Impulsive systems and behaviors in the theory of linear dynamical systems

Abstract

A linear dynamical system resulting from the interconnection of subsystems is considered. Assuming that this interconnection is ‘‘temporal’’, i.e. starting at a given initial time in the continuous-time case and ending at a given final time in the discrete-time case, such a system is also said to be ‘‘temporal’’. Temporal interconnections generate ‘‘uncontrollable impulsive behaviors’’ which are not found in the classical theory, though they have been studied for more than 20 years in the case of systems with constant coefficients. Determining the structure of the impulsive behavior of a temporal system is a key problem in the theory of linear dynamical systems. It is addressed here, using module theory, for systems with time-varying coefficients, in both the continuous- and discrete-time cases. These two cases are merged into a general framework. The impulsive behavior of a temporal system satisfying a suitable regularity condition has a structure which is fully elucidated. It turns out that the determination of this structure in practice is an algebraic—not an analytic—problem, which makes the calculations simpler and easier to computerize. The theory is illustrated through several examples.