Exact modelling and identifiability of linear systems

In this paper we consider the question of identifiability of a system on the basis of observed data. In particular we present a framework for exact modelling and identifiability of finite dimensional, linear, time invariant systems, on the basis of time series of infinite or finite length. Systems are characterized in terms of their behaviour, i.e. the set of time series which are allowed by the system. In this behavioural approach, no a priori distinction is made between input and output variables, and also non-controllable systems can be considered. We describe identification procedures for exact modelling of multivariable time series of infinite or finite length. These procedures are based on the concept of corroboration of models by data, reflecting that the data should incorporate evidence for the acceptability of the identified models. The assumed a priori information on the data generating system consists of the qualitative behavioural properties of linearity, time invariance, and completeness. No knowledge is required concerning quantitative properties like the number of input and output variables, the number of state variables, or the observability indices. A complete characterization is obtained of the minimal number of time series required for system identifiability. The class of systems identifiable by one time series contains the controllable systems as a strict subclass.

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