基於增益與相位邊限規格之控制器設計

Design of Controllers Based on Gain and Phase Margin Specifications

林建存(Jian-cun Lin)

指導教授 ： 莊堯棠

國立中央大學/資訊電機學院/電機工程學系/碩士(2007年)

摘要

我們都知道當系統有適當的相位邊限以及增益邊限的話，將會使系統有良
好的強健性，因此在本研究中，我們主要是將探討如何以相位邊限以及增益邊
限這兩個頻域規格來設計控制器，而其中所探討的控制器類型包含了比例-微
分控制器(Proportional-Derivative Controller) 、比例- 積分控制器
(Proportional-Integral Controller) 、比例- 積分- 微分控制器
(Proportional-Integral-Derivative Controller)、以及相位領先或落後補
償器(Phase Lead or Lag Compensator)這幾種形式。
由於整個控制器的設計流程中，為了要使得整個系統同時達到我們所要求
的相位邊限以及增益邊限的規格將會是非常複雜且需要不斷以嘗試錯誤(try
and error)的方式來設計，因此我們利用增益-相位邊限測試方法(Gain-Phase
Margin Tester Method)來實現。增益-相位邊限測試方法是一種非常快速且直
覺的控制器設計方法，可以將控制器的解轉換成二維平面，只要讀取圖上的值
即可找出解來。而在一些情況下，控制器的解可能會超過二維平面，因此我們
利用穩態誤差(steady-state error)的條件限制，進而不增加控制器設計上的
複雜度。最後我們也利用幾個簡單的例子來作模擬及測試。

關鍵字

控制器設計 ； 補償器設計 ； 相位邊限 ； 增益邊限

並列摘要

In control theorems, gain margin and phase margin are important specifications
in the frequency domain for the analysis and design of practical control systems and
have served as important measures of robustness analysis. In this thesis, we will
discuss how to design of controllers to satisfy required gain and phase margin
specifications, and the controllers are proportional-integral (PI) controller,
proportional-derivative (PD) controller, proportional-integral-derivative (PID)
controller, and phase lead or lag compensator.
In the procedure of controller design, it is very hard and complicated to let the
system achieve our required gain and phase margin specifications, simultaneously.
So we utilize gain-phase margin tester method to achieve this goal. The gain-phase
margin tester method can transform the solutions of the controller into
two-dimension surface, and we can find solutions on the figure. In some cases, the
solutions of the controller will exceed two-dimension surface, so we utilize the
condition of steady-state error to constrain the dimension of solutions, and avoid to
increase the complexity of controller design. Finally, the effectiveness of the
method is demonstrated by examples.

並列關鍵字

gain margin ； controller design ； compensator design ； phase margin

參考文獻

[1] B. C. Kuo, “Automatic Control Systems”, Addison-Wesley, 8th ed., 2002.

[2] C. H. Chang, and K. W. Han, “Gain Margin and Phase Margin Analysis of a Unclear Reactor Control System with Multiple Transport Lags”, IEEE Transactions on Nuclear Science., vol. 36, no. 4, pp. 1418-1425, 1989.

[4] H. W. Fung, Q. G. Wang, and T. H. Lee, “PI tuning in Terms of Gain and Phase Margins,” Automatica, vol. 34, No. 9, pp. 1145-1149, 1998.

[5] J. H. Lee, “A New Phase-Lead Design Method Using the Root Locus Diagrams”, IEEE Trans. On Automatic Control, vol. 50, No. 11, pp. 1887-1891, 2005.

[6] K. S. Yeung, K. Chen, ”A Non-Trial-and-Error Method for Lag-Lead Compensator Design,” IEEE Transaction on Education, Vol. 41, pp.76-80, Feb. 1998