Tryphon T. Georgiou

Stability theory for linear time-invariant plants with periodic digital controllers

This paper considers the control of a linear time-invariant plant by a digital controller composed of a sampler. a periodic discrete-time component, and a zero-order hold. The stability of such a configuration is analyzed in detail. It is shown how closed-loop zeros can be placed using such a scheme. As a consequence. it is proved that the gain margin can be arbitrarily assigned by suitable choice of sampling time and digital controller. The design procedure is constructive, using state-space methods.

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.

Noninvasive estimation of tissue temperature via high-resolution spectral analysis techniques

We address the noninvasive temperature estimation from pulse-echo radio frequency signals from standard diagnostic ultrasound imaging equipment. In particular, we investigate the use of a high-resolution spectral estimation method for tracking frequency shifts at two or more harmonic frequencies associated with temperature change. The new approach, employing generalized second-order statistics, is shown to produce superior frequency shift estimates when compared to conventional high-resolution spectral estimation methods Seip and Ebbini (1995). Furthermore, temperature estimates from the new algorithm are compared with results from the more commonly used echo shift method described in Simon et al. (1998).

The interpolation problem with a degree constraint

The author previously (1983, 1987) showed that there is a correspondence between nonnegative (Hermitian) trigonometric polynomials of degree /spl les/n and solutions to the standard Nevanlinna-Pick-Caratheodory interpolation problem with n+1 constraints, which are rational and also of degree /spl les/n. It was conjectured that the correspondence under suitable normalization is bijective and thereby, that it results in a complete parametrization of rational solutions of degree /spl les/n. The conjecture was proven by Byrnes et al. (1995), along with a detailed study of this parametrization. However, Byrnes et al. used a slightly restrictive assumption that the trigonometric polynomials are positive and accordingly, the corresponding solutions have positive real part. The purpose of the present note is to extend the result to the case of nonnegative trigonometric polynomials as well. We present the arguments in the context of the general Nevanlinna-Pick-Caratheodory-Fejer interpolation.

The structure of state covariances and its relation to the power spectrum of the input

We study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances, and identify the structure of positive-semidefinite matrices that qualify as state covariances of the filter. This structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices. We then characterize all input power spectra consistent with any specific state covariance. The parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous, but not equivalent, to a matricial Nehari problem.

Spectral analysis based on the state covariance: the maximum entropy spectrum and linear fractional parametrization

Input spectra, which are consistent with a given state covariance of a linear filter, correspond to solutions of an analytic interpolation problem. We derive an explicit formula for the power spectrum with maximal entropy, and provide a linear fraction parametrization of all solutions.

Robust stabilization in the gap metric: controller design for distributed plants

The problem of robustness in the gap metric for infinite-dimensional systems is considered. The problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are H/sub infinity / functions with continuous boundary values is studied. The underlying Hankel and related operators, which are important in the gap optimization problem, are studied, and relations between their singular values and vectors are established. A computational approach to the optimal robustness problem is developed for single-input/single-output systems whose transfer function is an inner function in H/sub infinity / times a rational function. The procedure is applied to a general first-order delay system and a closed-form formula is obtained for the optimal controller. The frequency response plots of the compensated system for various values of time delay are examined. >

Relative entropy and the multivariable multidimensional moment problem

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The best known are the von Neumann entropy |(rho):=-trace(rhologrho) and a generalization of the Kullback- Leibler distance S(rhoparsigma):=trace(rhologrho-rhologsigma), referred to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore | and S as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application, we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work

Distances and Riemannian Metrics for Spectral Density Functions

We introduce a differential-geometric structure for spectral density functions of discrete-time random processes. This is quite analogous to the Riemannian structure of information geometry, which is used to study perturbations of probability density functions, and which is based on the Fisher information metric. Herein, we introduce an analogous Riemannian metric, which we motivate with a problem in prediction theory. It turns out that this problem also provides a prediction theoretic interpretation to the Itakura distortion measure, which relates to our metric. Geodesies and geodesic distances are characterized in closed form and, hence, the geodesic distance between two spectral density functions provides an explicit, intrinsic (pseudo)metric on the cone of density functions. Certain other distortion measures that involve generalized means of spectral density functions are shown to lead to the same Riemannian metric. Finally, an alternative Riemannian metric is introduced, which is motivated by an analogous problem involving smoothing instead of prediction.