Subspace identification of 1D spatially-varying systems using Sequentially Semi-Separable matrices

We consider the problem of identifying 1D spatially-varying systems that exhibit no temporal dynamics. The spatial dynamics are modeled via a mixed-causal, anti-causal state space model. The methodology is developed for identifying the input-output map of e.g a 1D flexible beam described by the Euler-Bernoulli beam equation and equipped with a large number of actuators and sensors. It is shown that the static input-output map between the lifted inputs and outputs possess a so-called Sequentially Semi-Separable (SSS) matrix structure. This structure is of key importance to derive algorithms with linear computational complexity for controller synthesis of large-scale systems. A nuclear norm subspace identification method of the N2SID class is developed for estimating these state space models from input-output data. To enable the method to deal with a large number of repeated experiments a dedicated Alternating Direction Method of Multipliers (ADMM) algorithm is derived. It is shown in this paper that a nuclear norm relaxation on the SSS structure can be imposed which improves the estimates of the system matrices.