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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

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
Full Text: DOI
[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, ()
[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 7017 Deveron Ridge Rd., Canoga Park, Calif. 91301
[9] Byers, R., Hamiltonian and symplectic algorithms for the algebraic Riccati equation, ()
[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, () · Zbl 0607.65014
[13] Hammarling, S., Newton’s method for solving the algebraic Riccati equation, ()
[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 New York
[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 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., {\sclinpack} users’ guide, (1979), 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, () · 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., {\sceispack} guide, (1974), Springer New York
[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 Oxford · Zbl 0258.65037
[29] Wonham, W., Linear multivariable control: A geometric approach, (1979), Springer New York · Zbl 0424.93001
[30] Yoo, R.; Denman, E., Uncoupling of constant coefficient canonical differential equations of optimal control, ()
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