Model reduction through an Hankel problem and finite state machines

Abstract

(cont.) computation required to produce the approximation is O((n2̂)/[epsilon]) where [epsilon] is the desired ℓ[omega] error and n is the original system order.

The problem of linear time invariant model reduction seeks to transform a given model into a model that has fidelity to the original but allows for easier completion of desired tasks such as controller design and simulation. Hankel-norm approximation consistently performs very well in terms of H[omega] error and provides lower bounds on how well any model can approximate the given model; however, it requires substantial and sometimes prohibitive computation to produce the reduced system and calculate the lower bounds. Here we present a Hankel like approximation problem that allows easier computation of lower bounds. It is shown that the lower bounds produced by the new method do a reasonable job of approximating the lower bounds produced by Hankel-norm approximation. On the negative side, It is also shown that, for the new Hankel problem, there can be no theorem analogous to the major theorem of Hankel-norm approximation that actually produces a reduced model. For nonlinear model reduction, model order does not always predict how difficult it is to perform desired tasks, so we introduce the idea of using finite state machines to approximate models. Lower state count for a finite machine indicates lower computational time to perform many tasks. First, we show, through finite state machine approximation of (1/s+1), that finite state machines are feasible as approximations. That is to say that the amount of states required to approximate a system does not blow up as desired fidelity is increased. We then show that for a given class of linear time invariant models we can set a desired ℓ[omega] error and then find the finite state machine with the minimal number of states that achieves the desired error level. Moreover, the