A Covariance-Free Strictly Complex-Valued Relevance Vector Machine for Reducing the Order of Linear Time-Invariant Systems

Abstract

Multiple-input multiple-output (MIMO) linear time-invariant (LTI) systems exhibit enormous computational costs for high-dimensional problems. To address this problem, we propose a novel approach for reducing the dimensionality of MIMO systems. The method leverages the Takenaka–Malmquist basis and incorporates the strictly complex-valued relevant vector machine (SCRVM). We refer to this method as covariance-free maximum likelihood (CoFML). The proposed method avoids the explicit computation of the covariance matrix. CoFML solves multiple linear systems to obtain the required posterior statistics for covariance. This is achieved by exploiting the preconditioning matrix and the matrix diagonal element estimation rule. We provide theoretical justification for this approximation and show why our method scales well in high-dimensional settings. By employing the CoFML algorithm, we approximate MIMO systems in parallel, resulting in significant computational time savings. The effectiveness of this method is demonstrated through three well-known examples.

Keywords

• model order reduction
• rational approximation
• Takenanka–Malmquist
• sparse Bayesian learning