Modifications of rational transfer matrices to achieve positive realness

Abstract

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 system, by means of stochastic subspace identification procedures. The modifications proposed preserve rationality and are reasonable in terms of systems theoretic properties expected of spectral density matrices. First, a stability problem related to stationarity of an underlying stochastic process is addressed and solved, making use of the reciprocal symmetry of spectral densities or alternatively Glover's optimal approximation. Then three methods are presented which compensate possible remaining positivity problems. The first two make use of the Kalman–Yakubovich–Popov lemma and the recent advances in semidefinite programming problems. The last method suggests corrections based on the asymptotic behaviour of generalized Schur parameters and algorithms related to maximum entropy extension and the backward Levinson algorithm.

Zusammenfassung

Wir entwickeln diverse Methoden, um eine nicht-positiv definite reale Übertragungsmatrix in eine positiv definite reale zu transformieren. Das Problem ist von praktischem, technischem Interesse, da es in Verbindung mit dem Versuch, eine lineare Beschreibung eines Systems zu identifizieren, steht. Dieses geschieht unter Gebrauch von stochastischen Unterraum-Identifikationsprozessen. Die vorgeschlagenen Änderungen erhalten die Rationalität und sind im Zusammenhang von systemtheoretischen Bedingungen, die von spektralen Dichtematrizen erwartet werden, sinnvoll. Zuerst wird ein Stabilitätsproblem, verwandt zu den darunterliegenden stochastischen, stationären Prozessen, angegangen und gelöst, dabei wird die reziproke Symmetrie von spektralen Dichten oder in andern Fällen Glover's optimale Näherung verwendet. Weiterhin werden drei Methoden präsentiert, die möglicherweise verbleibende Probleme angehen, im Falle, daß die erhaltene Matrix nicht positiv real sein sollte. Die ersten beiden Methoden verwenden das Kalman–Yakubovich–Popov–Lemma und die jüngsten Fortschritte in semidefiniten Programmierproblemen. Die letzte Methode schlägt Korrekturen vor, die auf dem asymptotischen Verhalten der generalisierten Schurparameter und Algorithmen, die zur maximalen Entropieausdehnung und dem rückwärts (backward) Levinsonalgorithmus in Beziehung stehen, basieren.

Résumé

Nous développons plusieurs méthodes permettant de transformer une matrice de transfert non-positive réelle en une matrice positive réelle. Le problème concerne le domaine de l'ingéniérie appliquée puisqu'il peut apparaı̂tre lorsque l'on essaie d'obtenir une description linéaire d'un système par des procédures d'identification aléatoire de sous-espaces stochastiques. Les modifications proposées préservent la rationalité et les propriétés du type systèmes théoriques des matrices de densité spectrale sont raisonnables. Dans un premier temps, nous resolvons un problème de stabilité lié á un procéssus aléatoire stationnaire sous-jacent, en utilisant soit la symétrie réciproque des densités spectrales, soit l'approximation optimale de Glover. Dans un deuxième temps, nous présentons trois méthodes pour corriger les problèmes de positivité qui pourraient continuer à exister. Les deux premières méthodes utilisent le lemme de Kalman–Yakubovich–Popov et les progrès récents en matière de programmation semi-définie. La dernière méthode propose des corrections basées sur le comportement asymptotique des paramètres généralisés de Schur, sur les algorithmes associés à l'extension du maximum d'entropie et sur l'algorithme pas á pas descendant de Levinson.

Introduction

Positive real functions are common in many areas of engineering. For example, consider any electrical two-port system in which the input is the current and the output is the voltage. If the product of current and voltage, the power, is always non-negative, the system is called passive. If this electrical system is also linear, then its driving point impedance, i.e. the transfer function from current to voltage, is a positive real function. Networks consisting of only resistors, inductors and capacitors are passive. Similarly, mechanical systems with only masses, springs and dashpots are also passive (they dissipate power) so positive real functions arise here too. Passivity has also become a fundamental concept in control theory, especially in the stability analysis of feedback systems.

In this paper, we essentially consider a converse problem in which a linear system known to be passive is identified, by any chosen identification mechanism, but the resulting system fails to be positive real. This can happen because of numerical errors or algorithmical assumptions not fulfilled with the data set used to identify. Hence, we investigate in which way the given transfer function may be modified in the least possible way (to be explained later) so as to become positive real.

Our main motivation to consider the problem of modifying transfer matrices arises in the theory and application of stochastic subspace identification methods, i.e. the modeling of vector-valued random time series. Hence, we shall deal in this paper with matrix transfer functions, systems in discrete time, and we shall often justify modifications with an eye on what seems reasonable in the context of stochastic systems identification.

Roughly speaking, the main objective of stochastic systems identification is that of finding a finite-dimensional, linear state-space representation of a system, with full free-component matrices, such that when excited by white noise the system would generate a vector-valued output stochastic process whose second-order moments (or covariances) approximate the moments estimated from the available data. This type of unstructured modeling of vector-valued time series effectively circumvents the problem of parametrizing the corresponding set of admissible transfer matrices (cf. maximum likelihood methods [28]) and is amenable to pure linear-algebra non-iterative solutions [3], [55]. At the same time it overlooks some less trivial theoretical issues associated with the positive and algebraic degrees of partial covariance sequences [35] which may lead to certain non-positive real transfer matrices. Even better known in the field is the fact that errors in the covariance estimation may also produce invalid state-space representations (equivalently, transfer matrices). In the scalar case, corrective mechanisms were suggested, e.g. in [48], [50] and in the matrix case after the issues mentioned above were pointed out in [34], partial solutions appeared in [36], [55]. Solutions in a spirit close to the ones here presented are also found in the numerical literature (e.g. [54], [10]) where matrices not fulfilling certain properties are perturbed in order to recover those properties.

In this paper we make no assumption regarding the origin of the transfer matrices considered. The set of (discrete-time) positive real transfer matrices is a subset of the matrix-valued functions analytic outside the open unit disc. Moreover, if V(z) is any such transfer matrix it must satisfy there that its real part be a positive semidefinite matrix. Assuming that V(z) fails to meet the positive real conditions, from a mathematical viewpoint one would ideally like to solve an optimal approximation problem, say with respect to the $\text{L}{}_{\mathrm{\infty }}$-norm on the unit circle, where the original non-positive real function is substituted by the closest positive real one. We are not aware of any theoretical solution to this problem, and even if there were one, it would most likely involve an important computational burden. Instead, we propose several suboptimal solutions obtained in turn as the concatenation of optimal or suboptimal partial solutions, computable with reasonable effort. To this end several algorithms are presented, some new and some known. We track down the algorithms so that they can be easily programmed with present-day software and indicate what routines from the SLICOT [46] or MATLAB [52] packages may be used. We also illustrate the theory with several examples. We hope by doing this to help the practitioner solve the problems he or she might meet.

The paper is organized as follows. In Section 2 we present the basic definitions and background necessary to understand the material in the rest of the paper. In Section 3 we state the problem and justify the modifications proposed to enforce positive realness. In Section 4 we consider the problem of approximating an unstable transfer matrix by a stable one. This is a problem of independent interest in itself. We present several reasonable solutions to this problem including one based on the classical Nehari problem. We give a detailed algorithmic account on how to obtain these solutions in a numerically reliable way. In Section 5 we present several methods to force a stable square transfer matrix to map the complement of the unit disc to matrices with real part in the cone of positive-semidefinite matrices. Two of the methods make use of linear matrix inequalities (LMI) and semidefinite programming. The third is based on the classical theory of Schur and recent advances stemming from the dynamics of fast filtering algorithms. In the Conclusions we compare the methods and suggest one which we deem strikes a nice balance between algorithmic effort and the quality of the results achieved.