Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004 POLYHEDRAL LYAPUNOV FUNCTIONS COMPUTATION FOR ROBUST AND GAIN SCHEDULED DESIGN Franco Blanchini * Stefano Miani **,1 Carlo Savorgnan ** * Dipartimento di Matematica e Informatica, Universita degli Studi di Udine, Via deLLe Scienze 208, 33100 Udine - Italy ** Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Universita degli Studi di Udine, Via deLLe Scienze 208,33100 Udine -Italy ELSEVIER IFAC PUBLICATIONS www.elsevier.com/locate/ifac Abstract: In this paper, the problem of efficiently computing polyhedral Lyapunov functions for polytopic time-varying linear discrete-time systems, is considered. Two strictly related problems are considered: the robust case, when the designer is unaware of the value of the uncertain parameter entering the system, and the gain scheduled case, when such value is available and exploitable for synthesis design. The relation between the two is evidenced, together with an efficient algorithm for the determination of the respective Lyapunov functions. Finally, it is shown how such algorithm can be used for the determination of proper domains of attraction when constraints on the state and/or input are present. Copyright © 2004 IFAC Keywords: Lyapunov function, robust, gain scheduling, algorithm, constrained control 1. INTRODUCTION Many synthesis design and analysis control problems can be carried out by means of Lyapunov functions. Although such fact is well-known, constructive Lya- punov techniques are not in general so common given the inherent difficulty in their determination and the lack of converse results. Indeed, when talking about "constructive Lyapunov function techniques" for a problem solution, some problems arise. The first one concerns the possibility of effectively having at dis- posal a converse result which guarantees that the prob- lem solvability is equivalent to the existence of a Lya- punov function. The second, assuming the above has a positive answer, concerns the knowledge of the class where to look for the Lyapunov function. The third, finally, assuming again that the above has a positive I Corresponding author. Tel: +39 0432 558262, fax. +39 0432 558251, email miani.stefano@uniud.it 835 answer, concerns the possibility of effectively comput- ing the Lyapunov function, say the numerics. For some classes of systems, such as those considered in this paper, the chain "converse theorem-Lyapunov class-numerics" holds and the aim of this work is to present one of the tools which can be used to solve stability analysis as well as synthesis design problems, namely polyhedral Lyapunov functions, together with an efficient numerical algorithm used for their compu- tation. Needless to say, this is one of the tools available, and we refer the reader to some of the many papers which have appeared in the literature (Blanchini, 1999), to get the flavour of the existing results and techniques available for Lyapunov function computation for the class of systems analyzed here, results and techniques which focus on Lyapunov functions in the class of polyquadratic forms (Daafouz and Bernussou, 200 1), polyhedral (Brayton and Tong, 1980; Gilbert and