TY - GEN

T1 - Singular decompositions of state-covariances

AU - Georgiou, Tryphon T.

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Singularities in second-order statistics reveal linear dependencies between correlation variables. This fact underlies techniques ranging from Gauss' least squares to modern subspace methods in system identification. In particular, the latter exploit a decomposition of covariance data according to a hypothesis that the stochastic process breaks up into a sum of a deterministic component plus white noise. In the present paper we continue our earlier studies [9], [10] on the correspondence between state-covariances and input power spectra in dynamical systems. We characterize state-covariances which correspond to deterministic inputs and develop formulae for the input spectrum of a singular state-covariance. We show that multivariable decomposition of a state-covariance in accordance with a "deterministic component + white noise" hypothesis for the input does not exist in general, and study a possible alternative where the "white noise" is replaced with a general "moving-average process" having "short-range correlation structure". The decomposition according to the range of their time-domain correlations is an alternative to the well-known (Carathéodory- Fejér-)Pisarenko decomposition of Toeplitz matrices with potentially great practical significance, and can be determined via convex optimization.

AB - Singularities in second-order statistics reveal linear dependencies between correlation variables. This fact underlies techniques ranging from Gauss' least squares to modern subspace methods in system identification. In particular, the latter exploit a decomposition of covariance data according to a hypothesis that the stochastic process breaks up into a sum of a deterministic component plus white noise. In the present paper we continue our earlier studies [9], [10] on the correspondence between state-covariances and input power spectra in dynamical systems. We characterize state-covariances which correspond to deterministic inputs and develop formulae for the input spectrum of a singular state-covariance. We show that multivariable decomposition of a state-covariance in accordance with a "deterministic component + white noise" hypothesis for the input does not exist in general, and study a possible alternative where the "white noise" is replaced with a general "moving-average process" having "short-range correlation structure". The decomposition according to the range of their time-domain correlations is an alternative to the well-known (Carathéodory- Fejér-)Pisarenko decomposition of Toeplitz matrices with potentially great practical significance, and can be determined via convex optimization.

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M3 - Conference contribution

AN - SCOPUS:34047200009

SN - 1424402107

SN - 9781424402106

T3 - Proceedings of the American Control Conference

SP - 3725

EP - 3730

BT - Proceedings of the 2006 American Control Conference

T2 - 2006 American Control Conference

Y2 - 14 June 2006 through 16 June 2006

ER -