Solving the algebraic Riccati equation with the matrix sign function. (English) Zbl 0611.65027
The algebraic Riccati equation \(G+A^ TX+XA-XFX=0\) is reduced to a linear matrix equation of the form \(MX=N\) where the matrices M and N are defined by the sign function, Sign(K), of the Hamiltonian matrix \(K=\left[ \begin{matrix} A^ T\quad G\\ F\quad -A\end{matrix} \right]\). An iterative refinement of the matrix-sign-function algorithm and a stopping criterion limiting the effects of rounding errors lead to a stable numerical procedure which compares favorably with current Schur vector- based algorithms [A. Laub, IEEE Trans. Autom. Control AC-24, 913- 921 (1979; Zbl 0424.65013)]. Comparative numerical experiments on three examples are also presented.
Reviewer: S.Mirica
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 15A24 | Matrix equations and identities |
Keywords:
numerical examples; comparison of methods; algebraic Riccati equation; linear matrix equation; Hamiltonian matrix; iterative refinement; stopping criterion; rounding errors; Schur vector-based algorithmsCitations:
Zbl 0424.65013Software:
LINPACKReferences:
| [1] | Anderson, B. D.O., Second order convergent algorithms for the steady state Riccati equation, Internat. J. Control, 28, 295-306 (1978) · Zbl 0385.49017 |
| [2] | Arnold, W. F., Numerical solution of algebraic matrix Riccati equations, (Report NWC TP 6521 (1984), Naval Weapons Center: Naval Weapons Center China Lake, Calif. 93555) |
| [3] | Attarzadeh, F., Block decomposition algorithm for time-invariant systems using the generalized matrix sign function, Internat. J. Systems Sci., 14, 1075-1085 (1983) · Zbl 0512.93025 |
| [4] | Balzer, L. A., Accelerated convergence of the matrix sign function, Internat. J. Control, 32, 1057-1078 (1980) · Zbl 0464.93029 |
| [5] | Barraud, A. Y., Investigations autour de la fonction signe d’une matrice. Application a l’equation de Riccati, RAIRO Automat./Systems Anal. and Control, 13, 335-368 (1979) · Zbl 0424.93062 |
| [6] | Barraud, A. Y., Produit étoile et fonction signe de matrice. Application à l’equation de Riccati dans le cas discret, RAIRO Automat./Systems Anal. and Control, 14, 55-85 (1980) · Zbl 0435.49010 |
| [7] | Beavers, A.; Denman, E., A new solution method for quadratic matrix equations, Math. Biosci., 20, 135-143 (1974) · Zbl 0278.65040 |
| [8] | Bierman, G. J., Computational aspects of the matrix sign function to the ARE, Report (1984), Factorized Estimation Applications, Inc: Factorized Estimation Applications, Inc 7017 Deveron Ridge Rd., Canoga Park, Calif. 91301 |
| [9] | Byers, R., Hamiltonian and Symplectic Algorithms for the Algebraic Riccati Equation, (Ph.D. Thesis (1983), Cornell Univ: Cornell Univ Ithaca, N.Y) |
| [10] | Denman, E.; Beavers, A., The matrix sign function and computations in systems, Appl. Math. Comput., 2, 63-94 (1976) · Zbl 0398.65023 |
| [11] | Denman, E.; Layva-Ramos, J., Spectral decomposition of a matrix using the generalized sign matrix, Appl. Math. Comput., 8, 237-250 (1981) · Zbl 0459.65015 |
| [12] | Higham, N., Computing the polar decomposition—with applications, (Numerical Analysis Report No. 94 (Nov. 1984), Dept. of Mathematics, Univ. of Manchester: Dept. of Mathematics, Univ. of Manchester Manchester M13 9PL, England) · Zbl 0607.65014 |
| [13] | Hammarling, S., Newton’s method for solving the algebraic Riccati equation, (Technical Report DICT 12/82 (1982), National Physics Lab: National Physics Lab Teddington, Middlesex) |
| [14] | Howland, J. L., The sign matrix and the separation of matrix eigenvalues, Linear Algebra Appl., 49, 221-232 (1980) · Zbl 0507.15007 |
| [15] | Kleinman, D., On an iterative technique for Riccati equation computations, IEEE Trans. Automat. Control, 13, 114-115 (1968) |
| [16] | Kwaadernaak, H.; Sivan, R., Linear Optimal Control Systems (1972), Wiley-Interscience: Wiley-Interscience New York · Zbl 0276.93001 |
| [17] | Laub, A., A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, 24, 913-925 (1979) · Zbl 0424.65013 |
| [18] | Lawson, C.; Hanson, R., Solving Least Squares Problems (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N.J · Zbl 0860.65028 |
| [19] | Levine, W.; Athans, M., On the optimal error regulation of a string of moving vehicles, IEEE Trans. Automat. Control, 11, 355-361 (1966) |
| [20] | Dongarra, J.; Moler, C.; Bunch, J.; Stewart, G., LINPACK Users’ Guide (1979), SIAM: SIAM Philadelphia |
| [21] | Lupas, L.; Popeea, C., Solution of differential matrix equations by the matrix sign function, Rev. Roumaine Sci. Techn. Sér. Électrotechn. Énergét, 22, 89-97 (1976) |
| [22] | Matheys, R., Stability analysis via the extended matrix sign function, Proc. Inst. Elec. Engrs., 125, 241-243 (1978) |
| [23] | Potter, J., Matrix quadratic solutions, SIAM J. Appl. Math., 14, 496-501 (1966) · Zbl 0144.02001 |
| [24] | Roberts, J., Linear model reduction and solution of algebraic Riccati equations by use of the sign function, (Engineering Report, CUED/B-Control, Tr-13 (1971), Cambridge Univ: Cambridge Univ Cambridge, England) · Zbl 0463.93050 |
| [25] | Roberts, J., Linear model reduction and solution of the algebraic Riccati equation by the use of the sign function, Internat. J. Control, 32, 677-687 (1980) · Zbl 0463.93050 |
| [26] | Smith, T.; Boyle, J.; Garbow, B.; Ikebe, Y.; Kema, V.; Moler, C., EISPACK Guide (1974), Springer: Springer New York · Zbl 0289.65017 |
| [27] | Stewart, G. W., Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev., 15, 727-764 (1973) · Zbl 0297.65030 |
| [28] | Wilkinson, J., The Algebraic Eigenvalue Problem (1965), Clarendon: Clarendon Oxford · Zbl 0258.65037 |
| [29] | Wonham, W., Linear Multivariable Control: A Geometric Approach (1979), Springer: Springer New York · Zbl 0424.93001 |
| [30] | Yoo, R.; Denman, E., Uncoupling of constant coefficient canonical differential equations of optimal control, (Report DEE (1974), Univ. of Houston: Univ. of Houston Houston, Tex) |
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