Stabilization with Optimal Performance for Dissipative Discrete-Time Impulsive Hybrid Systems
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Research Article Stabilization with Optimal Performance for Dissipative Discrete-Time Impulsive Hybrid Systems Lamei Yan1 and Bin Liu2, 3 1
School of Printing Engineering, Hangzhou Dianzi University, Hangzhou 310018, China Department of Information Engineering, The Australian National University, ACT 0200, Australia 3 College of Science, Hunan University of Technology, Zhuzhou 412008, China 2
Correspondence should be addressed to Bin Liu, [email protected] Received 14 September 2009; Accepted 16 April 2010 Academic Editor: Jianshe S. Yu Copyright q 2010 L. Yan and B. Liu. 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. 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
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Advances in Difference Equations
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 stabilizati
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