A numerical evaluation of solvers for the periodic Riccati differential equation. (English) Zbl 1192.65096

Summary: 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.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C25 Periodic solutions to ordinary differential equations
65K10 Numerical optimization and variational techniques
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI Link


[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 · Zbl 1027.93001
[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
[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) · Zbl 1055.65053
[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) · Zbl 1008.93035
[7] Bittanti, S., Colaneri, P.: Periodic Systems: Filtering and Control. Springer, Berlin (2009). ISBN 978-1-84800-910-3 · Zbl 1163.93003
[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) · Zbl 0689.93066
[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) · Zbl 0631.93010
[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 · Zbl 0656.93067
[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)
[12] Calvo, M., Sanz-Serna, J.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)
[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) · Zbl 0946.93038
[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) · Zbl 1061.93061
[15] Dieci, L.: Numerical integration of the differential Riccati equation and some related issues. SIAM J. Numer. Anal. 29(3), 781–815 (1992) · Zbl 0768.65037
[16] Dieci, L., Eirola, T.: Positive definiteness in the numerical solution of Riccati differential equations. Numer. Math. 67, 303–313 (1994) · Zbl 0791.65050
[17] Franco, J., Gómez, I.: Fourth-order symmetric DIRK methods for periodic stiff problems. Numer. Algorithms 32, 317–336 (2003) · Zbl 1058.65072
[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)
[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)
[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 · Zbl 1368.93106
[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) · Zbl 1113.65039
[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) · Zbl 1146.65029
[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 · Zbl 1094.65125
[26] Hairer, E., McLachlan, R., Razakarivony, A.: Achieving Brouwer’s law with implicit Runge-Kutta methods. BIT 48(2), 231–243 (2008) · Zbl 1148.65058
[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) · Zbl 0804.93037
[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) · Zbl 0820.93022
[29] Hu, G.: Symplectic Runge-Kutta methods for the Kalman-Bucy filter. IMA J. Math. Control Info. (2007) · Zbl 1146.93040
[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) · Zbl 0859.65036
[33] Kano, H., Nishimura, T.: Periodic solutions of matrix Riccati equations with detectability and stabilizability. Int. J. Control 29(3), 471–487 (1979) · Zbl 0409.93037
[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) · Zbl 0424.65013
[36] Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004). ISBN 0-521-77290-7 · Zbl 1069.65139
[37] Löfberg, J.: YALMIP homepage. Automatic Control Laboratory, ETH Zurich, Switzerland (2009). http://control.ee.ethz.ch/\(\sim\)joloef/wiki/pmwiki.php
[38] Lust, K.: psSchur homepage. Department of Mathematics, K.U. Leuven, Belgium (2009). http://perswww.kuleuven.be/\(\sim\)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) · Zbl 0746.93001
[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) · Zbl 1117.93052
[41] Reid, W.: Riccati Differential Equations. Academic Press, San Diego (1972) · Zbl 0254.34003
[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) · Zbl 0863.65038
[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) · Zbl 1231.93060
[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) · Zbl 1195.93065
[49] Varga, A.: On solving periodic Riccati equations. Numer. Linear Algebra Appl. 15(9), 809–835 (2008) · Zbl 1212.65254
[50] Yakubovich, V.: Linear-quadratic optimization problem and frequency theorem for periodic systems. Sib. Math. J. 27(4), 181–200 (1986) · Zbl 0648.93037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.