Nonlinear control under delays that depend on delayed states

Abstract We recently introduced a design methodology for the stabilization of nonlinear systems with input and state delays that depend on the current state of the plant. In the present paper we consider nonlinear systems with delays that depend on the delayed state of the plant, that is, the delay is defined implicitly as a nonlinear function of the state at a past time which depends on the delay itself. Since the prediction horizon and the delay depend on the state of the plant, the key design challenges are how to compute the predictor state and the delay (since the delay needs to be available in order to compute the predictor). We resolve these challenges and we establish closed-loop stability with the aid of a strict Lyapunov functional that we construct. We also design a predictor feedback law for systems with state delays that depend on delayed states. We present an example of a strict-feedforward nonlinear system with input delay.

[1]  Iasson Karafyllis,et al.  Delay-robustness of linear predictor feedback without restriction on delay rate , 2013, Autom..

[2]  Miroslav Krstic,et al.  Compensation of Time-Varying Input and State Delays for Nonlinear Systems , 2012 .

[3]  P. Olver.  Nonlinear Systems , 2013 .

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

[5]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[6]  Iasson Karafyllis,et al.  Nonlinear Stabilization Under Sampled and Delayed Measurements, and With Inputs Subject to Delay and Zero-Order Hold , 2012, IEEE Transactions on Automatic Control.

[7]  I. Karafyllis.  Stabilization by Means of Approximate Predictors for Systems with Delayed Input , 2009, SIAM J. Control. Optim..

[8]  Delphine Bresch-Pietri,et al.  Prediction-based feedback control of a class of processes with input-varying delay , 2012, 2012 American Control Conference (ACC).

[9]  Frédéric Mazenc,et al.  Generating positive and stable solutions through delayed state feedback , 2011, Autom..

[10]  Karl Henrik Johansson,et al.  Nonlinear state-dependent delay modeling and stability analysis of internet congestion control , 2010, 49th IEEE Conference on Decision and Control (CDC).

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

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

[13]  Miroslav Krstic,et al.  Lyapunov stability of linear predictor feedback for time-varying input delay , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[14]  Miroslav Krstic,et al.  Feedback linearizability and explicit integrator forwarding controllers for classes of feedforward systems , 2004, IEEE Transactions on Automatic Control.

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

[16]  Miroslav Krstic,et al.  Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems , 2010, IEEE Transactions on Automatic Control.

[17]  Mrdjan Jankovic,et al.  Cross-Term Forwarding for Systems With Time Delay , 2009, IEEE Transactions on Automatic Control.

[18]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

[19]  Delphine Bresch-Pietri,et al.  Halanay inequality to conclude on closed-loop stability of a process with input-varying delay , 2012 .

[20]  Iasson Karafyllis,et al.  Finite-Time Global Stabilization by Means of Time-Varying Distributed Delay Feedback , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[21]  Iasson Karafyllis,et al.  Predictor-Based Output Feedback for Nonlinear Delay Systems , 2011, ArXiv.

[22]  Michael Malisoff,et al.  Asymptotic stabilization for feedforward systems with delayed feedbacks , 2013, Autom..

[23]  Miroslav Krstic,et al.  Robustness of nonlinear predictor feedback laws to time- and state-dependent delay perturbations , 2012, Autom..

[24]  Sabine Mondié,et al.  Global asymptotic stabilization of feedforward systems with delay in the input , 2004, IEEE Transactions on Automatic Control.

[25]  S. P. Travis,et al.  A One-Dimensional Two-Body Problem of Classical Electrodynamics , 1975 .

[26]  Michael Malisoff,et al.  Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements , 2010, Autom..

[27]  Joon-Young Choi,et al.  Global asymptotic stability of FAST TCP in the presence of link dynamics , 2009 .