Regional Stability and Performance Analysis for a Class of Nonlinear Discrete-Time Systems 1 D. F. Coutinho †, M. Fu ‡, A. Trofino † † Department of Automation and Systems, Universidade Federal de Santa Catarina, PO BOX 476, 88040-900, Florian´ opolis, SC, Brazil  On leave from Department of Electrical Engineering, PUC-RS, Brazil. ‡ School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, N.S.W. 2308, Australia.  Currently on leave at School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore. emails:coutinho(trofino)@das.ufsc.br, eemf@ee.newcastle.edu.au Abstract This paper deals with the problem of regional stabil- ity and performance analysis for a class of nonlinear discrete-time systems with uncertain parameters. We use polynomial Lyapunov functions to derive stability conditions and performance criteria in terms of linear matrix inequalities (LMIs). Although the use of poly- nomial Lyapunov functions is common for continuous- time systems as a way to reduce the conservatism in analysis, we point out that direct generalization of such an approach to discrete-time systems leads to in- tractable solutions because it results in a large number of LMIs. We introduce a novel approach to reduce the computational complexity by generalizing a result of Oliveira et. al. on robust stability analysis for discrete- time systems with parameter uncertainties. We point out that the proposed method can lead to less conserva- tive results when compared with results using quadratic Lyapunov functions. 1 Introduction The last decade or so has witnessed active research work in the area of robust control of continuous-time nonlinear systems in the framework of linear matrix in- equalities (LMIs). Design approaches range from using quadratic Lyapunov functions ([1, 2]) to those based on polynomial Lyapunov functions ([3, 4]). In general, non-quadratic Lyapunov functions are less conservative for dealing with uncertain and nonlinear systems than quadratic Lyapunov functions at the expense of extra computation [5]. However, most of the robust control 1 This work was partially supported by ’CAPES’, Brazil, under grant BEX0784/00-1 and ’CNPq’, Brazil, under grant 147055/99-7. results using non-quadratic Lyapunov functions are for continuous-time systems. The fundamental difficulty with non-quadratic Lya- punov functions for discrete-time systems lies in the fact that the difference between the Lyapunov func- tions at time k + 1 and k is highly nonlinear. To make this point clear, we consider the following system: x(k + 1) = A(x(k),δ)x(k) (1) and the Lyapunov function V (x, δ) = x  P (x, δ)x, where δ represents (constant) uncertain parameters, and the matrices A(x, δ) and P (x, δ) depend on x and δ. The Lyapunov function difference is given by the fol- lowing inequality (which we will refer to as a Lyapunov inequality): V (x(k + 1),δ) - V (x(k),δ)= x  (k)A  (x(k),δ) P (A(x(k),δ)x(k),δ)A(x(k),δ)x(k) (2) -x  (k)P (x(k),δ)x(k) which is typically a highly nonlinear function of x(k) and δ. In contrast, if we considered a similar continuous-time system ˙ x(t)= A(x(t),δ)x(t) (3) and a similar Lyapunov function, we would have the derivative of the Lyapunov function given by the fol- lowing Lyapunov inequality: ˙ V (x(t),δ(t)) = x   A  (x, δ)P (x, δ)+ P (x, δ)A(x, δ) + n  i=1 ∂P (x, δ) ∂x i e  i A(x, δ)x  x (4) where x i is the ith element of x and e i is the ith col- umn of an identity matrix. It is obvious from the above