AN EXTENDED BLOCK ARNOLDI ALGORITHM FOR LARGE-SCALE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATI ON ∗

We present a new iterative method for the computation of approximate solutions to large-scale continuous-time algebraic Riccati equations. The proposed method is a projection method onto an extended block Krylov subspace, which can be seen as a sum of two block Krylov subspaces in A and A 1 . We give some the- oretical results and present numerical experiments for large and sparse problems. These numerical tests show the efficiency of the proposed scheme as compared to the block Arn oldi and Newton-ADI methods.

[1]  P. Benner,et al.  An Implicitly Restarted Symplectic Lanczos Method for the Hamiltonian Eigenvalue Problem , 1997 .

[2]  I. Jaimoukha,et al.  Krylov subspace methods for solving large Lyapunov equations , 1994 .

[3]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[4]  Volker Mehrmann,et al.  A symplectic orthogonal method for single input or single ouput discrete time optimal quadratic control problems , 1988 .

[5]  A. Antoulas.  5. Linear Dynamical Systems: Part 2 , 2005 .

[6]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[7]  Khalide Jbilou,et al.  An Arnoldi based algorithm for large algebraic Riccati equations , 2006, Appl. Math. Lett..

[8]  Ralph Byers,et al.  Hamiltonian and symplectic algorithms for the algebraic riccati equation , 1983 .

[9]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[10]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[11]  A. Laub,et al.  Matrix-sign algorithms for Riccati equations , 1992 .

[12]  R. Byers.  Solving the algebraic Riccati equation with the matrix sign function , 1987 .

[13]  Valeria Simoncini,et al.  A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations , 2007, SIAM J. Sci. Comput..

[14]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[15]  E. Flytzanis.  Linear dynamical systems , 1976 .

[16]  Peter Benner,et al.  An exact line search method for solving generalized continuous-time algebraic Riccati equations , 1998, IEEE Trans. Autom. Control..

[17]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[18]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[19]  W. Wonham.  On a Matrix Riccati Equation of Stochastic Control , 1968 .

[20]  B. Anderson,et al.  Linear Optimal Control , 1971 .

[21]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[22]  Jan G. Korvink,et al.  Model Order Reduction for Linear Convective Thermal Flow , 2004 .

[23]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[24]  S. Bittanti,et al.  The Riccati equation , 1991 .

[25]  Khalide Jbilou,et al.  Block Krylov Subspace Methods for Large Algebraic Riccati Equations , 2003, Numerical Algorithms.

[26]  B. Datta.  Numerical methods for linear control systems : design and analysis , 2004 .

[27]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[28]  J. D. Roberts,et al.  Linear model reduction and solution of the algebraic Riccati equation by use of the sign function , 1980 .

[29]  J. Demmel,et al.  Using the Matrix Sign Function to Compute Invariant Subspaces , 1998, SIAM J. Matrix Anal. Appl..

[30]  Chun-Hua Guo,et al.  Analysis and modificaton of Newton's method for algebraic Riccati equations , 1998, Math. Comput..

[31]  Thilo Penzl,et al.  A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations , 1998, SIAM J. Sci. Comput..

[32]  P. Dooren.  A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[33]  Peter Benner,et al.  On the Parameter Selection Problem in the Newton-ADI Iteration for Large Scale Riccati Equations , 2007 .