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Empirical Detection of Time Scales in LTI Systems using Sparse Optimization Techniques

Identification and control of systems with multiple time scales is in general known to be quite challenging than that of single scale systems. The foremost and crucial step in the development of any method for data-driven multiscale analysis is to determine whether the system warrants a multiscale treatment or otherwise. Existing works in the associated literature either assume that this non-trivial knowledge is readily available or do not necessarily assess the need for a multiscale analysis. This letter is aimed at addressing this fundamental issue. The main objectives of this letter are to develop a method for: 1) detecting the presence and 2) counting the number of time scales for linear time-invariant systems solely from input-output data. The problem is formulated in the sparse optimization framework with a non-orthogonal dictionary consisting of filtered inputs resulting from finite-order Laguerre basis filter expansions of transfer functions over a highly redundant pole grid. A clustering of the most informative poles selected by the sparse optimizer reveals the presence of multiscale nature and aids in the cardinality of time scales. The proposed method can be applied to multi-input, multi-output systems under open-loop conditions. Simulation studies demonstrate the success of the proposed methodology.