Stochastic stability and stabilization of a class of piecewise-homogeneous Markov jump linear systems with mixed uncertainties

Abstract

This paper investigates the stochastic stability and stabilization problem for a general class of uncertain, continuous-time Markov jump linear systems (MJLSs). The system under consideration is a piecewise-homogenous Markovian structure subject to piecewise-constant time-varying transition rates (TRs). The time variation of the TRs is characterized by a high-level Markovian signal, which is independent from the low-level Markovian mechanism that governs the switching between the system dynamics. It is assumed that the structure is subject to mixed uncertainties in the form of additive norm-bounded terms. The uncertainties help to consider the effect of imperfections induced by modeling errors for the system dynamics and the TRs of Markovian signals of both levels. This new uncertain, two-level Markovian jump linear system is a more general model than the existing ones and is applicable to more practical situations. Besides, it is capable of being specialized to uncertain piecewise-homogeneous MJLS with arbitrarily varying TRs, as well as the uncertain time-homogeneous MJLS. The stability/stabilizability of this system is first examined by constructing a Lyapunov function which depends on both switching signals. Then, based on the analysis results, the corresponding robust controller gains are synthesized through solving a set of linear matrix inequalities (LMIs). Finally, simulation results for an industrial stirred tank reactor (CSTR) are used to demonstrate the applicability and potentials of the proposed theoretical method. Comparative simulations are also provided to show the superiority of the presented approach to the existing ones. Remark 4The controller gains (22) need to a perfect measure of the switching modes of both levels. It should be fairly admitted that, practically, the assumption of perfectly measured Markov modes is very idealistic and restrictive, and considering the additive uncertain terms for the TR matrices of both 901 PIECEWISE-HOMOGENEOUS MARKOV JUMP LINEAR SYSTEMS and S(i, m) and R(i, m) are defined by (11) and (13). Thus, the controller is designed as K(i,m) = Y(i, m)X À1 (i,m). 902 M. FARAJI-NIRI, M.-R. JAHED-MOTLAGH AND M. BARKHORDARI-YAZDIThe condition of Eq. (34) is nonlinear in P(i,m) and K(i,m). In order to find controller gains it is desired to transform (34) into an LMI form, so let X(i,m) = P À1 (i,m).Pre-and post-multiplying Eq. (34) by X(i,m) gives Eq. (35). 903PIECEWISE-HOMOGENEOUS MARKOV JUMP LINEAR SYSTEMS

Observer‐based H∞ control for Markovian jump systems with time‐varying delays and incomplete transition rates

“…Then, premultiplying and postmultiplying, respectively, both sides of inequalities (40) and (42) , I, I, I, I} and its transpose, one can obtain (32), which can lead to LMIs (12) and (14) by using the Schur complement lemma. According to Theorem 1, the proof is completed.…”

“…Recently, the problems of the stability analysis and controller design for MJSs have became the hot issues, such as robust H ∞ control with saturating actuator, stability analysis with singular MJSs, and H ∞ filters design . On the hypothesis of completely known transition rates, some new results on H ∞ control, sliding mode control, stability, stabilization, and passivity have been reported in other works . On the other hand, in many engineering systems, it is difficult to measure or estimate all transition rates.…”

“…7,8 On the hypothesis of completely known transition rates, some new results on H ∞ control, sliding mode control, stability, stabilization, and passivity have been reported in other works. [9][10][11][12][13] On the other hand, in many engineering systems, it is difficult to measure or estimate all transition rates. To reduce the gap between theory and practical applications, some results with incomplete transition rates were stated in other works.…”

Boundary control of stochastic reaction‐diffusion systems with Markovian switching

“…For example, pth moment stability of stochastic Markovian functional differential equations, 3 exponential stability of stochastic singular delay systems with Markovian switchings, 14 and stochastic stability of Markov jump linear systems. 15 In this article, we will consider the general SRDSs with Markovian switching.…”

Positive filter design for positive piecewise homogeneous Markovian jump T–S fuzzy system

“…On the other hand, in the case of the time‐varying transition rate, Zhang [33] has investigated the estimation for the discrete‐time piecewise homogeneous Markovian jump system. In recent years, the analysis and control synthesis for the continuous‐time piecewise homogeneous Markovian jump system can be found in [34–36]. However, surprisingly, when the considered system is positive, the problems on the positive piecewise homogeneous Markovian jump system have not yet been reported up until now.…”