Anders Lindquist

On the Partial Realization Problem

Abstract In this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra, most of which were originally developed in other contexts. We approach the partial realization problem from several different angles and explore the connections to such topics as factoriza- tion of Hankel matrices, block tridiagonalization, generalizations of the Lanczos process for biorthogonalization, the Euclidean algorithm and the principal-part con- tinued fractions of Arne Magnus, the Pade table, and the Berlekamp-Masseyalgo- rithm. In this way we are able to clarify some previous results by Rissanen, Kalman, and others and place them in a broader context. This leads to several results and concepts which we think are new. Our analysis is restricted to the scalar case, but some definitions and formulations have been rigged to facilitate an extension to the matrix case.

A generalized entropy criterion for Nevanlinna-Pick interpolation with degree constraint

We present a generalized entropy criterion for solving the rational Nevanlinna-Pick problem for n+1 interpolating conditions and the degree of interpolants bounded by n. The primal problem of maximizing this entropy gain has a very well-behaved dual problem. This dual is a convex optimization problem in a finite-dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n. The criterion requires a selection of a monic Schur polynomial of degree n. It follows that this class of monic polynomials completely parameterizes all such rational interpolants, and it therefore provides a set of design parameters for specifying such interpolants. The algorithm is implemented in a state-space form and applied to several illustrative problems in systems and control, namely sensitivity minimization, maximal power transfer and spectral estimation.

A new approach to spectral estimation: a tunable high-resolution spectral estimator

Traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates. Here, we present a new approach to spectral estimation, which is based on the use of filter banks as a means of obtaining spectral interpolation data. Such data replaces standard covariance estimates. A computational procedure for obtaining suitable pole-zero (ARMA) models from such data is presented. The choice of the zeros (MA-part) of the model is completely arbitrary. By suitable choices of filter bank poles and spectral zeros, the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum.

On the Stochastic Realization Problem

Given a mean square continuous stochastic vector process y with stationary increments and a rational spectral density

A Convex Optimization Approach to the Rational Covariance Extension Problem

\Phi

Canonical correlation analysis, approximate covariance extension, and identification of stationary time series

such that

Realization theory for multivariate stationary Gaussian processes

\Phi (\infty )

From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

is finite and nonsingular, consider the problem of finding all minimal (wide sense) Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman–Busy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algori...

A New Algorithm for Optimal Filtering of Discrete-Time Stationary Processes

In this paper we present a convex optimization problem for solving the rational covariance extension problem. Given a partial covariance sequence and the desired zeros of the modeling filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgious seminal existence result and his conjecture, a stronger version of which we have resolved in [Byrnes et al., IEEE Trans. Automat. Control, AC-40 (1995), pp. 1841--1857].

Generalized interpolation in H-infinity with a complexity constraint

Abstract In this paper we analyze a class of state space identification algorithms for time-series, based on canonical correlation analysis, in the light of recent results on stochastic systems theory. In principle, these so called “subspace methods” can be described as covariance estimation followed by stochastic realization. The methods offer the major advantage of converting the nonlinear parameter estimation phase in traditional ARMA models identification into the solution of a Riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity. The reason for this is that an essential part of the problem is equivalent to the well-known rational covariance extension problem. Therefore, the usual deterministic arguments based on factorization of a Hankel matrix are not valid for generic data, something that is habitually overlooked in the literature. We demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension, unless some rather stringent assumptions are made which, in general, are not explicitly reported. In this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory. We review the basic ideas of stochastic realization theory in the context of identification, discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation. The model reduction method of Desai and Pal (1982) [A realization approach to stochastic model reduction. Proc. 1st Decision and Control Conf., pp. 1105–1112.], based on truncated balanced stochastic realizations, is partially justified, showing that the reduced system structure has a positive covariance sequence but is in general not balanced. As a byproduct of this analysis we obtain a theorem prescribing conditions under which the ‘subspace identification’ methods produce bona fide stochastic systems.