Realization theory for linear hybrid systems, part I: Existence of realization

The paper is the first part of a series of papers which deal with realization theory for linear hybrid systems. Linear hybrid systems are hybrid systems in continuous-time without guards whose continuous dynamics is determined by linear control systems and whose the discrete dynamics is determined by a finite state automaton. In Part I of the current series of papers we will formulate necessary and sufficient conditions for the existence of a linear hybrid system realizing a specified set of input-output maps. We will also sketch a realization algorithm for computing a linear hybrid system from the input-output data. In Part II we will present conditions for observability and span-reachability of linear hybrid systems and we will show that minimality is equivalent to observability and span-reachability; we will also discuss algorithms for checking observability and span-reachability and for transforming a linear hybrid system to a minimal one.