Identifiability in the Behavioral Setting

The behavioral approach to system identification starts from a given time series without a priori separation of the variables into inputs and outputs. The available identifiability conditions however require persistency of excitation of an input component of the time series which implicitly assumes that an input/output partitioning of the variables is given. In addition, a standard identifiability assumption is that the true data generating system is controllable. The conditions of controllability and persistency of excitation are sufficient but not necessary. Motivated by the need to infer linear-time invariant models from rank deficient Hankel matrices and to use such matrices as data-driven predictors in signal processing and control, we derive necessary and sufficient identifiability conditions that do not require a priori given input/output partitioning of the variables nor controllability of the true system. The prior knowledge needed for identifiability is the number of inputs, lag, and order of the true system. The results are based on a modification of the notion of a most powerful unfalsified model for finite data and a novel algorithm for its computation. The results in the paper are derived assuming exact data, however, low-rank approximation allows their application in the case of noisy data. We compare empirically low-rank approximation of the Hankel, Page, and trajectory matrices in the errors-invariables setting. Although the Page and trajectory matrices are unstructured, the parameter estimates obtained from them are less accurate than the one obtained from the Hankel matrix.