Model reduction of linear time-varying systems over finite horizons

We consider the problem of approximating a linear time-varying p×mp×m discrete-time state space model SS of high dimension by another linear time-varying p×mp×m discrete-time state space model Sˆ of much smaller dimension, using an error criterion defined over a finite time interval. We derive the gradients of the norm of the approximation error and show how this can be solved via a fixed point iteration. We compare this to the classical H2H2 norm approximation problem for the infinite horizon time-invariant case and show that our solution extends this to the time-varying and finite horizon case.