Riemannian Optimal Model Reduction of Stable Linear Systems

In this paper, we develop a method for solving the problem of minimizing the H
²
error norm between the transfer functions of the original and reduced systems on the product set of the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all the asymptotically stable linear systems. However, it is difficult to develop an algorithm for solving this problem, because the set of stable matrices is highly non-convex. To overcome this issue, we show that the problem can be transformed into a tractable Riemannian optimization problem on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The asymptotic stability of the reduced systems constructed using optimal solutions to our problem is preserved. To solve the reduced problem, the Riemannian gradient and Hessian are derived, and a Riemannian trust-region method is developed. The initial point in the proposed approach is selected using the output from the balanced truncation (BT) method. The numerical experiments demonstrate that our method considerably improves the results given by BT and other methods in terms of the H
²
norm and also provides the reduced systems that are globally near-optimal solutions to the problem of minimizing the H
^∞
error norm. Moreover, we show that our method provides a better reduced model than the BT and other methods from the viewpoint of the frequency response.