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


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.


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati equations. In: Control and Systems Theory. Birkhäuser, Basel (2003). ISBN 3-7643-0085-X Google Scholar
  2. 2.
    Anderson, B., Feng, Y.: An iterative algorithm to solve periodic Riccati differential equations with an indefinite quadratic term. In: Proc. of the 47th IEEE Conference on Decision and Control, CDC’08, Cancun, Mexico (2008) Google Scholar
  3. 3.
    Anderson, B., Moore, J.: Optimal Control: Linear Quadratic Methods. Dover, New York (2007). ISBN 0486457664 Google Scholar
  4. 4.
    Arnold, W., Laub, A.: Generalized eigenproblem algorithms and software for algebraic Riccati equations. In: Proc. IEEE, vol. 72, pp. 1746–1754 (1984) Google Scholar
  5. 5.
    Benner, P., Byers, R.: Evaluating products of matrix pencils and collapsing matrix products. Numer. Linear Algebra Appl. 8, 357–380 (2001) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benner, P., Byers, R., Mayo, R., Quintana-Orti, E.S., Hernandez, V.: Parallel algorithms for LQ optimal control of discrete-time periodic linear systems. J. Parallel Distrib. Comput. 62, 306–325 (2002) MATHCrossRefGoogle Scholar
  7. 7.
    Bittanti, S., Colaneri, P.: Periodic Systems: Filtering and Control. Springer, Berlin (2009). ISBN 978-1-84800-910-3 MATHGoogle Scholar
  8. 8.
    Bittanti, S., Colaneri, P., De Nicolao, G.: A note on the maximal solution of the periodic Riccati equation. IEEE Trans. Automat. Control 34(12), 1316–1319 (1989) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bittanti, S., Colaneri, P., Guardabassi, G.: Analysis of the periodic Lyapunov and Riccati equations via canonical decomposition. SIAM J. Control Optim. 24(6), 1138–1149 (1986) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bittanti, S., Colaneri, P., De Nicolao, G.: The periodic Riccati equation. In: Bittanti, S., Laub, A.J., Willems, J.C. (eds.) The Riccati Equation, pp. 127–162. Springer, Berlin (1991), Chap. 6 Google Scholar
  11. 11.
    Bojanczyk, A., Golub, G.H., Van Dooren, P.: The periodic Schur decomposition; algorithm and applications. In: Luk, F.T. (ed.) Proc. SPIE Conference, vol. 1770, pp. 31–42. SPIE, Bellingham (1992) CrossRefGoogle Scholar
  12. 12.
    Calvo, M., Sanz-Serna, J.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994) MATHGoogle Scholar
  13. 13.
    Chen, Y.Z., Chen, S.B., Liu, J.Q.: Comparison and uniqueness theorems for periodic Riccati differential equations. Int. J. Control 69(3), 467–473 (1998) MATHCrossRefGoogle Scholar
  14. 14.
    Chu, E., Fan, H., Lin, W., Wang, C.: Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations. Int. J. Control 77, 767–788 (2004) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dieci, L.: Numerical integration of the differential Riccati equation and some related issues. SIAM J. Numer. Anal. 29(3), 781–815 (1992) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Dieci, L., Eirola, T.: Positive definiteness in the numerical solution of Riccati differential equations. Numer. Math. 67, 303–313 (1994) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Franco, J., Gómez, I.: Fourth-order symmetric DIRK methods for periodic stiff problems. Numer. Algorithms 32, 317–336 (2003) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Freidovich, L., Gusev, S., Shiriaev, A.: LMI approach for solving periodic matrix Riccati equation. In: Proc. of the 3rd IFAC Workshop on Periodic Control Systems, PSYCO’07, St. Petersburg, Russia (2007) Google Scholar
  19. 19.
    Freidovich, L., Johansson, R., Robertsson, A., Sandberg, A., Shiriaev, A.: Virtual-holonomic-constraints-based design of stable oscillations of Furuta pendulum: Theory and experiments. IEEE Trans. Robot. 23(4), 827–832 (2007) CrossRefGoogle Scholar
  20. 20.
    Freidovich, L., La Hera, P., Mettin, U., Shiriaev, A.: New approach for swinging up the Furuta pendulum: Theory and experiments. Mechatronics 19(8), 1240–1250 (2009) CrossRefGoogle Scholar
  21. 21.
    Freidovich, L., Gusev, S., Shiriaev, A.: Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Trans. Automat. Contr. (2010, in press). doi:10.1109/TAC.2010.2042000
  22. 22.
    Furuta, K., Yamakita, M., Kobayashi, S.: Swing up control of inverted pendulum. In: Proc. of IECON’91, Kobe, Japan (1991) Google Scholar
  23. 23.
    Granat, R., Kågström, B.: Direct eigenvalue reordering in a product of matrices in periodic Schur form. SIAM J. Matrix Anal. Appl. 28(1), 285–300 (2006) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Granat, R., Kågström, B., Kressner, D.: Matlab tools for solving periodic eigenvalue problems. In: Proc. of the 3rd IFAC Workshop, PSYCO’07, St. Petersburg, Russia (2007) Google Scholar
  25. 25.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006). ISBN 3-540-30663-3 MATHGoogle Scholar
  26. 26.
    Hairer, E., McLachlan, R., Razakarivony, A.: Achieving Brouwer’s law with implicit Runge-Kutta methods. BIT 48(2), 231–243 (2008) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hench, J.J., Laub, A.J.: Numerical solution of the discrete-time periodic Riccati equation. IEEE Trans. Automat. Contr. 39(6), 1197–1209 (1994) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hench, J.J., Kenney, C.S., Laub, A.J.: Methods for the numerical integration of Hamiltonian systems. Circuits Syst. Signal Process. 13(6), 695–732 (1994) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Hu, G.: Symplectic Runge-Kutta methods for the Kalman-Bucy filter. IMA J. Math. Control Info. (2007) Google Scholar
  30. 30.
    Johansson, S.: Tools for control system design—stratification of matrix pairs and periodic Riccati differential equation solvers. Ph.D. Thesis, Report UMINF 09.04, Department of Computing Science, Umeå University, Sweden (2009). ISBN 978-91-7264-733-6 Google Scholar
  31. 31.
    Johansson, S., Kågström, B., Shiriaev, A., Varga, A.: Comparing one-shot and multi-shot methods for solving periodic Riccati differential equations. In: Proc. of the 3rd IFAC Workshop on Periodic Control Systems, PSYCO’07, St. Petersburg, Russia (2007) Google Scholar
  32. 32.
    Kågström, B., Poromaa, P.: Computing eigenspaces with specified eigenvalues of a regular matrix pair (A,B) and condition estimation: theory, algorithms and software. Numer. Algorithms 12, 369–407 (1996) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Kano, H., Nishimura, T.: Periodic solutions of matrix Riccati equations with detectability and stabilizability. Int. J. Control 29(3), 471–487 (1979) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kressner, D., Mehrmann, V., Penzl, T.: CTDSX—a collection of benchmark examples for state-space realizations of continuous-time dynamical systems. SLICOT Working Note 1998-9, WGS (1998) Google Scholar
  35. 35.
    Laub, A.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr. AC-24, 913–921 (1979) CrossRefMathSciNetGoogle Scholar
  36. 36.
    Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004). ISBN 0-521-77290-7 MATHGoogle Scholar
  37. 37.
    Löfberg, J.: YALMIP homepage. Automatic Control Laboratory, ETH Zurich, Switzerland (2009).
  38. 38.
    Lust, K.: psSchur homepage. Department of Mathematics, K.U. Leuven, Belgium (2009).
  39. 39.
    Mehrmann, V.: The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution. Lecture Notes in Control and Information Sciences, vol. 163. Springer, Berlin (1991) MATHGoogle Scholar
  40. 40.
    Perram, J., Robertsson, A., Sandberg, A., Shiriaev, A.: Periodic motion planning for virtually constrained mechanical system. Syst. Control Lett. 55(11), 900–907 (2006) MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Reid, W.: Riccati Differential Equations. Academic Press, San Diego (1972) MATHGoogle Scholar
  42. 42.
    SeDuMi homepage. Advanced Optimization Laboratory, McMaster University, Canada (2009).
  43. 43.
    Sima, V.: Algorithms for Linear-Quadratic Optimization. Pure and Applied Mathematics, vol. 200. Dekker, New York (1996) MATHGoogle Scholar
  44. 44.
    SLICOT homepage. Germany (2008).
  45. 45.
    Sturm, J.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones (updated for version 1.05). Tech. Rep., Department of Econometrics, Tilburg University, Tilburg, The Netherlands (2001) Google Scholar
  46. 46.
    Tan, S., Zhong, W.: Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation. Appl. Math. Mech. 28(3), 277–287 (2007) MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Varga, A.: On solving periodic differential matrix equations with applications to periodic system norms computation. In: Proc. of CDC’05, Seville, Spain (2005) Google Scholar
  48. 48.
    Varga, A.: A periodic systems toolbox for MATLAB. In: Proc. of 16th IFAC World Congress, Prague, Czech Republic (2005) Google Scholar
  49. 49.
    Varga, A.: On solving periodic Riccati equations. Numer. Linear Algebra Appl. 15(9), 809–835 (2008) CrossRefMathSciNetGoogle Scholar
  50. 50.
    Yakubovich, V.: Linear-quadratic optimization problem and frequency theorem for periodic systems. Sib. Math. J. 27(4), 181–200 (1986) MATHMathSciNetGoogle Scholar

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

Personalised recommendations