Model order reduction for discrete-time LTI systems using the alignment distance

The alignment distance is a recently introduced (differential-geometric) distance on the manifold of LTI systems of fixed order n and output-input dimension (p,m). In this paper, we formulate model order reduction for discrete-time LTI (MIMO) systems in terms of the alignment distance. The intuition behind our formulation is to consider systems of orders lower than n as boundary points of the mentioned manifold in an appropriate ambient space, and the goal is to find a system of order at most r (on the boundary) closest to a given system of order n, where closeness is measured in the alignment distance. We introduce an algorithm for this minimization problem and give some a-priori error bounds in terms of the Hankel singular values of the system. Interesting relations and resemblances emerge with the popular balanced truncation reduction, which is a method not based on any optimality criterion. We show that in certain cases (but not always) balanced truncation provides a good approximation to reduction based on the alignment distance. In fact, our approach can be considered as a principled attempt to put balanced truncation in an optimization framework, and in doing so we allude to a shortcoming of balanced truncation that highlights an advantage of our approach. The proposed approach is general and can be extended to other classes of systems.