#### Abstract

We introduce the concept of generalized -pairs which is related to generalized -invariant subspaces and generalized -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 -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., [4โ11]). On the other hand, from the practical viewpoint, Ghosh [12] and Otsuka [13] studied the concepts of simultaneous -pairs and of generalized -pairs, respectively, for finite-dimensional systems, and the parameter-insensitive disturbance-rejection problems for uncertain linear systems were studied. Then, Otsuka and Inaba [14โ16] extended the concepts of simultaneous invariant subspaces and simultaneous -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 -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 -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 -pairs

First, we give some notations used throughout this investigation. Let denote the set of all bounded linear operators from a Hilbert space into another Hilbert space ; for notational simplicity, we write for . For a linear operator the domain, the image, the kernel, and the -semigroup generated by are denoted by , Im, , and , respectively. Further, the dimension and the orthogonal complement of a closed subspace are denoted by dim() and , respectively.

Next, consider the following linear systems defined in a Hilbert space :

where , , are the state, the input, and the measurement output, respectively. Operators , and are unknown in the sense that they are represented as the forms: where :=, :=, :=, is the infinitesimal generator of a -semigroup on , , and . Here, in the system and mean the nominal system model and a specific uncertain perturbation, respectively.

Since are bounded linear operators, we remark that always generates a -semigroup and has the domain for all . Further, from the practical viewpoint it is assumed that the dimensions of input and output are finite.

Now, introduce a compensator defined in a Hilbert space of the form : where is the infinitesimal generator of a -semigroup on a Hilbert space with the domain , , and

If a compensator of the form is applied to the system , the resulting closed-loop system with the extended state space is easily seen to be where means the direct sum of and . For the closed-loop system , define with domain D

For the system , we give the following invariant subspaces.

*Definition 2.1. *Let be a closed subspace of .

(i) is said to be a generalized -invariant if there exists an such that
Also for all

(ii) is said to be a generalized -invariant if there exists an such that
Also is a generalized -invariant and is contained in a given closed subspace

(iii) is said to be a generalized -invariant if there exists a such that
Also for all

(iv) is said to be a generalized -invariant if there exists a such that
Also is a generalized -invariant and contains a given closed subspace . for all and all

*Remark 2.2. *(i) For the system a generalized -invariant subspace has the property that if an arbitrary initial state , then there exists a state feedback which is independent of and such that the state trajectory for all .

(ii) If is a bounded linear operator on (i.e., ), then the statements (i), (ii) and (iii), (iv) in Definition 2.1 are equivalent, respectively. Further, in this case and

Theorem 2.3 (see [17, 18]). *Suppose that are arbitrary fixed real numbers such that and . Then, the following three statements are equivalent.*(i)* is a generalized -invariant.*(ii)*There exists an such that and and *(iii)*There exists an such that
for all and *(iv)*There exists an such that
for all and *

The following theorem is the dual version of Theorem 2.3.

Theorem 2.4 (see [17, 18]). *Suppose that are arbitrary fixed real numbers such that and . Then, the following three statements are equivalent.*(i)* is a generalized -invariant.*(ii)*There exists a such that and and *(iii)*There exists a such that
for all and *(iv)*There exists a such that
for all and *

For finite-dimensional systems, Schumacher [3] first introduced the concept of -pair. The following definition is a generalized and infinite-dimensional version of -pair.

*Definition 2.5. *Let and be closed subspaces of . A pair of subspaces is said to be a generalized -pair if the following three conditions hold.(i) is a generalized -invariant.(ii) is a generalized -invariant.(iii)

For closed-loop system , we give the following definition.

*Definition 2.6. *Let be a closed subspace of .(i) is said to be a generalized -invariant if for all (ii) is said to be a generalized -invariant if for all and all

The following lemma was shown by Zwart.

Lemma 2.7 (see [11]). *Let be a closed subspace of and the following three subspaces are introduced:
**
where is the projection operator from onto along . Then, the following statements hold.*(i)(ii)(iii)*If ( then is a closed subspace of and *

Lemma 2.8 (see [4, 6, 11]). *Suppose that is the infinitesimal generator of a -semigroup on , and is a closed subspace of and Then, the following statements hold.*(i)*If for all , then *(ii)*If D() and , then for all *(iii)*If for all , then *(iv)*If there exists a such that for all and then for all *(v)*If there exists a such that for all and then for all and all *

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

Lemma 2.9. *If a pair of subspaces of is a generalized -pair such that
**
then there exist , , , and such that
**
for all *

*Proof. *Suppose that a pair is a generalized -pair satisfying the stated above conditions. Since we remark that *Claim 1. * for all and

To prove Claim 1, choose an arbitrary element Then, by Lemma 2.8(iii) there exists an such that
Now, noticing that and we have
Since dim that is a closed subspace. Further, noticing that are bounded operators,
which proves Claim 1.

Next, Claims 2 and 3 hold as follows.*Claim 2. *There exists a such that

To prove Claim 2, choose a and such that and with Define a linear map by Then, for some
which proves Claim 2.*Claim 3. *There exists a and such that for all

To prove Claim 3, let be a basis of Then, it follows from Claim 2 that there exists an and such that
Define linear maps and by
Then,
which proves Claim 3.*Claim 4. *There exists an such that Im

In fact, it follows from Claim 2 and hypotheses of this lemma that there exists a such that

Let be an arbitrary element. Then, there exist and such that Define such that
Hence, which implies Then, we can easily obtain
which proves Claim 4.

Now, choose and define such that

Then, the following claim holds.*Claim 5. *One has for all and

In fact, at first, is obvious. Therefore, we prove the first one. Since implies , it follows from Lemma 2.8(v) that it suffices to show in order to prove .

Now, let be an arbitrary element. Then, there exists and such that Then, we have
Now, it follows from Lemma 2.8(iii) that there exists a such that .

Hence,
Noticing that is bounded linear operator, it follows from (2.28) and Claim 4 that
which proves Claim 5.

Finally, define a bounded linear operator such that Then, the following claim holds. *Claim 6 (). *In fact,
which proves Claim 6. This completes the proof of Lemma 2.9.

Lemma 2.10. *If a pair of subspaces of is a generalized -pair such that
**
then there exist a compensator on and a subspace of such that , , and is generalized -invariant, where and are given in Lemma 2.7.*

*Proof. *Suppose that there exists a pair of subspaces satisfying the stated above conditions. Since and are closed subspaces and we have Define and Let be a bounded linear operator such that Ker and Im Then, there exists a such that which implies (see [2, p.95]). Further, define
Then, it follows from Lemma 2.7 that = and = . Further, it follows from Lemma 2.9 that there exist , , , , and such that
for all

Define and such that ,

Noticing that D it is easily shown that
Hence, since and , we have Ker Ker which is equivalent to that there exists an such that
Then, the following two Claims hold.*Claim 1. * for all

In fact, let be an arbitrary element of :
*Claim 2. * for all .

In fact,
Finally, we have the following claim.*Claim 3. * for all .

Choose an arbitrary element of

Since we have Then,
which proves that is generalized -invariant.

Since , it follows from Lemma 2.8(ii) that is generalized -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 defined in a Hilbert space : where , and are the state, the input, the measurement output, the controlled output, and the disturbance which is a Hilbert space valued locally integrable function, respectively. It is assumed that coefficient operators have the following unknown parameters: where are the same as system in Section 2, , and Further, from the practical viewpoint, we assume that uncertain parameters satisfy where , , and are given real numbers.

In system and (,,,,) represent the nominal system model and a specific uncertain perturbation, respectively.

If a compensator of the form is applied to system the resulting closed-loop system with the extended state space is easily obtained as

For convenience, we set Then, our disturbance-rejection problem with dynamic compensator is to find a compensator of such that for all which is a set of all locally square integrable functions on , all , and all parameters

This problem can be formulated as follows.

*Parameter Insensitive Disturbance-Rejection Problem with Dynamic Compensator (PIDRPDC)*

Given , find (if possible) a compensator of (2.3) such that
for all parameters ,, where and the over bar indicate the linear subspace generated by the set and the closure in , respectively.

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

Theorem 3.1. *If there exists a generalized -pair such that
**
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 on and a subspace of such that and is generalized -invariant. Further, it can be easily shown that Im Then,
for all parameters which implies the PIDRPDC is solvable.

Corollary 3.2. *Assume that and have the minimal element and the maximal element , respectively. If , and then the PIDRPDC is solvable.*

#### 4. An Illustrative Example

Consider the following system with uncertain parameters :
where is the temperature distribution of a bar of unit length at position and time . Moreover, , **,** and are the input, the disturbance, and the controlled output, respectively, and , for .

Now, let a Hilbert space which is a set of all square integrable functions on . Then, we remark that is an orthonormal basis of . Further, define the following operators as

where means the inner product in .

Then, it is easily seen that is the infinitesimal generator of a -semigroup on and that for each is an eigenvector of belonging to an eigenvalue . Moreover, the operators , and are all bounded. Then, by using these operators we can rewrite the above system as

Since for all , one can see that the controlled output of the original system (4.3) is influenced by disturbunces .

Let us define two closed subspaces and of as If we introduce an operator as then the condition (ii) of Theorem 2.4 for the system (4.3) is satisfied, and hence it follows from Theorem 2.4 that is a generalized -invariant. Moreover, if we introduce an operator as then it follows from Theorem 2.3 that is a generalized -invariant. Thus, we can see that the is a generalized -pair and it is easily shown that the pair satisfies the conditions of Theorem 3.1 for the system (4.3), that is,

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 in a Hilbert space described by where and

which solves the PIDRPDC.

#### 5. Concluding Remarks

In this paper we studied the concept of generalized -pairs and its properties for infinite-dimensional systems. This concept is an extension of generalized -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 and the maximal element 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.