On embeddings and inverse embeddings of input design for regularized system identification

Abstract

Input design is an important problem for system identification and has been well studied for the classical system identification, i.e., the maximum likelihood/prediction error method. For the emerging regularized system identification, the study on input design has just started, and it is often formulated as a non-convex optimization problem minimizing a scalar measure of the Bayesian mean squared error matrix subject to certain constraints. Among the state-of-art input design techniques for regularized system identification is the so-called quadratic mapping and inverse embedding (QMIE) method. Based on the quadratic mapping between the input and its covariance, the QMIE method is first to obtain the optimal autocovariance by solving a transformed convex optimization problem and then to find all the inputs corresponding to the optimal autocovariance by the time domain inverse embedding (TDIE). In this paper, we report some new results on the embeddings/inverse embeddings of the QMIE method. Firstly, we present a general result on the frequency domain inverse embedding (FDIE) that is to find the inverse of the quadratic mapping described by the discrete-time Fourier transform. Then we show the relation between the TDIE and the FDIE from a graph signal processing perspective. Finally, motivated by this perspective, we further propose a graph induced embedding and its inverse, which include the previously introduced embeddings as special cases. This deepens our understanding of input design from a broader perspective beyond the time domain and frequency domain viewpoints.