engleski

On the invariant subspace approach to solving the Riccati equation

Finding the solution of the continuous algebraic Riccati equation A'X + XA + Q - XGX = 0 is of great interest to the control theory community, and current applications require efficient algorithms in cases where A is a large sparse matrix and Q = C'C, G = BB' are positive semidefinite low-rank matrices. In particular, one is interested in obtaining the stabilizing solution X+, which is the unique positive semidefinite solution that makes the closed-loop matrix A-GX stable. There are several competitive methods to tackle this problem, designed to exploit the expected low-rank structure of the solution. These methods include the Newton-ADI (Alternate Direction Implicit) and the various projection-type methods, usually based on approximations using the Krylov or rational Krylov subspaces generated by the matrices A' and the inverse of A', and the initial (block-)vector C'. In this talk, we follow up on the approach introduced in [Benner97, Ferng97]. They suggested computing a low-dimensional stable invariant subspace of the Hamiltonian matrix H via a symplectic Lanczos procedure and using it for approximating the stabilizing solution of the Riccati equation. We discuss the properties of the Riccati equation that imply the rapid decay in the singular values of its solution, and justify the existence of a low-rank invariant subspace of H that yields a good approximation. We address the questions on how to construct such an approximation, and which are the eigenvalues the Lanczos procedure should be steered towards. Finally, we relate the Krylov methods for computing Hamiltonian eigenspaces to the aforementioned projection-type methods for solving the Riccati equation. This gives some new insights on the latter, in particular on the shift selection in the rational Krylov method.

matrix equations; algebraic Riccati equations; Hamiltonian matrices; invariant subspaces; Krylov subspaces; ADI iteration

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