使用多項式近似法實現Preisach 模型並應用於微壓電致動器

Abstract

近年來由於工程應用上對精準度要求的增加, 磁滯非線性系統的建模已受到相當的注意。在現今已發展的數種描述磁滯系統的模型, Preisach 模型是最廣為人知的一種。磁滯行為的描述可以被Preisach 模型的無限多組可計算的一階逆曲線(First Order Reversal Curves, FORC) 表示。實現方面,傳統的方法是以資料表格實現Preisach 模型; 其資料相應FORC 的有限數量的取樣點,而透過表格資料的線性內插計算模型輸出。然而, 這種方法有兩個缺點: 為了準確地預測磁滯行為需要大量的記憶體,以及發展有效的方法調整表格資料以反映磁滯元件的老化或時間效應是困難的,或說甚至是不可能的。為了克服這些缺點,本論文提出使用一組多項式取代資料表格來實現Preisach模型。這樣的方式明顯地減少所需的記憶體因為只需儲存多項式的少量係數。再者, 多項式的係數可以使用最小平方法或是適應性鑑別演算法取得, 因此可以追蹤磁滯系統的參數。我們提出的方法已經被驗證在微壓電致動器的位移預測與追跡控制上。我們應用最小平均平方的方法發展一種適應性的演算法來鑑別微壓電致動器的多項式係數;以此結果實現並與表格實現的方式相比較,在模型準確度與減少所需記憶體大小方面有顯著的改善。

Modeling of systems with hysteresis nonlinearities has received considerable attention recently due to the increasing accuracy requirement in engineering applications. Several models have been developed to characterize systems with hysteresis, and the most popular one is the Preisach model. The Preisach model to characterize the hysteresis behavior can be represented by infinite but countable first order reversal curves (FORC). In practice, the conventional approach to realize the Preisach model uses a data table; the data of which correspond with the samples of a finite number of FORC, then the model output is evaluated via the linear interpolation from the table data. This approach, however, suffers from two drawbacks: it requires a large amount of memory in order to obtain an accurate prediction of hysteresis behavior and it is difficult or even impossible to derive efficient ways to modify the data table in order to reflect the aging or timing effect of elements with hysteresis. To overcome these drawbacks, this thesis proposes to use a set of polynomials instead of a data table for realization of the Preisach model. The proposed approach reduces significantly the required memory because it only requires to store a small number of polynomial coefficients. Furthermore, the polynomial coefficients can be obtained using the least-square approximation or the adaptive identification algorithm such that the tracking of hysteresis model parameters is possible. The proposed approach has been verified by the displacement prediction and the tracking control of the micro piezoelectric actuator. We apply the least mean square method to develop an adaptive algorithm for identification of the polynomial coefficients for the micro piezoelectric actuator; the resulting realization compared with the realization via table method yields significant improvement in model accuracy as well as the reduction in the required memory size.