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

DOI: 10.1007/s10543-010-0257-5

Cite this article as:
Gusev, S., Johansson, S., Kågström, B. et al. Bit Numer Math (2010) 50: 301. doi:10.1007/s10543-010-0257-5

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 systemsPeriodic Riccati differential equationsOrbital stabilizationPeriodic real Schur formPeriodic eigenvalue reorderingHamiltonian systemsLinear matrix inequalitiesNumerical methods

Mathematics Subject Classification (2000)

15A2115A3934K1349N0565F1565P1070M2070Q0590C22

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