On Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks

We analyze a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions are both known. We prove that the closed loop systems are input-to-state stable (ISS) relative to actuator errors when small time delays are introduced in the feedbacks. We explicitly construct ISS Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the known Lyapunov functions for the original undelayed closed-loop dynamics. We also provide a general result on ISS for cascade systems with delays. We demonstrate the efficacy of our results using a generalized pendulum dynamics and other examples.

[1]  Kolmanovskii,et al.  Introduction to the Theory and Applications of Functional Differential Equations , 1999 .

[2]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[3]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[4]  O. Sename,et al.  A LMI approach to robust observer design for linear time-delay systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[5]  Sophie Tarbouriech,et al.  Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs , 2000, IEEE Trans. Autom. Control..

[6]  Wim Michiels,et al.  A perturbation approach to the stabilization of nonlinear cascade systems with time-delay , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[7]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[8]  Frédéric Mazenc,et al.  Backstepping design for time-delay nonlinear systems , 2006, IEEE Transactions on Automatic Control.

[9]  I. Karafyllis,et al.  Lyapunov Theorems for Systems Described by Retarded Functional Differential Equations , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[10]  A. Teel.  Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem , 1998, IEEE Trans. Autom. Control..

[11]  Michael Malisoff,et al.  On strict Lyapunov functions for rapidly time-varying nonlinear systems , 2006, 2006 American Control Conference.

[12]  Mrdjan Jankovic,et al.  Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems , 2001, IEEE Trans. Autom. Control..

[13]  S. Niculescu.  H∞ memoryless control with an α-stability constraint for time-delay systems: an LMI approach , 1998, IEEE Trans. Autom. Control..

[14]  Zongli Lin,et al.  On Input-to-State Stability for Nonlinear Systems with Delayed Feedbacks , 2007, 2007 American Control Conference.