Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 10 (2003) 647-662 Copyright c 2003 Watam Press ROBUST H ∞ FILTERING FOR A CLASS OF LINEAR JUMPING DISCRETE-TIME DELAY SYSTEMS M. S. Mahmoud 1 , P. Shi 2 and A. Ismail 1 1 Electrical Engineering Department UAE University, Al-Ain, PO Pox 17555, United Arab Emirates 2 Land Operations Division Defense Science and Technology Organization PO Box 1500, Salisbury SA 5108,Australia Abstract. In this paper, we examine the robust H∞ filtering problem for a class of linear, uncertain discrete delay systems with Markovian jump parameters. The uncertainties are time-varying and norm-bounded parametric uncertainties and the delay factor is arbitrary constant. We provide initially a robust stochastic stability result with a prescribed perfor- mance measure. Then we design a linear causal filter using algebraic inequality procedure which ensures a prescribed disturbance attenuation level from the disturbance signal to the filtering error. Keywords. Discrete delay systems, Markovian jump parameters, Robust H ∞ filtering, Norm-bounded parametric uncertainties, Linear causal filter. 1 Introduction Filtering is perhaps one of the oldest problems studied in systems theory [1]. In recent years, robust filtering arose out of the desire to determine estimates of unmeasurable state variables for dynamical systems with uncertain param- eters. From this perspective, robust filtering can be viewed as an extension of the celebrated Kalman filter [1] to uncertain dynamical systems. The past decade has witnessed major developments in robust filtering problem using various approaches [2,5-7,10,11,16,22]. Of particular interest to our work is the H ∞ filtering in which the design is based on minimizing the H ∞ -norm of the system. This design reflects a worst-case gain of the transfer function from the disturbance inputs to the estimation error output. In addition, H ∞ filtering is superior to standard H 2 filtering since no statistical assumption is made on the input signals. On another front of research, state-space modeling of industrial and engineer- ing systems frequently encounter delay effects in processing state, input or related variables. In connection with system measurements and/or informa- tion flow amongst different parts of dynamical systems, time-delay arise quite naturally [9,12,21]. Thus, the class of dynamical systems with time-delay has