Jorge Mari

Vector ARMA estimation: a reliable subspace approach

A parameter estimation method for finite-dimensional multivariate linear stochastic systems, which is guaranteed to produce valid models approximating the true underlying system in a computational time of a polynomial order in the system dimension, is presented. This is achieved by combining the main features of certain stochastic subspace identification techniques with sound matrix Schur restabilizing procedures and multivariate covariance fitting, both of which are formulated as linear matrix inequality problems. All aspects of the identification method are discussed, with an emphasis on the two issues mentioned above, and examples of the overall performance are provided for two different systems.

Experimental evidence showing that stochastic subspace identification methods may fail

It is known that certain popular stochastic subspace identification methods may fail for theoretical reasons related to positive realness. In fact, these algorithms are implicitly based on the assumption that the positive and algebraic degrees of a certain estimated covariance sequence coincide. In this paper, we describe how to generate data with the property that this condition is not satisfied. Using these data we show through simulations that several subspace identification algorithms exhibit massive failure.

Stability Analysis of a Class of PWM Systems

This note considers stability analysis of a class of pulsewidth modulated (PWM) systems that incorporates several different switched mode dc-dc converters. The systems of the class typically have periodic solutions. A sampled data model is developed and used to prove stability of these solutions. Conditions for global and local exponential stability are derived using quadratic and piecewise quadratic Lyapunov functions. The state space is partitioned and the stability conditions are verified by checking a set of coupled linear matrix inequalities (LMIs).

MA estimation in polynomial time

The parameter estimation of moving-average (MA) signals from second-order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible. In this paper we show how the problem of MA parameter estimation from sample covariances can be formulated as a semidefinite program which can be solved in polynomial time as efficiently as a linear program. The method proposed relies on a specific (over)parameterization of the MA covariance sequence, whose use makes the minimization of the covariance fitting criterion a convex problem. The MA estimation algorithm proposed here is computationally fast, statistically accurate, and reliable (i.e. it never fails). None of the previously available algorithms for MA estimation (methods based on higher-order statistics included) shares all these desirable properties. Our method can also be used to obtain the optimal least squares approximant of an invalid (estimated) MA spectrum (that takes on negative values at some frequencies), which was another long-standing problem in the signal processing literature awaiting a satisfactory solution.

Modifications of rational transfer matrices to achieve positive realness

Abstract We develop several methods to transform a non-positive real transfer matrix into a positive real one. The problem is of practical engineering interest, since it might arise when trying to identify a linear description of a system, by means of stochastic subspace identification procedures. The modifications proposed preserve rationality and are reasonable in terms of systems theoretic properties expected of spectral density matrices. First, a stability problem related to stationarity of an underlying stochastic process is addressed and solved, making use of the reciprocal symmetry of spectral densities or alternatively Glovers optimal approximation. Then three methods are presented which compensate possible remaining positivity problems. The first two make use of the Kalman–Yakubovich–Popov lemma and the recent advances in semidefinite programming problems. The last method suggests corrections based on the asymptotic behaviour of generalized Schur parameters and algorithms related to maximum entropy extension and the backward Levinson algorithm.

A counterexample in power signals space

The present author points out that in the paper by Vidyasagar (1993) on nonlinear system analysis, some of the results obtained are false. The purpose of this paper is to give a counterexample. It is shown that the set of power signals is not a vector space.

A covariance extension approach to identification of time series

In this paper we consider a three-step procedure for identification of time series, based on covariance extension and model reduction, and we present a complete analysis of its statistical convergence properties. A partial covariance sequence is estimated from statistical data. Then a high-order maximum-entropy model is determined, which is finally approximated by a lower-order model by stochastically balanced model reduction. Such procedures have been studied before, in various combinations, but an overall convergence analysis comprising all three steps has been lacking. Supposing the data is generated from a true finite-dimensional system which is minimum phase, it is shown that the transfer function of the estimated system tends in H^~ to the true transfer function as the data length tends to infinity, if the covariance extension and the model reduction is done properly. The proposed identification procedure, and some variations of it, are evaluated by simulations.

Stability analysis of a class of PWM systems using sampled-data modeling

Stability analysis is considered for a class of pulse-width modulated (PWM) systems. A procedure is developed for systematic search for Lyapunov functions. The duty ratios that determine the switching times are used to partition the state space in such a way that stability is verified if a set of coupled linear matrix inequalities (LMIs) is feasible. Global stability as well as the computation of local regions of attraction is considered.

A Semidefinite Programming Approach to ARMA Estimation

Abstract A method for estimation of scalar ARMA models is proposed. In a first step the AR polynomial is estimated using a novel subspace fitting method. By using the estimated AR polynomial to filter the original data the MA polynomial is determined via an MA-covariance fitting method. Both estimation steps are formulated as quadratic optimization problems with LMI constraints which guarantee valid ARMA solutions and are solved as semidefinite programs.

Vector ARMA estimation: an enhanced subspace approach

A parameter estimation method for finite-dimensional multivariate linear stochastic systems is presented which is guaranteed to produce valid models close enough to the true underlying model, in a computational time of at most a polynomial order of the system dimension. This is achieved by combining the main features of certain stochastic subspace identification techniques together with sound statistical order estimation methods, matrix Schur restabilization procedures and multivariate covariance fitting, the latter formulated as linear matrix inequality problems. In this paper we make emphasis on the last issues mentioned, and provide an example of the overall performance for a multivariable case.