BIT Numerical Mathematics

, Volume 50, Issue 2, pp 301–329

A numerical evaluation of solvers for the periodic Riccati differential equation

  • Sergei Gusev
  • Stefan Johansson
  • Bo Kågström
  • Anton Shiriaev
  • Andras Varga
Article

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 of an 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 reformulation leads to a semidefinite programming problem with linear matrix inequality constraints admitting an effective numerical 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.

Keywords

Periodic systems Periodic Riccati differential equations Orbital stabilization Periodic real Schur form Periodic eigenvalue reordering Hamiltonian systems Linear matrix inequalities Numerical methods 

Mathematics Subject Classification (2000)

15A21 15A39 34K13 49N05 65F15 65P10 70M20 70Q05 90C22 

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Copyright information

© Springer Science + Business Media B.V. 2010

Authors and Affiliations

  • Sergei Gusev
    • 1
  • Stefan Johansson
    • 2
  • Bo Kågström
    • 2
  • Anton Shiriaev
    • 3
    • 4
  • Andras Varga
    • 5
  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of Computing Science and HPC2NUmeå UniversityUmeåSweden
  3. 3.Department of Applied Physics and ElectronicsUmeå UniversityUmeåSweden
  4. 4.Department of Engineering CyberneticsNorwegian University of Science and TechnologyTrondheimNorway
  5. 5.Institute of Robotics and Mechatronics, German Aerospace CenterDLROberpfaffenhofenGermany

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