Automatica 41 (2005) 339 – 344 www.elsevier.com/locate/automatica Technical communique A note on asymptotic stabilization of linear systems by periodic, piecewise constant, output feedback  J.C. Allwright, A. Astolfi ∗ , H.P. Wong Electrical and Electronic Engineering Department, Imperial College London, Exhibition Road, London SW7 2BT, UK Received 19 June 2003; received in revised form 7 September 2004; accepted 15 September 2004 Abstract This note studies the asymptotic stabilization problem for controllable and observable, single-input single-output, linear, time-invariant, continuous-time systems by means of memoryless output feedback of the form u(t) = k(t)y(t), with k(t) periodic and piecewise constant. A necessary and sufficient condition, together with a simpler to test sufficient condition, given in terms of a bilinear matrix inequality, is presented. A few illustrative examples complete the paper.  2004 Elsevier Ltd. All rights reserved. Keywords: Periodic output feedback; Output feedback stabilization; Linear systems 1. Introduction Consider a controllable and observable, single-input single-output, linear, time-invariant, continuous-time sys- tem described by equations of the form ˙ x(t) = Fx(t) + gu(t), y(t) = hx(t), (1) with state x(t) ∈ R n , output y(t) ∈ R and input u(t) ∈ R. If the uncontrolled system ˙ x(t) =Fx(t) is not asymptotically stable, then it is natural to address the feedback stabilization problem. Classically, this problem has been addressed in two ways. If the state of the system is measurable then, by controllability, there exists a static state feedback control law asymptotically stabilizing the system. If only the output is available for feedback, then the problem can be studied from several points of view. The simplest approach is to select a static output feed- back control law, i.e. a control law described by equations  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Middleton under the direction of Editor P. Van den Hof. ∗ Corresponding author. E-mail addresses: j.allwright@ic.ac.uk (J.C. Allwright), a.astolfi@ic.ac.uk (A. Astolfi), macross1031@yahoo.co.uk (H.P. Wong). 0005-1098/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.09.009 of the form u(t) = ky(t). However, it is well known, see Syrmos, Abdallah, Dorato, and Grigoriadis (1997) for de- tail, that such an approach is in general inadequate, i.e. the set of systems which are stabilizable by static out- put feedback is non-generic. A second possible approach is to use a classical observer based design, which is fea- sible by controllability and observability of the system. Alternatively, one can use generalized sampled-data out- put feedback with a suitable sampling period. It has been shown that the use of such control laws enables one to deal with problems otherwise unsolvable with time- invariant output feedback controllers, such as pole as- signment, simultaneous stabilization of a finite number of plants and gain margin improvement, see Kabamba (1987). However, such control laws cannot be considered as static output feedback controllers: indeed they are dynami- cal systems. The study of benefits and inherent limitations of generalized sampled-data control systems are on their way and some aspects are still to be understood (Feuer & Goodwin, 1994). Finally, it has been shown in Moreau and Aeyels (1999) that the use of time-varying memory- less output feedback provides a simple stabilization tool which possesses more flexibility than static output feed- back: there are systems which are not stabilizable by static output feedback, but which are stabilizable by time-varying memoryless output feedback.