Abstract
This paper deals with the problem of passivity analysis and passivity-based synthesis for continuous-time stochastic hybrid singular systems (SHSSs). First, a set of linear matrix inequalities for the stochastic admissibility and passivity, which is also known as the positive real lemma, is obtained for continuous-time SHSSs. The proposed condition successfully holds a necessary and sufficient condition for the positive realness of SHSSs, whereas the existing papers in the literature have been handled only sufficient conditions. Next, the passivity-based control synthesis problem is also considered based on the new positive real lemma. The passivity-based stabilization criterion for the closed-loop system with its mode-dependent state-feedback control is expressed in terms of matrix inequality. Thus, by introducing an additional slack matrix to the non-convex conditions, the feasible conditions for the control gain are obtained in terms of linear matrix inequalities. Finally, a numerical example is provided to show the effectiveness of the result.
Highlights
The state-space systems which consist of system states and their first order differential equations are widely used in the field of control theory
The main objectives of this paper are obtaining the equivalent condition of the positive real lemma (PRL) for stochastic hybrid singular systems (SHSSs) in terms of linear matrix inequality (LMI), and constructing the mode-dependent passivity-based control based on the proposed PRL
Based on the proposed PRL, state-feedback stabilization criterion will be derived in the subsection
Summary
The state-space systems which consist of system states and their first order differential equations are widely used in the field of control theory. This paper proposes a new PRL for the continuous-time SHSSs. To obtain the necessary and sufficient condition for the passivity analysis, a mode-dependent Lyapunov function for stochastic admissibility and passivity is used. The main objectives of this paper are obtaining the equivalent condition of the PRL for SHSSs in terms of LMIs, and constructing the mode-dependent passivity-based control based on the proposed PRL. (Necessary) Since the hybrid system is stochastically stable and passive, i.e., Dw(i) + DTw(i) > 0, there exist a positive definite solution P (i) for the following algebraic Riccati equation (ARE): sym{AT (i)P (i)} + λijP (j) + (P (i)Bw(i) − CT (i)). Remark 1: The passivity condition for stochastic hybrid systems has been considered in many papers in the literature. Based on the proposed PRL, state-feedback stabilization criterion will be derived in the subsection
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