Riemannian optimal model reduction of stable linear systems

TL;DR

This paper introduces a Riemannian optimization approach for optimal model reduction of stable linear systems, improving upon balanced truncation by minimizing the $H^2$ error norm while preserving stability.

Contribution

It transforms the non-convex stable matrix set into a Riemannian manifold, enabling efficient optimization for better reduced models compared to traditional methods.

Findings

Significantly improves $H^2$ norm reduction over balanced truncation.

Provides globally near-optimal solutions for $H^{ }$$ error minimization.

Achieves better frequency response in reduced models.

Abstract

In this paper, we develop a method for solving the problem of minimizing the $H^2$ error norm between the transfer functions of original and reduced systems on the set of stable matrices and two Euclidean spaces. That is, we develop a method for identifying the optimal reduced system from all 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 on the product manifold of the set of skew-symmetric matrices, the manifold of the symmetric positive-definite matrices, and two Euclidean spaces. The stability of the reduced systems constructed using the 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. Numerical experiments demonstrate that our method considerably improves the results given by BT in the sense of the $H^2$ norm, and also provides reduced systems that are globally near-optimal solutions to the problem of minimizing the $H^{\infty}$ error norm. Moreover, we show that our method provides a better reduced model than BT from the viewpoint of the frequency response.