Model Order Reduction in the Alignment Distance and Metrization of the Kalman Decomposition

In this paper, we formulate the problem of model order reduction for LTI (MIMO) dynamical systems in terms of the alignment distance. The significance of this formulation is that it establishes a natural quantitative link between the “Kalman canonical decomposition” and “model order reduction,” and fills an existing gap in this regard. The alignment distance includes a large class of state-space based distances on the manifold of systems of fixed minimal order n and output-input dimension (p,m); it is a natural distance associated with the quotient space structure of this manifold. The intuition behind our formulation is to consider systems of orders lower than n as points on the boundary 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. In materializing this idea, certain theoretical and computational challenges arise, which will be addressed (e.g., while we have to extend the alignment distance to the boundary, the entire of the boundary is not metrizable; hence, we pass to a subset of the boundary, called diagonalizable s-balanced systems, and establish its metrizability). Ultimately, a computationally-friendly problem is formulated, for which, using methods of optimization on manifolds, we introduce an efficient algorithm called Align, Truncate, and Project (ATP). We also give some a-priori error bounds in terms of the Hankel singular values of the system. Interesting connections emerge with the popular balanced truncation method, which is a method not based on any optimality criterion. Our approach is applicable to both stable and unstable systems, and we establish robustness of feedback stability in the alignment distance.