變動區間系統PID控制器之設計

Abstract

摘 要 以往的古典控制理論，乃是利用一些基本方法如根軌跡圖(Root Locus)、羅斯穩定準則(Routh Stability Criterion)、奈氏圖(Nyquist Plot)、波德圖(Bode Plot)及尼可圖(Nichols Chart)來判斷固定參數受控體(Plant)的穩定性，並依據所需規格(時域或頻域)來設計控制器。 本論文先介紹變動區間系統的相關理論，包括Kharitonov定理、Mikhailov準則、剔零原理(Zero Exclusion Principle)及極值系統(Extremal System)等，並討論變動區間多項式的穩定度。 然後將變動區間系統套入奈氏圖、波德圖及尼可圖中，討論該系統的穩定度，並定義增益邊限(Gain Margin，GM)及相位邊限(Phase Margin，PM)。 最後則是以基本的二階變動區間系統為主，配合增益波德圖的上邊界(upper bound)及相位波德圖的下邊界(lower bound)，透過簡單的數學推導，來設計符合所需規格的PID控制器，並舉例說明如何設計，再以奈氏圖、波德圖及尼可圖來檢查受控系統補償後的穩定度。並且說明本論文所提方法優點及可行性。

ABSTRACT In classic control theory, we can usually apply methods including root locus, Routh stability criterion, Nyquist plot, Bode plot and Nichols chart to design the controllers such that the resulting systems meet the specifications of time or frequency domain. In the thesis, we first introduce some theory about interval system including Kharitonov theory, Mikhailov criterion, zero exclusion principle and extremal system to discuss the stability of interval parametric polynomial. Then, we use the methods of Nyquist plot, Bode plot and Nichols chart to check the stability of an interval system and define the gain and phase margins. Lastly, we use the upper bound of Bode plot in gain and the lower bound of Bode plot in phase to design a PID controller of an interval system. We also use Nyquist plot, Bode plot and Nichols chart to check the system stability after compensation.