二階動態系統模型化簡的新數值方法

Abstract

摘要：本研究計畫將提出一種對二階動態系統模型化簡的新的數值方法。二階動態系統廣泛地出現在許多工程領域的問題中，例如微機電系統利用修正節點分析(Modified Nodal Analysis)技術可以用二階動態系統來描述。而在結構力學模型上經過有限元素法也會導出二階動態系統。對於有極大數量狀態變數的系統，在有限的計算資源限制下，直接去模擬這樣的系統是不可行的。因此我們運用模型化簡的技巧將原始系統的狀態變數降至一合理的數目而得到簡化系統。這樣的簡化系統必須符合下列條件： (1)精準反應原始系統的現象 。 (2)保有原始系統的物理性質，如穩定性跟被動性。 (3)保有二階結構。 對於線性動態系統的模型化簡在過去二十年來有下列兩種主流的方法 (1) 動差匹配法 (Moment Matching)模型化簡，這個數值方法是保證簡化系統的傳輸函數的泰勒展開式低階系數將與原始系統的傳輸函數之相對應的系數相同。其優點是計算過程僅需要矩陣向量乘法，大幅降低計算所需資源。 (2) 平衡截斷法 (Balanced Truncation)的做法是藉由解Lyapunov 方程找出一個新的座標，在新的座標下所描述的系統其狀態向量是同等可控制跟可觀察，而簡化系統則是揚棄較不容易控制(觀察)的部份。本方法的優點是可以做到全域近似並得到簡化系統跟原始系統差異的上界。 將動差匹配法應用到二階動態系統在過去十年間已在許多文獻上提出有效的算法，但是將平衡截斷法應用到二階動態系統上尚有許多值得探討的問題。舉例來說，如何求得一個具有二階結構的簡化模型且計算出全域近似的上界。本計畫主要是發展出兩階段的座標轉換技巧來找出適當的簡化模型來解決這個問題。

In this project, we will consider a general framework for macromodeling a second-order dynamical system. Nowadays, second-order dynamical systems arise in many fields. For example, many microelectromechanical systems can be described by second-order dynamical systems through a modified nodal analysis approach. Modeling of the structural dynamics by finite element method also leads to second-order dynamical systems. For systems of very large state space dimension, it is impractical or even prohibited to directly solve such systems. So it is necessary to construct a reduced-order model (ROM) which retains the following properties: 1) The input/output (I/O) behavior of the ROM is close to the I/O behavior of the original system. 2) Preserve the properties of the original systems (e.g. stability, passivity). 3) Obtain a second-order form reduced model. During the past two decades, there are two important approaches for macromodeling a ``linear dynamical system’’: 1) Moment matching method: This approach constructs a ROM whose lower moments match the corresponding moments of the original system. The main computational advantage of this approach is that it only requires matrix vector multiplications. 2) Balanced truncation method: This approach first tries to find a new coordinate system such that each state in this new coordinate system is equally controllable and observable. Subsequently, an ROM is obtained by truncating weak states. The advantage of this approach over the moment matching method is that the error bound between the original system and the ROM is computable. It is natural to ask if we can apply similar approaches on the second-order dynamical systems. During the past decade, there are several moment matching based methods for macromodeling second-order dynamical systems proposed on renowned journals. However there exist numerous issues about applying balanced truncation based approaches on the second-order dynamical systems. For instance, can we keep the second-order structure in the ROM and simultaneously guarantee the difference between the original system and the ROM ? In this project, we will propose a two-step coordinate transformation strategy to solve the proposed issue.