Stability Analysis Of Nonlinear Time-Delay Systems And Nonlinear Two-Dimensional Systems

This work deals with the robust stability and input-to-state stability of nonlinear time-delay systems and Lyapunov-type stability of general two-dimensional (2-D) nonlinear systems. The main contributions are given as follows:1. For time-delay systems with nonlinear perturbations, a quasi-one-sided Lipschitz condition is introduced. The quasi-one-sided Lipschitz condition includes much useful information of the nonlinear part and contributes to get delay-independently and delay-dependently robust stability criteria with less conservatism. Furthermore, it should be noted that the stability criteria can be presented in the form of linear matrix inequalities, and are still available even if the system parameter matrix is unstable.2. For nonlinear time-delay systems with arbitrary measurable, locally essentially bounded disturbances, the coverse Lyapunov theorem of robust stability is investigated. It is shown that the nonlinear time-delay system is robustly globally asymptotically stable if and only if it admits a locally Lipschitz continuous Lyapunov functional. Furthermore, the result is essential to the converse Lyapunov theorem of input-to-state stability for nonlinear time-delay control systems.3. Based on the locally Lipschitz continuous converse Lyapunov theorem of robust stability, the Lyapunov characterizations of input-to-state stability property are obtained for nonlinear time-delay control systems. That is to say, it is shown tha the input-to-state stability, robust stability, weakly robust stability and locally Lipschitz continous Lyapunov functional are equivalent.4. For general 2-D time-varying nonlinear FM second model, the Lyapunov-type stability problem is considered. At first, three comparison lemmata are given and play an important role in the proof of the corresponding Lyapunov-type stability results. Next, three Lyapunov-type stability theorems are presented to sufficiently guarantee the uniform stability, uniformly asymptotical stability and exponential stability, respectively.5. For general 2-D time-invariant nonlinear FM second model, a Lyapunov-type stability theorem is presented to sufficiently guarantee the stability and (globally) asymptotical stability. Then, for the globally asymptotical stability, it is further improved to lessen the conservatism of the stability theorem. More importantly, the improved theorem can derive global stability criteria which have the form of linear matrix inequalities.6. Applied the classic and improved Lyapunov-type stability theorems of general 2-D time-invariant nonlinear systems, some robust stability criteria are gotten for 2-D time-invariant FM II model with nonlinear perturbation (or overflow nonlinearites). Furthermore, based on the two Lyapunov-type stability theorems, some absolute stability criteria are gotten for 2-D time-invariant FM second model with sector-bounded nonlinearity. By comparison we can see the advantage of the improved stability theorem.