Generic properties and control of linear structured systems

In this paper we deal with generic properties and control of linear systems taking into account prior structural knowledge available from physical considerations. 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 fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed, most 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 specific properties like controllability, the finite and infinite 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 a wonderful modeling and analysis tool. We emphasize the fact that this modeling 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 axe easy to check by hand for small systems, and by means of well-known combinatorial techniques for larger systems. Copyright (C) 2001 IFAC.