3Optimal SMC of Markovian Jump Singular Systems with Time Delay

3.1 Introduction

It is recognized that the sliding mode of an SMC system is invariant to system perturbations and external disturbances, only if the perturbations/disturbances satisfy the so-called ‘matching condition’. Although many researchers have paid considerable attention to the mismatched uncertainties in SMC design, the obtained results are very conservative. If the undesired uncertainties/disturbances can not be eliminated in the sliding mode, it is possible to attenuate its effect on the system performance. In this chapter, we will consider the disturbance attenuation problem in sliding mode with performance. For this purpose, we design an integral switching function. The plant considered in this chapter is the Markovian jump singular time-delay system, which is a typical kind of hybrid systems of high complexity (including system parameter jumping, time delay in states, and singularity). How to establish a less conservative stability condition is a key issue in SMC design. As is well known, the slack matrix technique [228, 243], usually combined with the Lyapunov–Krasovskii approach, has been proved to be an effective tool to establish less conservative stability conditions for time-delay systems. Unfortunately, little progress has been made in dealing with singular time-delay systems by this technique, ...

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