Augusto Ferrante

HERMITIAN SOLUTIONS OF THE EQUATION X = Q + NX-1N

We consider the matrix equation X = Q + NX−1N∗. Its Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil. We show that the equation admits both a largest and a smallest solution. The largest solution corresponds to the unique positive definite solution. The smallest solution is the unique negative definite solution if and only if N is nonsingular. If N is singular, no negative definite solution exists. An interesting relation between the given equation and a standard algebraic Riccati equation of Kalman filtering theory is also obtained. Finally, we present an algorithm which converges to the positive definite solution for a wide range of initial conditions.

Hellinger Versus Kullback–Leibler Multivariable Spectrum Approximation

In this paper, we study a matricial version of a generalized moment problem with degree constraint. We introduce a new metric on multivariable spectral densities induced by the family of their spectral factors, which, in the scalar case, reduces to the Hellinger distance. We solve the corresponding constrained optimization problem via duality theory. A highly nontrivial existence theorem for the dual problem is established in the Byrnes-Lindquist spirit. A matricial Newton-type algorithm is finally provided for the numerical solution of the dual problem. Simulation indicates that the algorithm performs effectively and reliably.

A Globally Convergent Matricial Algorithm for Multivariate Spectral Estimation

In this paper, we first describe a matricial Newton-type algorithm designed to solve the multivariable spectrum approximation problem. We then prove its global convergence. Finally, we apply this approximation procedure to multivariate spectral estimation, and test its effectiveness through simulation. Simulation shows that, in the case of short observation records, this method may provide a valid alternative to standard multivariable identification techniques such as Matlabs PEM and Matlabs N4SID.

Time and Spectral Domain Relative Entropy: A New Approach to Multivariate Spectral Estimation

The concept of spectral relative entropy rate is introduced for jointly stationary Gaussian processes. Using classical information-theoretic results, we establish a remarkable connection between time and spectral domain relative entropy rates. This naturally leads to a new spectral estimation technique where a multivariate version of the Itakura-Saito distance is employed. It may be viewed as an extension of the approach, called THREE, introduced by Byrnes, Georgiou, and Lindquist in 2000 which, in turn, followed in the footsteps of the Burg-Jaynes Maximum Entropy Method. Spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate. The corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework. Indeed, it is equal to the one featured by THREE in the scalar case. The solution is computed via a globally convergent matricial Newton-type algorithm. Simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks, especially in the case of short data records.

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

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 Σ of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate Σ, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate Σο 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.

A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions

A generalization of the finite-horizon linear quadratic regulator problem is proposed for LTI continuous-time controllable systems. In particular, a formulation of the linear quadratic (LQ) problem is considered, with affine constraints on the initial and the terminal states and with general quadratic costs in the initial and terminal states. The solution presented is simple and attractive from a computational point of view, and is based on the solutions of an algebraic Riccati equation and of a Lyapunov equation, that enable all the solutions of the Hamiltonian differential equation to be parametrized in closed form.

The generalised discrete algebraic Riccati equation in linear-quadratic optimal control

This paper investigates the properties of the solutions of the generalised discrete algebraic Riccati equation arising from the classic infinite-horizon linear quadratic (LQ) control problem. In particular, a geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation and the output-nulling subspaces of the underlying system and the corresponding reachability subspaces. This analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem, which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil. In establishing the main results of this paper, several ancillary problems on the discrete Lyapunov equation and spectral factorisation are also addressed and solved.

On the Georgiou-Lindquist approach to constrained Kullback-Leibler approximation of spectral densities

We consider the Georgiou-Lindquist constrained approximation of spectra in the Kullback-Leibler sense. We propose an alternative iterative algorithm to solve the corresponding convex optimization problem. The Lagrange multiplier is computed as a fixed point of a nonlinear matricial map. Simulation indicates that the algorithm is extremely effective.

A -spectral factorization approach for H/spl infin/ estimation problems in discrete time

This note deals with the H ∞ filtering, prediction, and smoothing problems for discrete-time linear systems. The J-spectral factorization approach is pursued and, for each of the three problems, a computationally efficient procedure to derive a J-spectral factor is presented. The construction of filter, smoother, and predictor hinges on the stabilizing solution of the same algebraic Riccati equation. This fact allows for the avoidance of any fictitious augmentation of the state-space dimensions.

A Maximum Entropy Solution of the Covariance Extension Problem for Reciprocal Processes

Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Nonstationary reciprocal processes have been extensively studied in the past especially by Jamison et al. The specialization of the nonstationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem.