Approximate balancing of large scale linear dynamic systems using Krylov subspaces

The present research considers the problem of computing approximate balanced realizations for large scale, linear time invariant dynamic systems. Balanced realization provides an attractive and rational framework for reducing the order of a linear dynamic system. However, it suffers from a major drawback in that it cannot be applied to large-scale dynamic systems. The primary reason being the computational bottleneck that is encountered when solving the pair of Lyapunov equations for the controllability and observability gramians. The present research addresses this problem by providing the theoretical basis and algorithmic details for employing Krylov subspace methods for computing approximate balanced realizations for large-scale, linear dynamic systems.; The computational bottleneck associated with solving the large-scale Lyapunov equations is handled by employing Krylov subspace methods to compute low-rank approximate solutions for it. It is shown that the Lyapunov equation can be solved exactly using the traditional Krylov subspaces typically employed for these problems. However, it is also shown that a variety of other Krylov subspaces and reduced-order Lyapunov equations may be employed to compute the exact solution of the Lyapunov equation. From the standpoint of computing the exact solution, it makes no difference which subspace and Lyapunov equations are employed. However, from the standpoint of approximating the solution, the choice of subspace and Lyapunov equation has a definite impact on the convergence rate and accuracy of the solution obtained. These alternate Krylov subspace methods often produce far better approximations for the gramian than can be obtained using the traditional Krylov subspace.; It is also shown that Krylov subspace methods can be utilized to compute the exact, minimal order, balanced realization. This provides the theoretical basis and justification for employing Krylov subspace methods for computing approximate balanced realizations. Approximations for the Hankel singular values and balanced vectors are obtained by solving low order Lyapunov equations whose solution can be computed using conventional methods. These low order equations are obtained by projecting the Lyapunov equations onto appropriately defined Krylov subspaces. The alternate Krylov subspace methods developed for the solution of the Lyapunov equation continue to apply here, and lead to a variety of methods for generating approximate balance realizations. It is found that far better approximations for the balanced reduced model are obtained when using the modified Krylov subspace methods over that obtained using the traditional Krylov subspace method.