使用平衡截取與被動降階互連建模法簡化向量擬合所得之模型;On The Use of BT and PRIMA for Model-Order Reducion After Vector Fitting

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Title:	使用平衡截取與被動降階互連建模法簡化向量擬合所得之模型;On The Use of BT and PRIMA for Model-Order Reducion After Vector Fitting
Authors:	劉晃福;Liu, Huang-Fu
Contributors:	電機工程學系
Keywords:	向量擬合;平衡截取;被動降階互連建模法;巨觀建模;模型降階;Vector Fitting;Balanced Truncation;PRIMA;Macro-modeling;Model Order Reduction
Date:	2023-07-25
Issue Date:	2023-10-04 16:10:26 (UTC+8)
Publisher:	國立中央大學
Abstract:	隨著信號越來越高速的趨勢，封裝與印刷電路板之間的信號完整性變得重要，為了能夠有效處理這些複雜的信號傳輸行為，可以使用建模法來輔助分析。向量擬合是一種常用的模型建構方法，它利用有理函數來描述系統的寬頻響應，此響應來源是模擬或是量測所得到的系統資料。向量擬合的方法所獲得的極點與留數，可以用來建構稀疏的狀態空間方程式，例如:對於一個極點數為300埠數為10的系統，其階數為3000，在時域模擬中，這可能需要很長的計算時間。為了縮短時域模擬的耗時，我們可以嘗試使用模型降階的方法，使模型階數變小，以利增加時域模擬速度。在本論文中，我們將使用平衡截取(BT)與被動降階互聯建模法(PRIMA)來實現系統的模型降階。BT是一種藉由平衡轉換，將系統狀態轉換成平衡系統，再藉由觀察漢克奇異值的分布，來截取(truncate)掉可觀性與可控性較小的狀態，來達成模型縮減目的的方法。PRIMA原先是一種白盒子巨觀建模的方法，在知道走線電路的元件內容後，建立一個狀態空間方程式，接著使用Block Arnoldi iteration 的方法，求得krylov 子空間的正交基底，並運用矩陣的映射在保有矩(moment)的情況下，來達成模型縮減的目的。論文中BT的部分，我們首先探討了在不同漢克奇異值比例下，使用極點與留數模型縮減後的誤差，直觀上，誤差會隨著漢克奇異值比例的降低，而變得較小。為了精確量化這層關係，我們使用H_∞ 與 H_2 兩種誤差指標進行分析，發現對於大部分的系統而言，H_∞ 與H_2 誤差正比於漢克奇異值的截取比例。在BT時域模擬方面，我們將經過BT縮減過後的模型，經過論文提出之等效路合成法，將狀態空間模型轉換成SPICE 可讀取之R 、C、 VCCS等元件，並觀察不同模型縮減大小，所耗費的時間關係。經驗證，有做模型縮減後，確實可以達到較快的模擬時間與同等為縮減前的90%的精度。在PRIMA 部分，我們則是探討在與BT相同的誤差值下，PRIMA 所能縮減模型的能力。經過比較PRIMA 有較差的模型縮減能力，其縮減能力遜於BT的3到30倍不等，並且針對PRIMA ，可以觀察到典型的moment matching 的特色 ;With the advancement of high-speed systems, signal integrity between packaging and printed circuit boards has become increasingly important. In order to effectively deal with these complex signal transmission behaviors, macro-modeling can be used to assist us in analysis. Vector Fitting is a commonly used method for model construction, which utilizes input parameters obtained from simulation or measurement of a system, and represents the possible broadband response of the system using rational functions. The poles and zeros obtained from Vector Fitting can be used to construct a sparse state-space equation, where the number of ports (P) and poles (N) determines the system order (NP). For example, if the number of ports is 10 and the number of poles is 300, the system order would be 3000. In time-domain simulations, this can require longer computation time. In this thesis, we employ Balanced Truncation (BT) and Passive Reduce-order Interconnect Macro-modeling Algorithm (PRIMA) for achieving model order reduction (MOR) of the system. BT is a technique that transforms system states into balanced states through balance transformation. It then truncates states with lower observability and controllability by observing the distribution of Hankel singular values, achieving the goal of model reduction. PRIMA was originally a white box macro modeling method. After knowing the component content of the routing circuit, it establishes a state-space equation. Then, using the Block Arnoldi iteration method, it calculates the orthogonal basis of the Krylov subspace. By applying matrix projection while preserving moments, it achieves the goal of model reduction. This paper does not use white box macro modeling, but instead utilizes a black box macro model established using VF. The PRIMA process is applied to it. In this thesis, the part about BT (Balanced Truncation) discusses the reduction of errors using the pole-residue model under different Hankel singular value ratios. The errors decrease as the Hankel singular value ratio decreases. The relationship between H_∞(H-infinity norm) and H_2 (H2 norm) with the Hankel singular values is thatH_∞ and H_2 are proportional to the Hankel singular values ratio. In terms of BT time-domain simulation, we propose a circuit synthesis method and apply it to the reduced model obtained through BT. This method converts the state-space model into SPICE-readable R,C,VCCS elements. We observe the time relationship consumed by different model reduction sizes and validate that after model reduction, faster simulation time can be achieved with an accuracy of 90% compared to the original model. In the PRIMA section, we explore the model reduction capabilities of PRIMA under the same error criterion as BT. Through comparison, it is found that PRIMA has inferior model reduction capabilities compared to BT, ranging from 3 to 30 times less reduction ability. Additionally, for PRIMA, typical characteristics of moment matching can be observed.
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