Abstract
In the present paper, we present numerical methods for the computation of approximate solutions to large continuous-time and discrete-time algebraic Riccati equations. The proposed methods are projection methods onto block Krylov subspaces. We use the block Arnoldi process to construct an orthonormal basis of the corresponding block Krylov subspace and then extract low rank approximate solutions. We consider the sequential version of the block Arnoldi algorithm by incorporating a deflation technique which allows us to delete linearly and almost linearly dependent vectors in the block Krylov subspace sequences. We give some theoretical results and present numerical experiments for large problems.
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References
B.D.O. Anderson and J.B. Moore, Linear Optimal Control (Prentice-Hall, Englewood Cliffs, NJ, 1971).
W.F. Arnold, III and A.J. Laub, Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proc. 72 (1984) 1746–1754.
Z. Bai and J. Demmel, Using the matrix sign function to compute invariant subspaces, SIAM J. Anal. Appl. 19 (1998) 205–225.
P. Benner and R. Byers, An exact line search method for solving generalized continous algebraic Riccati equations, IEEE Trans. Automat. Control 43(1) (1998) 101–107.
P. Benner and H. Faβbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl. 263 (1997) 75–111.
P. Benner, A. Laub and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Technical Report SPC 95-22, Fak. f. Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany (1995).
A. Bunse-Gerstner and V. Mehrmann, A symplectic QR-like algorithm for the solution of the real algebraic Riccati equations, IEEE Trans. Automat. Control 43(1) (1998) 1104–1113.
R. Byers, A Hamiltonian QR-algorithm, SIAM J. Sci. Statist. Comput. 7 (1986) 212–229.
R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl. 85 (1987) 267–279.
J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis, State-space solutions to standar \({\mathcal{H}}_2\) and \({\mathcal{H}}_\infty\) control problems, IEEE Trans. Automat. Control 34(8) (1989) 831–846.
E.J. Grimme, D.C. Sorenson and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numer. Algorithms 12 (1996) 1–31.
J.J. Hench and A.J. Laub, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Automat. Control 39 (1994) 1197–1209.
I.M. Jaimoukha and E.M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal. 31 (1994) 227–251.
C.S. Kenny, A.J. Laub and P. Papadopoulos, Matrix sign function algorithms for Riccati equations, in: Proc. of IMA Conf. on Control: Modelling, Computation, Information, Southend-on-Sea (IEEE Computer Soc. Press, Los Alamitos, CA, 1992) pp. 1–10.
D.L. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Automat. Control 13 (1968) 114–115.
P. Lancaster and L. Rodman, The Algebraic Riccati Equations (Clarendon Press, Oxford, 1995).
A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control 24 (1979) 913–921.
V. Mehrmann, The Autonomous Linear Quadratic Problem, Theory and Numerical Solution, Lecture Notes in Control and Information Sciences, Vol. 163 (Springer, Heidelberg, 1991).
D.J. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control 32 (1998) 677–687.
A. Ruhe, Rational Krylov sequence methods for eigenvalue computations, Linear Algebra Appl. 58 (1984) 391–405.
Y. Saad, Numerical solution of large Lyapunov equations, in: Signal Processing, Scattering, Operator Theory and Numerical Methods, Proc. of the Internat. Symposium MTNS-89, Vol. 3, eds. M.A. Kaashoek, J.H. van Schuppen and A.C. Ran (Birkhäuser, Boston, 1990) pp. 503–511.
M. Sadkane, Block Arnoldi and Davidson methods for unsymmetric large eigenvalue problems, Numer. Math. 64 (1993) 687–706.
D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13(1) (1992) 357–387.
G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory (Academic Press, New York, 1990).
J.-G. Sun, Perturbation theory for algebraic Riccati equation, SIAM J. Matrix Anal. Appl. 19(1) (1998) 39–65.
J.-G. Sun, Residual bounds of approximate solutions of the algebraic Riccati equation, Numer. Math. 76 (1997) 249–263.
P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Statist. Comput. 2 (1981) 121–135.
C.F. Van Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl. 16 (1984) 233–251.
W.M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control 6 (1968) 681–697.
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Jbilou, K. Block Krylov Subspace Methods for Large Algebraic Riccati Equations. Numerical Algorithms 34, 339–353 (2003). https://doi.org/10.1023/B:NUMA.0000005349.18793.28
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DOI: https://doi.org/10.1023/B:NUMA.0000005349.18793.28