1058 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009 Analysis of Polynomial Systems With Time Delays via the Sum of Squares Decomposition Antonis Papachristodoulou, Matthew M. Peet, and Sanjay Lall Abstract—We present a methodology for analyzing robust indepen- dent-of-delay and delay-dependent stability of equilibria of systems described by nonlinear Delay Differential Equations by algorithmically constructing appropriate Lyapunov-Krasovskii functionals using the sum of squares decomposition of multivariate polynomials and semidefinite programming. We illustrate the methodology using an example from population dynamics. Index Terms—Linear matrix inequality (LMI), Lyapunov-Krasovskii, sum of squares (SOS), time delay. I. INTRODUCTION Delay Differential Equations (DDEs) are used to model systems that involve transport and propagation of data; examples include networked systems [1] and modeling maturation and growth in population dy- namics [2]. The analysis and control of such systems is important [3], [4], as the presence of delays may induce performance degradation or even instabilities. DDEs fall in the category of Functional Differential Equations (FDEs), which differ from Ordinary Differential Equations (ODEs) because the system state belongs to an infinite dimensional space. Assuming local existence and uniqueness of solutions, appropriate Lyapunov functions can be used for stability analysis. However, while for the case of ODEs these are functions, in the case of DDEs they are functionals as the state belongs in a function space itself. For linear DDEs, the form of these functionals that is necessary and sufficient for Delay-Dependent (DD) and strong Independent-Of- Delay (IOD) stability is known [5]–[7], but these conditions are diffi- cult to test algorithmically. Under restrictions on their structure, convex optimization was used to construct them with conservative results on the delay interval guaranteeing stability [8], [9]. This is because con- structing the functional that is necessary and sufficient for stability amounts to parameterizing the set of positive operators on an infinite- dimensional space. Lyapunov functionals with piecewise-linear kernels can be constructed by solving a set of LMIs whose size depends on the discretization level [10], and as the discretization level is decreased, delay values closer to the boundary of stability can be tested. In [11] a new approach was proposed which uses an explicit parametrization of positive operators and uses the Sum of Squares (SOS) decomposition and semidefinite programming for computation. As far as nonlinear time delay systems are concerned, the only methodologies centre on the construction of simple Lyapunov certifi- cates for systems of low dimension through a judicious choice for a candidate Lyapunov function [2]. This is the case even for systems described by ODEs, where constructing Lyapunov functions is usually based on system structure and its properties (Volterra, gradient systems Manuscript received July 31, 2007; revised March 14, 2008. Current version published May 13, 2009. This work was supported in part by the Engineering and Physical Sciences Research Council Grant EP/E05708X/1. Recommended by Guest Editors G. Chesi and D. Henrion. A. Papachristodoulou is with the Department of Engineering Science, Uni- versity of Oxford, Oxford OX1 3PJ, U.K. (e-mail: antonis@eng.ox.ac.uk). M. M. Peet is with the Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: mpeet@iit.edu). S. Lall is with the Department of Aeronautics and Astronautics, Stanford Uni- versity, Stanford, CA 94305 USA (e-mail: lall@stanford.edu). Digital Object Identifier 10.1109/TAC.2009.2017168 etc.). Recently, however, a computational methodology based on the SOS decomposition has been proposed [12]–[14]. In this technical note we present an algorithmic methodology for constructing L-K functionals to assess IOD and DD stability for poly- nomial time delay systems. Preliminary results have been presented in [15], [16]. The present technical note offers significant improvements on the way these functionals can be constructed. Applications of this approach to Internet congestion control problems have appeared in [17] and preliminary results on state feedback stabilization have appeared in [18]. The methodology unifies local and robust DD and IOD sta- bility, but only the single-delay case is presented in this technical note in order to simplify the exposition: the case of multiple, incommensu- rate delays can be treated in a unified way. Section II outlines the proposed methodology and Section III shows how this can be used for the nominal, robust and local IOD and DD sta- bility analysis of polynomial delayed systems, followed by an example from population dynamics. A. Notation denotes the reals and the -dimensional Euclidean space. For , is the ring of polynomials in with real coefficients and the vector of monomials in of degree or less. is the Banach space of continuous functions mapping the interval [ , 0] into with the topology of uniform convergence. The norm on is where is the infinity norm. Suppose , and ; then for any , define by , . Symbolic independent variables will reference state and delayed state variables: will reference where and will denote the row vector of ’s, . Also, will be used for , for and vectors and are similarly defined. Finally, we will use and . II. THE PROPOSED METHODOLOGY Background theory on stability and Lyapunov theory for DDEs can be found in [19]. Consider a polynomial time delay system with a single delay of the form (1) where with is such that a unique solution exists from an appropriate initial condition close to 0. The Lyapunov functional (2) can be used to verify the DD stability of the zero steady-state; its deriva- tive takes the form (3) where the map , to will be presented in later sections. 0018-9286/$25.00 © 2009 IEEE Authorized licensed use limited to: Illinois Institute of Technology. Downloaded on May 20, 2009 at 19:09 from IEEE Xplore. Restrictions apply.