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A numerical evaluation of solvers for the periodic Riccati differential equation

BIT Numerical Mathematics volume 50, pages 301–329 (2010)Cite this 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.

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References

  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. 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)

  3. Anderson, B., Moore, J.: Optimal Control: Linear Quadratic Methods. Dover, New York (2007). ISBN 0486457664

    Google Scholar 

  4. Arnold, W., Laub, A.: Generalized eigenproblem algorithms and software for algebraic Riccati equations. In: Proc. IEEE, vol. 72, pp. 1746–1754 (1984)

  5. Benner, P., Byers, R.: Evaluating products of matrix pencils and collapsing matrix products. Numer. Linear Algebra Appl. 8, 357–380 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  7. Bittanti, S., Colaneri, P.: Periodic Systems: Filtering and Control. Springer, Berlin (2009). ISBN 978-1-84800-910-3

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  12. Calvo, M., Sanz-Serna, J.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dieci, L.: Numerical integration of the differential Riccati equation and some related issues. SIAM J. Numer. Anal. 29(3), 781–815 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dieci, L., Eirola, T.: Positive definiteness in the numerical solution of Riccati differential equations. Numer. Math. 67, 303–313 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Franco, J., Gómez, I.: Fourth-order symmetric DIRK methods for periodic stiff problems. Numer. Algorithms 32, 317–336 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Furuta, K., Yamakita, M., Kobayashi, S.: Swing up control of inverted pendulum. In: Proc. of IECON’91, Kobe, Japan (1991)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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

    MATH  Google Scholar 

  26. Hairer, E., McLachlan, R., Razakarivony, A.: Achieving Brouwer’s law with implicit Runge-Kutta methods. BIT 48(2), 231–243 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hu, G.: Symplectic Runge-Kutta methods for the Kalman-Bucy filter. IMA J. Math. Control Info. (2007)

  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

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kano, H., Nishimura, T.: Periodic solutions of matrix Riccati equations with detectability and stabilizability. Int. J. Control 29(3), 471–487 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  35. Laub, A.: A Schur method for solving algebraic Riccati equations. IEEE Trans. Automat. Contr. AC-24, 913–921 (1979)

    Article  MathSciNet  Google Scholar 

  36. Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004). ISBN 0-521-77290-7

    MATH  Google Scholar 

  37. Löfberg, J.: YALMIP homepage. Automatic Control Laboratory, ETH Zurich, Switzerland (2009). http://control.ee.ethz.ch/~joloef/wiki/pmwiki.php

  38. Lust, K.: psSchur homepage. Department of Mathematics, K.U. Leuven, Belgium (2009). http://perswww.kuleuven.be/~u0006235/ACADEMIC/r_psSchur.html

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  41. Reid, W.: Riccati Differential Equations. Academic Press, San Diego (1972)

    MATH  Google Scholar 

  42. SeDuMi homepage. Advanced Optimization Laboratory, McMaster University, Canada (2009). http://sedumi.ie.lehigh.edu/

  43. Sima, V.: Algorithms for Linear-Quadratic Optimization. Pure and Applied Mathematics, vol. 200. Dekker, New York (1996)

    MATH  Google Scholar 

  44. SLICOT homepage. Germany (2008). http://www.slicot.org

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  47. Varga, A.: On solving periodic differential matrix equations with applications to periodic system norms computation. In: Proc. of CDC’05, Seville, Spain (2005)

  48. Varga, A.: A periodic systems toolbox for MATLAB. In: Proc. of 16th IFAC World Congress, Prague, Czech Republic (2005)

  49. Varga, A.: On solving periodic Riccati equations. Numer. Linear Algebra Appl. 15(9), 809–835 (2008)

    Article  MathSciNet  Google Scholar 

  50. Yakubovich, V.: Linear-quadratic optimization problem and frequency theorem for periodic systems. Sib. Math. J. 27(4), 181–200 (1986)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia

    Sergei Gusev

  2. Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden

    Stefan Johansson & Bo Kågström

  3. Department of Applied Physics and Electronics, Umeå University, Umeå, Sweden

    Anton Shiriaev

  4. Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway

    Anton Shiriaev

  5. Institute of Robotics and Mechatronics, German Aerospace Center, DLR, Oberpfaffenhofen, Germany

    Andras Varga

Authors
  1. Sergei Gusev
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  2. Stefan Johansson
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  3. Bo Kågström
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  4. Anton Shiriaev
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  5. Andras Varga
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Corresponding author

Correspondence to Stefan Johansson.

Additional information

Communicated by Axel Ruhe.

Financial support has been provided in part by the Swedish Foundation for Strategic Research (the frame program grant A3 02:128), by the Swedish Research Council (the grant 2008-5243), EU Mål2 Structural Funds (UMIT-project), and by the Russian Federal Agency for Science and Innovation (the project 02.740.11.5056).

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Gusev, S., Johansson, S., Kågström, B. et al. A numerical evaluation of solvers for the periodic Riccati differential equation. Bit Numer Math 50, 301–329 (2010). https://doi.org/10.1007/s10543-010-0257-5

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  • DOI: https://doi.org/10.1007/s10543-010-0257-5

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