Skip to main content

Theory and Modern Applications

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

## 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 [1â€“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 [7â€“16]), switched systems, discrete-time systems, and discontinuous systems, see [17â€“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 [26â€“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 .

Consider the following controlled DIHS:

(21)

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

(22)

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

Remark 2.1.

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:

(23)
1. (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 ,

(24)

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

(25)

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

(26)

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.

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

1. (i)

there exist -functions such that for any ,

(31)
1. (ii)

there exists a satisfying and

(32)

where is the identity function: for any ;

1. (iii)

for , there exists -function such that

(33)
1. (iv)

there exists a sufficient large , such that

(34)

Then, DIHS (2.1) with is GUAS.

Proof.

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

(35)

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

(36)

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

(37)

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

(38)

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)â€“(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

(39)

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

(310)

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

1. (i)

there exists a constant satisfying and

(311)
1. (ii)

for , there exists a constant such that

(312)
1. (iii)

one of the following cases holds.

Case 1.

There exists a sufficient large , such that

(313)

Case 2.

There exists a sufficient large , such that

(314)

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 [26â€“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

1. (i)

there exist with satisfying (3.4) and

(315)
(316)
1. (ii)

the following equations and inequalities are satisfied:

(317)

where and.

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

(318)

Specially,

Proof.

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

(319)

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

(320)

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

(321)

Denote . From (3.21), we have

(322)

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

(323)

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

(324)

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

(325)

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*)â€‚ there exist with satisfying (3.10) and

(326)
(327)

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.

1. (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.

1. (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.

Consider the following linear DIHS:

(329)

with the hybrid quadratic performance functional:

(330)

where are matrices with appropriate dimensions and and .

The quadratic supply rate is given by

(331)

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:

(332)

Moreover, if satisfy

(333)
(334)

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

(335)

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

(336)

Proof.

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

(337)

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 ,

(338)

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

(339)

Hence, by , we get

(340)

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

(341)

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:

(342)

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.

Consider DIHS in form of (3.29) where

(41)

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

(42)

By solving (3.32), we obtain

(43)

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

(44)

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

(45)

## 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.

## References

1. Willems JC: Dissipative dynamical systemsâ€”part I: general theory. Archive for Rational Mechanics and Analysis 1972, 45: 321-351. 10.1007/BF00276493

2. Willems JC: Dissipative dynamical systemsâ€”part II: linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis 1972,45(5):352-393. 10.1007/BF00276494

3. Hill D, Moylan P: The stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control 1976,21(5):708-711. 10.1109/TAC.1976.1101352

4. 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)90026-5

5. 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)90080-7

6. 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. 10.1109/9.100932

7. Lakshmikantham V, BaÄ­nov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.

8. Yang T: Impulsive Control Theory, Lecture Notes in Control and Information Sciences. Volume 272. Springer, Berlin, Germany; 2001:xx+348.

9. 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.

10. 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.2005.852187

11. 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.2005.851462

12. 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)00094-2

13. 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.2008.2006210

14. Liu B, Liu X, Teo KL: Feedback stabilization of dissipative impulsive dynamical systems. Information Sciences 2007,177(7):1663-1672. 10.1016/j.ins.2006.09.019

15. 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.1286

16. 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.2008.05.020

17. Zhao J, Hill DJ: Dissipativity theory for switched systems. IEEE Transactions on Automatic Control 2008,53(4):941-953. 10.1109/TAC.2008.920237

18. Haddad WM, Chellaboina V, Kablar NA: Nonlinear impulsive dynamical systemsâ€”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.

19. Haddad WM, Chellaboina V, Kablar NA: Nonlinear impulsive dynamical systemsâ€”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.

20. 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/S1687183904310071

21. 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.

22. 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.2009.04.006

23. 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/S1024123X03204014

24. Lei M, Liu B: Robust impulsive synchronization of discrete dynamical networks. Advances in Difference Equations 2008, 2008:-17.

25. LaSalle LP, Lefschetz S: Stability by Lyapunov's Direct Method. Academic Press, New York, NY, USA; 1961.

26. 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.2008.924897

27. 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.2007.892395

28. Liu B, Liu X: Uniform stability of discrete impulsive systems. International Journal of Systems Science 2008,39(2):181-192. 10.1080/00207720701748380

29. 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.2008.09.003

30. Cardoso RTN, Takahashi RHC: Solving impulsive control problems by discrete-time dynamic optimization methods. TendÃªncias em MatemÃ¡tica Aplicada e Computacional 2008,9(1):21-30.

31. Boyd SP, El Ghaoui L, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, Pa, USA; 1994.

Download references

## 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).

## Author information

Authors

### Corresponding author

Correspondence to Bin Liu.

## Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

## About this article

### Cite this article

Yan, L., Liu, B. Stabilization with Optimal Performance for Dissipative Discrete-Time Impulsive Hybrid Systems. Adv Differ Equ 2010, 278240 (2010). https://doi.org/10.1155/2010/278240

Download citation

• Received:

• Accepted:

• Published:

• DOI: https://doi.org/10.1155/2010/278240