Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.

On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valued derivative” of vector Lyapunov functions is introduced, some generalized comparison principles on discontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.

Uniformly ultimate boundedness for discontinuous systems with time-delay

The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.

State feedback stabilization of cascaded nonlinear systems with discontinuous connection

In practical engineering, many phenomena are described as a discontinuous function of a state variable, and the discontinuity is usually the main reason for the degradation of the control performance. For example, in the set-point control problem of mechanical systems, the static friction （described by a sgn function of velocity of the contacting faces） causes undesired positioning error. In this paper, we will investigate the stabilization problem for a class of nonlinear systems that consist of two subsystems with cascaded connection. We will show the basic idea with a special case first, and then the result will be extended to more general cases. Some interesting numerical examples will be given to demonstrate the effectiveness of the proposed design approach.

New approaching condition for sliding mode control design with Lipschitz switching surface

In this paper, we concern the approaching condition of sliding mode control （SMC） with a Lipschitz switching surface that may be nonsmooth. New criteria on the relation between phase trajectories and an arbitrary Lipschitz continuous surface are examined firstly. Filippov＇s differential inclusion is adopted to describe the dynamics of trajectories of the closed-loop system with SMC. Compared with Filippov＇s criteria for only smooth surface, new criteria are proposed by utilizing the cone conditions that allow the surface to be nonsmooth. This result also yields a new approaching condition of SMC design. Based on the new approaching condition, we develop the sliding mode controller for a class of nonlinear single-input single-output （SISO） systems, of which the switching surface is designed Lips- chitz continuous for the nonsmooth sliding motion. Finally, we provide a numerical example to verify the new design method.

Robust stability of mixed Cohen–Grossberg neural networks with discontinuous activation functions

Purpose–The purpose of this paper is to develop a method for the existence,uniqueness and globally robust stability of the equilibrium point for Cohen–Grossberg neural networks with time-varying delays,continuous distributed delays and a kind of discontinuous activation functions.Design/methodology/approach–Basedonthe Leray–Schauderalternativetheoremand chainrule,by using a novel integral inequality dealing with monotone non-decreasing function,the authors obtain a delay-dependent sufficient condition with less conservativeness for robust stability of considered neural networks.Findings–Itturns out thattheauthors’delay-dependent sufficientcondition canbeformed intermsof linear matrix inequalities conditions.Two examples show the effectiveness of the obtained results.Originality/value–The novelty of the proposed approach lies in dealing with a new kind of discontinuous activation functions by using the Leray–Schauder alternative theorem,chain rule and a novel integral inequality on monotone non-decreasing function.