Vector-exponential time-series modeling for polynomial J-spectral factorization

An iterative algorithm to perform the J-spectral factorization of a para-Hermitian matrix is presented. The algorithm proceeds by computing a special kernel representation of an interpolant for a sequence of points and associated directions determined from the spectral zeroes of the to-be factored matrix

Solution of polynomial Lyapunov and Sylvester equations

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation

Linear Boundary Port-Hamiltonian Systems with Implicitly Defined Energy

In this paper we extend the previously introduced class of boundary port-Hamiltonian systems to boundary control systems where the variational derivative of the Hamiltonian functional is replaced by a pair of reciprocal differential operators. In physical systems modelling, these differential operators naturally represent the constitutive relations associated with the implicitly defined energy of the system and obey Maxwell's reciprocity conditions. On top of the boundary variables associated with the Stokes-Dirac structure, this leads to additional boundary port variables and to the new notion of a Stokes-Lagrange subspace. This extended class of boundary port-Hamiltonian systems is illustrated by a number of examples in the modelling of elastic rods with local and non-local elasticity relations. Finally it shown how a Hamiltonian functional on an extended state space can be associated with the Stokes-Lagrange subspace, and how this leads to an energy balance equation involving the boundary variables of the Stokes-Dirac structure as well as of the Stokes-Lagrange subspace.Comment: 23 page

A two-variable approach to solve the polynomial Lyapunov equation

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov equation is proposed. Lifting the problem from the one-variable to the two-variable context allows to use Faddeev-type recursions in order to solve the polynomial Lyapunov equation in an iterative fashion. The method is especially suitable for applications requiring exact or symbolic computation

Algorithmes numériques pour les matrices polynomiales avec applications en \ud commande\ud

Dans cette thèse nous développons de nouveaux algorithmes de calcul numérique pour les matrices polynomiales. Nous abordons le problème du calcul de la structure propre (rang, espace nul, structures finie et infinie) d'une matrice polynomiale et nous appliquons les résultats obtenus au calcul de la factorisation J-spectrale des matrices polynomiales. Nous présentons également quelques applications de ces algorithmes en théorie de la commande. Tous les nouveaux algorithmes décrits ici sont basés sur le calcul d'espaces nuls constants de matrices bloc Toeplitz associées à la matrice polynomiale analysée. Pour calculer ces espaces nuls nous utilisons des méthodes standard de l'algèbre linéaire numérique comme la décomposition en valeurs singulières ou la factorisation QR. Nous étudions aussi l'application de méthodes rapides comme la méthode généralisée de Schur pour les matrices structurées. Nous analysons les algorithmes présentés au niveau complexité algorithmique et stabilité numérique, et effectuons des comparaisons avec d'autres algorithmes existants dans la littérature.\ud In this thesis we develop new numerical algorithms for polynomial matrices. We tackle the problem of computing the eigenstructure (rank, null-space, finite and infinite structures) of a polynomial matrix and we apply the obtained results to the matrix polynomial J-spectral factorization problem. We also present some applications of these algorithms in control theory. All the new algorithms presented here are based on the computation of the constant null-spaces of block Toeplitz matrices associated to the analysed polynomial matrix

New algorithms for polynomial J-spectral factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix.