LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties

Rajchakit and Rajchakit Advances in Difference Equations 2012, 2012:106 http://www.advancesindifferenceequations.com/content/2012/1/106 RESEARCH Open Access LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties M Rajchakit and G Rajchakit* * Correspondence: Major of Mathematics and Statistics, Faculty of Science, Maejo University, Chiangmai, 50290, Thailand Abstract This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC: 15A09; 52A10; 74M05; 93D05 Keywords: robust stability and stabilization; nonlinear uncertain discrete-time systems; convex polytopic uncertainties; Lyapunov-Krasovskii functional; linear matrix inequality 1 Introduction Since the time delay is frequently viewed as a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, networked control systems, etc., the study of delay systems has received much attention and various topics have been discussed over the past years. The problem of stability and stabilization of dynamical systems with time delays has received considerable attention, and lots of interesting results have reported in the literature, see [–] and the references therein. Some delay-dependent stability criteria for discrete-time systems with time-varying delay are investigated in [, , –], where the discrete Lyapunov functional method are employed to prove stability conditions in terms of linear matrix inequalities (LMIs). A number research works for dealing with asymptotic stability problem for discrete systems with interval time-varying delays have been presented in [–]. Theoretically, stability analysis of the systems with time-varying delays is more complicated, especially for the case where the system matrices belong to some convex polytope. In this case, the parameter-dependent Lyapunov-Krasovskii functionals are constructed as the convex combination of a set of functions assures the robust stability of the nominal systems and the stability conditions must be solved upon a grid on the parameter space, which results in testing a finite number of LMIs [, , ]. To the best of the authors’ © 2012 Rajchakit and Rajchakit; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rajchakit and Rajchakit Advances in Difference Equations 2012, 2012:106 http://www.advancesindifferenceequations.com/content/2012/1/106 Page 2 of 14 knowledge, the stability for linear discrete-time systems with both time-varying delays and polytopic uncertainties has not been fully investigated. The articles [, ] propose sufficient conditions for robust stability of discrete and continuous polytopic systems without time delays. More recently, combining the ideas in [, ], improved conditions for D stability and D -stabilization of linear polytopic delay-difference equations with constant delays have been proposed in []. In this article, we consider polytopic nonlinear uncertain discrete-time equations with interval time-varying delays. Using the parameter-dependent Lyapunov-Krasovskii functional combined with LMI techniques, we propose new criteria for the robust stability of the nonlinear uncertain system. The delay-dependent stability conditions are formulated in terms of LMIs, being thus solvable by the numeric technology available in the literature to date. The result is applied to robust stabilization of nonlinear uncertain discrete-time control systems. Compared to other results, our result has its own advantages. First, it deals with the nonlinear uncertain delay-difference system, where the state-space data belong to the convex polytope of uncertainties and the rate of change of the state depends not only on the current state of the nonlinear systems but also its state at some times in the past. Second, the time-delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and the upper bounds for the time-varying delay are available. Third, our approach allows us to apply in robust stabilization of the nonlinear uncertain discrete-time system subjected to polytopic uncertainties and external controls. Therefore, our results are more general than the related previous results. The article is organized as follows. In Section , introduces the main notations, definitions, and some lemmas needed for the development of the main results. In Section , sufficient conditions are derived for robust stability, stabilization of nonlinear uncertain discrete-time systems with interval time-varying delays and polytopic uncertainties. They are followed by some remarks. Illustrative examples are given in Section . 2 Preliminaries The following notations will be used throughout this article. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product ·, · and the vector norm  · ; Rn×r denotes the space of all matrices of (n × r)-dimension. AT denotes the transpose of A; a matrix A is symmetric if A = AT , a matrix I is the identity matrix of appropriate dimension. Matrix A is semi-positive definite (A ≥ ) if Ax, x ≥ , for all x ∈ Rn ; A is positive definite (A > ) if Ax, x >  for all x = ; A ≥ B means A – B ≥ . Consider a nonlinear uncertain delay-difference systems with polytopic uncertainties of the form       x(k + ) = A(ξ ) + A(k) x(k) + D(ξ ) + D(k) x k – h(k)    + f k, x k – h(k) , k = , , , . . . , x(k) = vk , k = –h , –h + , . . . , , (ξ ) Rajchakit and Rajchakit Advances in Difference Equations 2012, 2012:106 http://www.advancesindifferenceequations.com/content/2012/1/106 Page 3 of 14 where x(k) ∈ Rn is the state, the system matrices are subjected to uncertainties and belong to the polytope  given by   = [A, D](ξ ) := p  ξi [Ai , Di ], i= p   ξi = , ξi ≥  , (.) i= where Ai , Di , i = , , . . . , p, are given constant matrices with appropriate dimensions. The nonlinear perturbations f (k, x(k – h(k))) satisfies the following condition           f T k, x k – h(k) f k, x k – h(k) ≤ β  xT k – h(k) x k – h(k) , (.) where (...truncated)