Stability And Sliding Mode Control For Classes Of Systems With Time-delay

The study of stability and stabilization of time-delay systems have receivedconsiderable attention in the past few decades. Due to the complex factors such asdelay or uncertainty as well as their impact on the system, further discussion about thestability and controller design issues for time-delay systems is still needed.In this note, the problem of stability and sliding mode variable structure control(SM-VSC) for time-delay systems is mainly studied, the according analysis andSM-VSC methods for classes of different time-delay systems are given.The main contents are included as follows.1. The development and research situation about time-delay systems issummarized.2. The robust sliding mode control is investigated for a class of discrete-timesystems with time-varying delay, which is based on a novel nonlinear time-varyingquasi-sliding surface design scheme. Firstly, the general form of the nonlinearswitching function with time-varying feature is given, which can dynamicallyimprove the quality of its motion. By using the free weighting matrix and linearmatrix inequality technique, we propose a design method for the nonlinearquasi-sliding surface, and obtain a sufficient condition of asymptotical stability for thenonlinear sliding motion. Secondly, the quasi-sliding mode controller is designedbased on the improved discrete reaching law to ensure the state of the system forreaching the quasi-sliding mode in a finite time. Thus, the quasi-sliding-modevariable-structure control (VSC) for such nonlinear system is extended to thenonlinear control with time-delay. Simulation validates that the dynamic behavior ofthe system is stable with rapid response, low overshoot, less settling time, whenemploying the proposed quasi-sliding surface and quasi-sliding mode VSC. 3. The problem of state feedback control design is investigated for a class ofdistributed parameter systems with both time-varying delay and continuousdistributed time-delay. By selecting an appropriate Lyapunov-Krasovskii function, asufficient condition is obtained to guarantee the asymptotical stability of theclosed-loop system with time-varying delay by using a linear matrix inequality (LMI)approach. A memoryless state feedback controller is designed, such that theclosed-loop system is stable when there exists a positive matrix. Finally, a numericalsimulation example is given to illustrate the feasibility and validity of the proposedmethod.4. This paper is focused on integral sliding mode control (ISMC) for a class ofparameter uncertain neutral-type systems with time-varying delays(NTSTD). Anintegral sliding surface functional is first constructed for the NTSTD. Then, adelay-dependent robust stability criterion is established for all admissibleuncertainties, which can guarantee the sliding mode dynamic system to be robustlyexponentially stable under a certain decay rate. In addition, by solving the generalizedeigenvalue problem (GEVP) presented in the subsequent work, the global maximalvalue of the decay rate as well as the condition of exponential stability for the slidingmode system are derived. Furthermore, a sliding mode control (SMC) law is thensynthesized to ensure the stability of the closed-loop system, and the proposed SMClaw can drive the state of controlled dynamic system into the predefined slidingsurface in finite time. Finally, a numerical example is provided to demonstrate theeffectiveness of the obtained results.