$$H_\infty $$ Sliding Mode Observer Design for Nonlinear Time Delay Systems

In this chapter, we investigate the \(H_\infty \) sliding mode observer (SMO) design problem for a class of discrete time delay nonlinear systems. The nonlinear descriptions quantify the maximum possible derivations from a linear model, and the system states are allowed to be immeasurable. A discrete-time discontinuous switched term is firstly proposed to make sure that the reaching condition holds. Then, by constructing a new Lyapunov–Krasovskii functional based on the idea of delay fractioning and introducing some appropriate free-weighting matrices, a sufficient condition is established to guarantee the desired performance of the error dynamics in the specified sliding mode surface by solving a minimization problem. This minimization problem involves linear objective and linear matrix inequalities that can be easily tested by means of standard numerical software.

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