Error bounded Padé approximation via bilinear conformal transformation

Abstract

Since Asymptotic Waveform Evaluation AWE was introduced in 5 , many i n terconnect model order reduction methods via Pad e approximation have been proposed. Although the stability and precision of model reduction methods have been greatly improved, the following important question has not been answered: "What is the error bound in the time domain?". This problem is mainly caused by the gap" between the frequency domain and the time domain, i.e. a good approximated transfer function in the frequency domain may not be a good approximation in the time domain. All of the existing methods approximate the transfer function directly in the frequency domain and hence can not provide error bounds in the time domain. In this paper, we present new moment matching methods which can provide guaranteed error bounds in the time domain. Our methods are based on the classic work by Teasdale in 1 which performs Pad e approximation in a transformed domain by the bilinear conformal transformation s = 1,z 1+z .

Laguerre-SVD reduced order modeling

“…It is seen that the overall convergence rate is dictated by the largest coefficient and hence the optimal may be found as the solution to the minimax problem (23) It should be noted that the optimal thus obtained is also the value that maximizes the radius of convergence of the power series in the r.h.s of (14), in agreement with the asymptotic theory developed in [17]. Since the minimum of is obtained for , it is a simple matter to show that the solution of the minimax problem (23) is given by , where is the solution to the discrete minimax problem (24) It has been indicated in [2] and [18] that , where is the bandwidth of the system. The relationship between the Laguerre parameter and the bandwidth can be understood as follows: suppose we truncate the Fourier-Laguerre expansion of the impulse response matrix to terms, i.e.,…”

“…In a sense we can say that the Laguerre technique in the Laplace domain is equivalent to the Padé technique inside the unit circle. A single-input/single-output (SISO) reduced-order modeling strategy presented recently [24] developed more or less the same idea, although without explicit reference to the Laguerre COB, but with an actual Laguerre parameter . A major difference with the approach in [24] and our approach, is that is in fact related to the bandwidth of the system-see (27).…”

Reduced-order modeling based on Prony's and Shank's methods via the bilinear transformation

“…This mapping has the property that it transforms the jΩ-axis in the s-plane onto the unit circle (z = e jω ) in the z-plane [11]. Moreover, the left-half s-plane (Re(s) < 0) is mapped inside the unit circle in the z-plane and the right-half s-plane (Im(s) > 0) is mapped outside the unit circle in the z-plane, thus preserving system stability.…”

“…First, the original large system transfer function is transformed from the s-domain into the z-domain via the bilinear transformation. It is well known that the bilinear transformation always preserves system stability, and can always be made to preserve the system frequency response characteristics for a specified frequency range [2,11,12]. In addition, working in the z-domain results in better approximations than pure s-domain approaches since the parameter α in the bilinear transformation places more emphasis on the frequency range of interest.…”

Model-order reduction based on Prony's method

“…First, the original large system transfer function is transformed from the s-domain into the z-domain via the bilinear transformation. It is well known that the bilinear transformation always preserves system stability, and can always be made to preserve the system frequency response characteristics for a specified frequency range [9,10]. Second, we propose to apply Prony's least-squares approximation method to reduce the order of the transformed system function.…”