Robust H_∞ Control For Nonlinear Descriptor System Based On LMI Methods

Descriptor systems are also referred to as singular systems, implicit systems, descriptor state-space systems, differential algebra systems or semi-state systems, which have more extensive form than normal systems. There may be some uncertain parameters in all kinds of physical, industrial and engineering systems, due to the simplification of model, change of running environment, aging of electrical elements and so on. At the same time, time-delay is frequently encountered in a variety of systems. In addition, there may be some nonlinear parameters in systems for the objective factor. Therefore, the analysis and synthesis of nonlinear delayed descriptor systems with uncertainties are important both in theory and practice. The main contribution of this paper is summarized as follows:Firstly, the H∞control problems are considered for a class of nonlinear descriptor systems. By means of generalized Lyapunov function, the stability is studied for the systems. A condition of linear matrix inequality (LMI) is presented such that the systems are zero solution asymptotically stable with a H∞norm constraint. Then, existence conditions of a state feedback H∞controller and a derivative and proportional state feedback H∞controller are given guaranteeing the above performance of the resulting closed-loop systems.Secondly, the robust H∞control problems are discussed for a class of nonlinear descriptor systems with linear-fractional parameter uncertainties. Depend on linear matrix inequality (LMI), a sufficient condition is obtained such that nonlinear descriptor systems with linear-fractional parameter uncertainties are zero solution asymptotically stable with a H∞norm constraint, then a state feedback robust H∞controller and a derivative and proportional state feedback robust H∞controller are designed to guarantee the H∞performance of the resulting closed-loop systems.Thirdly, the H∞control the problems are explored for a class of nonlinear delayed descriptor systems. The stability is studied for the systems by choosing a appropriate generalized Lyapunov function. A condition of linear matrix inequality (LMI) is presented such that the systems are zero solution asymptotically stable and satisfy a prescribed H∞norm constraint for every delay d>0. Then, existence conditions of a state feedback H∞ controller and a derivative and proportional state feedback H∞controller are given guaranteeing the above performance of the resulting closed-loop systems.Finally, the robust H∞control problems are discussed for a class of nonlinear delayed descriptor systems with linear-fractional parameter uncertainties. A condition of linear matrix inequality (LMI) is presented such that the systems are zero solution asymptotically stable with a H∞norm constraint for every delay d>0 and admissible uncertainties. Then a state feedback robust H∞controller and a derivative and proportional state feedback robust H∞controller are designed to guarantee the performance of the resulting closed-loop systems.The examples are provided to demonstrate the effectiveness of the proposed methods.