Fractional Exponent-Based Looped-Lyapunov Functional for Mode-Dependent Sampled-Data Control of T–S Fuzzy Markovian Jump Systems With $\mathscr {H}_{\infty }$ Performance

In this article, a novel fractional exponent (FE)-based looped-Lyapunov functional (FEBLLF) is proposed to analyze the stochastic stability (SS) criteria of a nonlinear Markovian jump systems (MJSs) under a mode-dependent sampled-data control through the Takagi–Sugeno (T–S) fuzzy approach. An FE, represented by an exponential function with a fractional parameter, is introduced to define looped FEs (LFEs) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ ({\mathsf {1-e}}^{-\hat{\upalpha} (\upepsilon_{1}(\mathsf{t}))})$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ ({\mathsf {1-e}}^{-\hat{\upalpha}(\upepsilon_{2}(\mathsf{t}))})$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ \upepsilon_{1}(\mathsf {t})= {{\mathsf{t}}-{{\mathsf{t}}_{\mathsf{k}}}},$</tex-math></inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol {\upepsilon_{2}} {(\mathsf {t})= {{{\mathsf{t}}_{\mathsf{k+1}}}-{\mathsf{t}}},} {\hat{\boldsymbol{\upalpha}} \in (0,1)}$</tex-math></inline-formula>. These LFEs are utilized to construct a novel FEBLLF and to partition the sampling interval into four distinct sampling subintervals, providing detailed information about the state within each subinterval. Subsequently, the sampling-dependent sufficient conditions are obtained as linear matrix inequalities to ensure the SS of the T–S fuzzy MJSs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {H}_{\infty }$</tex-math></inline-formula> performance level <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\upgamma$</tex-math></inline-formula> and to verify the effectiveness of the proposed results, a nonlinear mass–spring system is assessed. In addition, comparative examples are discussed to illustrate the better conservative results of the proposed approaches.