L2 optimal model reduction

Abstract

This paper deals with the problem of computing an L2-optimal reduced-order model for a given stable multivariable linear system. By way of orthogonal projection, the problem is formulated as that of minimizing the L2 model reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation. The closed form formula for the gradient of the cost over the manifold is derived, from which a gradient flow is formed as an ordinary differential equation. A number of nice properties about such a flow are obtained. Among them are the decreasing property of the cost along the ODE solution and the convergence of the flow from any starting point in the manifold. Furthermore, an explicit iterative convergent algorithm is developed from the flow and inherits the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model-reduction cost is decreasing to minimums along the iterates.