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

- Lamei Yan
^{1}and - Bin Liu
^{2, 3}Email author

**2010**:278240

**DOI: **10.1155/2010/278240

© L. Yan and B. Liu. 2010

**Received: **14 September 2009

**Accepted: **16 April 2010

**Published: **27 April 2010

## Abstract

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.

## 1. Introduction

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.

## 2. Preliminaries

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 .

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 .

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

- (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:(23)

- (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.

Lemma 2.4.

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

is GUAS. Moreover, can minimize .

## 3. Main Results

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.

- (i)there exist -functions such that for any ,(31)

- (ii)there exists a satisfying and(32)

- (iii)for , there exists -function such that(33)

- (iv)there exists a sufficient large , such that(34)

Then, DIHS (2.1) with is GUAS.

Proof.

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

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

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.

- (i)there exists a constant satisfying and(311)

- (ii)for , there exists a constant such that(312)

- (iii)
one of the following cases holds.

Case 1.

Case 2.

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.

- (i)there exist with satisfying (3.4) and(315)

- (ii)the following equations and inequalities are satisfied:(317)

where and .

Specially,

Proof.

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

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:

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.

- (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.

*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(328)

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.

where are matrices with appropriate dimensions and and .

where are matrices with appropriate dimensions and are symmetric matrices.

Denote and .

Theorem 3.8.

Proof.

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.

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 .

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

Corollary 3.9.

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.

## 4. Examples

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

Example 4.1.

## 5. Conclusions

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.

## Declarations

### Acknowledgments

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).

## Authors’ Affiliations

## References

- Willems JC:
**Dissipative dynamical systems—part I: general theory.***Archive for Rational Mechanics and Analysis*1972,**45:**321-351. 10.1007/BF00276493MathSciNetView ArticleMATHGoogle Scholar - Willems JC:
**Dissipative dynamical systems—part II: linear systems with quadratic supply rates.***Archive for Rational Mechanics and Analysis*1972,**45**(5):352-393. 10.1007/BF00276494MathSciNetView ArticleMATHGoogle Scholar - Hill D, Moylan P:
**The stability of nonlinear dissipative systems.***IEEE Transactions on Automatic Control*1976,**21**(5):708-711. 10.1109/TAC.1976.1101352MathSciNetView ArticleMATHGoogle Scholar - Hill DJ, Moylan PJ:
**Dissipative dynamical systems: basic input-output and state properties.***Journal of the Franklin Institute*1980,**309**(5):327-357. 10.1016/0016-0032(80)90026-5MathSciNetView ArticleMATHGoogle Scholar - Byrnes CI, Isidori A:
**New results and examples in nonlinear feedback stabilization.***Systems & Control Letters*1989,**12**(5):437-442. 10.1016/0167-6911(89)90080-7MathSciNetView ArticleMATHGoogle Scholar - Byrnes CI, Isidori A, Willems JC:
**Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems.***IEEE Transactions on Automatic Control*1991,**36**(11):1228-1240. 10.1109/9.100932MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Baĭnov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar - Yang T:
*Impulsive Control Theory, Lecture Notes in Control and Information Sciences*.*Volume 272*. Springer, Berlin, Germany; 2001:xx+348.Google Scholar - Li Z, Soh Y, Wen C:
*Switched and Impulsive Systems: Analysis, Design and Application, Lecture Notes in Control and Information Sciences*.*Volume 313*. Springer, Berlin, Germany; 2005:xviii+271.Google Scholar - Zhang Y, Sun J:
**Stability of impulsive linear differential equations with time delay.***IEEE Transactions on Circuits and Systems II*2005,**52**(10):701-705. 10.1109/TCSII.2005.852187View ArticleGoogle Scholar - Guan Z, Hill DJ, Shen X:
**On hybrid impulsive and switching systems and application to nonlinear control.***IEEE Transactions on Automatic Control*2005,**50**(7):1058-1062. 10.1109/TAC.2005.851462MathSciNetView ArticleGoogle Scholar - Basin MV, Pinsky MA:
**On impulse and continuous observation control design in Kalman filtering problem.***Systems & Control Letters*1999,**36**(3):213-219. 10.1016/S0167-6911(98)00094-2MathSciNetView ArticleMATHGoogle Scholar - Chen W-H, Zheng WX:
**Global exponential stability of impulsive neural networks with variable delay: an LMI approach.***IEEE Transactions on Circuits and Systems I*2009,**56**(6):1248-1259. 10.1109/TCSI.2008.2006210MathSciNetView ArticleGoogle Scholar - Liu B, Liu X, Teo KL:
**Feedback stabilization of dissipative impulsive dynamical systems.***Information Sciences*2007,**177**(7):1663-1672. 10.1016/j.ins.2006.09.019MathSciNetView ArticleMATHGoogle Scholar - Chen W-H, Wang J-G, Tang Y-J, Lu X:
**Robust**control of uncertain linear impulsive stochastic systems.*International Journal of Robust and Nonlinear Control*2008,**18**(13):1348-1371. 10.1002/rnc.1286MathSciNetView ArticleMATHGoogle Scholar - Chen W-H, Zheng WX:
**Robust stability and**-control of uncertain impulsive systems with time-delay.*Automatica*2009,**45**(1):109-117. 10.1016/j.automatica.2008.05.020MathSciNetView ArticleMATHGoogle Scholar - Zhao J, Hill DJ:
**Dissipativity theory for switched systems.***IEEE Transactions on Automatic Control*2008,**53**(4):941-953. 10.1109/TAC.2008.920237MathSciNetView ArticleGoogle Scholar - Haddad WM, Chellaboina V, Kablar NA:
**Nonlinear impulsive dynamical systems—part I: stability and dissipativity.***Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), December 1999, Phoenix, Ariz, USA***5:**4404-4422.Google Scholar - Haddad WM, Chellaboina V, Kablar NA:
**Nonlinear impulsive dynamical systems—part II: feedback interconnections and optimality.***Proceeedings of the 38th IEEE Conference on Decision and Control (CDC '99), December 1999, Phoenix, Ariz, USA***5:**5225-5234.Google Scholar - Haddad WM, Hui Q, Chellaboina V, Nersesov S:
**Vector dissipativity theory for discrete-time large-scale nonlinear dynamical systems.***Advances in Difference Equations*2004,**2004**(1):37-66. 10.1155/S1687183904310071MathSciNetView ArticleMATHGoogle Scholar - Haddad WM, Chellaboina V, Nersesov SG:
*Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics*. Princeton University Press, Princeton, NJ, USA; 2006:xvi+504.Google Scholar - Haddad WM, Hui Q:
**Dissipativity theory for discontinuous dynamical systems: basic input, state, and output properties, and finite-time stability of feedback interconnections.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):551-564. 10.1016/j.nahs.2009.04.006MathSciNetMATHGoogle Scholar - Liu B, Liu X, Liao X:
**Robust dissipativity for uncertain impulsive dynamical systems.***Mathematical Problems in Engineering*2003,**2003**(3-4):119-128. 10.1155/S1024123X03204014MathSciNetView ArticleMATHGoogle Scholar - Lei M, Liu B:
**Robust impulsive synchronization of discrete dynamical networks.***Advances in Difference Equations*2008,**2008:**-17.Google Scholar - LaSalle LP, Lefschetz S:
*Stability by Lyapunov's Direct Method*. Academic Press, New York, NY, USA; 1961.Google Scholar - Liu B, Hill DJ:
**Comparison principle and stability of discrete-time impulsive hybrid systems.***IEEE Transactions on Circuits and Systems I*2009,**56**(1):233-245. 10.1109/TCSI.2008.924897MathSciNetView ArticleGoogle Scholar - Liu B, Liu X:
**Robust stability of uncertain discrete impulsive systems.***IEEE Transactions on Circuits and Systems II*2007,**54**(5):455-459. 10.1109/TCSII.2007.892395View ArticleGoogle Scholar - Liu B, Liu X:
**Uniform stability of discrete impulsive systems.***International Journal of Systems Science*2008,**39**(2):181-192. 10.1080/00207720701748380MathSciNetView ArticleMATHGoogle Scholar - Azhmyakov V, Boltyanski VG, Poznyak A:
**Optimal control of impulsive hybrid systems.***Nonlinear Analysis: Hybrid Systems*2008,**2**(4):1089-1097. 10.1016/j.nahs.2008.09.003MathSciNetMATHGoogle Scholar - Cardoso RTN, Takahashi RHC:
**Solving impulsive control problems by discrete-time dynamic optimization methods.***Tendências em Matemática Aplicada e Computacional*2008,**9**(1):21-30.MathSciNetMATHGoogle Scholar - Boyd SP, El Ghaoui L, Feron E, Balakrishnan V:
*Linear Matrix Inequalities in System and Control Theory*. SIAM, Philadelphia, Pa, USA; 1994.View ArticleMATHGoogle Scholar

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