H₂ order-reduction for bilinear systems based on Grassmann manifold

Abstract

In this paper, we focus on optimal H₂ model reduction methods for bilinear systems. The optimal H₂ model reduction problem can be viewed as a minimization problem on Grassmann manifold. By utilizing the geometry of Grassmann manifold, a gradient descent model reduction algorithm and a conjugate gradient model reduction algorithm based on Grassmann manifold are given. These two algorithms can simulate well the behavior of the original system and the order-reduced systems generated by our algorithms can satisfy the minimal stability requirement. Numerical examples are given to illustrate the effectiveness of our methods.

Introduction

The model reduction methods, which approximate a complex dynamical system with higher order by a lower order system, have received considerable attention in recent years. For linear systems, the problem is well studied. Generally speaking, there are two types of methods: one is based on projection methods, such as Krylov subspace methods [1], [2]; the other is the optimization-based methods, such as the H₂ and L₂ optimal model reduction methods [3], [4] and the $H_{\infty }$ optimal model reduction method [5]. In many applications, the linear models are often insufficient to describe the behavior of some physical processes. Recently, more and more attention has been paid to nonlinear systems. However, bilinear systems have been pointed out to be an interesting interface between linear and fully nonlinear dynamical systems. They can represent an important class of nonlinear systems [6], [7], [8].

It is known that bilinear systems exhibit close relationship with linear systems, whose many concepts known from linear order-reduced systems have shown to possess bilinear analogs. Various efficient model reduction methods for bilinear systems have been proposed. The methods based on balancing were suggested in [9], [10], which require to solve large-scale Lyapunov equations. Another method is based on the interpolation algorithm [11], [12]. However, compared with the balanced truncation method, they often perform worse approximation. Similar to linear systems, there exists optimal H₂ model reduction method to reduce bilinear systems, which was first introduced in [13] and was further studied in [14]. Recently, the fast algorithms based on Grassmann manifold were presented to reduce linear systems [15].

Inspired by them, for bilinear systems, we propose a couple of H₂ model reduction algorithms based on Grassmann manifold. Specifically, the optimal H₂ model reduction problem is viewed as an unconstrained minimization problem on Grassmann manifold. And we describe the geodesic equation on Grassmann manifold. Then, a gradient descent model reduction algorithm based on Grassmann manifold is proposed. In the process of solving the H₂ minimization problem, the benefits of the gradient descent algorithm can be fully reflected. In order to improve the convergence speed, we propose a conjugate gradient model reduction algorithm based on Grassmann manifold. The formula for parallel translation along geodesic is given. On Grassmann manifold, these algorithms perform minimization along the geodesic. It should be noted that the order-reduced systems generated by our algorithms can satisfy the minimal stability requirement.

This paper is organized as follows. In Section 2, we give a brief review on the H₂ model reduction method for bilinear systems. In Section 3, for bilinear systems, we focus on the H₂ model reduction algorithms based on Grassmann manifold. A gradient descent model reduction algorithm and a conjugate gradient model reduction algorithm based on Grassmann manifold are proposed. In Section 4, numerical examples are given. Finally, some conclusions are drawn in Section 5.