Compressive topology identification of interconnected dynamic systems via Clustered Orthogonal Matching Pursuit

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In this paper, we consider topology identification of large-scale interconnected dynamical systems. The system topology under study has the structure of a directed graph. Each edge of the directed network graph represents a Finite Impulse Response (FIR) filter with a possible transport delay. Each node is a summer, whose inputs are the signals from the incoming edges, while the output of the summer is sent to outgoing edges. Edges of the graph can be of different unknown orders and delays. Both the graph topology and the FIR filters and delays that make up the edges are unknown. We aim to do the topology identification from the smallest possible number of node observations when there is limited data available and for this reason, we call this problem Compressive Topology Identification (CTI). Inspired by Compressive Sensing (CS) which is a recent paradigm in signal processing for sparse signal recovery, we show that in cases where network interconnections are suitably sparse (i.e., the network contains sufficiently few links), it is possible to perfectly identify the network topology along with the filter orders and delays from small numbers of node observations, even though this leaves an apparently ill-conditioned identification problem. The main technical novelty of our approach is in casting the identification problem as the recovery of a clustered-sparse signal z ∈ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> from the measurements b = Az ∈ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sup> with M <; N, where the measurement matrix A is a block-concatenation of Toeplitz matrices. To this end, we introduce a greedy algorithm called Clustered Orthogonal Matching Pursuit (COMP) that tackles the problem of recovering clustered-sparse signals from few measurements. In a clustered-sparse model, in contrast to block-sparse models, there is no prior knowledge of the locations or the sizes of the clusters. We discuss the COMP algorithm and support our discussions with simulations.