A Survey of Model Reduction by Balanced Truncation and Some New Results

Abstract Balanced truncation is one of the most common model reduction schemes. In this note, we present a survey of balancing related model reduction methods and their corresponding error norms, and also introduce some new results. Five balancing methods are studied: (1) Lyapunov balancing, (2) stochastic balancing, (3) bounded real balancing, (4) positive real balancing and (5) frequency weighted balancing. For positive real balancing, we introduce a multiplicative-type error bound. Moreover, for a certain subclass of positive real systems, a modified positive-real balancing scheme with an absolute error bound is proposed. We also develop a new frequency-weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians. Two numerical examples are illustrated to verify the efficiency of the proposed methods. Acknowledgement This work was supported in part by NSF through Grants DMS-9972591, CCR-9988393 and ACI-0082645. Notes † Originally stochastic balancing was introduced as a spectral factor-based algorithm, i.e. given a positive real function G, the method approximates the spectral factor V of Φ where Φ = VV ∼ = G + G ∼ which results in solving two Riccati equations. Later the method is generalized and the stochastic balanced reduction is defined as approximating V given V, which results in solving one Lyapunov and one Riccati equation (see Zhou Citation1995, Zhou et al. Citation1999, Varga Citation2000, Benner et al. Citation2001). In this note, by stochastic balancing, we mean the latter which only requires that the original model is square and invertible. We will discuss the former version of the stochastic balancing, which requires solving two Riccati equations, under the name positive real balancing. These issues will be clarified throughout the text. ‡ G(s) in (Equation1) is called asymptotically stable if , and is called stable if where denotes the real part of λ. § G(s) in (Equation1) is called minimal if the pair (A, B) is reachable and the pair (C, A) is observable. † For better comparison, the highest singular values, i.e. σ1, π1 and are normalized to 1.