Abstract
A Newton-type algorithm and line search strategies for solving generalized discrete-time algebraic Riccati equations are dealt with. The conceptual algorithm is presented, and its main computational steps are discussed. Evaluation of residuals and closed-loop matrices at each iteration, determination of the step size, and the use of line search with backtracking, are addressed in detail. Algorithmic and implementation issues taken into account in the developed solver are described. An extensive performance investigation on a large collection of examples has been performed, and the results are summarized. Both usual line search and line search with backtracking, and either identity or diagonal performance index matrices are considered. Difficult examples are included. The results often show significantly improved accuracy, measured in terms of normalized and relative residuals, in comparison with the state-of-the-art MATLAB function. The new solver is strongly recommended for improving the solutions computed by other solvers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abels J, Benner P (1999) CAREX—A collection of benchmark examples for continuous-time algebraic Riccati equations (Version 2.0). SLICOT Working Note 1999-14. http://www.slicot.org
Abels J, Benner P (1999) DAREX—A collection of benchmark examples for discrete-time algebraic Riccati equations (Version 2.0). SLICOT Working Note 1999-16. http://www.slicot.org
Anderson BDO (1978) Second-order convergent algorithms for the steady-state Riccati equation. Int J Contr 28(2):295–306. https://doi.org/10.1080/00207177808922455
Anderson BDO, Moore JB (1971) Linear Optimal Control. Prentice-Hall, Englewood Cliffs
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK users’ guide: third edition. Software \(\cdot \) Environments \(\cdot \) Tools. SIAM, Philadelphia
Armstrong ES, Rublein GT (1976) A stabilization algorithm for linear discrete constant systems. IEEE Trans Autom Contr AC-21(4):629–631. https://doi.org/10.1109/TAC.1976.1101295
Arnold WF III, Laub AJ (1984) Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc IEEE 72(12):1746–1754. https://doi.org/10.1109/PROC.1984.13083
Balzer LA (1980) Accelerated convergence of the matrix sign function method of solving Lyapunov, Riccati and other matrix equations. Int J Contr 32(6):1057–1078. https://doi.org/10.1080/00207178008910040
Benner P (1997) Contributions to the numerical solution of algebraic Riccati equations and related eigenvalue problems. Dissertation, Fakultät für Mathematik, Technische Universität Chemnitz–Zwickau, D–09107 Chemnitz, Germany
Benner P (1998) Accelerating Newton’s method for discrete-time algebraic Riccati equations. In: Beghi A, Finesso L, Picci G (eds) Mathematical Theory of Networks and Systems, Il Poligrafo, Padova, Italy, pp 569–572. https://doi.org/10.1.1.20.7467
Benner P, Byers R (1998) An exact line search method for solving generalized continuous-time algebraic Riccati equations. IEEE Trans Autom Contr 43(1):101–107. https://doi.org/10.1109/9.654908
Benner P, Sima V (2003) Solving algebraic Riccati equations with SLICOT. In: The 11th Mediterranean conference on control and automation, Rhodes, Greece. https://doi.org/10.1.1.89.3409
Benner P, Mehrmann V, Sima V, Van Huffel S, Varga A (1999) SLICOT—a subroutine library in systems and control theory. In: Datta BN (ed) Applied and computational control, signals, and circuits, vol 1, Birkhäuser, Boston, MA, pp 499–539. https://doi.org/10.1007/978-1-4612-0571-5_10
Benner P, Kressner D, Sima V, Varga A (2010) Die SLICOT-Toolboxen für Matlab. at—Autom 58(1):15–25. https://doi.org/10.1524/auto.2010.0814
Benner P, Sima V, Voigt M (2016) Algorithm 961: Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems. ACM Trans Math Softw 42(3):1–26. https://doi.org/10.1145/2818313
Bini DA, Iannazzo B, Meini B (2012) Numerical solution of algebraic Riccati equations. SIAM, Philadelphia
Bunse-Gerstner A, Mehrmann V (1986) A symplectic QR like algorithm for the solution of the real algebraic Riccati equation. IEEE Trans Autom Contr AC–31(12):1104–1113. https://doi.org/10.1109/TAC.1986.1104186
Byers R (1987) Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl 85(1):267–279. https://doi.org/10.1016/0024-3795(87)90222-9
Chu EW, Fan HY, Lin WW (2005) A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations. Linear Algebra Appl 386:55–80. https://doi.org/10.1016/j.laa.2004.10.010
Ciubotaru BD, Staroswiecki M (2009) Comparative study of matrix Riccati equation solvers for parametric faults accommodation. In: 10th European control conference, Budapest, Hungary, pp 1371–1376. https://doi.org/10.23919/ECC.2009.7074597
Dongarra JJ, Du Croz J, Duff IS, Hammarling S (1990) Algorithm 679: a set of level 3 basic linear algebra subprograms. ACM Trans Math Softw 16(1):1–17, 18–28. https://doi.org/10.1145/77626.77627
Gardiner JD, Laub AJ (1986) A generalization of the matrix sign function solution for algebraic Riccati equations. Int J Contr 44:823–832. https://doi.org/10.1109/CDC.1985.268700
Giftthaler M, Neunert M, Stäuble M, Buchli J (2018) The control toolbox—an open-source C++ library for robotics, optimal and model predictive control. https://arxiv.org/abs/1801.04290. https://doi.org/10.1109/SIMPAR.2018.8376281
Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore
Guo C, Laub AJ (2000) On a Newton-like method for solving algebraic Riccati equations. SIAM J Matrix Anal Appl 21(2):694–698. https://doi.org/10.1137/S0895479898348519
Guo CH, Iannazzo B, Meini B (2007) On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation. SIAM J Matrix Anal Appl 29(4):1083–1100. https://doi.org/10.1137/060660837
Guo PC (2016) A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory. Numer Algorithms 71(3):541–552. https://doi.org/10.1007/s11075-015-0008-4
Guo XX, Lin WW, Xu SF (2006) A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer Math 103(3):393–412. https://doi.org/10.1007/s00211-005-0673-7
Hammarling SJ (1982) Newton’s method for solving the algebraic Riccati equation. NPC report DIIC 12/82, National Physics Laboratory, Teddington, Middlesex TW11 OLW, U.K
Hewer GA (1971) An iterative technique for the computation of the steady state gains for the discrete optimal regulator. IEEE Trans Autom Contr AC–16(4):382–384. https://doi.org/10.1109/TAC.1971.1099755
Jónsson GF, Vavasis S (2004) Solving polynomials with small leading coefficients. SIAM J Matrix Anal Appl 26(2):400–414. https://doi.org/10.1137/S0895479899365720
Kenney C, Laub AJ, Wette M (1989) A stability-enhancing scaling procedure for Schur-Riccati solvers. Syst Contr Lett 12:241–250. https://doi.org/10.1016/0167-6911(89)90056-X
Kleinman DL (1968) On an iterative technique for Riccati equation computations. IEEE Trans Autom Contr AC 13:114–115. https://doi.org/10.1109/TAC.1968.1098829
Lancaster P, Rodman L (1980) Existence and uniqueness theorems for the algebraic Riccati equation. Int J Contr 32:285–309. https://doi.org/10.1080/00207178008922858
Lancaster P, Rodman L (1995) The algebraic Riccati equation. Oxford University Press, Oxford
Lancaster P, Ran ACM, Rodman L (1986) Hermitian solutions of the discrete algebraic Riccati equation. Int J Contr 44:777–802. https://doi.org/10.1080/00207178608933632
Lancaster P, Ran ACM, Rodman L (1987) An existence and monotonicity theorem for the discrete algebraic matrix Riccati equation. Linear Multilinear Algebra 20:353–361. https://doi.org/10.1080/03081088708817768
Lanzon A, Feng Y, Anderson BDO, Rotkowitz M (2008) Computing the positive stabilizing solution to algebraic Riccati equations with an indefinite quadratic term via a recursive method. IEEE Trans Autom Contr AC 53(10):2280–2291. https://doi.org/10.1109/TAC.2008.2006108
Laub AJ (1979) A Schur method for solving algebraic Riccati equations. IEEE Trans Autom Contr AC 24(6):913–921. https://doi.org/10.1109/CDC.1978.267893
Leibfritz F, Lipinski W (2004) COMPl\(_e\)ib 1.0 – User manual and quick reference. Tech. rep., Department of Mathematics, University of Trier, D–54286 Trier, Germany
Mehrmann V (1991) The autonomous linear quadratic control problem. Theory and numerical solution. In: Thoma M, Wyner A (eds) LNCIS, vol 163. Springer-Verl, Heidelberg
Mehrmann V, Tan E (1988) Defect correction methods for the solution of algebraic Riccati equations. IEEE Trans Autom Contr AC 33(7):695–698. https://doi.org/10.1109/9.1282
Pappas T, Laub AJ, Sandell NR (1980) On the numerical solution of the discrete-time algebraic Riccati equation. IEEE Trans Autom Contr AC 25(4):631–641. https://doi.org/10.1109/TAC.1980.1102434
Roberts J (1980) Linear model reduction and solution of the algebraic Riccati equation by the use of the sign function. Int J Contr 32:667–687. https://doi.org/10.1080/00207178008922881
Sima V (1996) Algorithms for linear-quadratic optimization. In: Taft EJ, Nashed Z (eds) Pure and applied mathematics: a series of monographs and textbooks, vol 200. Marcel Dekker, Inc., New York
Sima V (2013) Solving discrete-time algebraic Riccati equations using modified Newton’s method. In: 6th International scientific conference on physics and control, San Luis PotosÃ, Mexico
Sima V (2013) Solving SLICOT benchmarks for algebraic Riccati equations by modified Newton’s method. In: 17th joint international conference on system theory, control and computing, Sinaia, Romania. IEEE, pp 491–496. https://doi.org/10.1080/00207178008922881
Sima V (2014) Efficient computations for solving algebraic Riccati equations by Newton’s method. In: Matcovschi MH, Ferariu L, Leon F (eds) 2014 18th joint international conference on system theory, control and computing, Sinaia, Romania. IEEE, pp 609–614. https://doi.org/10.1109/ICSTCC.2014.6982483
Sima V (2015) Computational experience with a modified Newton solver for continuous-time algebraic Riccati equations. In: Ferrier JL, Gusikhin O, Madani K, Sasiadek J (eds) Informatics in control, automation and robotics, LNEE, vol 325. Springer International Publishing, pp 55–71. https://doi.org/10.1007/978-3-319-10891-9_3
Sima V, Benner P (2006) A SLICOT implementation of a modified Newton’s method for algebraic Riccati equations. In: 14th Mediterranean conference on control and automation, Ancona, Italy, Omnipress. https://doi.org/10.1109/MED.2006.328740
Sima V, Benner P (2008) Experimental evaluation of new SLICOT solvers for linear matrix equations based on the matrix sign function. In: 9th IEEE international symposium on computer-aided control systems design, San Antonio, Texas, U.S.A., Omnipress, pp 601–606. https://doi.org/10.1109/CACSD.2008.4627361
Sima V, Benner P (2014) Numerical investigation of Newton’s method for solving continuous-time algebraic Riccati equations. In: Ferrier JL, Gusikhin O, Madani K, Sasiadek J (eds) 11th international conference on informatics in control, automation and robotics, Vienna, Austria, vol 1. SciTePress, Portugal, pp 404–409. https://doi.org/10.5220/0005117004040409
Sima V, Benner P (2015) Solving SLICOT benchmarks for continuous-time algebraic Riccati equations by Hamiltonian solvers. In: 2015 19th international conference on system theory, control and computing, Cheile Gradistei - Fundata Resort, Romania. IEEE, pp 1–6. https://doi.org/10.1109/ICSTCC.2015.7321260
Sima V, Benner P (2018) Numerical investigation of Newton’s method for solving discrete-time algebraic Riccati equations. In: Madani K, Gusikhin OY (eds) 15th international conference on informatics in control, automation and robotics, Porto, Portugal, vol 1. SciTePress, pp 66–75
Van Dooren P (1981) A generalized eigenvalue approach for solving Riccati equations. SIAM J Sci Stat Comput 2(2):121–135. https://doi.org/10.1137/0902010
Van Huffel S, Sima V, Varga A, Hammarling S, Delebecque F (2004) High-performance numerical software for control. IEEE Contr Syst Mag 24(1):60–76. https://doi.org/10.1109/MCS.2004.1272746
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Sima, V., Benner, P. (2020). Computational Experience with a Modified Newton Solver for Discrete-Time Algebraic Riccati Equations. In: Gusikhin, O., Madani, K. (eds) Informatics in Control, Automation and Robotics. ICINCO 2018. Lecture Notes in Electrical Engineering, vol 613. Springer, Cham. https://doi.org/10.1007/978-3-030-31993-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-31993-9_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-31992-2
Online ISBN: 978-3-030-31993-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)