H \(_\infty \) Control of Linear Systems Case

This chapter investigates the asynchronous \(H_{\infty }\) control problem for linear discrete hidden Markov jump singularly perturbed systems (HMJSPSs). Since the mode state of the original system is not directly available, the hidden Markov model (HMM) is used to detect the system mode information. Considering the limitation that it is difficult to obtain all the probabilistic information in the HMM, this chapter adopts the HMM with partially known probabilities, where the partially known probabilities exist in both the transition probability matrix and the observation probability matrix. A suitable asynchronous controller is designed using the detection modes. Then some sufficient conditions for the existence of the asynchronous controller gain are given using Lyapunov functions, which ensure the stochastic stability of the HMJSPSs with satisfactory H \(\infty \) performance. Finally, the effectiveness of the proposed controller is verified by a numerical example.This chapter aims to give the stability analysis criteria for linear discrete HMJSPSs.