INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control (2009) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1507 A BMI approach for H ∞ gain scheduling of discrete time-varying systems Renato A. Borges 1,2 , Ricardo C. L. F. Oliveira 1 , Chaouki T. Abdallah 2 and Pedro L. D. Peres 1, ∗, † 1 School of Electrical and Computer Engineering, University of Campinas—UNICAMP, 13083-970, Campinas, SP, Brazil 2 Electrical and Computer Engineering Department, MSC01 1100 1 University of New Mexico, Albuquerque, NM 87131-0001, U.S.A. SUMMARY The problem of gain-scheduled state feedback control for discrete-time linear systems with time-varying parameters is considered in this paper. The time-varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time-delayed inputs, allowing a simplified closed-loop analysis by means of parameter-dependent Lyapunov functions. A gain-scheduled state feedback controller that minimizes an upper bound to the H ∞ performance of the closed-loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler’s lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi-definite programming algorithm to solve the problem. Robust controllers for time-invariant uncertain parameters, as well as gain-scheduled controllers for arbitrarily time-varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed-loop H ∞ performance. Copyright 2009 John Wiley & Sons, Ltd. Received 3 September 2008; Revised 18 May 2009; Accepted 6 July 2009 KEY WORDS: discrete-time systems; time-varying parameters with bounded rates; gain scheduling; parameter-dependent Lyapunov functions; linear matrix inequalities 1. INTRODUCTION One cannot deny the fact that gain scheduling has become an important topic within control system theory [1, 2]. As shown in [3], this technique can extend the validity of the linearization approach of non-linear ∗ Correspondence to: Pedro L. D. Peres, School of Electrical and Computer Engineering, University of Campinas—UNICAMP, 13083-970, Campinas, SP, Brazil. † E-mail: peres@dt.fee.unicamp.br Contract/grant sponsor: Brazilian agencies FAPESP, CAPES and CNPq systems to a range of operating points. Consequently, gain-scheduled controllers are guaranteed to work in a bigger region instead of only in a certain neighbor- hood of a single operating point. The main idea is to model the system in such a way that the different oper- ating points are parametrized by one or more variables, commonly called scheduling variables [3]. The stability is then assured by a closed-loop Lyapunov function and a family of linear controllers, whose parameters are changed in accordance with the scheduling rules. Although there are other articles addressing the topic of gain scheduling [4–6] can be considered as pioneering works. The use of linear parameter varying (LPV) struc- tures to model certain classes of non-linear systems has Copyright 2009 John Wiley & Sons, Ltd.