Mean Square Stability Analysis of Hybrid Jump Linear Systems using a Markov Kernel Approach

Abstract

Abstract-Hybrid Jump Linear Systems (HJLS's) were recently introduced to study the fundamental properties of supervisory control systems. In this paper, their mean square (MS) stability is analyzed through a modified lifting technique adapted from the literature on Markov jump linear systems. The original technique tests whether ρ(A), the spectral radius of a second moment transition matrix A (which contains the transition probabilities of the Markov chain driving the jump linear system), is less than 1. Here, the test is adapted to yield two new sufficient MS stability conditions. The first condition applies to jump linear systems driven by general finite-state stochastic processes. It requires computing and testing the spectral radius of the matrix AMθ , which is not a second moment transition matrix, using estimated upper bounds of the transition probabilities associated with the processes driving the system. The second condition applies to a large subclass of HJLS's and makes use of the Markov kernel associated with HJLS's to test ρ(AM z ), a particular instance of ρ(AMθ ). These results are illustrated through the stability analysis of an AFTI-F16 aircraft deployed on a computer platform equipped with an advanced error recovery mechanism.

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

“…Proof: The proof parallels that of [23, Th. 2.1], so only a sketch of the proof of (i) is offered here (see [5]): implies that for any set of real, symmetric, positive definite matrices there exists a set of real, symmetric, positive definite matrices such that (10) for every [17]. Thus, the Lyapunov function is such that (the equality is attained when is a Markov chain; see [23, Eq.…”

“…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].…”

“…For these cases stability results already exist (e.g., see [13]- [18]). Thus, the second goal of the paper is to pursue a different solution by building on a theorem from [5] to produce a sufficient mean square stability (MSS) test for these HJLS's. The test employs upper bounds on the one-step transition probabilities of the switching signal process, which is not necessarily Markov.…”