Oblique Production Methods for Large Scale Model Reduction

The aim of this paper is to consider approximating a linear transfer function $F( s )$ of McMillan degree N, by one of McMillan degree m in which $N \gg m$ and where N is large. Krylov subspace methods are employed to construct bases to parts of the controllability and observability subspaces associated with the state space realisation of $F( s )$. Low rank approximate grammians are computed via the solutions to low dimensional Lyapunov equations and computable expressions for the approximation errors incurred are derived. We show that the low rank approximate grammians are the exact grammians to a perturbed linear system in which the perturbation is restricted to the transition matrix, and furthermore, this perturbation has at most rank = 2. This paper demonstrates that this perturbed linear system is equivalent to a low dimensional linear system with state dimension no greater than m. Finally, exact low dimensional expressions for the $\mathcal{L}^\infty $ norm of the errors are derived. The model reduction of discrete time linear systems is considered via the use of the same Krylov schemes. Finally, the behaviour of these algorithms is illustrated on two large scale examples.