Federico Ramponi

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.

Stochastic receding horizon control with output feedback and bounded controls

We study the problem of receding horizon control for stochastic discrete-time systems with bounded control inputs and incomplete state information. Given a suitable choice of causal control policies, we first present a slight extension of the Kalman filter to estimate the state optimally in mean-square sense. We then show how to augment the underlying optimization problem with a negative drift-like constraint, yielding a second-order cone program to be solved periodically online. We prove that the receding horizon implementation of the resulting control policies renders the state of the overall system mean-square bounded under mild assumptions. We also discuss how some quantities required by the finite-horizon optimization problem can be computed off-line, thus reducing the on-line computation.

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.

Attaining Mean Square Boundedness of a Marginally Stable Stochastic Linear System With a Bounded Control Input

We construct control policies that ensure bounded variance of a noisy marginally stable linear system in closed-loop. It is assumed that the noise sequence is a mutually independent sequence of random vectors, enters the dynamics affinely, and has bounded fourth moment. The magnitude of the control is required to be of the order of the first moment of the noise, and the policies we obtain are simple and computable.

On the connections between PCTL and dynamic programming

Probabilistic Computation Tree Logic (PCTL) is a well-known modal logic which has become a standard for expressing temporal properties of finite-state Markov chains in the context of automated model checking. In this paper, we consider PCTL for noncountable-space Markov chains, and we show that there is a substantial affinity between certain of its operators and problems of Dynamic Programming. We prove some basic properties of the solutions to the latter. We also provide two examples and demonstrate how recovery strategies in practical applications, which are naturally stated as reach-avoid problems, can be viewed as particular cases of PCTL formulas.

On mean square boundedness of stochastic linear systems with bounded controls

Abstract We start by summarizing the state of the art in stabilization of stochastic linear systems with bounded inputs and highlight remaining open problems. We then report two new results concerning mean-square boundedness of a linear system with additive stochastic noise. The first states that, given any nonzero bound on the controls, it is possible to construct a policy with bounded memory requirements that renders a marginally stable stabilizable system mean-square bounded in closed-loop. The second states that it is not possible to ensure mean-square boundedness in closed-loop with a bounded control policy for systems affected by unbounded noise and having at least one eigenvalue outside the unit circle.

On the Convergence of an Efficient Algorithm for Kullback–Leibler Approximation of Spectral Densities

This paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem à la Byrnes/Georgiou/Lindquist. The approximation is pursued with respect to the Kullback-Leibler pseudo-distance, which gives rise to a convex optimization problem. After developing the variational analysis, we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem, based on the iteration of a nonlinear map in a bounded subset of the dual space. Our main result is the proof of local convergence of the latter, established as a consequence of the central manifold theorem. Supported by numerical evidence, we conjecture that, in the mentioned bounded set, the convergence is actually global.

Stochastic receding horizon control with output feedback and bounded control inputs

We study the problem of receding horizon control of stochastic discrete-time systems with bounded control inputs and incomplete state information. Given a suitable choice of causal control policies, we first present a slight extension of the Kalman filter to estimate the state optimally in mean-square sense. We then show how to augment the underlying optimization problem with a negative drift-like constraint, yielding a second-order cone program to be solved periodically online. Finally, we prove that the receding horizon implementation of the resulting control policies renders the state of the overall system mean-square bounded under mild assumptions.

Further Results on the Byrnes-Georgiou-Lindquist Generalized Moment Problem

In this paper, we consider the problem of finding, among solutions of a moment problem, the best Kullback-Leibler approximation of a given a priori spectral density. We present a new complete existence proof for the dual optimization problem in the Byrnes-Lindquist spirit. We also prove a descent property for a matricial iterative method for the numerical solution of the dual problem. The latter has proven to perform extremely well in simulation testbeds.

On the well-posedness of multivariate spectrum approximation and convergence of high-resolution spectral estimators.

Abstract In this paper, we establish the well-posedness of the generalized moment problems recently studied by Byrnes–Georgiou–Lindquist and coworkers, and by Ferrante–Pavon–Ramponi. We then apply these continuity results to prove the almost sure convergence of a sequence of high-resolution spectral estimators indexed by the sample size.