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

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 one-step 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.

I. Introduction

Significant progress has been achieved in the design of adaptive control systems that employ a supervisor monitoring the closed-loop system performance to switch between a finite number of controllers (see, e.g., [1] and [2]). in general, the supervisor can switch not only the controller, but also the active model used to represent the plant's dynamics. for the purposes of this paper, the supervisor will simply switch between closed-loop dynamics. Implementation of a theoretically designed supervisor is facilitated if it is represented as a finite state machine (FSM). It can then be naturally implemented in most sequential digital logic devices, including programmable logic devices. Work toward the development of such an automated conversion is presented in [3]. The interconnection of a FSM model of a supervisor with the switched closed-loop linear models is called a hybrid jump linear system (HJLS) model (see also [4]–[6]). The first goal of this paper is to develop these HJLS models that include a FSM, so that they can be analyzed and used to synthesize a supervisor. in particular, to monitor performance, it is assumed that the supervisor can measure the state vector of the closed-loop system and generate a symbol sequence that serves as an input to the FSM. To determine the effect of stochastic switching as introduced, for example, by the random occurrence of failures, a Markovian stochastic process is included as an exogenous input to the FSM. a third FSM input that could be used for plant estimation purposes can also be added as in [7], but that option is not employed in the non-estimator based supervisor considered here. These hybrid jump linear systems can be considered discrete-time piecewise deterministic Markov processes (with no reset maps) [8], [9] or simplified discrete-time stochastic hybrid systems [10]. They can also be viewed as autonomous discrete stochastic hybrid automata [9]. However, unlike other models, HJLS's represent the dynamics of both the logical supervisor and the closed-loop system in the same analytical framework: randomly switched difference equations. This facilitates the study of their joint properties such as mean square (MS) stability. HJLS models are also useful in the analysis and design of fault tolerant control systems as in [5], [11]. Another benefit of developing an FSM-based model of the supervisor is that it becomes easier to augment it with fault detection and identification capabilities to monitor the occurrence of soft faults in the digital logic circuits [12].