Robust Stabilization of Nonlinear Time Delay Systems Using Convex Optimization

We address the problem of robust, global, delay-dependent and delay-independent stabilization of nonlinear time-delay systems with memory state feedback. The methodology we use is based on a linear-like representation of the time-delay system for which we construct appropriate Lyapunov-Krasovskii functionals. The resulting conditions take the form of infinite-dimensional state-dependent Linear Matrix Inequalities which can be treated as sum of squares matrices. The sum of squares program that emerges can then be solved using semidefinite programming and SOSTOOLS, which results in an algorithmic construction of the control law and the Lyapunov-Krasovksii functional. An example is presented that shows the effectiveness of the methodology.

[1]  Yuanqing Xia,et al.  Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals , 2003, Syst. Control. Lett..

[2]  Jean-Michel Dion,et al.  Robust stabilization of uncertain linear systems with input delay , 1997, 1997 European Control Conference (ECC).

[3]  A. Papachristodoulou.  Analysis of nonlinear time-delay systems using the sum of squares decomposition , 2004, Proceedings of the 2004 American Control Conference.

[4]  Masayuki Fujita,et al.  Output feedback control synthesis for linear time-delay systems via infinite-dimensional LMI approach , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[5]  Sabine Mondié,et al.  Global asymptotic stabilization for chains of integrators with a delay in the input , 2003, IEEE Trans. Autom. Control..

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

[7]  Wook Hyun Kwon,et al.  Delay-dependent guaranteed cost control for uncertain state-delayed systems , 2005, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[8]  Jack K. Hale,et al.  Functional differential equations: Basic theory , 1993 .

[9]  A. Papachristodoulou,et al.  Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach , 2004, 2004 5th Asian Control Conference (IEEE Cat. No.04EX904).

[10]  Ian R. Petersen,et al.  Optimal quadratic guaranteed cost control of a class of uncertain time-delay systems , 1997 .

[11]  V. Kolmanovskii,et al.  On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems , 1999 .

[12]  M. Jankovic.  Control of nonlinear systems with time delay , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  Derong Liu.  The Mathematics of Internet Congestion Control , 2005, IEEE Transactions on Automatic Control.

[14]  Takehito Azuma,et al.  An approach to solving parameter-dependent LMI conditions based on finite number of LMI conditions , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[15]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[16]  J. Cloutier.  State-dependent Riccati equation techniques: an overview , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[17]  Chaouki T. Abdallah,et al.  Stabilization of linear and nonlinear systems with time delay , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[18]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[19]  Pablo A. Parrilo,et al.  Introducing SOSTOOLS: a general purpose sum of squares programming solver , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[20]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[21]  A. Papachristodoulou,et al.  Analysis of Non-polynomial Systems using the Sum of Squares Decomposition , 2005 .

[22]  Xi Li,et al.  Delay-dependent robust H control of uncertain linear state-delayed systems , 1999, Autom..

[23]  X. Nian,et al.  Guaranteed-cost control of a linear uncertain system with multiple time-varying delays: an LMI approach , 2003 .

[24]  K. Gu.  Discretized LMI set in the stability problem of linear uncertain time-delay systems , 1997 .

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

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

[27]  Qing-Long Han,et al.  Controller design for time-delay systems using discretized Lyapunov functional approach , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).