Model reduction by balancing from a behavioral point of view

We illustrate a framework for model reduction by balancing for systems described by high-order differential equations. This point of view is particularly significant in that it allows the computation of a reduced order state space model of a complex system described by higher-order linear differential equations, without the need to compute an intermediate state space model. We do not go into the details in this note, but it can be shown that classical Lyapunov and Riccati balanced model reduction are special cases of the approach pursued here. Moreover, in these two cases many of the computations necessary to obtain a balanced state map can be done polynomially