Decay of Hankel singular values of analytic control systems

We show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a Hankel operator that belongs to the Schatten class S-p for all positive p. This implies that the Hankel singular values converge to zero faster than any polynomial rate. This in turn implies fast convergence of balanced truncations. As a corollary, decay rates for the eigenvalues of the controllability and observability Gramians are also provided. Applications to the heat equation and a plate equation are given

Model Reduction of Linear PDE Systems: A Continuous Time Eigensystem Realization Algorithm

The Eigensystem Realization Algorithm (ERA) is a well known system identification and model reduction algorithm for discrete time systems. Recently, Ma, Ahuja, and Rowley (Theoret. Comput. Fluid Dyn. 25(1) : 233-247, 2011) showed that ERA is theoretically equivalent to the balanced POD algorithm for model reduction of discrete time systems. We propose an ERA for model reduction of continuous time linear partial differential equation systems. The algorithm differs from other existing approaches as it is based on a direct approximation of the Hankel integral operator of the system. We show that the algorithm produces accurate balanced reduced order models for an example PDE system

Problèmes de benchmark pour l'identiifcation de modèles à temps continu: conception, résultats et perspectives

International audienceThe problem of estimating continuous-time model parameters of linear dynamical systems using sampled time-domain input and output data has received considerable attention over the past decades and has been approached by various methods. The research topic also bears practical importance due to both its close relation to first principles modeling and equally to linear model-based control design techniques, most of them carried in continuous time. Nonetheless, as the performance of the existing algorithms for continuous-time model identification has seldom been assessed and, as thus far, it has not been considered in a comprehensive study, this practical potential of existing methods remains highly questionable. The goal of this brief paper is to bring forward a first study on this issue and to factually highlight the main aspects of interest. As such, an analysis is performed on a benchmark designed to be consistent both from a system identification viewpoint and from a control-theoretic one. It is concluded that robust initialization aspects require further research focus towards reliable algorithm development.Ce papier traite de benchmarking de l'identification de modèles à temps continu qui sont très utilisés dans l'ingiénerie

Problèmes de benchmark pour l'identiifcation de modèles à temps continu: conception, résultats et perspectives

International audienceThe problem of estimating continuous-time model parameters of linear dynamical systems using sampled time-domain input and output data has received considerable attention over the past decades and has been approached by various methods. The research topic also bears practical importance due to both its close relation to first principles modeling and equally to linear model-based control design techniques, most of them carried in continuous time. Nonetheless, as the performance of the existing algorithms for continuous-time model identification has seldom been assessed and, as thus far, it has not been considered in a comprehensive study, this practical potential of existing methods remains highly questionable. The goal of this brief paper is to bring forward a first study on this issue and to factually highlight the main aspects of interest. As such, an analysis is performed on a benchmark designed to be consistent both from a system identification viewpoint and from a control-theoretic one. It is concluded that robust initialization aspects require further research focus towards reliable algorithm development.Ce papier traite de benchmarking de l'identification de modèles à temps continu qui sont très utilisés dans l'ingiénerie

On the approximability of Koopman-based operator Lyapunov equations

Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws such as the well-known policy iteration. In this manuscript, the focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman or composition operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup, in a specially weighted $\mathrm{L}^p$-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay which justifies finite rank approximations with numerical methods. The potential benefit for numerical computations will be demonstrated with two short examples.Comment: Lyapunov equations, Koopman operator, infinite dimensional systems, semigroup

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance

Mini-Workshop: Wellposedness and Controllability of Evolution Equations

This mini-workshop brought together mathematicians engaged in partial differential equations, operator theory, functional analysis and harmonic analysis in order to address a number of current problems in the wellposedness and controllability of infinite-dimensional systems

Model reduction by balanced truncation for systems with nuclear Hankel operators

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26(1998), pp. 863-898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the Hankel operator in a Sobolev space. We also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation