Mixed $$H_\infty$$ H ∞ and Passive Filtering for A Class of Nonlinear Switched Systems with Unstable Subsystems

Abstract

This paper investigates the mixed $$H_\infty$$ and passive filtering problem for a class of nonlinear switched systems with unstable subsystems. The mixed weighted $$H_\infty$$ and passivity performance index is proposed for switched systems. This new performance index covers the weighted $$H_\infty$$ performance index and the passivity performance index as special cases. Therefore, based on this new performance index, the weighted $$H_\infty$$ filtering problem and the passive filtering problem for nonlinear switched systems with unstable subsystems are solved in a unified framework. The Takagi–Sugeno (T–S) fuzzy model is an effective tool in approximating most complex nonlinear systems, which utilizes local linear system description for each rule. Using the T–S fuzzy model to approximate each nonlinear subsystem, the nonlinear switched systems are modeled as the switched T–S fuzzy systems. The states of the filtering error system constructed by the augmentation technique will be divergent when unstable subsystems are activated. To overcome this problem, a set of mode-dependent filters of a Luenberger-like observer type are constructed in this paper. The multiple Lyapunov functions approach and the average dwell time technique are employed to solve the filtering problem. New sufficient conditions for the existence of mixed $$H_\infty$$ and passive filters are developed, which ensure the filtering error system to be asymptotically stable with a prescribed mixed $$H_\infty$$ and passivity performance index. Moreover, the desired mixed $$H_\infty$$ and passive filters can be constructed by solving a set of linear matrix inequalities. Finally, two numerical examples are given to demonstrate the effectiveness and advantage of the obtained results.