Stable Feedback Control For Singularly Perturbed Systems

Singular perturbation techniques for time scale systems have found wide applications in the area of analysis and synthesis of control processes, which has been studied recently in different set-ups by many researchers. On the other hand, the linear matrix inequality (LMI) approach has been attracting more attention due to its extensive applications in solving various control problems. However, up to the present, the problem of robust control for singularly perturbed systems through LMI approach has not been solved. Considering this, in this dissertation, attention is focused on the development of an LMI approach to solving robust control problems for singularly perturbed systems.The main results obtained in this dissertation are as follows:i) Quadratic stability is extended to singularly perturbed continuous and discrete systems, respectively. It is shown that quadratic stability of a singularly perturbed system is equivalent to that of the slow and fast system. Using the LMI approach, we obtain sufficient conditions of quadratic stability and quadratic stabilizability for singularly perturbed systems. An iterative algorithm is proposed for the design of desired quadratic stabilizing feedback controllers.ii) A sub-optimal regulator is designed for nonstandard discrete-time singularly perturbed systems through the LMI method; it is different from the general fast-slow decomposition method. By introducing a solution matrix with perturbed parameters, sub-optimal controllers are designed. A simulation result is presented to demonstrate the effectiveness of the proposed design method.iii) By a similar method to the inverse optical quadratic control, the problem of sub-optical disturbance attenuation subject to a quadratic performance index is discussed by the LMI approach.iv) The ε-independent H∞ control problem for singularly perturbed continuous and discrete systems is investigated, respectively. Sufficient conditions are obtained and sub-optimal controllers are designed, respectively. By the positive real lemmas for singular systems and regular systems, we present a sufficient and necessary condition for the solvability of positive real control problem for singularly perturbed systems.v) State feedback design for singularly perturbed systems is discussed in the Delta(d) domain. State feedback controllers of slow and fast singularly perturbed subsystems are designed, respectively, such that the closed-loop system has desired poles. The derived result can be viewed as a unified one for continuous and discrete time systems in the unified Delta framework. An example is presented to demonstrate the validity and effectiveness of our design method.Furthermore, the problem of stability analysis for singularly perturbed discrete systems is considered. A necessary and sufficient condition is obtained for singularly perturbed discrete systems with two kinds perturbations. This is also extended to the case of two-dimensional singularly perturbed Roesser model.