×
Huang, Tsung-Ming; Lin, Wen-Wei

Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations. (English) Zbl 1169.65038

Linear Algebra Appl. 430, No. 5-6, 1452-1478 (2009).
The authors present structured doubling algorithms for the computation of the weakly stabilizing Hermitian solutions of the continuous- and discrete-time algebraic Riccati equations (CARE and DARE), respectively. It is assumed that the partial multiplicities of purely imaginary and unimodular eigenvalues of the associated Hamiltomian and symplectic pencil, respectively, are all even and the C/DARE and the dual C/DARE have weakly stabilizing Hermitian solutions with property (P). Under these assumptions, it is proved that if the algorithms do not break down, they converge to the desired Hermitian solutions globally and linearly. The effectiveness of the algorithms is tested with some numerical experiments.
Reviewer: Sonia Pérez Díaz (Madrid)

Cited in 16 Documents

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities

Keywords:

algebraic Riccati equation; Hermitian solution; structured doubling algorithm; purely imaginary eigenvalue; unimodular eigenvalue; global and linear convergence; Hamiltonian and symplectic pencil; algorithms; numerical experiments

Software:

HQR3; Matlab; mctoolbox; HQR3 EXCHNG
PDF BibTeX XML Cite
\textit{T.-M. Huang} and \textit{W.-W. Lin}, Linear Algebra Appl. 430, No. 5--6, 1452--1478 (2009; Zbl 1169.65038)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Ammar, G.; Mehrmann, V., On Hamiltonian and symplectic Hessenberg forms, Linear algebra appl., 149, 55-72, (1991) · Zbl 0724.65042
[2] Anderson, B.D.O., Second-order convergent algorithms for the steady-state Riccati equation, Int. J. control, 28, 295-306, (1978) · Zbl 0385.49017
[3] B.D.O. Anderson, S. Vongpanitlerd, Network analysis and synthesis, Lecture Notes in Control and Information Science, Prentice-Hall, Englewood Cliffs, NJ, 1973.
[4] Bai, Z.; Demmel, J.; Gu, M., An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems, Numer. math., 76, 279-308, (1997) · Zbl 0876.65021
[5] Balzer, L., Accelerated convergence of the matrix sign function, Int. J. control, 21, 1057-1078, (1980) · Zbl 0464.93029
[6] P. Benner, Contributions to the numerical solutions of algebraic Riccati equations and related eigenvalue Problems. PhD thesis, Fakultät für Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany, 1997. · Zbl 0896.65047
[7] Benner, P.; Byers, R., Evaluating products of matrix pencils and collapsing matrix products, Numer. linear algebra appl., 8, 357-380, (2001) · Zbl 1055.65053
[8] Byers, R., Solving the algebraic Riccati equation with the matrix sign function, Linear algebra appl., 85, 267-279, (1987) · Zbl 0611.65027
[9] Cardoso, J.R.; Silva Leite, F., The moser – veselov equation, Linear algebra appl., 360, 237-248, (2003) · Zbl 1020.15016
[10] Chu, D.; Liu, X.; Mehrmann, V., A numerical method for computing the Hamiltonian Schur form, Numer. math., 105, 375-412, (2007) · Zbl 1116.65043
[11] Chu, E.K.-W.; Fan, H.-Y.; Lin, W.-W., A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations, Linear algebra appl., 396, 55-80, (2005) · Zbl 1151.93340
[12] Chu, E.K.-W.; Fan, H.-Y.; Lin, W.-W.; Wang, C.-S., Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations, Int. J. control, 77, 767-788, (2004) · Zbl 1061.93061
[13] Clements, D.; Glover, K., Spectral transformations via Hermitian pencils, Linear algebra appl., 123, 797-846, (1989) · Zbl 0678.93006
[14] Denman, E.; Beavers, R., The matrix sign function and computations in systems, Appl. math. comput., 2, 63-94, (1976) · Zbl 0398.65023
[15] Francis, B.A.; Doyle, J.C., Linear control theory with an \(H^\infty\)-optimality criterion, SIAM J. control optim., 25, 815-844, (1987) · Zbl 0628.93011
[16] Gardiner, J.; Laub, A.J., A generalization of the matrix-sign-function solution to the algebraic Riccati equations, Int. J. control, 44, 823-832, (1986) · Zbl 0598.15012
[17] Glover, K.; Limebeer, D.J.N.; Doyle, J.C.; Kasenally, E.M.; Safonov, M.G., A characterization of all solutions to the fourblock general distance problem, SIAM J. control optim., 39, 283-324, (1991) · Zbl 0736.93015
[18] Gohberg, I.; Lancaster, P.; Rodman, L., On the Hermitian solution of the symmetric algebraic Riccati equation, SIAM J. control optim., 24, 1323-1334, (1986) · Zbl 0607.93013
[19] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), The Johns Hopkins University Press · Zbl 0865.65009
[20] Guo, C.-H., Newton’s method for discrete algebraic Riccati equations with the closed-loop matrix has eigenvalues on the unit circle, SIAM J. matrix anal. appl., 20, 279-294, (1998) · Zbl 0923.65018
[21] Guo, C.-H.; Lancaster, P., Analysis and modificaton of newton’s method for algebraic Riccati equations, Math. comput., 67, 1089-1105, (1998) · Zbl 0903.65047
[22] Guo, X.-X.; Lin, W.-W.; Xu, S.-F., A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation, Numer. math., 103, 393-412, (2006) · Zbl 1097.65055
[23] Higham, N.J., Accuracy and stability of numerical algorithms, (2002), SIAM Philadelphia · Zbl 1011.65010
[24] Hwang, T.-M.; Chu, E.K.-W.; Lin, W.-W., A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations, Int. J. control, 78, 1063-1075, (2005) · Zbl 1090.93014
[25] Ionescu, V.; Weiss, M., Two Riccati formulae for the discrete-time \(H^\infty\)-control problem, Int. J. control, 57, 141-195, (1993) · Zbl 0774.93026
[26] Kimura, M., Convergence of the doubling algorithm for the discrete-time algebraic Riccati equation, Int. J. syst. sci., 19, 701-711, (1988) · Zbl 0645.93012
[27] Kimura, M., Doubling algorithm for continuous-time algebraic Riccati equation, Int. J. syst. sci., 20, 191-202, (1989) · Zbl 0667.93035
[28] Lancaster, P.; Ran, A.C.M.; Rodman, L., Hermitian solutions of the discrete algebraic Riccati equation, Int. J. control, 44, 777-802, (1986) · Zbl 0598.15011
[29] Lancaster, P.; Rodman, L., Existence and uniqueness theorems for the algebraic Riccati equation, Int. J. control, 32, 285-309, (1980) · Zbl 0439.49011
[30] Laub, A.J., A Schur method for solving algebraic Riccati equations, IEEE trans. autom. control, 24, 913-921, (1979) · Zbl 0424.65013
[31] Lin, W.-W., A new method for computing the closed-loop eigenvalues of a discrete-time algebraic riccatic equation, Linear algebra appl., 96, 157-180, (1987) · Zbl 0649.65040
[32] Lin, W.-W.; Mehrmann, V.; Xu, H., Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear algebra appl., 302-303, 469-533, (1999) · Zbl 0947.15004
[33] Lin, W.-W.; Wang, C.-S., On computing stable largangian subspaces of Hamiltonian matrices and symplectic pencils, SIAM J. matrix anal. appl., 18, 590-614, (1997) · Zbl 0876.47005
[34] Lin, W.-W.; Wang, C.-S.; Xu, Q.-F., Numerical computation of the minimum \(H_\infty\) norm of the discrete-time output feedback control problem, SIAM J. numer. anal., 38, 515-547, (2000) · Zbl 0969.93015
[35] Lin, W.-W.; Xu, S.-F., Convergence analysis of structure-preserving doubling algorithms for Riccati-tye matrix equations, SIAM J. matrix anal. appl., 28, 1, 26-39, (2006) · Zbl 1116.65051
[36] Malyshev, A.N., Parallel algorithm for solving some spectral problems of linear algebra, Linear algebra appl., 188/189, 489-520, (1993) · Zbl 0782.65056
[37] MathWorks, MATLAB user’s guide (for UNIX Workstations), The Math Works, Inc., 2002.
[38] Mehrmann, V., The autonomous linear quadratic control problem, (1991), Springer-Verlag
[39] Mehrmann, V., A step toward a unified treatment of continuous and discrete time control problems, Linear algebra appl., 241-243, 749-779, (1996) · Zbl 0859.93013
[40] Moser, J.; Veselov, A.P., Discrete versions of some classical integrable systems and factorization of matrix polynomials, Commun. math. phys., 139, 217-243, (1991) · Zbl 0754.58017
[41] Paige, C.; Van Loan, C., A Schur decomposition for Hamiltonian matrices, Linear algebra appl., 41, 11-32, (1981) · Zbl 1115.15316
[42] Pappas, T.; Laub, A.J.; Sandel, N.R., On the numerical solution of the discrete-time algebraic Riccati equation, IEEE trans. autom. control, 25, 631-641, (1980) · Zbl 0456.49010
[43] Stewart, G.W., HQR3 and EXCHNG: Fortran subroutines for calulating and ordering the eigenvalues of a real upper Hessenberg matrix, ACM trans. math. software, 2, 275-280, (1976)
[44] Stewart, G.W.; Sun, J.-G., Matrix perturbation theory, (1990), Academic Press New York
[45] Sun, J.-G., Perturbation theory for algebraic Riccati equations, SIAM J. matrix anal. appl., 19, 39-65, (1998) · Zbl 0914.15009
[46] Sun, J.-G., Condition numbers of algebraic Riccati equations in the Frobenius norm, Linear algebra appl., 350, 237-261, (2002) · Zbl 1003.15004
[47] Van Der Schaft, A.J.; Willems, J.C., A new procedure for stochastic realization of spectral density matrices, SIAM J. control optim., 22, 845-855, (1984) · Zbl 0551.93059
[48] Van Dooren, P., A generalized eigenvalue approach for solving Riccati equations, SIAM J. sci. stat. comput., 2, 121-135, (1981) · Zbl 0463.65024
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.