Control Synthesis of Hidden Semi-Markov Uncertain Fuzzy Systems via Observations of Hidden Modes

This paper investigates the stability analysis and fuzzy control problems for a class of discrete-time fuzzy systems with hidden semi-Markov stochastic uncertainties. The nonlinear plant is described via the Takagi-Sugeno (T-S) fuzzy model, and the parameter uncertainties are represented by a hidden semi-Markov chain. Owing to the semi-Markov kernel (SMK), the probability density functions (PDFs) of sojourn time for different modes in describing the stochastic uncertainties can address different types of distributions according to different target modes. A novel Lyapunov function that depends on the hidden mode and the observed mode with the elapsed time is proposed to analyze the stability and the H
_∞
performance of the fuzzy system. Then, the sufficient criteria for the elapsed time-dependent and observed-mode-dependent fuzzy controller are achieved by exploiting the observations of hidden modes, ensuring that the closed-loop system is σ-error mean square stable with guaranteed H
_∞
performance. A cart-pendulum system is used to demonstrate the effectiveness and applicability of the proposed theoretical results.