A relaxed Lyapunov-Krasovskii condition for global exponential stability of Lipschitz time-delay systems (original) (raw)

Input-to-State Stability of Time-Delay Systems: A Link With Exponential Stability

IEEE Transactions on Automatic Control, 2008

The main contribution of this technical note is to establish a link between the exponential stability of an unforced system and the input-to-state stability (ISS) via the Liapunov-Krasovskii methodology. It is proved that a system which is (globally, locally) exponentially stable in the unforced case is (globally, locally) input-to-state stable when it is forced by a measurable and locally essentially bounded input, provided that the functional describing the dynamics in the unforced case is (globally, on bounded sets) Lipschitz and the functional describing the dynamics in the forced case satisfies a Lipschitz-like hypothesis with respect to the input. Moreover, a new feedback control law is provided for delay-free linearizable and stabilizable time-delay systems, whose dynamics is described by locally Lipschitz functionals, by which the closed-loop system is ISS with respect to disturbances adding to the control law, a typical problem due to actuator errors.

New sufficient conditions for stability analysis of time delay systems using dissipativity theory

Proceedings of the American Control Conference, 2004

In this paper, we extend the concepts of dissipativity and exponential dissipativity to provide new sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite-dimensional linear dynamical system and an infinitedimensional time delay operator, we show that the time delay operator is dissipative. As a special case of this result we show that the storage functional of the dissipative delay operator involves an integral term identical to the integral term appearing in standard Lyapunov-Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop new sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides an explicit framework for constructing Lyapunov-Krasovskii functionals as well as deriving new sufficient conditions for stability analysis of asymptotically stable time delay dynamical systems based on the dissipativity properties of the time delay operator.

Input-to-state stability and exponential stability for time-delay systems: Further results

2007

The main contribution of this paper is to establish a link between the exponential stability of an unforced system and the Input-to-State Stability (ISS) via the Liapunov-Krasovskii methodology. It is proved that a system which is (globally, locally) exponentially stable in the unforced case is (globally, locally) input-to-state stable when it is forced by a measurable and locally essentially bounded input, provided that the functional describing the dynamics in the unforced case is (globally, on bounded sets) Lipschitz and the functional describing the dynamics in the forced case satisfies a Lipschitzlike hypothesis with respect to the input. Moreover, a new feedback control law is provided for delay-free linearizable and stabilizable time-delay systems, whose dynamics is described by locally Lipschitz functionals, by which the closed loop system is ISS with respect to disturbances adding to the control law, a typical problem due to actuator errors.

A dissipative dynamical systems approach to stability analysis of time delay systems

2005

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite-dimensional linear dynamical system and an infinite-dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov-Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov-Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay.

A novel approach to exponential stability of nonlinear systems with time-varying delays

Journal of Computational and Applied Mathematics, 2011

In this paper, the stability of nonlinear systems with time-varying delays is investigated by means of the concepts of generalized Dahlquist constant, generalized relative Dahlquist constant and relative minimal Lipschitz constant. In detail, two sufficient conditions are derived for the exponential stability of nonlinear systems with time-varying delays and the exponential decay of the solutions is also estimated. Compared with some existing results, our stability conditions are less conservative. Some examples are given to illustrate the effectiveness of the obtained results.

Exponential stability in time-delay systems

In this paper, we defined exponential stability for nonlinear time-delay systems with delayed impulses. We derive the Lyapunov-based sufficient conditions for exponential stability. We show that the nonlinear impulsive time-delay system without impulse input delays is exponentially stable under the conditions. It is shown that the stable nonlinear impulsive time-delay system. It is a magnitude of the delayed impulses is sufficiently small, under the same conditions. The delayed impulses do not destroy the stability of the sizes of the impulse input delays.

New Lyapunov–Krasovskii Functionals for Global Asymptotic Stability of Delayed Neural Networks

IEEE Transactions on Neural Networks, 2000

This brief deals with the problem of global asymptotic stability for a class of delayed neural networks. Some new Lyapunov-Krasovskii functionals are constructed by nonuniformly dividing the delay interval into multiple segments, and choosing proper functionals with different weighting matrices corresponding to different segments in the Lyapunov-Krasovskii functionals. Then using these new Lyapunov-Krasovskii functionals, some new delay-dependent criteria for global asymptotic stability are derived for delayed neural networks, where both constant time delays and time-varying delays are treated. These criteria are much less conservative than some existing results, which is shown through a numerical example. Index Terms-Linear matrix inequality (LMI), Lyapunov-Krasovskii functional, neural networks, stability, time delay.