Modifications of rational transfer matrices to achieve positive realness
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 -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.
Section snippets
Background and notation
In this section we define the notation and concepts used in the rest of the paper. Most of the material is standard (see e.g. [44], [17]). Throughout the paper, stands for the open unit disc in the complex plane, . The unit circle or border of the unit disc is , and the exterior of the closed unit disc is .
Problem formulation
Let the matrices (A,C,G,J) describe an arbitrary transfer matrix not fulfilling the positive real conditions. This typically means that the condition does not hold and hence that Φ(z)=V(z)+VT(1/z) when restricted to the unit circle need not be a positive-semidefinite matrix function in . But, it may also happen that the matrix A is not (Schur) stable, so that V(z) may fail to be in , as it should. As a matter of fact, in [14] we report examples of identified
The stability issue
The definition of positive realness appearing in Section 2, requires that V(z) be analytic outside the unit disc. Hence, if the matrix A is not Schur stable, it should be altered to become stable. For simplicity, we only consider the generic case where A has no eigenvalues on the unit circle, and concentrate exclusively on the stability problem first. In the next section we deal with the lack of positivity.
Consider the characteristic polynomial of the matrix A,and suppose it has
The positivity issue
From now on, we shall assume that A is a Schur stable matrix, and that the real part of V(z) is not positive semidefinite on the unit circle. Typically, this means that is indefinite, since one expects just a slight error, for example as a result of the main step in an identification procedure. A few of the methods in the literature may recognize that the condition in is not fulfilled and attempt an ad hoc remedy to recover. Such is the case of Algorithm 3 in [55,
Conclusions
In this paper we have suggested several ways to modify rational transfer matrices which are not positive real so that they become positive real after the modification. The changes are effected in such a way that the modified transfer matrix is as close as possible to the original transfer matrix. The proximity criteria have mainly been justified by the connections to the field of stochastic identification, although the methods themselves are not bound to them. An important aspect common to all
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