Automatica 37 (2001) 91}97 Brief Paper Generalized Popov theory applied to state-delayed systems Vlad Ionescu, Silviu-Iulian Niculescu*, Jean-Michel Dion, Luc Dugard, Huaizhong Li Univ. **Politehnica++ Bucharest, Spl. Independentei 313, 77206 Bucharest, Romania HEUDIASYC (UMR CNRS 6599), Univ. de Technologie de Compie % gne, BP 20529, 60205 Compie % gne, Ced ! ex, France Laboratoire d'Automatique de Grenoble, ENSIEG, BP 46, 38402 Saint-Martin-d'He % res, France School of Engineering, Rockingham Campus, Murdoch University, Murdoch, WA 6150, Australia Received 28 April 1997; revised 19 January 2000; received in "nal form 29 May 2000 Abstract This paper deals with the generalized Popov theory applied to linear delay systems. Su$cient conditions for memoryless stabilization as well as for coerciveness of an appropriate quadratic cost are given in terms of algebraic properties of some matrix pencil. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Delay systems; Riccati theory; Matrix pencil; ¸-control 1. Introduction and problem statement Constructing memoryless controllers is one of the "rst way to handle control problems for linear continuous systems including delays in their models, (see, e.g. Ikeda & Ashida, 1979) due to their simplicity and possibility to use existing xnite-dimensional techniques for numerical treatment (Shen, Chen & Kung, 1991). Of recent interest is the stabilizing problem including supplementary per- formance requirements such as: -attenuation and/or robustness with respect to parameter uncertainties (Xie & de Souza, 1993). As the investigations in this area are incipient, memoryless controllers were under re- search attention prior to other compensation techniques. However, it is worth-while to be mentioned that in spite of the simplicity of the formulation of the corresponding problem, technical machineries as Riccati inequality and LMI techniques have been thoroughly involved, see Niculescu (1997) and the references therein. In this paper an attempt has been made in order to apply the generalized Popov theory (see Ionescu, Oara & Weiss, 1998) for achieving simultaneously both closed-loop stability and -attenuation for systems This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Logemann under the direction of Editor Roberto Tempo. * Corresponding author. E-mail address: niculescu@hds.utc.fr (S. -I. Niculescu). described in terms of linear delay di!erential equations. This paper provides an uni"ed approach allowing simple and nice interpretations of some control problems for delay systems in terms of Popov triplets and associated objects. An illustrative example is included. Although the obtained results are of delay-independent type, the pro- posed technique may be applied to derive delay-depen- dent conditions, as well as to time-varying or multiple delays. Consider now the following state-delayed generalized system: x (t)"Ax(t)#A x(t!)#B u (t)#B u (t), y (t)"C x(t)#C x(t!)#D u (t)#D u (t), (1) y (t)"C x(t)#C x(t!)#D u (t), where x(t)3R is the vector state, u (t)3R , u (t)3R are the disturbance and control inputs, y (t)3R , y (t)3R are the controlled and measured outputs, etc. We are interested in "nding memoryless controllers of the form u (t)"F x(t) (2) that simultaneously stabilizes (1) and achieves -at- tenuation property, i.e., ¹ ( where ¹ is the ¸-linear bounded input}output operator de"ned by the closed-loop con"guration obtained by coupling (2) to (1). When u is accessible for measurement, (2) can be 0005-1098/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 2 6 - 6