Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction

Abstract

We consider the solution of the balancing-related frequency-weighted model and controller reduction problems using accuracy enhanced numerical algorithms. We propose ÿrst new stability-enforcing choices of the frequency-weighted grammians which can guarantee the stability of reduced models for two-sided frequency weights. Then we show that for the frequency-weighted controller reduction problems with standard stability and performance-enforcing frequency weights the computation of the frequency-weighted grammians can be done by solving reduced order Lyapunov equations. For both frequency-weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their Cholesky factors. This allows the extension of the square-root and balancing-free accuracy-enhancing techniques to the frequency-weighted case.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…Rational interpolation methods have been extended to bilinear [24,36,37,39,58,92,98,192] and quadratic-in-state systems [37,38,97,114]. Even though these methods have proven effective for a wide range of problem settings, they are most widely used in circuit theory, such as [23,44,90,184,195], e.g., to analyze and predict signal propagation and interference in electric circuits; in structural mechanics, such as [53,106,174,198,211], to study, e.g., vibration suppression in large structures or behavior of micro-electromechanical systems; and in (optimal) control and controller reduction, such as [11,21,126,185,215,231], e.g., in LQR/LQG control design.…”

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.…”

“…In this way we consider the effect of controller disceretization on grammians (2-7) and (2)(3)(4)(5)(6)(7)(8). At first consider the s-plane to z -plane mapping with bilinear transformation…”

“…Therefore, controller order reduction is one of the main research topics. Controller reduction methods can be divided in two classes: direct [1,2] and indirect [3][4][5][6]. The common philosophy in direct method is to seek to minimize a quadratic performance index subject to the constraint that the controller has fixed degree.…”

Weighted Model Reduction via Interpolation

“…The weighted model reduction problem is tackled most often using variants of frequency-weighted balanced truncation; see Varga and Anderson [2003], Anderson and Liu [2002], Schelfhout and De Moor [2002], Gugercin and Antoulas [2004], Enns [1984], Wang et al [2002], Lin and Chiu [1992], Wang et al [1999], Sreeram and Ghafoor [2005] and references therein. For approaches that seek to minimize a weighted-H 2 error, see Halevi [1992] and Spanos et al [1992].…”