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Abstract A frequency weighted passivity preserving technique for balanced model order reduction is proposed. The technique yields passive reduced order models even for the case when both input and output weightings are present. The numerical examples show the usefulness and effectiveness of the proposed technique. 1. Introduction Model order reduction (MOR) has played a significant role in the analysis, design and simulation of large complex systems over the last few decades. Many practical systems (such as chip design, fluid flow, mechanical systems simulation, systems with interconnects) are complex in nature. The idea of MOR is to provide an equivalent reduced order model (ROM) which not only yields good approximation, but also preserves key characteristics of the original system such as stability, passivity and input–output behaviour. Balanced truncation (see Moore, 1981) is the most commonly used MOR technique which ensures stability in the ROMs and yields a priori error bounds (see Enns, 1984). Enns extended balanced truncation technique to incorporate frequency weighting. However, Enns technique does not guarantee stability in the presence of double sided weighting. To address this issue several frequencies weighted MOR techniques have been proposed in the literature (these include Lin & Chiu, 1992; Wang et al., 1999; Varga & Anderson, 2003; Ghafoor & Sreeram, 2008; Imran et al., 2014) to preserve stability in the presence of double sided weightings. Besides stability, passivity is important for systems with interconnects in order to avoid artificial oscillations during transient simulations. Note that, a system is guaranteed to be stable if it is passive, however, the converse is not guaranteed. Various passivity preserving techniques (without frequency weighting) appear in the literature (including Odbasioglu et al. 1998; Phillips et al., 2003; Unneland et al., 2007; Yan et al., 2007; etc). Heydari & Pedram (2006) introduced frequency weighted passivity preserving technique for strictly proper systems. Muda et al. (2010) technique modified the existing frequency weighted MOR techniques (Enns, 1981; Lin & Chiu, 1992; Wang et al., 1999) to preserve passivity of ROMs. Recently, Muda et al. (2015) pointed out that the technique proposed by Heydari & Pedram (2006) neither retains passivity nor stability in the presence of double sided weighting case. This is due to the simultaneous diagonalization of Gramians corresponding to two different systems instead of the one system which is being reduced. Following the same argument, the results in Muda et al. (2010) work also do not guarantee passivity in ROMs. To the author’s best knowledge, uptil now, no technique has been reported in the literature, which guarantees to preserve passivity in ROMs for two sided frequency weighted MOR case. So there is a need to introduce a technique which can preserve passivity in double sided frequency weighted MOR scenario. In this work, a new technique is developed for frequency weighted balanced MOR that ensures passivity, even for the double sided weighting case. Numerical examples show the usefulness of proposed technique. 2. Main Results The technique proposed by Heydari & Pedram (2006) does not guarantee passivity in the presence of double sided weighting (Muda et al., 2015), however, passivity is always guaranteed in the case of single sided weighting. By taking the advantage of preservation of passivity of single sided weighting case, a hierarchical balancing technique is proposed. Two sided passivity preserving frequency weighted balancing problem is considered as two one sided problems in sequence, such that the balanced realization of the first one sided is used for augmentation with the other side by simultaneously using the Lyapunov and Lur’e equations. This work is motivated from Ghafoor & Imran (2015), where a stability preserving frequency weighted balanced MOR technique was proposed. Consider a passive full order original system with transfer function $$G(s) = C(sI - A)^{-1} B + D$$, an input weighting system with input transfer function $$V_i(s) = C_i (sI-A_i)^{-1} B_i + D_i$$ and an output weighting system with output transfer function $$W_o(s) = C_o (sI-A_o)^{-1} B_o + D_o$$ respectively. Let the augmented system given by G(s)Vi(s)=C¯i(sI−A¯i)−1B¯i+D¯i (2.1) satisfy the following Lyapunov controllability equation: A¯iP¯i+P¯iA¯iT+B¯iB¯iT=0, (2.2) where By writing (1,1) block of equation (2.2) we have: AP11+P11AT=−BCiP12T−P12CiTBT−BDiDiTBT. (2.3) Let $$Q$$ be the observability Gramian satisfy the following Lur’e equation: ATQ+QA=−KoTKoQB−CT=−KoTJoJoTJo=D+DT. (2.4) Simultaneously diagonalizing these Gramians, we get Tb−TQTb−1=TbP11TbT=diag(σi), (2.5) where $$ \sigma_i$$ are the Hankel singular values such that $$ \sigma_i \geq \sigma_{i+1}, i= 1,2,\ldots,n-1 $$. Transforming the original system using balancing transformation $$T_b$$, the input weighted balanced realization can be formed as: Ab=Tb−1ATb, Bb=Tb−1B,Cb=CTb Remark 2.1 For input weighted case only, the ROMs are obtained by partitioning and truncating the realization $$\{A_b, B_b, C_b, D\}$$ upto the desired order. Remark 2.2 The weighted input Gramian is obtained using Lyapunov equation and output Gramian is obtained using a Lur’e equation. Remark 2.3 The ROMs obtained using input/one side weighting case are passive. This follows from section II-B of Muda et al. (2015). Now consider the output passivity preserving frequency weighted portion. Let the output augmented system be: Wo(s)G(s)=C¯o(sI−A¯o)−1B¯o+D¯o, (2.6) where A¯o=[Ab0BoCbAo ],B¯o=[BbBoDb]C¯o=[DoCbCo],D¯o=DoD (2.7) Remark 2.4 Note that, the system realization $$\left\{A_b,B_b,C_b,D\right\}$$ is used here for finding the augmented system $$W(s)G(s)$$. Let the frequency weighted observability Gramian $$\bar{Q}_o$$ Q¯o=[Q11Q12Q12TQW] satisfy the Lyapunov equation: A¯oTQ¯o+Q¯oA¯o+C¯oTC¯o=0, (2.8) where the (2,2) block of (2.8) is AbTQW+QWAb=−CbTBoTQ12T−Q12BoCb−CbTDoTDoCb Let the controllability Gramian $$P$$ for transformed system be obtained by solving the Lur’e equation: AbP+PAbT=−KcKcTPCbT−Bb=−KcJcTJcJcT=D+D (2.9) Simultaneously diagonalizing the Gramians $$P$$ and $$Q_{W}$$, we get T−TQWT−1=TPTT=diag(σi) (2.10)$$ \sigma_i \geq \sigma_{i+1}, i= 1,2,\ldots,n-1 $$ and $$ \sigma_r > \sigma_{r+1} $$. Applying the similarity transformation to the input weighted balanced realization, the two sided frequency weighted balanced realization is obtained as following: \begin{align*}\ \end{align*} (2.11) \begin{align*}\ \end{align*} (2.12) The ROM can be obtained by partitioning and truncating up to the desired order. Remark 2.5 The similarity transformation matrix $$T_b$$ has been constructed using the positive semidefinite matrices $${Q}$$ and $$P_{11}$$. The matrix $$T_b$$ is then used in calculation of final transformation matrix, which is also constructed using positive definite matrices $$Q_{W}$$ and $$P$$. The proposed technique ensures the diagonalization of Gramians matrices $$\{{Q}$$, $$P_{11}\}$$ and $$\{Q_{W}, P$$} to guarantee the passivity of ROMs (Muda et al., 2015). Remark 2.6 For higher order systems with $$D+D^T > 0$$, the solution of equations (2.4) and (2.9) are obtained by solving the following algebraic Riccati equation A¯TQ+QA¯+QB(D+DT)−1BTQ+CT(D+DT)−1C=0 (2.13) A^P+PA^T+PCbT(D+DT)−1CbP+Bb(D+DT)−1BbT=0, (2.14) where $$\bar{A} = AB(D+D^T)^{-1}C$$ and $$\hat{A} = A_bB_b(D+D^T)^{-1}C_b$$. Remark 2.7 For a strictly proper system (i.e. $$D=0$$), Lur’e equations (2.4) and (2.9) reduce to ATQ+QA=−KoTKoQB=CbT (2.15) AbP+PAbT=−KcKcTPCbT=B (2.16) which are solved using the method given in Sadegh et al. (1997). 2.1. Algorithm I Given a passive original system and the input and output weighting systems, Step (1) Compute the augmented realization from equation (2.1). Step (2) Compute $$P_{11}$$ and $$Q$$ from equations (2.3) and (2.4), respectively. Step (3) Compute Cholesky factor $$R_{11}$$ of $$P_{11}$$ i.e. $$P_{11}=R_{11}^TR_{11}$$. Step (4) Compute singular value decomposition of $$R_{11}QR_{11}^T = U_{11}\sum_b^2 U_{11}^T$$. Step (5) Compute transformation matrix as $$T_b = R_{11}^TU_{11}\sum_b^{-1/2}$$. Step (6) Compute the augmented realization using equation (2.6). Step (7) Compute $$Q_W$$ and $$P$$ using equations (2.9) and(2.9), respectively. Step (8) Compute Cholesky factor $$R_{22}$$ of $$P$$ i.e. $$P=R_{22}^TR_{22}$$. Step (9) Compute singular value decomposition of $$R_{22}Q_WR_{22}^T = U_{22}\sum ^2 U_{22}^T$$. Step (10) Compute transformation matrix as $$T = R_{22}^TU_{22}\sum ^{-1/2}$$ where $$\sum = diag\{\sigma_1, \sigma_2, \cdots \sigma_n\}$$. Step (11) ROMs are obtained by partitioning and truncating realization in equation (2.12) up to the desired order. Theorem 2.1 The ROMs obtained by Algorithm I are passive. Proof: Since the Algorithm I involves two one sided frequency weighted passivity preserving problems in sequence, therefore, passivity is preserved in ROMs. 3. Numerical Examples Example 3.1 Consider a sixth order single lossy line modelled by ladder RLC circuit (Muda et al., 2015) with parameter values as $$R_s = 1 {\it{\Omega}}$$, $$R_i = 1 {\it{\Omega}}$$, $$L_i = 0.01 H$$, $$C_i = 1 F$$. A system realization $$G(s)$$ is given by: A=[000100−10000000100000−11000−100−10000−100100100−10000−10000100−10000]B=[00100000]TC=[001000] with the following input and output weighting functions Vi(s)=Wo(s)=s+4s+1 Figure 1 shows the Nyquist plot for the original and fifth order ROM obtained by the Heydari & Pedram (2006) and proposed techniques. It can be seen that the Nyquist plot for the ROM lies in the left half plane for Heydari & Pedram (2006), thus passivity of the ROM is not guaranteed in this case. On the other hand, Nyquist plot for the ROM lies in the right half plane for the proposed technique, which guarantees the passivity of the ROM (Tan et al., 2007). Example 3.2 Consider a 100th order original passive system of a transmission line modelled by a 50 ladder RLC section (Muda et al., 2010) with parameter values as $$R_L = 0.1 {\it{\Omega}}$$, $$R_C = 1 {\it{\Omega}}$$, $$L_i = 0.1 H$$ with the following input and output weighting functions Vi(s)=Wo(s)=s+1s+6 Figure 2 shows the Nyquist plot for the original and third order ROM obtained by the proposed technique. It can be seen that the Nyquist plot for the ROM lies in the right half plane which guarantees the passivity of the ROM (Tan et al. 2007). Fig. 1. View largeDownload slide Nyquist plot of original and ROM. Fig. 1. View largeDownload slide Nyquist plot of original and ROM. Fig. 2. View largeDownload slide Nyquist plot of original and ROM. Fig. 2. View largeDownload slide Nyquist plot of original and ROM. 4. Conclusion A frequency weighted MOR technique is presented. The ROMs obtained in the presence of input, output and double sided weightings are passive. References Enns, D. F. 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IMA Journal of Mathematical Control and Information – Oxford University Press
Published: Sep 21, 2018
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