3.1 Introduction

Model order reduction methods for linear and non-linear dynamic systems in general can be classified into two categories [6]:

1. Singular-value-decomposition (SVD) based approaches

2. Krylov-subspace-based approaches.

Krylov-subspace-based methods have been reviewed in Chapter 2. In this chapter, we focus on the SVD-based reduction methods. Singular value decomposition is based on the lower rank approximation, which is optimal in the 2-norm sense. The quantities for deciding how a given system can be approximated by a lower-rank system are called singular values, which are the square roots of the eigenvalues of the product of the system matrix and its adjoint. The major advantage of SVD-based approaches over Krylov subspace methods lies in their ability to ensure the errors satisfying an a-priori upper bound. Also, SVD-based methods typically lead to optimal or near optimal reduction results as the errors are controlled in a global way. However, SVD-based methods suffer the scalability issue as SVD is a computational intensive process and cannot deal with very large dynamic systems in general. In contrast, Krylov-subspace-based methods can scale to reduce vary large systems due to efficient computation methods for moment vectors and their orthogonal forms.

SVD-based approaches consist of several reduction methods [6]. In this chapter, we mainly focus on the truncated-balanced-realization (TBR) approach and its variants, which were first introduced by Moore [81]. In the TBR method, a system is mapped onto a basis where the states that are difficult to reach are also difficult to observe. Then, the reduced...