼₁𝐴₁+⋯+𝛼_𝑝𝐴_𝑝)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=0 (2.37)Claim 2. (𝑀𝐶(𝛾)+𝑁𝑅)𝒱₂=𝑅(𝐴(𝛼)+𝐵(𝛽)𝐹(𝛾))𝒱₂ for all 𝛼,𝛽,𝛾.
In fact, (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,(byClaim1). (2.38) Finally, we have the following claim.Claim 3. 𝐴^𝑒(𝛼,𝛽,𝛾)𝒱^𝑒⊂𝒱^𝑒 for all 𝛼,𝛽,𝛾.
Choose an arbitrary element [^𝑥_𝑅𝑥] of 𝒱^𝑒(𝑥∈𝒱₂).
Since 𝑅^†𝑅𝑥−𝑥∈Ker𝑅=𝒱₁, we have 𝐿𝑅𝑥=𝐹₀𝑥. Then, 𝑥==𝐴(𝛼)+𝐵(𝛽)𝐾𝐶(𝛾)𝐵(𝛽)𝐿𝑀𝐶(𝛾)𝑁𝑅𝑥(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,(byClaim=2)(𝐴(𝛼)+𝐵(𝛽)𝐹(𝛾))𝑥𝑅(𝐴(𝛼)+𝐵(𝛽)𝐹(𝛾))𝑥∈𝒱^𝑒,∀(𝛼,𝛽,𝛾), (2.39) which proves that 𝒱^𝑒 is generalized 𝐴^𝑒-invariant.
Since 𝒱_𝑒⊂𝐷(𝐴^𝑒_{𝛼,𝛽,𝛾}), it follows from Lemma 2.8(ii) that 𝒱^𝑒 is generalized 𝑆_𝐴^𝑒(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 system 𝑆(𝛼,𝛽,𝛾,𝛿,𝜎) defined in a Hilbert space 𝒳: 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), (3.1) where 𝑥(𝑡)∈𝒳,𝑢(𝑡)∈𝒰∶=𝐑^𝑚,𝑦(𝑡)∈𝒴∶=𝐑^ℓ,𝑧(𝑡)∈𝒵∶=𝐑^𝜇, and 𝜉(𝑡)∈𝐿₁^𝑙𝑜𝑐((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: 𝐴(𝛼)=𝐴₀+𝛼₁𝐴₁+⋯+𝛼_𝑝𝐴_𝑝∶=𝐴₀+Δ𝐴(𝛼),𝐵(𝛽)=𝐵₀+𝛽₁𝐵₁+⋯+𝛽_𝑞𝐵_𝑞∶=𝐵₀+Δ𝐵(𝛽),𝐶(𝛾)=𝐶₀+𝛾₁𝐶₁+⋯+𝛾_𝑟𝐶_𝑟∶=𝐶₀+Δ𝐶(𝛾),𝐷(𝛿)=𝐷₀+𝛿₁𝐷₁+⋯+𝛿_𝑠𝐷_𝑠∶=𝐷₀+Δ𝐷(𝛿),𝐸(𝜎)=𝐸₀+𝜎₁𝐸₁+⋯+𝜎_𝑡𝐸_𝑡∶=𝐸₀+Δ𝐸(𝜎), (3.2) where 𝐴_𝑖,𝐵_𝑖,𝐶_𝑖 are the same as system 𝑆(𝛼,𝛽,𝛾) in Section 2, 𝐷_𝑖∈𝐁(𝒳;𝐑^𝜇),𝐸_𝑖∈𝐁(𝒬;𝒳), and 𝛼∶=(𝛼₁,…,𝛼_𝑝),𝛽∶=(𝛽₁,…,𝛽_𝑞),𝛾∶=(𝛾₁,…,𝛾_𝑟),𝛿∶=(𝛿₁,…,𝛿_𝑠),𝜎∶=(𝜎₁,…,𝜎_𝑡). Further, from the practical viewpoint, we assume that uncertain parameters satisfy 𝛼_𝑖∈[𝑝_𝑖,𝑞_𝑖](𝑖=1,…,𝑝),𝛽_𝑖∈[𝑟_𝑖,𝑠_𝑖](𝑖=1,…,𝑞),𝛾_𝑖∈[𝑡_𝑖,𝑢_𝑖𝛿](𝑖=1,…,𝑟),_𝑖∈[ℎ_𝑖,𝑗_𝑖](𝑖=1,…,𝑠),𝜎_𝑖∈[𝑘_𝑖,𝑙_𝑖](𝑖=1,…,𝑡) (3.3) where 𝑝_𝑖,𝑞_𝑖(𝑖=1,…,𝑝), 𝑟_𝑖,𝑠_𝑖(𝑖=1,…,𝑞)𝑡_𝑖,𝑢_𝑖(𝑖=1,…,𝑟)ℎ_𝑖,𝑗_𝑖(𝑖=1,…,𝑠), and 𝑘_𝑖,𝑙_𝑖(𝑖=1,…,𝑡) are given real numbers.

In system 𝑆(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 form Σ is applied to system 𝑆(𝛼,𝛽,𝛾,𝛿,𝜎), the resulting closed-loop system with the extended state space 𝒳^𝑒∶=𝒳⊕𝒲 is easily obtained as

𝑑=⋅+0.𝑑𝑡𝑥(𝑡)𝑤(𝑡)𝐴(𝛼)+𝐵(𝛽)𝐾𝐶(𝛾)𝐵(𝛽)𝐿𝑀𝐶(𝛾)𝑁𝑥(𝑡)𝑤(𝑡)𝐸(𝜎)𝜉(𝑡),𝑧(𝑡)=𝐷(𝛿)0𝑥(𝑡)𝑤(𝑡) (3.4) For convenience, we set 𝑥^𝑒(𝑡)∶=𝑥(𝑡)𝑤(𝑡),𝐴^𝑒_{𝛼,𝛽,𝛾}∶=𝐴(𝛼)+𝐵(𝛽)𝐾𝐶(𝛾)𝐵(𝛽)𝐿𝑀𝐶(𝛾)𝑁,𝐸^𝑒0,𝐷(𝜎)∶=𝐸(𝜎)_𝑒(𝛿)∶=𝐷(𝛿)0. (3.5) Then, our disturbance-rejection problem with dynamic compensator is to find a compensator (𝐾,𝐿,𝑀,𝑁) of Σ such that 𝐷^𝑒(𝛿)^𝑡₀𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(𝑡−𝜏)𝐸^𝑒(𝜎)𝜉(𝜏)𝑑𝜏=0 (3.6) for all 𝜉(⋅)∈𝐿₁^𝑙𝑜𝑐(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)
Given 𝐴_𝑖,𝐵_𝑖,𝐶_𝑖,𝐷_𝑖,𝐸_𝑖, find (if possible) a compensator (𝐾,𝐿,𝑀,𝑁) of (2.3) such that <𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(⋅)∣Im𝐸^𝑒(𝜎)>∶=ℒ(_𝑡≥0𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(𝑡)(Im𝐸^𝑒(𝜎)))⊂Ker𝐷^𝑒(𝛿) (3.7) 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 𝒳^𝑒, respectively.

The following results are extensions of the results of Otsuka [13] to infinite-dimensional systems.

Theorem 3.1. If there exists a generalized 𝑆(𝐶,𝐴,𝐵)-pair (𝒱₁,𝒱₂) such that {^𝑞_𝑖=1Im𝐵_𝑖+^𝑡_𝑖=0Im𝐸_𝑖}⊂𝒱₁⊂𝒱₂⊂{^𝑟_𝑖=1Ker𝐶_𝑖∩^𝑠_𝑖=0Ker𝐷_𝑖𝐴},_𝑖𝒱₂⊂𝒱₁(𝑖=1,…,𝑝),𝒱₂⊂D(𝐴₀), (3.8) then the PIDRPDC is solvable.

Proof. Suppose that the stated above conditions are satisfied. Then, it follows from Lemma 2.10 that there exist a compensator (𝐾,𝐿,𝑀,𝑁) on 𝒲∶=(𝒱₂∩𝒱^⟂₁) and a subspace 𝒱^𝑒 of 𝒳^𝑒 such that 𝒱₁=𝑆₁,𝒱₂=𝑆₂ and 𝒱^𝑒 is generalized 𝑆_𝐴^𝑒(𝑡)-invariant. Further, it can be easily shown that Im𝐸^𝑒(𝜎)⊂𝒱^𝑒⊂Ker𝐷^𝑒(𝛿). Then, <𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(⋅)∣Im𝐸^𝑒(𝜎)>⊂<𝑆_𝐴^𝑒_{𝛼,𝛽,𝛾}(⋅)∣𝒱^𝑒>=𝒱^𝑒⊂Ker𝐷^𝑒(𝛿) (3.9) for all parameters 𝛼,𝛽,𝛾,𝛿,𝜎 which implies the PIDRPDC is solvable.

Corollary 3.2. Assume that 𝓥(∑^𝑡_𝑖=0Im𝐸_𝑖;𝐶,𝐴) and 𝓥(𝐴,𝐵;⋂^𝑟_𝑖=0Ker𝐷_𝑖) have the minimal element 𝒱_1∗ and the maximal element 𝒱^∗₂, respectively. If ∑^𝑞_𝑖=1Im𝐵_𝑖⊂𝒱_1∗⊂𝒱^∗₂⊂⋂^𝑟_𝑖=1Ker𝐶_𝑖,𝐴_𝑖𝒱^∗₂⊂𝒱_1∗(𝑖=1,…,𝑝), and 𝒱^∗₂⊂D(𝐴₀), then the PIDRPDC is solvable.

4. An Illustrative Example

Consider the following system with uncertain parameters 𝛼,𝛽,𝛾∈𝐑: 𝜕𝑥(𝑡,𝜂)=𝜕𝜕𝑡²𝑥(𝑡,𝜂)𝜕𝜂²+{𝛼¹₀𝑥(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)=¹₀𝑥(𝑡,𝜇)(2𝜙₁(𝜇)−𝜙₂(𝜇))𝑑𝜇+𝛾¹₀𝑥(𝑡,𝜇)(𝜙₁(𝜇)+𝜙₂(𝜇)−2𝜙₃(𝜇))𝑑𝜇,𝑧(𝑡)=¹₀𝑥(𝑡,𝜇)(𝜙₁(𝜇)+𝜙₂(𝜇)−2𝜙₃(𝜇))𝑑𝜇, (4.1) 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 𝜙_𝑘(𝜂)=√2sin(𝑘𝜋𝜂), for 𝜂∈(0,1),𝑘≥1.

Now, let a Hilbert space 𝒳∶=𝐿²(0,1) which is a set of all square integrable functions on (0,1). Then, we remark that {𝜙_𝑘;𝑘≥1} is an orthonormal basis of 𝒳=𝐿²(0,1). Further, define the following operators as

𝐴(𝛼)∶=𝐴₀+𝐴₁,𝛼∈𝐑,(𝐴₀𝑑𝑥)(𝜂)∶=²𝑥(𝜂)𝑑𝜂²,𝑥∈D(𝐴₀)∶={𝑥=^∞_𝑘=1𝑥_𝑘𝜙_𝑘∈𝒳∣^∞_𝑘=1|𝜆_𝑘𝑥_𝑘|²<∞},(𝐴₁𝑥)∶=⟨𝑥,𝜙₁+𝜙₂+𝜙₃⟩(𝜙₁+𝜙₂+𝜙₃),𝑥∈D(𝐴₁)∶=𝒳,𝐵(𝛽)∶=𝐵₀+𝛽𝐵₁𝐵,𝛽∈𝐑,₀𝑢∶=(𝜙₁−2𝜙₂)𝑢,𝑢∈D(𝐵₀𝐵)∶=𝐑,₁𝑢∶=(𝜙₁+𝜙₂+𝜙₃)𝑢,𝑢∈D(𝐵₁)∶=𝐑,𝐶(𝛾)∶=𝐶₀+𝛾𝐶₁𝐶,𝛾∈𝐑,₀∶=⟨𝑥,2𝜙₁−𝜙₂⟩,𝑥∈D(𝐶₀𝐶)∶=𝒳,₁∶=⟨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, (4.2) where ⟨⋅,⋅⟩ means the inner product in 𝒳.

Then, it is easily seen that 𝐴₀ is the infinitesimal generator of a 𝐶₀-semigroup {𝑆_𝐴0(𝑡);𝑡≥0} on 𝒳 and that for each 𝑘≥1𝜙_𝑘 is an eigenvector of 𝐴₀ belonging to an eigenvalue 𝜆_𝑘=−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 𝑑𝑑𝑡𝑥(𝑡)=𝐴(𝛼)𝑥(𝑡)+𝐵(𝛽)𝑢(𝑡)+𝐸𝜉(𝑡),𝑦(𝑡)=𝐶(𝛾)𝑥(𝑡),𝑧(𝑡)=𝐷𝑥(𝑡). (4.3)

Since 𝑆_𝐴(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 subspaces 𝒱₁ and 𝒱₂ of 𝒳 as 𝒱₁∶={𝑎(𝜙₁+𝜙₂+𝜙₃)∣𝑎∈𝐑},𝒱₂∶={𝑎(𝜙₁+𝜙₂+𝜙₃)+𝑏(𝜙₁−𝜙₂)∣𝑎,𝑏∈𝐑}. (4.4) If we introduce an operator 𝐺∈𝐁(𝐑;𝒳) as 𝐺𝑠∶={(𝜆₂+𝜆₃)𝜙₁+(𝜆₁+𝜆₃)𝜙₂+(𝜆₁+𝜆₂)𝜙₃}𝑠,𝑠∈𝐑, (4.5) 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 𝐹𝑥∶=⟨𝑥,𝜆₁𝜙₁+𝜆₂𝜙₂−2𝜆₃𝜙₃⟩,𝑥∈𝒳, (4.6) 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, {Im𝐵₁+Im𝐸}⊂𝒱₁⊂𝒱₂⊂{Ker𝐶₁∩Ker𝐷},𝐴₁𝒱₂⊂𝒱₁,𝒱₂⊂D(𝐴₀). (4.7)

Therefore, it follows from Theorem 3.1 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 𝑑Σ∶𝑑𝑡𝑤(𝑡)=𝑁𝑤(𝑡)+𝑀𝑦(𝑡),𝑢(𝑡)=𝐿𝑤(𝑡)+𝐾𝑦(𝑡), (4.8) where 𝒲∶=𝒱₂∩𝒱^⟂₁={𝑎𝜓;𝜓=𝜙₁−𝜙₂,𝑎∈𝐑} and

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, (4.9) which solves the PIDRPDC.

5. Concluding Remarks

In this paper we studied the concept of generalized 𝑆(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 Theorem 3.1 is not easy to check. As future studies it is useful to investigate the existence conditions and computational algorithms of the minimal element 𝒱_1∗ and the maximal element 𝒱^∗₂ 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.