Compressive system identification of LTI and LTV ARX models: The limited data set case

In this paper, we consider identifying Auto Regressive with eXternal input (ARX) models for both Linear Time-Invariant (LTI) and Linear Time-Variant (LTV) systems. We aim at doing the identification from the smallest possible number of observations. This is inspired by the field of Compressive Sensing (CS), and for this reason, we call this problem Compressive System Identification (CSI). In the case of LTI ARX systems, a system with a large number of inputs and unknown input delays on each channel can require a model structure with a large number of parameters, unless input delay estimation is performed. Since the complexity of input delay estimation increases exponentially in the number of inputs, this can be difficult for high dimensional systems. We show that in cases where the LTI system has possibly many inputs with different unknown delays, simultaneous ARX identification and input delay estimation is possible from few observations, even though this leaves an apparently ill-conditioned identification problem. We discuss identification guarantees and support our proposed method with simulations. We also consider identifying LTV ARX models. In particular, we consider systems with parameters that change only at a few time instants in a piecewise-constant manner where neither the change moments nor the number of changes is known a priori. The main technical novelty of our approach is in casting the identification problem as recovery of a block-sparse signal from an underdetermined set of linear equations. We suggest a random sampling approach for LTV identification, address the issue of identifiability and again support our approach with illustrative simulations.