Asynchronous Nonfragile Mixed $H_\infty$ and $L_{2}-L_\infty$ Control of Switched Fuzzy Systems With Multiple State Impulsive Jumps
Qunxian Zheng, Shengyuan Xu, Baozhu Du
Abstract
This article investigates the asynchronous nonfragile mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</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">$L_{2}-L_\infty$</tex-math></inline-formula> dynamical output feedback (DOF) control problem for switched Takagi–Sugeno (T–S) fuzzy systems with multiple state impulsive jumps. The novelties lie in the following three aspects. First, the novel mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</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">$L_{2}-L_\infty$</tex-math></inline-formula> performance index is adopted, which can include the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}-L_\infty$</tex-math></inline-formula> performance indices as special cases. Second, besides the switching DOF control law for every subsystem, an additional controller state jump rule is employed at controller switching instant. Third, the “multiple state impulsive jumps” means that not only the system states, but also the controller states will jump, and these two different types of switching perform asynchronously. The average dwell time approach is used to design the switching law. By employing a new type of Lyapunov-like functions permitting to increase during the asynchronous period and jump at system switching and controller switching instants, new criteria are established to guarantee the asymptotical stability with a mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</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">$L_{2}-L_\infty$</tex-math></inline-formula> performance index of the switched T–S fuzzy systems with multiple state impulsive jumps and asynchronous switching. Then, controller design conditions are obtained in the form of linear matrix inequalities. Finally, two examples are provided to illustrate the effectiveness of the derived results.