Analysis and Control of Semi-Markov Jump Linear Systems Under Persistent Disturbances via Full Utilization of Fragmentary Kernel

This article treats the problems of the stability, boundedness, and stabilizing control of discrete-time semi-Markov jump systems (SMJSs) with fragmentary semi-Markov kernel (SMK) under persistent disturbances. Since the statistical characteristics of stochastic processes are difficult to describe precisely and comprehensively, the available SMK information may be fragmentary, and only a portion of the information is known. Regarding this problem, we propose new approaches that leverage all the known SMK information and derive new criteria for analysis and control. The feasibility therein can be enhanced compared to the existing approaches with inadequate utilization of the known SMK information. Additionally, a polytopic approach is proposed to approximate the unknown portion of the SMK information to enrich the information available for subsequent analysis and control design. This is achieved through constructing a polytopic quadratic Lyapunov-like function (LF), which further improves the feasibility. In this way, both the available information and the approximated unknown part about the SMK are incorporated. Meanwhile, the ultimate boundedness of the closed-loop semi-Markov jump linear system (SMJLS) is ensured in the mean-square sense without requiring the deviation between the state and its nominal one to converge at all times. We illustrate the validity and superiority of the proposed approach through a numerical example and a simulated chemical process example using a machine learning-based surrogate model.