Abstract
This chapter addresses the problems of stability and stabilization for a class of discrete-time nonlinear systems with semi-Markov stochastic uncertainties (which cause the variations of the system modes to be subject to a semi-Markov chain). The underlying systems are considered to be approximated by Takagi–Sugeno (T–S) fuzzy models. By means of the semi-Markov kernel, the probability density function of the sojourn time for different modes in describing the underlying semi-Markov stochastic uncertainties can be mode-dependent. Both the time-invariant and time-varying Lyapunov functions are used, by which the stability criteria are obtained in terms of the \(\sigma \)-error mean square stability concept, and the latter is demonstrated to be less conservative and more practical. Then, control synthesis problem is investigated and the sufficient conditions on the existence of admissible mode-dependent state-feedback stabilizing controller are developed. A numerical example and a cart-pendulum system are given to show the effectiveness and potential of the new design techniques.
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