Matrix output extension of the tensor network Kalman filter with an application in MIMO Volterra system identification

Abstract

This article extends the tensor network Kalman filter to matrix outputs with an application in recursive identification of discrete-time nonlinear multiple-input-multiple-output (MIMO) Volterra systems. This extension completely supersedes previous work, where only l scalar outputs were considered. The Kalman tensor equations are modified to accommodate for matrix outputs and their implementation using tensor networks is discussed. The MIMO Volterra system identification application requires the conversion of the output model matrix with a row-wise Kronecker product structure into its corresponding tensor network, for which we propose an efficient algorithm. Numerical experiments demonstrate both the efficacy of the proposed matrix conversion algorithm and the improved convergence of the Volterra kernel estimates when using matrix outputs.

Tensor Network Kalman Filter for LTI Systems

“…It adapts the Kalman filter equations to the required multilinear algebra. The TN Kalman filter is introduced in [2] and [3].…”

“…For completeness, the definition of the remaining matrices and vectors are given as S ∈ R p×p , R ∈ R p×p and v ∈ R p . The specific implementation for the SISO and MIMO filter is elaborated in [2] and [3] respectively.…”

“…One possibility for a large scale Kalman filter is the Tensor Network (TN) Kalman filter as developed in [2], [3]. Both concern the system identification of Multiple-Input Multiple-Output (MIMO) Volterra systems using Kalman filter where the latter one uses a batch of multiple measurements.…”

“…The Volterra systems are rewritten in LTI system form and therefore the described implementation can equally be used for dynamic linear systems. Hence, in the remainder of the paper the method in [2] will be denoted as Single-Input Single-Output (SISO) and the method in [3] as MIMO TN Kalman filter. This filter makes use of special TNs [12] in Tensor Train (TT) format [13] without explicitly constructing the underlying exponentially large matrices and vectors.…”

“…This results in longer computation times for larger TN ranks. Additionally, the implementation of the MIMO TN Kalman filter in [3] has a complexity of O p 3 .…”