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

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

Identification of Piecewise Linear Models of Complex Dynamical Systems

The paper addresses the realization and identification problem or a subclass of piecewise-affine hybrid systems. The paper provides necessary and sufficient conditions for existence of a realization, a characterization of minimality, and an identification algorithm for this subclass of hybrid systems. The considered system class and the identification problem are motivated by applications in systems biology

Canonical lossless state-space systems: Staircase forms and the Schur algorithm

A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold

Time-Series Analysis Using Third-Order Recurrence Plots

Higher-order recurrence plots may enable us to reveal more structure than what is possible with traditional recurrence plots (RPs). While RPs attempt to detect recurrence relations by pair-wise comparison of time-delayed embeddings, given a time series, higher-order RPs may detect recurrences by comparing multiple time-delayed embeddings simultaneously. In this work, we limit ourselves to third-order recurrence plots (TORPs) for time series analysis, as they can still be graphed straightforwardly, and propose future directions

Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm

In this report, the connections are investigated between two different approaches towards the parametrization of multivariable stable all-pass systems in discrete-time. The first approach involves the tangential Schur algorithm, which employs linear fractional transformations. It stems from the theory of reproducing kernel Hilbert spaces and enables the direct construction of overlapping local parametrizations using Schur parameters and interpolation points. The second approach proceeds in terms of state-space realizations. In the scalar case, a balanced canonical form exists that can also be parametrized by Schur parameters. This canonical form can be constructed recursively, using unitary matrix operations. Here, this procedure is generalized to the multivariable case by establishing the connections with the first approach. It gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential Schur algorithm

Lossless scalar functions: boundary interpolation, Schur algorithm and Ober's canonical form

International audienceIn Ober (1987) a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In this paper, a connection is established between Ober's canonical form and a Schur algorithm builts from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober's form, as interpolation values at infinity, and a recursive construction of the balanced realization

Parametrization of matrix-valued lossless functions based on boundary interpolation

International audienceThis paper is concerned with parametrization issues for rational lossless matrix valued functions. In the same vein as previous works, interpolation theory with metric constraints is used to ensure the lossless property. We consider here boundary interpolation and provide a new parametrization of balanced canonical forms in which the parameters are angular derivatives. We finally investigate the possibility to parametrize orthogonal wavelets with vanishing moments using these results

Continuous-time lossless systems, boundary interpolation and pivot structures

International audienceBalanced realizations of lossless systems can be generated from the tangential Schur algorithm using linear fractional transformations. In discrete-time, canonical forms with pivot structures are obtained by interpolation at in finity. In continuous-time case interpolation at infi nity is on the stability boundary, leading to an angular derivative interpolation problem. In the scalar case, the balanced canonical form of Ober can be recovered in this way. Here we generalize to the multivariable case. It is shown that boundary interpolation can be regarded as a limit of classical interpolation with interpolation points tending to the imaginary axis. Some pivot structures can be generated, but no complete atlas is obtained. However, for input normal pairs an atlas of admissible pivot structures can be generated in a closely related way