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A numerical evaluation of solvers for the periodic Riccati differential equation

Gusev, Sergei and Johansson, Stefan and Kagström, Bo and Shiriaev, Anton and Varga, Andreas (2010) A numerical evaluation of solvers for the periodic Riccati differential equation. BIT Numerical Mathematics. Springer. DOI: 10.1007/s10543-010-0257-5.

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Abstract

Efficient and accurate structure exploiting numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, to design periodic feedback controllers for periodic control systems. Three recently proposed methods for solving the PRDE are presented and evaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first two methods are of the type multiple shooting and rely on computing the stable invariant subspace on associated Hamiltonian system. The stable subspace is determined using either algorithms for computing an ordered periodic real Schur form of a cyclic matrix sequence, or a recently proposed method which implicitly constructs a stable deflating subspace from an associated lifted pencil. The third method reformulates the PRDE as a convex optimization problem where the stabilizing solution is approximated by its truncated Fourier series. As known, this reformulations leads to a semi-definite programming problem with linear matrix inequality constraints admitting an effective nuermical realization. The numerical evaluation of the PRDE methods, with focus on the number of states (n) and the length of the period (T) of the periodic systems considered, includes both quantitative and qualitative results.

Document Type:Article
Title:A numerical evaluation of solvers for the periodic Riccati differential equation
Authors:
AuthorsInstitution or Email of Authors
Gusev, SergeiSt. Petersburg State University
Johansson, StefanUmea University
Kagström, BoUmea University
Shiriaev, AntonNorwegian University of Science and Technology
Varga, AndreasAndreas.Varga@dlr.de
Date:February 2010
Journal or Publication Title:BIT Numerical Mathematics
Refereed publication:Yes
In Open Access:No
In SCOPUS:Yes
In ISI Web of Science:Yes
DOI:10.1007/s10543-010-0257-5
Publisher:Springer
Status:Published
Keywords:Periodic systems, Periodic Riccati differential equations, Orbital stabilization, Periodic real Schur form, Periodic eigenvalue reordering, Hamiltonian systems, Linear matrix inequalities, Numerical methods
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Aeronautics
HGF - Program Themes:Aircraft Research
DLR - Research area:Aeronautics
DLR - Program:L AR - Aircraft Research
DLR - Research theme (Project):L - Systems & Cabin
Location: Oberpfaffenhofen
Institutes and Institutions:Institute of Robotics and Mechatronics > System Dynamics and Control (former Control Design Engineering)
Deposited By: Monika Klauer
Deposited On:28 Apr 2010 14:54
Last Modified:07 Feb 2013 21:00

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