On the Georgiou–Lindquist Approach to Constrained Kullback–Leibler Approximation of Spectral Densities

Pavon, Michele; Ferrante, Augusto

doi:10.1109/tac.2006.872755

Cited by 42 publications

(42 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, it follows from (11) and (14) that γ 2n = α n,0 0 α n,1 0 · · · α n,n−1 0 T = α 0 0 α 1 0 · · · α n−1 0 T (16) = e 2n = γ 2n−1 = 2n−1 e 1 = 2n e 1…”

Section: Characterisation Of Matrix Anti-exponential Functionmentioning

confidence: 99%

State response for continuous‐time antilinear systems

Zhang

Liu

et al. 2015

IET Control Theory & Applications

View full text Add to dashboard Cite

In this study, the authors investigate the state response for continuous-time antilinear dynamical systems. First, they propose the concept of matrix anti-exponential function, and then derived some nice properties of this new function. With the matrix anti-exponential function as an effective tool, they obtain a closed-form expression for the state response of continuous-time antilinear systems. Finally, they derived an expression with finite terms for the proposed matrix antiexponential function, which will be useful for numerical implementation. IntroductionIn linear systems theory, matrix exponential function plays an important role in the analysis of dynamical systems. It is well known that the state response of a continuous time-invariant linear system can be expressed explicitly in terms of matrix exponential functions [1,2]. In addition, the solution of differential Lyapunov matrix equation arising from the stability analysis [2] can be expressed in terms of the matrix exponential function [3]. The stability of a continuous time-invariant linear system can be characterised by the limit of the matrix exponential function of the system matrix [1, 2]. Furthermore, for a class of special time-varying linear systems, the state transition matrix can be expressed as a product of many matrix exponential functions [4]. Recently, some computing methods were given in [5] for matrix exponentials of special matrices. With the aid of the results in [6], the methods in [5] can be applied to more cases. In addition, the exponential function is also needed when the lifting technique is used to discretise a continuous-time system [7]. The concepts of controllability and observability are very fundamental in systems theory. In linear time-invariant systems, the tests for controllability and observability can be carried out using the controllability Gramian and observability Gramian respectively, which are expressed by matrix exponential functions [8]. In [9], the frequency domain conditions for controllability and observability of multivariable linear systems are obtained by using a Jordan-based decomposition of the matrix exponential functions. In [10], a characterisation of the matrix exponential function is given using finite terms of coefficients obtained by a differential equation. Also in [11], an explicit formula is derived to compute the matrix exponential function of a matrix in terms of its annihilation polynomial. It should be pointed out that some problems are also investigated for linear systems. For example, a novel coupledleast-squares parameter identification algorithm was introduced for estimating the parameters of the multiple linear regression models [12], where the algorithm can avoid the matrix inversion; in [13], some two-stage identification methods were derived for BoxJenkins systems. In [14], some iterative algorithms were proposed for some matrix equations related to control systems.In this paper, we will investigate the state response of antilinear systems as described in (4). First, complex dynam...

show abstract

“…In addition, it follows from (11) and (14) that γ 2n = α n,0 0 α n,1 0 · · · α n,n−1 0 T = α 0 0 α 1 0 · · · α n−1 0 T (16) = e 2n = γ 2n−1 = 2n−1 e 1 = 2n e 1…”

Section: Characterisation Of Matrix Anti-exponential Functionmentioning

confidence: 99%

State response for continuous‐time antilinear systems

Zhang

Liu

et al. 2015

IET Control Theory & Applications

View full text Add to dashboard Cite

show abstract

“…The generalization to the multivariable case of the Prior-THREE algorithm, however, is more challenging since the variational analysis cannot be carried through. In [31] and [32], this generalization has been successfully carried out in a different metric induced by a Hellingertype distance.…”

Section: Application To Spectral Estimationmentioning

confidence: 99%

A Maximum Entropy Enhancement for a Family of High-Resolution Spectral Estimators

Ferrante

Pavon

Zorzi

2012

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

Abstract-Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes-Georgiou-Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance 6 of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate6, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate 6 is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances.

show abstract

“…In fact, the use of generalized statistics, which relates to beamspace processing, was explored in [7], [15] as a way to improve resolution in power spectral estimation over selected frequency bands. More recent work addresses spectral estimation with priors, computational issues, as well as important multivariate generalizations [3], [5], [9], [10], [11], [17], [18], [20], [21], [38], [40], [43], [44].…”

mentioning

confidence: 99%

Uncertainty Bounds for Spectral Estimation

Karlsson

Georgiou

2013

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

Abstract-The purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second-order statistics. The family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the "true" power spectrum. Our aim is to quantify the size of this uncertainty set using suitable notions of distance, and in particular, to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the "true" power spectrum. Since the uncertainty set may contain power spectra with lines and discontinuities, it is natural to quantify distances in the weak topology-the topology defined by continuity of moments. We provide examples of such weakly-continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty. We then consider certain high resolution techniques which utilize filter-banks for pre-processing, and compute worst-case a priori uncertainty bounds solely on the basis of the filter dynamics. This allows the a priori tuning of the filter-banks for improved resolution over selected frequency bands.Index Terms-Robust spectral estimation, uncertainty set, spectral distances, geometry of spectral measures, THREE filter design.

show abstract

On the Georgiou–Lindquist Approach to Constrained Kullback–Leibler Approximation of Spectral Densities

Cited by 42 publications

References 16 publications

State response for continuous‐time antilinear systems

State response for continuous‐time antilinear systems

A Maximum Entropy Enhancement for a Family of High-Resolution Spectral Estimators

Uncertainty Bounds for Spectral Estimation

Contact Info

Product

Resources

About