H∞ control of singularly perturbed systems using deficient hidden semi-Markov model

This paper deals with the $H_{\infty }$ control of a class of stochastic multi-timescale systems, called Markov jump singularly perturbed systems. The hidden semi-Markov model is introduced to handle the situation when system modes are unavailable in semi-Markov systems. Such a model is assumed deficient, that is, it lacks knowledge about the emission probability, transition probability, and probability density function of the sojourn time. It is a more general case compared with works conducted with perfect transition information. Depending on whether a fast or slow sampling rate is used, the resulting discrete-time singularly perturbed system is modeled differently, for both of which the controller design is conducted. Furthermore, criteria expressed in terms of linear matrix inequalities (LMIs) are developed that guarantee the $\delta$-error mean-square stability. An approach to estimate the upper bound on $\delta$-error with incomplete information is provided, meanwhile, the relationship between system performance and the upper of singular perturbation parameter is also presented. Finally, two simulation examples using real-world systems are provided to corroborate the validity as well as the practical merits of the results.