On the well-posedness of multivariate spectrum approximation and convergence of high-resolution spectral estimators☆
Introduction
Consider a linear, time invariant system with the transfer function where is a stability matrix, is full column rank, and is a reachable pair. Suppose that the system is fed with a -dimensional, zero-mean, wide-sense stationary process having a spectrum . The asymptotic state covariance of the system (1) satisfies: Here and in the following, , and integration takes place over the unit circle with respect to the normalized Lebesgue measure . Let be the family of bounded, coercive, -valued spectral density functions on the unit circle. Hence, if and only if . Given a Hermitian and positive-definite matrix , consider the problem of finding that satisfies (3), i.e., that is compatible with . This is a particular case of a moment problem. In the last ten years, much research has been produced, mainly by the Byrnes–Georgiou–Lindquist school, on generalized moment problems [1], [2], [3], [4], [5], and analytic interpolation with complexity constraint [6], and their applications to spectral estimation [7], [8], [9] and robust control [10]. It is worth recalling that two fundamental problems of control theory, namely the covariance extension problem and the Nevanlinna–Pick interpolation problem of robust control, can be recast in this form [5].
Eq. (3), where the unknown is , is also a typical example of an inverse problem. Recall that a problem is said to be well posed, in the sense of Hadamard, if it admits a solution, such a solution is unique, and the solution depends continuously on the data. Inverse problems are typically not well posed. In our case, there may well be no solution , and when a solution exists, there may be (infinitely) many. It was shown in [11], that the set of solutions is nonempty if and only if there exists such that When (4) is feasible with , and there are infinitely many solutions to (3). To select a particular solution it is natural to introduce an optimality criterion. For control applications, however, it is desirable that such a solution be of limited complexity. It should namely be rational and with an a priori bound on its MacMillan degree. One of the great accomplishments of the Byrnes–Georgiou–Lindquist approach is having shown that the minimization of certain entropy-like functionals leads to solutions that satisfy this requirement. In [11], Georgiou provided an explicit expression for the spectrum that exhibits a maximum entropy rate among the solutions of (3).
Suppose now that some a priori information about is available in the form of a spectrum . Given , , and , we now seek a spectrum , which is the closest to in a certain metric, among the solutions of (3). Paper [5] deals with such an optimization problem in the case when is a scalar process. The criterion there is the Kullback–Leibler pseudo-distance from to . A drawback of this approach is that it does not seem to generalize to the multivariable case. This motivated us to provide a suitable extension of the so-called Hellinger distance with respect to which the multivariable version of the problem is solvable (see [12], [9]).
The main result of this paper is contained in Section 3. We show there that, under the feasibility assumption, the solution to the spectrum approximation problem with respect to both the scalar Kullback–Leibler pseudo-distance and the multivariable Hellinger distance depends continuously on , thereby proving that these problems are well-posed. In Section 4 we deal with the case when only an estimate of is available. By applying the continuity results of Section 3, we prove a consistency result for the solutions to both approximation problems.
Section snippets
Spectrum approximation problems
In this section, we collect some background material on the spectrum approximation problems. The reader is referred to [11], [5], [12], [9] for a more detailed treatment.
Well-posedness of the approximation problems
In this section, we show that both the dual problems (12), (23) are well-posed, since their unique solution is continuous with respect to a small perturbation of . The well-posedness of the respective primal problem then easily follows. All these continuity properties rely on the following basic result. Theorem 3.1 Let be an open and convex subset of a finite-dimensional Euclidean space . Let be a strictly convex function, and suppose that a minimum point of exists. Then, for all ,
Consistency
So far we have shown that both the approximation problems admit a unique solution for all , and that the solution is continuous with respect to the variations . The necessity of a restriction to becomes crucial in the case when we only have an estimate of .
In line with the Byrnes–Georgiou–Lindquist theory, and following an estimation procedure we have sketched in [9], we want to use the above theory to provide an estimate of the true spectrum of the process .
Let
Conclusion and future work
In this paper, we have considered the constrained spectrum approximation problems with respect to both the Kullback–Leibler pseudo-distance (scalar case) and the Hellinger distance (multivariable case). The range of the operator is the subspace of the Hermitian matrices that conveys all the structure that is needed from a positive-definite matrix in order to be an asymptotic covariance matrix of the system with transfer function . As such, it is also a natural subspace to which
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Partially supported by the Ministry of Education, University, and Research of Italy (MIUR), under Project 2006094843: New techniques and applications of identification and adaptive control.