Systems & Control Letters 60 (2011) 871–876 Contents lists available at SciVerse ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle H ∞ filtering for nonuniformly sampled systems: A Markovian jump systems approach Ghulam Mustafa ∗ , Tongwen Chen Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada article info Article history: Received 26 April 2010 Received in revised form 10 July 2011 Accepted 26 July 2011 Keywords: Nonuniform sampling Markovian jump systems Stochastic stability H ∞ filtering abstract This paper is concerned with the design of an H ∞ filter for a sampled-data system, where the measurement is sampled nonuniformly and randomly but the input is updated uniformly at a fast rate. The process of nonuniform measurement sampling is modeled using a Markov chain; therefore, the discrete- time system together with the Markov chain is a Markovian jump system. A mode-dependent, full-order H ∞ filter is constructed. Sufficient LMI conditions are derived to ensure stochastic stability and H ∞ disturbance attenuation for the error system. The case of unknown measurement sampling probabilities is also considered. Finally, the effectiveness of the proposed approach is demonstrated through simulation results and a comparison with a general time-varying H ∞ filter. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The theory of sampled-data control systems has been well- developed during the last two decades. Many excellent references, e.g., [1–3], are available for the analysis and design of these systems. The conventional sampled-data control theory assumes a periodic operation of the sample-and-hold devices. However, there are many applications where the uniform sampling of measurement may not be a natural or feasible choice and, hence, neither the periodic operation of these devices. Examples of such applications include cross-directional control of paper machine systems [4], brushless DC servo systems [5], and networked and embedded control systems [6,7]. Therefore, it is important to investigate systems with nonuniform sampling [8–10]. Nonuniformly sampled systems arise when the measurement sampling and input updating periods are time-varying. We consider a class of nonuniformly sampled systems where the input updating period is fast and uniform, but the measurement sampling period is slow and nonuniform. The state estimation problem for this class of systems has been considered in [11,12,5]. In [11], an oversampled Kalman filter is presented. In [12,5], a Luenberger type variable sampling rate observer is designed. But, to the authors’ best knowledge, the H ∞ filter design for this class of discrete-time systems has not yet been considered. On the other hand, Markovian jump systems are an active area of research. These are systems with discrete or continuous ∗ Corresponding author. Tel.: +1 780 492 3368; fax: +1 780 492 1811. E-mail addresses: g.mustafa@ece.ualberta.ca (G. Mustafa), tchen@ece.ualberta.ca (T. Chen). dynamics and a Markov chain, governing the transitions between different system modes. The behavior of these systems is primarily determined by the transition probabilities of jumping processes. The H ∞ filtering problem for these systems has been formulated under many different situations, and with known and unknown transition probabilities, see for instance [13–15]. In this paper, a novel approach is used to model the process of nonuniform measurement sampling using a Markov chain. The resulting discrete-time system together with the Markov chain is a Markovian jump system. A mode-dependent, full-order H ∞ filter is designed. The case of unknown measurement sampling probabilities (or Markov chain’s transition probabilities) is also considered. It is noted that Markov chains have already been used, e.g., in networked control systems [16,17], to model random networked induced delays; however, their use to model the nonuniform sampling process is a new idea. The rest of this article is organized as follows: Section 2 presents the formulation of an H ∞ filtering problem for a nonuniformly sampled system. Section 3 describes the main results for the analysis and synthesis of the proposed filter. Simulation results are given in Section 4 to show the effectiveness of the proposed approach. Notation: In this paper R n denotes the n-dimensional Euclidean space and P > 0 (≥ 0) means P is a positive definite (semi- definite) matrix. In a block matrix, a ∗ represents the corresponding transposed block induced by symmetry; l 2 [0, ∞) represents the space of square summable infinite sequences with norm ‖.‖ 2 . Finally, E {.} represents the mathematical expectation. 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.07.005