Addition and intersection of linear time-invariant behaviors

Abstract

We define and analyze the operations of addition and intersection of linear time-invariant systems in the behavioral setting, where systems are viewed as sets of trajectories rather than input–output maps. The classical definition of addition of input–output systems is addition of the outputs with the inputs being equal. In the behavioral setting, addition of systems is defined as addition of all variables. Intersection of linear time-invariant systems was considered before only for the autonomous case in the context of “common dynamics” estimation. We generalize the notion of common dynamics to open systems (systems with inputs) as intersection of behaviors. This is done by proposing trajectory-based definitions. The main results of the paper are (1) characterization of the link between the complexities (number of inputs and order) of the sum and intersection systems, (2) algorithms for computing their kernel and image representations and (3) a duality property of the two operations. Our approach combines polynomial and numerical linear algebra computations.