Disturbance decoupling problem for a class of descriptor systems with delay via systems over rings

In recent years, a considerable amount of research has been devoted to linear time-invariant descriptor systems (called also singular or differential–algebraic) and to delay differential systems because of their extensive applications. The presence of time delays, in fact, arises in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. and may cause undesirable system transient response or even instability. Several results in literature are devoted to stability issues for descriptor systems with delays under the hypothesis that the considered system is ‘impulse free’, namely that the solution does not possess dynamical infinite modes nor impulsive behaviour (see, for instance, Liu & Xie, 1996; Fridman, 2002; Xu et al., 2002, 2005; Ma & Cheng, 2005). In this paper, under the same hypotheses, we will investigate the problem of finding a feedback law such that disturbances do not affect the output. The approach we follow here relies on the possibility of using mathematical models with coefficients in a suitable ring, or systems over a ring, for studying and analysing delay differential dynamical systems (see Conte & Perdon, 1982; Kamen, 1991 and references therein). The use of systems over rings is motivated by the fact that it avoids the necessity of dealing with infinite-dimensional vector spaces, like the state space of classical delay differential systems, and, in place, it allows the use of finite-dimensional modules. Although ring and module algebra is more rich and complicated than linear algebra, this makes possible to extend a number of techniques from the framework of linear dynamical systems, in particular the geometric approach. The geometric approach for the analysis and synthesis of linear systems with coefficients in a field (see Basile & Marro, 1969, 1992; Wonham, 1985) proved very effective in the solution of decoupling problems. The disturbance decoupling problem (DDP) for linear time-invariant singular systems (without delay) over a field was first formulated and solved by Fletcher & Aasaraai (1989) and, more recently, in Wang et al. (2004) by different methods. A geometric theory for dynamical systems over a ring,