Abstract
This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem.
Highlights
The class of Markovian jump linear systems is very appropriate for modeling a plant the structure of which is subject to random abrupt changes
We study the robust stability and robust stabilization problems, under multi-perturbations, for a class of continuous-time Markovian jump linear systems affected by parametric uncertainties of multiplicative white noise type with unknown intensity
We first provide a lower bound of the stability radius in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations
Summary
The class of Markovian jump linear systems is very appropriate for modeling a plant the structure of which is subject to random abrupt changes. A second characterization of a lower bound of the stability radius is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities This second result is very useful for the robust control synthesis problem as it allows the formulation of the feedback gains computation as a convex optimization problem under the Linear Matrix Inequalities (LMIs) paradigm. (ii) Scaling techniques have been used in order to effectively address the multi-perturbations case This allows us to provide a lower bound of the stability radius in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations;. Reasoning as in the proof of Theorem 1.1 in Chapter 5 from [12], we obtain the following result regarding the existence of the solution of a stochastic differential equation of type (1) for an arbitrary function ∆kp, ̈, iq, satisfying the assumption of Hypothesis 2. Since the functions ∆kpt, zk, iq ” 0 satisfy the assumption of Hypothesis 2, we deduce that the result in Proposition 1 is applicable in the case of the nominal system (4)
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