Efficient Solution of Coupled Lyapunov Equations via Matrix Sign Function Iteration
Peter Benner, José Manuel Claver, Enrique S. Quintana–Ort́ı · 1998
The controllability and observability Gramians of timeinvariant linear systems are given by the solutions of two coupled Lyapunov equations. These Gramians play an essential role in many areas of control theory such as computing Hankel singular values, system balaning, and related model reduction algorithms. We investigate the numerical solution of coupled Lyapunov equations by iterations for the matrix sign function, and demonstrate by numerical examples the efficiency of the resulting algorithm on modern computer architectures. As the Cholesky factors of the Gramians are often required rather than the Gramians themselves, special emphasis is given on the direct computation of these factors without forming the Gramians explicitly.