Stability Analysis of Stochastic Hybrid Jump Linear Systems Using a Markov Kernel Approach

Abstract

Abstract-In this paper, the state dynamics of a supervisor implemented with a digital sequential system are represented with a finite state machine (FSM). The supervisor monitors a symbol sequence derived from a linear closed-loop system's performance and generates a switching signal for the closed-loop system. The effect of random events on the performance of the closed-loop system is analyzed by adding an exogenous Markov process input to the FSM, and by appropriately augmenting a switched system representation of the supervisor and the closed-loop system. For this class of hybrid jump linear systems, the switching signal is, in general, a non-Markovian process, making it hard to analyze its stability properties. This is ameliorated by introducing a sufficient mean square stability test that uses only upper bounds on the onestep transition probabilities of the switching signal. These bounds are explicitly derived from a Markov kernel associated with the hybrid system model. This stability test becomes necessary and sufficient when the switching signal is Markovian. To determine tighter stability bounds, procedures to determine the upper-bound transition probability matrices when the FSM has a Moore or a Mealy type output map are presented. Two examples illustrate the applicability of the presented results.Index Terms-Finite state machine (FSM), hybrid systems, Markov chains, Markov jump systems, Markov kernel, mean square stability, stochastic systems.

Finite-Time Stabilization for Discrete-Time Delayed Markovian Jump Systems with Partially Delayed Actuator Saturation

“…As we know, Markovian jump systems (MJSs) are very appropriate to describe dynamical systems experiencing changes in their structures or parameters randomly. Up to now, many important research topics of this kind of system, such as stability analysis [1][2][3][4], stabilization [5][6][7], ∞ control and filtering [8][9][10][11], output control [12,13], state estimation [14,15], adaptive control [16,17], and synchronization [18,19], have been considered. By investigating most results on the stability in the literature, it is seen that the classical Lyapunov stability guaranteeing the stability in an infinite-time interval was usually considered.…”

Stability and performance analysis of a jump linear control system subject to digital upsets

“…Correspondingly, stability analyses for the stochastic jump linear systems were also developed by many researchers. [14][15][16][17][18][19][20] The present paper employs mean square stability to analyze the jump linear system stability characteristics. Compared with the previous work, this paper makes two contributions.…”