Positive Forms and Stability of Linear Time-Delay Systems

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that stability implies that there exists a quadratic Lyapunov function on the state space, although this is in general infinite dimensional. We give an explicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions. Positivity of this class of functions is enforced using sum-of-squares polynomial matrices. This allows the computation to be formulated as a semidefinite program

[1]  K. L. Cooke,et al.  Hereditary Differential Systems with Constant Delays , .

[2]  S. Lall,et al.  Constructing Lyapunov Functions for nonlinear delay-differential equations using semidefinite programming , 2004 .

[3]  M. Kojima.  Sums of Squares Relaxations of Polynomial Semidefinite Programs , 2003 .

[4]  Bruce Reznick,et al.  Real zeros of positive semidefinite forms. I , 1980 .

[5]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[6]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[7]  Michel C. Delfour,et al.  Hereditary Differential Systems with Constant Delays. I. General Case , 1972 .

[8]  Graziano Chesi,et al.  On the Gap Between Positive Polynomials and SOS of Polynomials , 2007, IEEE Transactions on Automatic Control.

[9]  Weehong Tan,et al.  Nonlinear Control Analysis and Synthesis using Sum-of-Squares Programming , 2006 .

[10]  C. W. Scherer,et al.  Asymptotically Exact Relaxations for Robust LMI Problems based on Matrix-Valued Sum-of-Squares , 2004 .

[11]  Graziano Chesi,et al.  Computing output feedback controllers to enlarge the domain of attraction in polynomial systems , 2004, IEEE Transactions on Automatic Control.

[12]  Emilia Fridman,et al.  An improved stabilization method for linear time-delay systems , 2002, IEEE Trans. Autom. Control..

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

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

[15]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[16]  P. Parrilo.  On a decomposition of multivariable forms via LMI methods , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[17]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[18]  Dazhong Wang.  Polynomial level-set methods for nonlinear dynamics and control , 2007 .

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

[20]  Vladimir L. Kharitonov,et al.  Exponential estimates for time delay systems , 2004, Syst. Control. Lett..

[21]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[22]  J. Hale,et al.  Stability of Motion. , 1964 .

[23]  Iu M Repin.  Quadratic liapunov functionals for systems with delay , 1965 .

[24]  Ju Hyun Park,et al.  Stability Analysis of Linear Time-delay Systems , 1996 .

[25]  Chien-Yu Lu,et al.  On improved delay-dependent robust stability criteria for uncertain systems with multiple-state delays , 2002 .

[26]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[27]  Young Soo Moon,et al.  Delay-dependent robust stabilization of uncertain state-delayed systems , 2001 .

[28]  A. Papachristodoulou,et al.  Constructing Lyapunov-Krasovskii functionals for linear time delay systems , 2005, Proceedings of the 2005, American Control Conference, 2005..

[29]  Antonis Papachristodoulou,et al.  Positivity of kernel functions for systems with communication delay , 2007, 2007 46th IEEE Conference on Decision and Control.

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

[31]  Onur Toker,et al.  Mathematics of Control , Signals , and Systems Complexity Issues in Robust Stability of Linear Delay-Differential Systems * , 2005 .

[32]  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..