Robust Stability Analysis And Feedback Control For Singularly Perturbed Systems

The basic property of singularly perturbed systems is two-time-scale structure, which often leads to the increased order and stiffness of systems. This poses formidable computational complexities for the analysis and control of the systems. To overcome these problems, the singular perturbation method was put forward, which has been a highly recognized and rapidly developed area in control systems in the last fourty years. Results now exist for both the continuous and discrete-time systems. Based on the recent research on the theory of singularly perturbed systems, this dissertation investigated the robust stability and stabilization problems for singularly perturbed systems and gives some new results. To sum up, the major works in this dissertation are as follows:1. Firstly, the robust stability and stabilization problem for a class of continuous and discrete-time singularly perturbed systems with nonlinear perturbations are investigated. By applying the fix-point principle and linear matrix inequality technique, sufficient conditions for the existence of the isolate root are presented. Furthermore, using two-time-scale decomposition technique, the input-to-state stability of the full-order system can be obtained based on the corresponding slow and fast subsystems. Based on the established results, a state feedback control law is also designed such that the resulting closed-loop system is input-to-state stable.2. H∞, analysis and control for discrete-time singularly perturbed systems with nonlinear disturbances are investigated. The H∞, performance of the full-order system is analyzed based on those of the corresponding slow and fast subsystems. Based on the established property, the state feedback is designed to make the resulting closed-loop system asymptotically stable with a prescribed H∞ performance.3. A strictly proper dynamic output feedback control problem for fast sampling discrete-time singularly perturbed systems using the singular perturbation approach is considered. Sufficient conditions in terms of linear matrix inequalities are presented to guarantee the existence of a dynamic output feedback controller for the slow and fast subsystems, respectively. The theoretical result demonstrates that the composite dynamic output feedback control through the corresponding dynamic output feedback controllers of the slow and fast subsystems can stabilize the original system, provided that the perturbation parameter is sufficiently small.4. When the dynamic output feedback control is proper but not strictly proper, the robustness of feedback control problem for fast sampling discrete-time singularly perturbed systems is addressed on the basis of reduced-order models. To obtain it. an auxiliary system is designed. Based on this, the design of the dynamic output feedback for the reduced order subsystem is reduced to the simultaneous design of static output feedback controller for the fast subsystem and strictly proper dynamic output feedback controller for the auxiliary system, respectively. We confirm that it is possible to generate the robustness for the proposed dynamic output feedback control. Thus, the restriction on the strict properness can be alleviated.5. Based on the linear matrix inequality technique, the H, control of Markovian jump linear singularly perturbed systems is revisited. A sufficient condition, different from ones in previous existing works, is given for designing H., control to overcome the difficulties in solving either Riccati equations or nonlinear matrix inequalities. Then. a jump Hf controller is obtained effectively by solving the proposed linear matrix inequalities. Furthermore, a method is developed for evaluating an allowable upper bound of the singular perturbation parameter, in which a prescribed H, performance is satisfied.