Robust Control Of Uncertain Linear Singular Systems

Because robust control theory can successfully solve robust stability and robust controller design, therefore it has been abroadly regarded and fully developed in control domain. Robust control problem is derived from analysis and complexity of the parameter uncertain linear systems, one situation is mainly that the systems are asymptotically stable for all parameter uncertainties under giving the robust stability conditions of the systems, and the other one is investigated by using memoryless state feedback controller, such that the obtained closed-loop system be asymptotically stable for all parameter uncertainties. This paper includes the above mentioned situations for investigating robust control problems of the uncertain singular systems.Robust control for uncertain linear systems is formulated perfectly by using Lyapunov inequalities, Riccati equations or inequalities. In uncertain singular systems, the former research has considered robust stability analysis and robust control problems under the generalized Riccati equations or inequalities. However, the solving of Riccati equations has difficulty. As the linear matrix inequality software is ripening, the linear matrix inequality theory has become a main technique to study robust control problems. In this paper, basing on the Lyapunov stability theory, the linear matrix inequalities processing method is used to discuss robust stability and robust stabilization problems of uncertain singular systems, as well as robust stability, robust stabilization and robust preserving performance control problems of uncertain singular time-delay systems, etc.The main contents and contributions in the paper are as follows:(1) Robust control problems of uncertain singular systems are discussed. Above all, we generalize the quadratic stability and quadratic stabilization concepts of uncertain linear systems to uncertain singular systems. The definitions of generalized quadratic stability or generalized stabilization for uncertain singular systems are given on the condition of parametric uncertainty F being norm bounded (||F|| ≤ 1). With these concepts, the strict linear matrix inequalities (LMI) are developed. According to the matrix Schur complement property, prove these LMIs such that the uncertain singular systems are regular, impulse free and stable for all admissible uncertainties, and state feedback controllers are formulated, therefore robust stabi-lization problems of the uncertain singular systems are solved;at last, we give some illustrative examples to demonstrate the truth of the proposed approaches.(2) Robust control problems of uncertain singular time-delay systems are considered. Firstly, We generalize the concepts on robust control to uncertain singular time-delay systems, and give the definitions of the generalized quadratic stability or generalized stabilization for uncertain singular time-delay systems;then using the Schur complement property to prove generalized quadratic stability, generalized quadratic stabilization of the parametric uncertain singular time-delay systems under the established matrix inequalities, therefore robust stability and robust stabilization problems of the systems are solved, and giving the preserving performance controlers and the upper bound of performance index for the uncertain singular time-delay systems;Finally, some examples are given to illustrate the applicability of the proposed approaches.