Adaptive Backstepping Controller Design for Stochastic Jump Systems

• Published in: IEEE transactions on automatic control Vol. 54; no. 12; pp. 2853 - 2859
• Main Authors: Yuanqing Xia, Yuanqing Xia, Mengyin Fu, Mengyin Fu, Peng Shi, Peng Shi, Zhaojing Wu, Zhaojing Wu, Jinhui Zhang, Jinhui Zhang
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-12-2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptative systems; Adaptive control; Adaptive control systems; Applied sciences; Backstepping; backstepping control; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control systems; Control theory. Systems; Design engineering; Dynamical systems; Dynamics; Exact sciences and technology; Linear matrix inequalities; linear matrix inequality; Markov processes; Markovian jump system; Probability theory; Programmable control; Randomness; Sliding mode control; Stability; stochastic stability; Stochastic systems; Stochasticity; Switching; Uncertainty; Markov process; Multimodel control; Markovian system; Probabilistic approach; Variable structure control; Markovian jump system; Sliding mode; Control synthesis; Continuous time; Linear time; Stochastic stability; Dynamical system; Recursive algorithm; Adaptive control; Backstepping control; Closed feedback; Uncertain system; Linear matrix inequality; Jump process; Jumping; Non deterministic system; Lyapunov function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2009.2033131

In this technical note, we improve the results in a paper by Shi et al., in which problems of stochastic stability and sliding mode control for a class of linear continuous-time systems with stochastic jumps were considered. However, the system considered is switching stochastically between different subsystems, the dynamics of the jump system can not stay on each sliding surface of subsystems forever, therefore, it is difficult to determine whether the closed-loop system is stochastically stable. In this technical note, the backstepping techniques are adopted to overcome the problem in a paper by Shi et al.. The resulting closed-loop system is bounded in probability. It has been shown that the adaptive control problem for the Markovian jump systems is solvable if a set of coupled linear matrix inequalities (LMIs) have solutions. A numerical example is given to show the potential of the proposed techniques.