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

• Published in: IEEE transactions on automatic control Vol. 53; no. 6; pp. 1526 - 1531
• Main Authors: Yeganefar, N., Pepe, P., Dambrine, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Actuators; Applied sciences; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control systems; Control theory. Systems; Delay lines; Disturbances; Dynamical systems; Dynamics; Error correction; Exact sciences and technology; Exponential stability; Feedback control; Force measurement; Hydraulic actuators; input-to-state stability (ISS); International Space Station; Law; Liapunov-Krasovskii theorem; Linear feedback control systems; Links; Nonlinear dynamical systems; nonlinear time-delay systems; Stability; Sufficient conditions; System theory; nonlinear time-delay systems; Control program; Non linear control; Feedback regulation; Delay system; Liapunov-Krasovskii theorem; Dynamical system; Exponential stability; Closed feedback; Input to state stability; input-to-state stability (ISS); Lipschitz function; Delay time; Actuator; Lyapunov function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2008.928340

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.