Journal of Applied Mathematics

Volume 2012, Article ID 182040, 24 pages

http://dx.doi.org/10.1155/2012/182040

## Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

^{1}Mathematics and Physics Department, Beijing Institute of Petrochemical Technology, Beijing 102617, China^{2}Center for Systems and Control, LTCS, and Department of Industrial Engineering and Management, Peking University, Beijing 100871, China^{3}School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China

Received 14 May 2012; Accepted 22 July 2012

Academic Editor: Junjie Wei

Copyright © 2012 Hong Shi and Guangming Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Impulsive differential systems are an important class of mathematical models for many practical systems in physics, chemistry, biology, engineering, and information science that exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This paper studies the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.

#### 1. Introduction

In recent years, there has been increasing interest in the analysis and synthesis of impulsive systems, or impulsive control systems, due to their significance both in theory and in applications [1–15].

Different from another type of systems associated with the impulses, that is, the singular systems or the descriptor systems, impulsive control systems are described by impulsive ordinary differential equations. Many real systems in physics, chemistry, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the continuous dynamical processes. This kind of impulsive behaviors can be modelled by impulsive systems.

Controllability and observability of impulsive control systems have been studied by a number of papers [4, 6, 12, 13, 15, 16]. Leela et al. [4] investigated the controllability of a class of time-invariant impulsive systems with the assumption that the impulses of impulsive control are regulated at discontinuous points. Lakshmikantham and Deo [12] improved Leela et al.’s [4] results. Then, George et al. [13] extended the results to the linear impulsive systems with time-varying coefficients and nonlinear perturbations. Benzaid and Sznaier [6] studied the null controllability of the linear impulsive systems with the control impulses only acting at the discontinuous points. Guan et al. [15] investigated the controllability and observability of linear time-varying impulsive systems. Sufficient and necessary conditions for controllability and observability are established and their applications to time-invariant impulsive control systems are also discussed. Xie and Wang [16] investigated controllability and observability of a simple class of impulsive systems. Necessary and sufficient conditions are obtained.

Controllability and observability are the two most fundamental concepts in modern control theory [17–19]. They have close connections to pole assignment, structural decomposition, quadratic optimal control and observer design, and so forth. In this paper, we aim to derive necessary and sufficient criteria for controllability and observability of linear piecewise constant impulsive control systems. We first investigate the reachability of such systems and a geometric type necessary and sufficient condition is established. Then, we investigate the controllability and an equivalent condition is established as well. Moreover, it is shown that the controllability is not equivalent to reachability for such systems in general case but is equivalent under some extra conditions. Next, we investigate the observability and determinability of such systems, and get similar results as the controllability and reachability case.

This paper is organized as follows. Section 2 formulates the problem and presents the preliminary results. Sections 3 and 4 investigate reachability and controllability, respectively. Observability and determinability are investigated in Section 5. Section 6 contains a numerical example. Finally, we provide the conclusion in Section 7.

#### 2. Preliminaries

Consider the piecewise linear impulsive system given by where , and are the known , , , , , and constant matrices; is the state vector, and the input vector, the output vector; , , and the discontinuity points are where and , which implies that the solution of (2.1) is left-continuous at .

First, we consider the solution of the system (2.1).

Lemma 2.1. *For any , the general solution of the system (2.1) is given by
**
where , .*

*Proof. *For , we have
For , we have
Similarly, for , , we have
And, for , we have
Thus, by (2.4), (2.5), (2.6), and (2.7), it is easy to verify (2.3).

If , then we are just concerned with a linear time-invariant system. Controllability and observability criteria can be found in standard text books [18, 19]. Thus, in the remainder of the paper, we will only be concerned with the case .

Now, we give some mathematical preliminaries as the basic tools in the following discussion.

Given matrices and , denote as the *range* of , that is, , and denote as the *minimal invariant subspace* of on , that is, . Given a linear subspace , denote as the *orthogonal complement* of , that is, .

The following lemma is a generalization of Theorem 7.8.1 in [17], which is the starting point for deriving the criteria of reachability and controllability.

Lemma 2.2. *Given matrices , for any , one has
*

*Proof. *See Appendix A.

Lemma 2.3. *Given two matrices , two scalars , and a vector , the following two statements are equivalent:*(a)*,*(b)*.*

*Proof. *See Appendix B.

Lemma 2.4. *Given a matrix and a linear subspace , the following two statements are equivalent:*(a)*,*(b)*.*

*Proof. *See Appendix C.

#### 3. Reachability

In this section, we first investigate the reachability of system (2.1).

*Definition 3.1 (reachability). *The system (2.1) is said to be (completely) reachable on if, for any terminal state , there exists a piecewise continuous input such that the system (2.1) is driven from to . Moreover, the set of all the reachable states on is said to be the *reachable set* on , denoted as .

Theorem 3.2. *For the system (2.1), the reachable set on , where , is given by
*

*Proof. *By Lemma 2.1, letting , we have
It follows that

By Lemma 2.2, we get
This is just (3.1).

Since we have obtained the geometric form of the reachable set, we can establish a geometric type criterion as follows.

Theorem 3.3. *The system (2.1) is reachable on , where , if and only if
*

*Proof. *Since
and the matrix is nonsingular, the proof directly follows from Theorem 3.2.

*Remark 3.4. *Theorem 3.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for , denote
for , denote
and, finally, denote
Then, it is easy to verify that
Thus, we get the following algebraic type criterion.

Corollary 3.5. *The system (2.1) is reachable on , where , if and only if
*

#### 4. Controllability

In this section, we investigate the controllability of system (2.1).

*Definition 4.1 (controllability). *The system (2.1) is said to be (completely) controllable on if, for any initial state , there exists a piecewise continuous input such that the system (2.1) is driven from to . Moreover, the set of all the controllable states on is said to be the *controllable set* on , denoted as .

First, we show the relationship between the controllable set and the reachable set.

Theorem 4.2. *For the system (2.1), if is nonsingular, for , then the controllable set on , where , satisfies
*

*Proof. *By Lemma 2.1, letting , we have
It is equivalent to
This implies that
Hence,

Based on Theorem 4.2, we can establish a criterion for controllability of the system (2.1) as follows.

Theorem 4.3. *The system (2.1) is controllable on , where , if and only if
*

*Proof. *First, it is easy to prove that (4.6) is equivalent to

Necessity: since the system is controllable, we have
Then, by Theorem 4.2, we get

Sufficiency: suppose that (4.7) holds. For any , we have
This implies that there exists a piecewise continuous function , such that
Then, we know that . Hence, the system (2.1) is controllable.

In the general case, for system (2.1), controllability is not equivalent to reachability. But under some mild conditions, we can show that they are equivalent.

Corollary 4.4. *For the system (2.1), if is nonsingular, , then the following statements are equivalent:*(a)*the system is reachable,*(b)*the system is controllable,*(c)*.*

*Proof. *Since is nonsingular, , we have that
is nonsingular. It follows that
It is easy to see that .

*Remark 4.5. *For system (2.1), assume that . Then, it is easy to see that Theorem 4.2 concludes the results of Theorem 3.4 in [15].

*Remark 4.6. *For system (2.1), assume that . Then, it is easy to see that Theorem 5 in [20] is a special case of Corollary 4.4.

#### 5. Observability and Determinability

In the above analysis, reference is made to reachability and controllability only. It should be noticed that the observability and determinability counterparts can be addressed dualistically. In this section, we outline the relevant concepts and the corresponding criteria.

*Definition 5.1 (observability). *The system (2.1) is said to be (completely) observable on () if any initial state can be uniquely determined by the corresponding system input and the system output , for .

*Definition 5.2 (determinability). *The system (2.1) is said to be (completely) determinable on () if any terminal state can be uniquely determined by the corresponding system input and the system output , for .

In order to investigate observability and determinability for the system (2.1), we first investigate those of the following zero input system: It is obvious that observability and determinability of the system (2.1) are equivalent to those of the system (5.1), respectively.

For the system (5.1), by Lemma 2.1, the output is given by

Theorem 5.3. *The system (5.1) is observable on , where , if and only if
*

*Proof. *We prove the complementary proposition of Theorem 5.3, that is, the system (5.1) is not observable on , where , if and only if

Necessity: if the system (5.1) is not observable on , where , then there exists , nonzero, such that . This means that
By Lemma 2.3, we get
It follows that
Then, we know that
This implies (5.4).

Sufficiency: on the contrary, if (5.4) holds, there exists , nonzero, such that
It follows that
By Lemma 2.3, we get
This means that . Thus, the system (5.1) is not observable.

*Remark 5.4. *Theorem 5.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for , denote
for , denote
and, finally, denote
Then, it is easy to verify that
Thus, we get the following algebraic type criterion.

Corollary 5.5. *The system (5.1) is observable on , where , if and only if
*

Next, we establish a criterion for determinability.

Theorem 5.6. *The system (5.1) is determinable on , where , if and only if
*

*Proof. *First, by Lemma 2.4, we know that (5.17) is equivalent to
Similar to the proof of Theorem 5.3, we prove the complementary proposition of Theorem 5.6, that is, the system (5.1) is not determinable on , where , if and only if
Necessity: if the system (5.1) is not determinable on , where , then there exists a terminal , nonzero, such that . Then, there exists a nonzero as the initial state such that the system is driven from to , that is, . This means that
By Lemma 2.3, we get
It follows that
This implies that
Since , we know that
It implies that
Hence, (5.19) holds.

Sufficiency: on the contrary, if (5.19) holds, then we know that
Then, there exists a nonzero satisfying
such that there exists a nonzero satisfying
It follows that
By Lemma 2.3, we get
This means that , . Thus, we find a nonterminal nonzero state such that the output remains zero. Hence, the system (5.1) is not determinable.

Similar to the controllability and reachability case, under some simple condition, we can show that for the system (5.1), observability is equivalent to determinability.

Corollary 5.7. *For the system (5.1), if is nonsingular, , then the following statements are equivalent:*(a)*the system is observable,*(b)*the system is determinable,*(c)*.*

*Proof. *If is nonsingular, , then we know that is nonsingular. Hence, we get (5.3) and (5.17) are equivalent.

*Remark 5.8. *For system (2.1), assume that . Then, it is easy to see that Theorem 4.3 concludes the results of Theorem 4.2 in [15].

*Remark 5.9. *For system (2.1), assume that . Then, it is easy to see that Theorem 2 in [20] is a special case of Corollary 5.7.

#### 6. Examples

In this section, we give two numerical examples to illustrate how to utilize our criteria.

*Example 6.1. *Consider a 3-dimensional linear piecewise constant impulsive system with
where , , , and .

Now, we try to use our criteria to investigate the reachability, controllability, observability, and determinability on , where , of the system in Example 6.1.

First, we consider the reachability. By a simple calculation, we have By Theorem 3.3, the system should not be reachable. In fact, for any piecewise continuous input , and any nonzero initial state , we have This fact shows that the system is indeed not reachable.

Next, we consider the controllability. By a simple calculation, we have It is easy to see that

By Theorem 4.3, the system should be controllable. In fact, we can take the piecewise constant input Then, for any nonzero initial state , we have Obviously, if , , then . This fact shows that the system is indeed controllable.

Next, we consider the observability. By a simple calculation, we have By Theorem 5.3, the system should not be observable. In fact, for any piecewise continuous input , and nonzero initial state , we have This fact shows that the system is indeed not observable.

Finally, we consider the determinability. By a simple calculation, we have It follows that By Theorem 5.6, the system should be determinable. In fact, for any nonzero terminal state , there must exist a nonzero initial state such that It follows that This means that and . It is easy to verify that, for any initial state satisfying , we have , . This fact shows that the system is indeed determinable.

*Example 6.2. *Consider a 3-dimensional linear piecewise constant impulsive system with
where , , , and .

Now, we try to use our criteria to investigate the reachability and controllability on , where , of the system in Example 6.2.

First, we consider reachability. By a simple calculation, we have By Theorem 3.3, the system should be reachable. In fact, we take the piecewise constant input Then, letting , for any nonzero terminal state , we have Obviously, we can select suitable , and such that is any state in . This fact shows that the system is indeed reachable.

Next, by Theorem 4.3, the system should be reachable. In fact, we take the piecewise constant input Then, for any nonzero initial state , letting , we have Obviously, we can select suitable , and such that is any state in . This fact shows that the system is indeed controllable.

Finally, according to the conclusion in Corollary 4.4, since the matrices , , and in Example 6.1 are singular, we know that the reachability might not be equivalent to the controllability in this example. However, the reachability should be equivalent to the controllability in Example 6.2 since the matrices , and in this example are nonsingular. From the above analysis, all these statements are correct indeed.

#### 7. Conclusion

This paper has studied the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability have been established, respectively. Moreover, it has been proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for the observability and determinability of such systems have been established, respectively. It has been also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example has been given to illustrate the utility of our criteria.

#### Appendices

#### A. Proof of Lemma 2.2

By Theorem 7.8.1 in [17], we have Thus, it is easy to see that Moreover, we have where . Then, for any , there exist a piecewise continuous function , , and such that Then, we can take such that This implies that It follows that By (A.2) and (A.8), we know that (2.8) holds.

#### B. Proof of Lemma 2.3

If , we get . Then, for , calculating the th derivative of with respect to at , we get Thus, we know that Hence, .

If , it follows that . That is Then, it is easy to prove that .

#### C. Proof of Lemma 2.4

Given a matrix and a linear subspace , the following two statements are equivalent:(a),(b).

() Assume that (). It is equivalent to . It follows that, for any . That is, . This implies that .

() Assume that . It follows that, for any , . That is, . This implies that .

#### Acknowledgments

The authors are grateful to Professor Xinghuo Yu, the Associate Editor, and the reviewers for their helpful and valuable comments and suggestions for improving this paper. This work is supported by National Natural Science Foundation (NNSF) of China (60774089, 10972003, and 60736022). This work is also supported by N2008-07 (08010702014) from Beijing Institute of Petrochemical Technology.

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