On the elimination of latent variables in L2 behaviors

Abstract-This paper considers the problem to eliminate latent variables from models in the class of linear shift-invariant L2 systems. Models in this class are assumed to relate manifest and latent variables by means of rational operators. The question is addressed when the induced manifest behavior of such a model again admits a representation as the L2 kernel of a rational operator. Necessary and sufficient conditions for eliminability in this class are given and are compared with earlier obtained results for classical C ∞ behaviors. We also provide an explicit state space algorithm for the construction of the induced manifest behavior, which is a result from the obtained relation between elimination of variables and disturbance decoupling problems. I. INTRODUCTION This paper deals with the question to completely eliminate latent variables from a model description in which manifest and latent variables are related. For general models, manifest variables are thought of as distinguished variables that are relevant for the purpose of the model, whereas latent variables are auxiliary variables that serve to represent the model. Models derived from first principles are usually represented in terms of equations that relate both manifest and latent variables. The partial or complete elimination of latent variables from a general model representation that relates manifest and latent variables is of evident interest from a general modeling point of view. It amounts to characterizing and removing the redundancy in the latent variables of the model representation. We believe that the behavioral approach is, actually, the most natural framework for studying this question. This means that we view systems as sets of trajectories that evolve over time. Earlier work on the elimination problem in continuous time and infinitely smooth linear systems has been studied i