LMI Optimization Approach to Robust H ∞ Filtering for Discrete-Time Nonlinear Uncertain Systems Masoud Abbaszadeh and Horacio J. Marquez Abstract— A new approach for the design of robust H ∞ filter for a class of discrete-time Lipschitz nonlinear systems with time-varying uncertainties is proposed based on linear matrix inequalities. 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 optimization. The resulting H ∞ observer guarantees exponential stability of the estimation error dynamics with guaranteed decay rate and is robust against time-varying parametric uncertainties. The proposed observer has also an extra important feature, robustness against nonlinear additive uncertainty. Explicit norm-wise and element- wise bounds on the tolerable nonlinear uncertainty are derived. I. I NTRODUCTION I N many practical situations, it is not possible to obtain accurate measurements of all the system states making the usage of state observers essential. In addition, due to model uncertainties and disturbances, the observer often needs to have some robustness properties. The problem of nonlinear observer design for uncertain systems has been tackled using various approaches [1], [2], [3], [4], [5]. To deal with the exogenous disturbances, the H ∞ filtering was introduced. In an H ∞ observer, the L 2 gain from the unknown norm-bounded exogenous disturbance to the observer error is guaranteed to be less than a prespecified value. The original studies in this area go back to the works of de Souza et. al. where the authors considered a class of continuous-time nonlinear systems with time-varying parametric uncertainty and obtained Riccati-based sufficient conditions for the stability of the proposed observer with guaranteed disturbance attenuation level, [1], [6]. These references also present general matrix inequalities helpful in solving this type of problems. In the discrete-time domain, Xie et. al. proposed a Riccati equation approach to the robust H ∞ observer design [2]. The class of discrete- time systems considered was described by a linear state space model with the addition of known state dependent 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 nonlinearly satisfying a global Lipschitz condition. In order to guarantee the robust stability of the observer in the presence of parameter uncertainties, the authors added the restrictive assumption that the “A” matrix of the linear part must be non-singular. Wang and Unbenhauen considered the robust observer design problem for the same class of discrete systems [7]. They eliminated the aforementioned restrictive assumption. However, the observer structure proposed in [7] involves parameter uncertainties, making the design of such an observer difficult in practical applications. A second shortcoming in the observer of reference [7] is that no disturbance attenuation (H ∞ performance) is guaranteed. In addition, in the Riccati approach, all the H ∞ regularity assumptions must be satisfied. The regularity assumptions in the Riccati approach can be relaxed using LMIs. An LMI solution for robust H ∞ filtering has been proposed for a class of Lipschitz nonlinear systems in which the Lipschitz constant is fixed and predetermined, [8]. The resulting observer is robust against time-varying parametric uncertainties in the linear part of the model with the guaranteed disturbance attenuation level. Recently, we have developed a new LMI optimization approach to the solution of this problem in the continues-time domain [9], [10]. In our method, the linear matrix inequalities are linear in the system Lipschitz constant making it one of the LMI variables. Therefore, the admissible Lipschitz constant can be the maximized through convex optimization. This optimization adds an important extra future to the H ∞ filter over the aforementioned features, making the proposed observer robust against some nonlinear uncertainty. In this paper, we extend the results to the discrete-time case. The discrete-time case of this problem has the merits to be studied independently since most modern control systems are implemented digitally. Besides, due to the structure of the Lyapunov difference, the LMI formulation of the solution in the discrete-time domain is more complicated. The proposed H ∞ filter is robust against time-varying parametric uncertainties as well as additive nonlinear uncertainty with the guaranteed disturbance attenuation level. We derive norm-wise and element-wise bounds on the tolerable nonlinear uncertainty. Thanks to the linearity of our proposed LMIs in both the admissible Lipschitz constant and the disturbance attenuation 2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008 ThAI01.5 978-1-4244-2079-7/08/$25.00 ©2008 AACC. 1905