Robust Static Output Feedback Stabilization of Discrete-Time Nonlinear Uncertain Systems with H ∞ Performance Masoud Abbaszadeh and Horacio J. Marquez Abstract— A new approach for the design of robust static output feedback controller for a class of discrete-time Lipschitz nonlinear systems with time-varying uncertainties is proposed based on linear matrix inequalities. The controller has also a guaranteed disturbance attenuation level (H∞ performance). Thanks to the linearity of the proposed LMIs in both the admissible Lipschitz constant of the system and the disturbance attenuation level, they can be simultaneously optimized through convex multiobjective optimization. The optimization over Lip- schitz constant adds an extra important and new feature to the controller, robustness against nonlinear uncertainty. The resulting controller is robust against both nonlinear additive uncertainty and time-varying parametric uncertainties. Explicit norm-wise and element-wise bounds on the tolerable nonlinear uncertainty are derived. I. I NTRODUCTION T HE static output feedback (SOF) stabilization problem is known to be a challenging task and in spite of receiving great attention it is still one of the most important open questions in the control theory. Over the past decays most advances in the control theory have been limited to the state feedback control and dynamic output feedback control. However, state feedback techniques, require either the measurement of every system state some of which expensive or even impossible to be measured or using the observer based controllers which makes the implementation task expensive and hard. On the other hand, dynamic output feedback designs, result in high order controllers which may not be desirable in many industrial applications again due to implementation difficulties. Controllers using static output feedback are less expensive to implement and are more reliable. Therefore, many researchers have tried to charac- terize the problem of finding a stabilizing SOF controller. A comprehensive survey on static output feedback is given in [1]. Despite abundant literature, due to the non-convexity of the SOF formulation, the problem is still open both an- alytically and numerically. While the necessary and suf- ficient conditions of the existence of a stabilizing SOF M. Abbaszadeh is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2V4, e-mail: masoud@ece.ualberta.ca H. J. Marquez is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2V4, The corresponding author, Phone: +1-780-492-3333, Fax:+1-780-492-8506, e-mail: marquez@ece.ualberta.ca controller based on the original non-convex formulation are not numerically tractable, the existing convex formulations often lead to restrictive sufficient conditions. The proposed results include methods based on structural pole assignment [2], [3], Riccati-based approaches [4], [5] and optimization formulation (min-max problem) [6]. The original non-convex problem can be directly formulated using bilinear matrix inequalities (BMIs). However, the numerical solution of BMI problems has been shown to be NP-hard [7]. Due to recent advances in linear matrix inequalities both theoretically and numerically, several works have been recently addressed in the literature attempting to cast the SOF problem into the convex LMI framework. Available methods include using ILMIs (iterative LMIs) [8], [9] where there is no guarantee for the convergence of iterations or imposing nonsingularity conditions over the state space realization matrices [10] or a submatrix of “A” [11]. Alternatively, is has been shown that in some cases the BMI problem can be converted into a semidefinite cone programming problem (SDP) [12]. The advantage gained through this conversion is that reliable algorithms exist to solve SDP problems numerically. In this work, we propose a novel method for robust static output feedback stabilization of a class of discrete-time nonlinear uncertain systems. The method proposed in this paper is non-iterative and not only provides a less restrictive solution but also extends the results to a more general class of systems where there are parametric uncertainties and Lipschitz nonlinearity in the model. In addition the proposed controller has a guaranteed disturbance attenuation level (H ∞ performance). Our goal is to develop linear matrix inequalities in which the Lipschitz constant is one the LMI variables in order to achieve the maximum admissible Lipschitz constant through convex op- timization. This optimization adds an important extra feature to the SOF controller making it robust against nonlinear uncertainties. Explicit bounds on the tolerable nonlinear uncertainty are derived through norm-wise and element-wise robustness analysis. Actually, thanks to the linearity of the proposed LMIs in both the admissible Lipschitz constant of the system and the disturbance attenuation level, they can be simultaneously optimized through convex multiobjective optimization. In the next stage, we show that the proposed solution can be modified as an SDP problem. In fact original BMI problem is converted into an SDP problem. The paper 16th Mediterranean Conference on Control and Automation Congress Centre, Ajaccio, France June 25-27, 2008 978-1-4244-2505-1/08/$20.00 ©2008 IEEE 226