實時結構鑑定 2015年10月23日，香港汲水門大橋遭到船隻碰撞，導 致大橋被封1.5小時。由於汲水門大橋是通往赤鱲角機 場的主幹道，這場事故造成了巨大的經濟損失。如果 可以進行實時結構鑑定，就可以立刻確定大橋是否安 全，而不用封橋之後請工程師進行檢測或其他線下結 構鑑定計算。這種實時鑑定既能迅速確定建築物的安 全狀況，又能將經濟損失降至最低。但是，實時結構 鑑定是個非常具有挑戰性的問題。首先，土木工程結 構的規模通常都很大，所以必須使用非常複雜的有限 元素模型。其次，由於取樣頻率通常是數百赫茲，每 秒鐘要將一個巨大的結構模型更新幾百次的難度有多 大可想而知。 模型類別選擇 進行結構健康監測一定要選擇適當類別的模型。這 一點非常重要。這個問題看似簡單，但實則不然。 讓我們舉一個例子來解釋。假設甲同學在物理課上 學到了F=ma這個公式。在實驗課上，他做了一些實 驗，獲得了十個數據點。然後他通過EXCEL擬合直 線獲得常量m，看這個常量是否和質量吻合。再假 設乙同學只做實驗沒有上課，所以他並不知道F=ma 這個公式。他用二階多項式在EXCEL擬合拋物線。 結果他的數據擬合比甲同學的數據擬合好，因為拋 物線為數據擬合提供更多靈活性。然後，丙同學 也進行了數據擬合，但用的是九階多項式。結果怎 麼樣？她的結果是零誤差，因為這個多項式擁有10 個可調整係數，所以可以毫無偏差的穿過10個點。 但是，九階多項式非常波動，絕不是有效的預測模 型。這個故事說明了甚麼？說明我們不能僅僅根據 擬合誤差來選擇模型類別。擁有太多可調整參數的 模型在擬合數據的細節（包括噪音）方面很有用， 但是卻會導致「過度擬合行為」。這類模型並不能 對未來做出可靠的預測。 Beck和Yuen（2004）發表了首篇在結構健康監測 領域中探討如何選擇模型類別的方法的論文。大量 的未知參數、沒有激發測量值可參考以及土木工程 結構的高度不確定性，使得這個問題在結構健康監 測領域特別具有挑戰性。這篇論文對模型的複雜性 加以量化，同時綜合考慮模型擬合能力及模型複雜 性，對不同類別的模型進行優劣排序。模型擬合能 Real-time Structural Identi" cation On 23 October 2015, a ship struck the Kap Shui Mun Bridge in Hong Kong. ! e bridge was closed for about 1.5 hours. Since it is a critical line to the Chap Lap Kok Airport, the economic loss of this incident was huge. ! erefore, if a structural identi# cation can work in a real-time fashion, it can help identify whether or not the structure is safe almost immediately, instead of closing the bridge and conducting an investigation by engineers, or by other o) ine structural identi# cation calculation. ! is will be very useful from the point of both safety and economic concerns. However, real-time structural identi# cation is a very challenging problem. First, the scale of civil engineering structures is huge so one can expect a very complicated # nite element model is necessary. Second, since the sampling frequency is usually hundreds Hz, it is straightforward to imagine the di& culty in updating a huge structural model several hundred times every second. Model Class Selection One important problem is to select a proper class of models for the purpose of structural health monitoring. ! is sounds to be a simple problem but it is indeed much more di& cult than it looks at # rst glance. For instance, let’s consider a simple problem for the sake of explanation. Imagine a student, called student A, who takes a physics class and learns F=ma. In the laboratory section, he conducts experiments and obtains 10 data points. ! en, he uses EXCEL to # t a line to obtain the constant ‘m’ to determine whether or not it matches with the mass. Unfortunately, student B does not attend the lecture, and therefore when he conducts the experiments he does not know the formula F=ma. He then uses EXCEL to # t a parabola, ie, second order polynomial. It turns out that his data # tting is better than student A because a parabola o" ers more $ exibility to # t the data better. ! en, student C does the same but with a ninth-order polynomial. Guess what? She got zero error because this polynomial, with ten adjustable coe& cients, can go through all ten points exactly. However, a ninth-order polynomial is very bumpy and it is by no means a good model for prediction in this case. What can we learn from this story? We cannot select a class of models solely due to the # tting errors. In general, a class of models with too many adjustable parameters has great power in # tting the details of the data, including the noise, but this will lead to the so-called over-# tting behaviour. Such models are not reliable for future prediction. Beck and Yuen (2004) presented the # rst paper to tackle the model class selection problem in the area of structural health monitoring. ! is problem is particularly di& cult in this # eld due to the large number of unknowns, the unavailability of the excitation measurements, and a high level of uncertainty in civil engineering structures. In this paper, model complexity was quanti# ed and the model class candidates are ranked according to the tradeo" between the model # tting power and its complexity, which is a measure of the model robustness. Model robustness 學院專欄• FACULTY COLUMN 54
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