The Research On Stability Analysis And Stabilization Of Nonlinear Systems With Delay

A lot of research work and achievement for nonlinear systems have been done by many domestic and international scholars. Because of the complexity of nonlinear systems, they failed to find a general method for the complicated nonlinear problems. Therefor, many scholars focus attention on nonlinear systems.Using Lyapunov stability theory, comparison principle method and differential equation theory, the robust stability of control systems with one or several nonlinear elements has been studied in this paper. At the same time uncertainty and time-delay existing in systems are considered too. The main work and research results of this paper are summarized as follows:(1) According to Lyapunov stability theory and real symmetrical matrix characteristic, the delay-independent sufficient conditions for absolute stability of Lurie time-delay control systems with several stationary elements is proposed. Utilizing Razumikhin's theorem, the delay-dependent robust stability conditions for this kind of Lurie systems is presented. The conditions obtained in this paper can be used to test the absolute stability of this kind of systems directly and to estimate the corresponding delay bounds.(2) Utilizing differential equation theory , matrix measure and comparison principle method, the sufficient conditions for Lipschitz nonlinear systems are deduced. At the same time a new optimal design method of the observer gain matrix is presented. Basing on above several results, the sufficient robust stability conditions and robust stabilization method for this kind of systems with uncertainty are put forward.(3) By means of Lyapunov function method and comparison principle (combining M matrix characteristic) respectively, the delay-independent stability conditions for large-scale interconnected linear systems with time-varying delays are obtained. The direct relationship between stability and Jordan form of each subsystem's matrix is established. In succession, the delay-independent robust stability conditions for a class of large-scale interconnected time-delay systems with uncertainty and nonlinear perturbations is derived. And the simple method to test the stability of such systems is provided.(4) By utilizing the minimum super-bound theorem of the real symmetric matrix,Hx theory and LMI method, a decentralized stabilizing state feedback controller with smaller feedback gains matrix for a class of large-scale interconnected time-delay systems with uncertainty and nonlinear perturbations is designed. The conclusion of this paper only requires that the uncertainty is the continuous function matrix in each time segment without any strict match condition. And there is not any restraint to the nonlinear perturbations existing in systems.(5) By applying the results obtained in this paper, the robust stability of two typical nonlinear systems is discussed. First of all, using the method of observer design for nonlinear systems, the robust stability of a manipulator arm with flexible joint is studied;and then, the robust stability and stabilization of nonlinear double inverted pendulum system is discussed. The simulation and practical results proved the validity of theoretical research. The conclusions of this paper provide a new way for robust stability analysis and stabilization of nonlinear systems with time-delay.