Positive Realness in Stochastic Subspace Identification: A Regularized and Reweighted Nuclear Norm Minimization Approach

Abstract

In this paper, we propose a method to transform a non-positive real transfer function matrix into a positive real one. This problem is of engineering interest and arises when a linear time-invarant dynamics is identified by stochastic subspace identification methods. Recent methods to tackle this problem are based on semi-definite programming schemes and as illustrated by numerical examples in this paper suffer from leakage effect at peak frequencies of the modified frequency response. The method proposed in this paper is inspired from the matrix rank minimization problem, which consists of finding a matrix of minimum rank satisfying given convex constraints. This NP-hard problem is then solved by an iteratively reweighted nuclear norm heuristic. We apply this heuristic to the problem considered in this paper. Numerical examples show that this method converges only in a few iterations and is effective in eliminating leakages at peak frequencies.