Modifications of rational transfer matrices to achieve positive realness

We develop several methods to transform a non-positive real transfer matrix into a positive real one. The problem is of practical engineering interest , since it might arise when trying to identify a linear description of a sy stem, by means of stochastic subspace identification procedures. The modifi cations proposed preserve rationality and are reasonable in terms of system s theoretic properties expected of spectral density matrices. First, a stab ility problem related to stationarity of an underlying stochastic process i s addressed and solved, making use of the reciprocal symmetry of spectral d ensities or alternatively Glover's optimal approximation. Then three method s are presented which compensate possible remaining positivity problems. Th e first two make use of the Kalman-Yakubovich-Popov lemma and the recent ad vances in semidefinite programming problems. The last method suggests corre ctions based on the asymptotic behaviour of generalized Schur parameters an d algorithms related to maximum entropy extension and the backward Levinson algorithm. (C) 2000 Elsevier Science B.V. All rights reserved.