An implicit systems characterization of a class of impulsive linear switched control processes. Part 1: Modeling

Our paper focuses on a fundamental structural property associated with a family of linear switched control systems in the presence of impulsive dynamics. We consider dynamic processes governed by piecewise linear ODEs with controlled location transitions. Using the newly developed implicit systems characterization technique, we rewrite the initially given impulsive switched system as an implicit dynamic model. This auxiliary representation of the original control system makes it possible to hide the switched phenomenon and to apply the conventional approach to the resulting implicit dynamic model. The proposed algebraic-based modeling framework follows the celebrated behavioral approach (see Polderman and Willems, 1998) and makes it possible to apply to the switched dynamics some classic techniques from the well-established time-invariant implicit systems theory. The analytic results of our paper constitute a formal theoretical extension of switched control systems methodology and can be used (as an auxiliary step) in a concrete control design procedure. The second part of our manuscript is devoted to control aspects.