Published in IET Control Theory and Applications Received on 4th August 2008 Revised on 7th June 2009 doi: 10.1049/iet-cta.2008.0332 ISSN 1751-8644 Design of reduced-order l 2 – l 1 filter design for singular discrete-time systems using strict linear matrix inequalities M.S. Mahmoud 1 Y. Xia 2 1 Systems Engineering Department, King Fahd University of Petroleum and Minerals, P. O. Box 985, Dhahran 31261, Saudi Arabia 2 Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, People’s Republic of China E-mail: msmahmoud@kfupm.edu.sa Abstract: In this study, the problem of designing regular robust l 2 – l 1 filter for a class of linear singular discrete- time systems with norm-bounded uncertainties is investigated. A class of regular linear filters is fully analysed, then necessary and sufficient conditions of robust admissibility are established. Guaranteed l 2 – l 1 performance of the filtered system is provided and expressed in terms of the solution of strict linear matrix inequalities (LMIs). The cases of full- and zeroth-order realisable filter are derived as limiting cases of design. Numerical examples are worked out to illustrate the theoretical development. 1 Introduction The purpose of the filtering problem is to estimate the states (or a linear combination of the states) of a dynamical system using the past measurements. The celebrated Kalman filter [1] provides a recursive algorithm to minimise the variance of the state estimation error when the power spectral density of the process and measurement noise is known. During the last four decades, Kalman filtering techniques have found widespread applications in aerospace guidance, navigation and control problems [2–9]. When a priori information on the external noise is not precisely known, the Kalman filtering approach does not provide the optimal estimate in the sense of minimum error variance. In such cases, H 1 filtering was introduced [10, 11], in which the input signal is assumed to be energy bounded and the main objective is to minimise the energy of the estimation error for the worst possible bounded energy disturbance. On the other hand, the objective of l 2 – l 1 filtering problem is to minimise the peak value of the estimation error for all possible bounded energy disturbances. Hence, the l 2 – l 1 (energy-to-peak) filtering can be considered as a deterministic formulation of the Kalman filter [12, 13]. The class of robust filtering arose out of the desire to determine estimates of non-measurable state variables for dynamical systems with uncertain parameters. The past decade has witnessed major development in robust-filtering problem using various approaches [11, 14 – 16]. On another research front, dynamical system representation in singular-modelling format has been shown to provide convenient and natural framework in the characterisation of several applications including large-scale systems, power systems, economic systems, to name a few [17–19]. Robust stability and robust stabilisation problems of singular systems have been extensively studied in the past years, see [18, 20–27] and their references. These results have illuminated the fact that robust stability problem for singular systems is more involved than the counterpart in ordinary state-space systems. Research into robust filtering for singular systems and related results are available in [21, 28]. Most of these results focus on the use of singular filters to match the structure of the original dynamical systems and employ H 1 filtering method. It appears that the issue of realisability of the developed filters has been overlooked. In addition, the l 2 – l 1 (energy-to-peak) filtering criteria have received less attention. Moreover, the available filtering results are quite often expressed in terms of non-strict linear matrix inequalities (LMIs), which suffer from computational difficulties. IET Control Theory Appl., 2010, Vol. 4, Iss. 3, pp. 509–519 509 doi: 10.1049/iet-cta.2008.0332 & The Institution of Engineering and Technology 2010 www.ietdl.org