A new frequency-weighted balanced truncation method and an error bound

Abstract

In this paper, a new frequency-weighted model reduction method with an a priori error bound is proposed. The method is a generalization of Enns technique and yields stable models even when both input and output weightings are included. The proposed method is compared with other existing methods using numerical examples.

Frequency Weighted Discrete-Time Controller Order Reduction Using Bilinear Transformation

“…Although our method is proposed for discrete-time systems, Tab. 1 compares the approximation errors ∥W (s)(K(s) − K r (s))V (s)∥ ∞ obtained using Enns [3] method, Wang et als [10] method, Varga and Andersons [5] method and our new method, both in s-plane and z -plane. For a practical closed loop control system, it is necessary to discretize the controller part in order to implement it in a digital hardware.…”

“…However, with both weightings, the method may yield unstable models. In order to overcome this problem, several methods such as [6,[8][9][10] were introduced. Karimaghaee et al [8] developed a method for model reduction based on definition of controllability and observability grammians in the frequency domain.…”

A survey of balancing methods for model reduction

“…The conclusion is that there is a trade-off between the guaranteed stability and the performance in the specified frequency interval. This is also valid for Wang's et al modification to Enns' methods to guarantee stability [18]. The H ∞ norms and the upper bounds are presented in Table 2-b.…”

“…Then, the reduced model is obtained by truncating the states which have this property. Besides the Lyapunov balancing method, other types of balancing exist, namely stochastic balancing, first proposed by Desai and Pal [4] for balancing stochastic systems and later generalized by Green [8,9]; bounded real balancing [16], applied to the bounded-real systems; positive real balancing [4], applied for model reduction of positive real (passive) systems; and frequency weighted balancing [5,12,18,20,6] applied to minimize the weighted error over a selected frequency region.…”

Simultaneous balancing and model reduction of switched linear systems

“…In addition, frequency weighted balancing has been developed to approximate the system over a range of frequencies. [3], [9], [14] and [15] provide different schemes for frequency weighted balancing. Another category of model reduction approaches is Krylov based methods which are based on moment matching.…”