Published in IET Control Theory and Applications Received on 8th May 2010 Revised on 20th July 2010 doi: 10.1049/iet-cta.2010.0249 ISSN 1751-8644 Quantised feedback stabilisation of interconnected discrete-delay systems M.S. Mahmoud 1 A.Y. Al-Rayyah 2 Y. Xia 3 1 Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 5067, Dhahran 31261, Saudi Arabia 2 Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 3 Department of Automatic Control, Beijing Institute of Technology, Beijing 100081, People’s Republic of China E-mail: msmahmoud@kfupm.edu.sa Abstract: The authors investigate in this study the problem of designing decentralised quantised H 1 feedback control for a class of linear interconnected discrete-time systems. The system has unknown-but-bounded couplings and interval delays. The quantiser has an arbitrary form that satisfies a quadratic inequality constraint. A linear matrix inequality (LMI)-based method using a decentralised quantised output-feedback controller is designed at the subsystem level to render the closed-loop system delay dependent asymptotically stable with guaranteed g-level. To show the generality of the authors’ approach, it is established that several cases of interest are readily derived as special cases. The authors illustrate the theoretical developments by numerical simulations. 1 Introduction In conventional feedback control theory, most data and/or signals are processed in a direct manner. With the emerging control systems including networks, all signals are transferred through network and this eventually gives rise to packet dropouts or data transfer rate limitations [1]. On the other hand, signal processing and signal quantisation always exist in computer-based control systems [2] and therefore recent research studies have been reported on the analysis and design problems for control systems involving various quantisation methods, see [3–7] and the references cited therein. In [3], a quantiser taking value in a finite set is defined and then quantised feedback stabilisation for linear systems is considered. In [4], the problem of stabilising an unstable linear system by means of quantised state feedback, where the quantiser assumes value in a countable set is addressed. It should be noted that the approach in [3] relies on the possibility of making discrete online adjustments of quantiser parameters, which was extended in [6] for more general non-linear systems with general types of quantisers involving the states of the system, the measured outputs and the control inputs. Recently in [5], a study of quantised and delayed state-feedback control systems under constant bounds on the quantisation error and the time-varying delay was reported. Based on [6], stabilisation of discrete-time linear time-invariant (LTI) systems with quantised measurement outputs is reported in [7]. Further related results are reported in [8–12]. On another research front, decentralised stability and feedback stabilisation of interconnected systems have been the topic of recurring interests and recent relevant results have been reported in [13 – 18]. In this paper, we investigate a generalised approach to quantised feedback control in a linear discrete-time system. We cast the problem under consideration as the problem of designing a decentralised H 1 feedback control for a class of linear interconnected discrete-time systems with quantised signals in the subsystem control channel. The system has unknown-but-bounded couplings and interval time delays. Within our formulation, we take the quantiser of arbitrary form that satisfies a quadratic inequality constraint in the state and the delayed state. We illustrated the generality of this quantiser structure. Based on quantised output measurements, a decentralised quantised output-feedback controller is designed at the subsystem level to render the overall closed-loop system delay dependent asymptotically stable with guaranteed g-level. To further illustrate the generality of the developed approach, it is established that several classes of quantised feedback control systems of interest are readily derived as special cases. These include the classes of interconnected time- delay and delay-free systems, single time-delay systems and single systems. Finally, we provide numerical simulations to illustrate the theoretical developments. Notations: In the sequel, the Euclidean norm | . | is used for vectors in the n-dimensional vector space < n and we denote by ‖ . ‖ the corresponding induced matrix norm in < n×n . We use W t and W −1 to denote the transpose and the inverse of any square matrix W, respectively. We use W . 0(≤0) to denote a symmetric positive-definite (negative-semi-definite) matrix W and I j to denote the n j × n j identity matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. In IET Control Theory Appl., 2011, Vol. 5, Iss. 6, pp. 795–802 795 doi: 10.1049/iet-cta.2010.0249 & The Institution of Engineering and Technology 2011 www.ietdl.org