Observer-Based Adaptive Sliding Mode Control for Markovian Jumping Fuzzy Systems With Fractional Brownian Motions

This article concerns innovative adaptive sliding mode control governed by an observer for a time-varying fuzzy system constrained by fractional Brownian motion with <inline-formula><tex-math notation="LaTeX">$H \in (\frac{1}{2},1)$</tex-math></inline-formula> under Markovian jump with uncertain transition rates. First, for an uncertain signal, a Takagi-Sugeno (T-S) observer is constructed. Furthermore, in accordance with the unique feature of the coefficient matrix for an uncertain block, the observation system is divided into two low-order subsystems through regular transformation. Subsequently, a common and fixed sliding surface function that does not change with jump is constructed; this is one of the characteristics that differ from the characteristics of the integral switching surface designed by many researchers who have studied the Markovian jump. An adaptive controller is also designed and finite-time reachability analysis is performed by using it. Additionally, a stochastic stability study is performed by constructing Lyapunov--Krasovskii functional, introducing slack matrices and utilizing linear matrix inequalities. Finally, a numerical simulation was conducted to verify the reliability of the proposed approach.