Available online at www.sciencedirect.com Automatica 39 (2003) 1125–1144 www.elsevier.com/locate/automatica Survey Paper Generic properties and control of linear structured systems: a survey  Jean-Michel Dion a , Christian Commault a ; * , Jacob van der Woude b a Laboratoire d’Automatique de Grenoble (UMR 5528-CNRS), ENSIEG, INPG, BP 46, 38402 Saint-Martin-d’Heres, France b Delft University of Technology, Faculty ITS, Mekelweg 4, 2628 CD Delft, The Netherlands Received 19 March 2001; received in revised form 15 July 2002; accepted 10 March 2003 Abstract In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A;B;C and D, is either a xed zero or a free parameter. The location of the xed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specic properties like controllability, the nite and innite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input–output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Linear systems; Structured systems; Graph theory; Genericity 1. Introduction Since the last World War a huge amount of literature has been dedicated to the theory of linear systems. Such sys- tems have been studied thanks to various approaches based on for instance state space models, transfer matrices, matrix pencils, polynomial factorizations, and so on (cf. Kailath, 1980; Rosenbrock, 1970; Wonham, 1985). These studies may be roughly divided into two parts. On the one hand, there are studies concerning system-specic properties, which often lead to the search for invariants of systems under some given transformations. In these studies, the  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Manfred Morari. * Corresponding author. E-mail addresses: jean-michel.dion@inpg.fr (J.-M. Dion), christian.commault@lag.ensieg.inpg.fr (C. Commault), j.w.vanderwoude @its.tudelft.nl (J. van der Woude). notion of zero (nite or innite) plays a fundamental role. On the other hand, there are studies which try to determine conditions under which the systems will behave in a speci- ed way when using some type of control. It turns out that both types of studies are intimately related, and in partic- ular, the solvability conditions for several classical control problems can be stated in terms of invariants of systems. Both types of studies have been performed starting from a specied system with given values for the parameters. In practice however, we are often faced with the fol- lowing situation when trying to model a physical system. The system may contain xed parameters representing the specic role that certain variables in the system play. This may for instance happen if the system is a composition of subsystems, like in a series connection. Another reason for xed parameters to show up are xed algebraic relations be- tween variables, like for instance one state variable being the derivative of another state variable. Finally, the absence of relations between variables gives rise to xed zero entries. 0005-1098/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00104-3