Stabilization with Optimal Performance for Dissipative Discrete-Time Impulsive Hybrid Systems

This paper studies the problem of stabilization with optimal performance for dissipative DIHS (discrete-time impulsive hybrid systems). By using Lyapunov function method, conditions are derived under which the DIHS with zero inputs is GUAS (globally uniformly asymptotically stable). These GUAS results are used to design feedback control law such that a dissipative DIHS is asymptotically stabilized and the value of a hybrid performance functional can be minimized. For the case of linear DIHS with a quadratic supply rate and a quadratic storage function, sufficient and necessary conditions of dissipativity are expressed in matrix inequalities. And the corresponding conditions of optimal quadratic hybrid performance are established. Finally, one example is given to illustrate the results.

In many engineering problems, it is needed to consider the energy of systems. The energy of a controlled system is often linked to the concept of dissipativity [1–4]. A dissipative system here is one for which the energy dissipated inside the dynamical system is less than the energy supplied from the external source. The "energy" storage function of a dissipative system which can be viewed as generalization of energy function is often used to be a Lyapunov function, and thus the stability of a dissipative system can be investigated. It is also known that a dissipative system may be unstable. If one hopes that a dissipative but unstable system will be stable, it is necessary to use the technique of stabilization.

Feedback stabilization and dissipativity theory as well as the connected Lyapunov stability theory has been studied for systems possessing continuous motions. Byrnes et al. started to study the dissipativity and stabilization of continuous systems based on geometric system theory in [5, 6] and relevant references cited therein. Recently, notions of classical dissipativity theory have been extended for CIHS (continuous-time impulsive hybrid systems; see [7–16]), switched systems, discrete-time systems, and discontinuous systems, see [17–24]. But these reports include very few results of feedback stabilization for dissipative CIHS. The traditional methods used in the study of feedback stabilization of dissipative continuous-time systems are those based on the LaSalle invariance principle [25]. But it is difficult to use it to analyze the feedback stabilization of dissipative CIHS because solutions of impulsive hybrid systems are no longer continuous. In [14], feedback stabilization of dissipative CIHS is studied by using Lyapunov-like function, which is derived from the "energy" storage function of a dissipative CIHS. However, to the best of our knowledge, no dissipativity and feedback stabilization results have been previously reported for DIHS (discrete-time impulsive hybrid systems, see [26–28]), in which the impulses occur in discrete-time systems. Recently, in [29, 30] and the relevant references cited therein, the optimal control issue is also reported for CIHS and the Pontryagin-type Maximum Principle for CIHS is established. However, there are fewer results reported for stabilization with optimal performance for dissipative CIHS or DIHS.

The objective of this paper is to study the stabilization with optimal performance problem for dissipative DIHS in the spirit of [14, 20]. By using the Lyapunov function and dwell time method, we propose some GUAS results for DIHS. Then these GUAS results are used to derive the conditions under which a dissipative DIHS is asymptotically stabilized and the hybrid performance functional is minimized.

The rest of this paper is organized as follows. In Section 2, we introduce some notations and definitions. In Section 3, we give the main results for DIHS. Then, we specialize the results to linear DIHS. Finally, in Section 4, we discuss one example to illustrate our results.

Let denote the -dimensional Euclidean space. Let and . A function is of class- () if it is continuous, zero at zero and strictly increasing. It is of class- if it is of class- and is unbounded. For satisfying , denote , , and . (), , means that matrix is a positive definite (nonnegative definite) and symmetric matrix. Let stand for the Euclidean norm in .

Consider the following controlled DIHS:

where is the state; are the outputs; are known continuous functions with ; and satisfies ; and satisfies ; are external control inputs with , here is the class of admissible hybrid control inputs; ; and the impulsive sequence satisfie: and , with and . Let be the solution of system (2.1) with initial condition . For the impulsive sequence and any satisfying , we denote the number of impulses during .

The hybrid performance functional of DIHS (2.1) is

where with , or , and are given and known functions.

Remark 2.1.

(i) If there exists a positive integer such that , then (2.1) becomes a normal discrete-time system with initial point . In this paper, we study the DIHS under the following assumption:

(ii) By (2.3), we get the fact that for any , if and only if .

Definition 2.2.

A function , where , with and , is called a supply rate of (2.1) if and are locally summable: for all input-output pairs and any with , satisfy

Definition 2.3.

DIHS (2.1) is said to be dissipative under supply rate if there exists a nonnegative continuous function with , called storage function, such that for all the following dissipation inequality holds for any with ,

Lemma 2.4.

DIHS (2.1) is dissipative under the supply rate if and only if there exists a nonnegative continuous function with such that

where .

Proof.

By using Definition 2.3, it is easy to get that (2.4) is equivalent to (2.5). The details are omitted here.

2.1. Stabilization with Optimal Performance Problem

For the dissipative DIHS (2.1) with hybrid performance functional (2.2), the stabilization with optimal performance problem is to design the state feedback control law , where with , such that the closed-loop system

is GUAS. Moreover, can minimize .

In this section, by using the Lyapunov function method, some GUAS criteria are established for DIHS. Then, these stability criteria are used to study the optimal stabilization issue for a dissipative DIHS with hybrid performance functional.

Theorem 3.1.

Let . Suppose (2.3) holds and furthermore assume that there exists a function such that

(i)there exist -functions such that for any ,

(ii)there exists a satisfying and

where is the identity function: for any ;

(iii)for , there exists -function such that

(iv)there exists a sufficient large , such that

Then, DIHS (2.1) with is GUAS.

Proof.

Denote . By condition (ii), we get that, for any ,

It follows from (3.5) and condition (iii) that

For any , there exists an such that . By (3.6) and conditions (i)–(iii), we have

Hence, for any , let ; then, when , we get from (3.7) that for any . Thus, the system (2.1) is GUS (globally uniformly stable).

Denote . It follows from (3.6) that

Since by condition (iv), for any , thus we get that the sequence is monotone decreasing and exists. If , then, . This contradiction implies , that is,

For any , by conditions (i)–(iii), we have It follows from Remark 2.1(ii) that . Hence, DIHS (2.1) with is uniformly attractive and hence it is GUAS. The proof is complete.

Theorem 3.2.

Let and suppose (2.3) holds and furthermore assume that there exists a satisfying conditions (i) and (iii) of Theorem 3.1 and

(ii*)there exists a such that

(iv*)there exists a sufficient large , such that

Then, DIHS (2.1) with is GUAS.

Proof.

By similar proof of Theorem 3.1 with , we obtain that the result holds. The detailed is omitted here.

Corollary 3.3.

Let and suppose (2.3) holds and assume that there exists a function satisfying (3.1) and

(i)there exists a constant satisfying and

(ii)for , there exists a constant such that

(iii)one of the following cases holds.

Case 1.

There exists a sufficient large , such that

Case 2.

There exists a sufficient large , such that

Then, DIHS (2.1) with is GUAS.

Proof.

The result is the direct consequence of Theorems 3.1 and 3.2, where in Case 1, let , while in Case 2, let for any .

Remark 3.4.

Theorems 3.1 and 3.2 and Corollary 3.3 give two kinds of GUAS properties of DIHS by using the method of Lyapunov function and maximal and minimal dwell times. For more detailed stability results of DIHS, please refer to the literature [26–28] and relevant references cited therein.

Theorem 3.5.

Suppose (2.3) holds and assume that under the given supply rate , DIHS (2.1) is dissipative with a storage function satisfying (3.1), and that there exist functions with , such that

(i)there exist with satisfying (3.4) and

(ii)the following equations and inequalities are satisfied:

where and.

Then, under , the closed-loop system (2.6) is GUAS, and

Specially,

Proof.

Since system (2.1) is dissipative under the supply rate , then, for , we get

From condition (i) and (3.19), we derive that

Thus, from (3.20) and Theorem 3.1, we obtain that the closed-loop system (2.6) is GUAS.

By condition (ii), for , we get

Denote . From (3.21), we have

Thus, from (3.22) condition (i), and the fact that the closed-loop system is GUAS, we obtain that

Now, we prove that minimizes . From condition (ii), we have

Thus, using (3.24), (3.22), and , we have

Hence, (3.18) holds and all the results hold.

Theorem 3.6.

Suppose (2.3) holds and assume that under the supply rate , system (2.1) is dissipative with a storage function satisfying (3.1), and that there exist functions with , such that (ii) of Theorem 3.5 holds while (i) of Theorem 3.5 is replaced by the following:

(i*) there exist with satisfying (3.10) and

Then, all results of Theorem 3.5 still hold.

Proof.

By similar proof of Theorem 3.5 and using the result of Theorem 3.2, we obtain that all results are true.

Remark 3.7.

(i) For a dissipative DIHS (2.1) with supply rate and "energy" storage function , if or is negative during some time interval or at some time instance, then it implies that the "energy" of system will be decreasing during this period or at this instance. These two kinds of dissipativity properties all help to achieve the stability for whole DIHS. In Theorem 3.5, the negative supply rate leads to the decreasing of "energy" of system during two consecutive impulses (see (3.15)) and thus it permits to some extend the increasing of "energy" at impulsive instances (see (3.16)) while the stability property of whole system will be kept. On the other hand, in Theorem 3.6, the negative supply rate leads to the decreasing of "energy" of system at impulse instances (see (3.27)) and thus it permits to some extend of increasing of "energy" during two consecutive impulses (see (3.26)) while the stability property of whole system can still be guaranteed.

In the literature, if the stability property is derived from the dissipativity of system, it often needs the condition of negative supply rate. But one can see from Theorems 3.5 and 3.6 that this condition is relaxed for DIHS.

(ii) By (3.17), if and are continuously differential in , respectively, then, for

which can be used to derive the hybrid state feedback control law .

At the end of section, we specialize the results obtained to the case of linear DIHS with a quadratic supply rate.

Consider the following linear DIHS:

with the hybrid quadratic performance functional:

where are matrices with appropriate dimensions and and .

The quadratic supply rate is given by

where are matrices with appropriate dimensions and are symmetric matrices.

Denote and .

Theorem 3.8.

Assume that , and for , the linear DIHS (3.29) is dissipative with storage function , where , if and only if the following LMIs are satisfied:

Moreover, if satisfy

where satisfy and the condition (iii) of Corollary 3.3, then, the state feedback control law

can stabilize system (3.29), and minimizes , that is,

Proof.

By (3.29), it is not difficult to get that

where for , , and , , .

Thus, by Lemma 2.4, we get that system (3.29) is dissipative if and only if . By Schur Complement Theorem [31], for , it is not hard to get that if and only if LMI holds. Hence, we obtain that system (3.29) is dissipative if and only if LMIs (3.32) hold.

Let , where ; then, it follows from (3.33) that for ,

Thus, by Corollary 3.3, the closed-loop system given by (3.29) and (3.35) is GUAS.

Now, we show that (3.35) also minimizes .

Denote: and Then, by (3.34), we obtain

Hence, by , we get

Clearly, if , then, by (3.39), we have

Then, by Theorem 3.5, the result of this theorem follows readily. The proof is complete.

Corollary 3.9.

Assume that , and there exists a matrix satisfying LMI (3.32), (3.34), and the following matrix inequalities:

where , , , and if , if ; and for ; and where constants satisfy the condition (iii) of Corollary 3.3.

Then, all the results of Theorem 3.8 still hold.

Proof.

By Schur Complement Theorem [31] and Theorem 3.8, the result of this corollary follows.

In this section, one example is solved to illustrate the obtained results.

Example 4.1.

Consider DIHS in form of (3.29) where

The matrices appeared in (3.31) and (3.30) are given by and

By solving (3.32), we obtain

Thus, by Theorem 3.8, this system is dissipative under the quadratic supply rate. Moreover, we see that and . And by solving (3.42), we get and

Thus, by (3.14) of Corollary 3.3, if satisfies , that is, , then, all the conditions of Corollary 3.9 are satisfied. Therefore, given by (4.4) can stabilize the closed-loop system and minimizes , that is, if , then

In this paper, by establishing the GUAS results for DIHS, we have obtained the conditions under which a dissipative DIHS with a hybrid performance functional can be asymptotically stabilized by a feedback control law and meantime the hybrid performance functional is optimized. For the case of linear DIHS with a quadratic supply rate and a quadratic hybrid performance functional, the corresponding sufficient conditions are changed into matrix inequalities. One example verifies the theoretic results obtained.

The authors would like to thank the Editor, Professor Jianshe S. Yu, and the anonymous referees for their helpful comments and suggestions. This work was supported by NSFC-China (no. 60874025) and ARC-Australia (no. DP0881391).

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