連續時變系統之辨識與其控制性能評估

Identification and Performance Assessment of Control Loops for Continuous Time-Varying Systems

古紹武(Shao-Wu Gu)

指導教授 ： 陳榮輝

中原大學/工學院/化學工程研究所/碩士(2016年)

https://doi.org/10.6840/cycu201600817

摘要

本研究以線性時變狀態空間模型為基礎發展出對於開迴路、回饋控制以及串級控制三種情況的非線性程序進行辨識的方法。藉由辨識出的模型能進一步用於以LQG最佳控制為基準的控制器性能評估。
由於過去常用的時變模型結構無法直接用於子空間方法來辨識，本研究以基礎函數的方法將線性時變的狀態空間模型，數學意義上的轉換成線性非時變的模型。因此，經由轉換的線性非時變模型能夠像過去子空間方法，先藉由輸入與輸出的數據求得擴展觀測矩陣，並利用此矩陣估算出系統狀態。然而，利用基礎函數近似的方法會使擴展觀測矩陣的維度隨著基礎函數的用量，以及動態窗口的長度指數型的膨脹，這造成了運算上龐大的負擔。在這邊則採用核方法進行運算，以減輕龐大維度造成的計算負擔。在使用本研究所提出的對於時變系統的子空間辨識方法中，矩陣中相依性的部分會造成進行矩陣運算時出現病態的問題，這造成辨識的結果反而隨著基礎函數的增加而變差。因此，在此辨識方法的最後提出多變數正交最小平方法，從相依性的矩陣中選出獨立的基礎函數來使用，進而解決此問題。
在連續操作的生產操作中，程序受到干擾的影響時通常仰賴回饋控制的方式將控制變數維持在設定點上。而利用先前所提出的時變子空間辨識方法於此回饋控制的環境下，系統輸入與過去干擾之間的共關性會造成計算上的誤差。對此本研究利用更新計算的方法，藉由估算程序的干擾，組成回歸運算子進行計算。由於未來的輸入與此干擾組成的回歸運算子之間是無相關的而增加了時變辨識方法的準確性。本研究提出的時變辨識方法將對回饋控制與串級控制迴路分別進行討論。
在過去利用線性二次高斯LQG控制定理為基準的控制器性能評估中，利用輸入與輸出訊號代入LQG目標方程式於不同權值可計算構成一條權衡曲線圖。然而，此權衡曲線圖無法提供一個量化的結果來評估控制系統的性能。同時，過去的評估方法忽略了輸入與輸出之間有相關性的部分。由於實際應用的程序中，可以藉由一個指定的權值來獲得對應的最佳控制定理。因此，本研究將利用指定的權值獲得LQG控制定理，並以此建立主成分分析模型來對回饋控制迴路與串級控制迴路進行控制器性能評估。最後將分別於開迴路用數學例子，回饋控制與串級控制以數學例子和工廠實際數據進行模擬測試。

關鍵字

控制器性能評估 ； 時變系統 ； 子空間方法辨識

並列摘要

Abstract
In this study, an identification method with a linear time varying (LTV) state space model is developed to describe nonlinear processes operated in open-loops, feedback loops or cascade control loops. Once the parameters of the LTV system are identified, the benchmark of LQG based controller performance assessment can be obtained.
In the conventional time varying model structure, the subspace identification method (SID) cannot be applied directly. In this work, the basis function approach is used so that the basis function can make the LTV subspace identification problem convert mathematically into the linear time-invariant (LTI) one. Thus, like the conventional LTI-SID approach, the same two-step procedure is taken. First, estimate the subspace spanned by the columns of the extended observability matrix from the system input-output data. Second, the system matrices are determined directly by the extended observability matrix. However, with basis function approximation, the dimension of the extended observability matrix grows rapidly, resulting in high computational burden. The kernel method is introduced and the computational burden is then alleviated. The other problem of the proposed LTV-SID approach is the identification problem in most practical cases is ill-posed because not all the basis functions are independent. Thus, the multivariate orthogonal least squares method is developed to select the independent basis functions in the input data matrix.
An operating system often relies on some forms of feedback control to maintain the controlled variables at their set points in spite of disturbances. The previously developed LTI-SID method would produce biased results in the presence of feedback because of the correlation between the system inputs and the past noise. The innovation estimation method is used to estimate the past innovation; with the past innovation, the accuracy of LTV-SID increases because the future inputs would be uncorrelated with the past innovation. The proposed LTV-SID method for the LTV system with the single feedback control loop and with the cascade control loop is respectively developed.
In conventional LQG for CPA, a relationship in the two-dimensional space of such criterion as output and input signal variances with different values of lambda is used to construct the trade-off curve. However, the tradeoff curve cannot provide the quantitative performance assessment of the controlled system. For a real application, the optimal control law can be obtained under the specified lambda value to balance the output and input signal variances. Moreover, in the conventional method, the correlations between the output and the input variables are ignored. A PCA based performance assessment for the LTV system with the single feedback control loop and with the cascade control loop is developed respectively. To demonstrate the potential applications of the proposed strategies, real industrial problems with the single feedback control loop and with the cascade control loop are used.

並列關鍵字

controller performance assessment ； time varying system ； subspace method identification

參考文獻

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