臺灣博碩士論文加值系統

　　本研究探討使用PID控制器來穩定二階時延的程序以及設計PID控制器參數，使得回饋控制之閉環路系統的穩定度達到最大。
　　使用PID控制器來控制帶時延程序的回饋閉環路系統的特徵函數為
f(s)=A(s)+(ki+kps+kds^2)B(s)e^-hs
而穩定化PID控制器參數集的定義為
Ks(sigma)={(kp,ki,kd)包含於R^3| f(s-sigma;kp,ki,kd)包含於Hs }，
其中Hs表示所有的根都在複數平面的左半平面。本文發展出一套可行有效的演算法，以 Mathematica 程式語言編寫，判斷Ks(sigma)是否存在，利用此演算法，配合轉軸程序針對以下三種二階帶時延的不穩定程序作探討：(a)帶時延雙重積分器Gp0(s)=e^-hs/s^2；(b)單一不穩定極點Gp1(s)=e^-hs/(s-1)(s+tau)；(c) 雙不穩定極點Gp2(s)=e^-hs/(s^2 -2*xi*s+1) 。本文的目的在於分別識別出在(a)程序時延軸線h>=0上；(b)程序參數平面h-tau之第一象限及(c)程序參數平面h-xi之第一象限的哪些區域或範圍內的程序是可以PD控制器來加以穩定。
　　在設計最大穩定度PID控制器時，本論文先將時延項e^-hs用Pade'近似式表示，得到一無時延的特徵多項式，再利用文獻上的方法，先求得一組次佳的控制器參數(kp*,ki*,kd*)，使用此組控制器參數於原時延程序求得穩定度sigma*，根據此次佳穩定度，本論文建構出PID控制器參數空間內包含Ks(sigma*)的一個最小長方體R*，並在此長方體內判斷Ks(sigma)，sigma>sigma*是否存在，以及使用二分法求得最大穩定度的PID控制器參數。

This thesis deals with the stabilization using PID controllers of second-order time-delay unstable systems and the design of maximum stability PID controllers for time=delay processes. In general, the closed-loop characteristic function of a PID-controlled time-delay process can be written as f(s)=A(s)+(ki+kps+kds^2)B(s)e^-hs, where A(s) and B(s) are polynomials in s and kp, ki, kd, are the proportional, integral, and derivative gains of a PID controller. The stabilizing PID controller parameter set Ks(sigma) is defined to be
Ks(sigma)={(kp,ki,kd) belong to R^3| f(s-sigma;kp,ki,kd) belong to Hs }，
where Hs denotes the set of functions where roots are all in the open left half of the complex plane.
In this study, we present an efficient algorithm for testing the existence of Ks(sigma). The algorithm is implemented with the Mathematica symbolic package. This algorithm is applied along with the pivot procedure to investigate the PID stabilizability for the following three types of unstable second-order time-delay processes; (a) double integrators plus time-delay process , Gp0(s)=e^-hs/s^2; (b) one unstable- pole second-order plus time-delay process, Gp1(s)=e^-hs/(s-1)(s+tau) ;(c) two unstable-pole second-order plus time-delay process, Gp2(s)=e^-hs/(s^2 -2*xi*s+1). We have identified in stabilizable region that the process parameter space in which the process can be stabilized with a PD controller.
The algorithm of testing the existence of stabilizing PID controller parameter set Ks(sigma) is also applied along with the bisection method to design maximum-stability PID controllers. To reduce the computational load, we first apply the Pade' approximation to delay transfer function e^-hs and use an existed method to find a suboptional maximum-stability PID controller (kp*,ki*,kd*). With this set of PID controller parameters, we then find from the original time-delay process to evaluate the corresponding degree of stability sigma*. We further construct an axis-paralleled minimum box R* that contains the stabilizing set Ks(sigma*). Starting with sigma*, we final apply the algorithm of testing the existence of Ks(sigma), sigma>sigma*, in the domain R* along with the bisection method to find the maximum-stability PID controller (kp,ki,kd).

摘要 II
Abstract III
目錄 V
圖目錄 VII
表目錄 VIII
第一章 緒論 1
1-1 時延系統 3
1-2 PID控制器 4
1-3 最大穩定度控制器設計 6
1-4 D-Partition的理論 8
1-5 組織與章節 10
第二章 穩定化PID控制器參數集之存在測試 12
2-1 前言 12
2-2 無時延程序之穩定化PID控制器的純測試 16
2-2-1 Ｋs之存在測試 21
2-3　曲線分割平面上有線區域形成多邊形之計算 23
2-4　測試Ｋs存在之演算法 29
2-5　時延程序之穩定化PID控制器參數集 31
2-6 結論 35
第三章 不穩定二階程序之PID控制器穩定化 36
3-1 前言 36
3-2 帶時延雙積分程序的PD穩定化 38
3-3 單一不穩定極點二階時延程序的PD穩定化 40
3-4 二不穩定極點二階時延程序的PD穩定化 46
3-4-1 計算方式的修正 46
3-4-1-1範例 53
3-4-2 PD可穩定化之程序參數區域 56
第四章 時延程序之最大穩定度PID控制器設計 62
4-1利用時延項 的 近似 63
4-2利用時延模式求最大穩定度PID控制器 69
4-3範例 72
第五章總結與未來展望 80
5-1 總結 80
5-2 未來展望 81
參考文獻 82
附錄A 88

 

圖目錄
Fig 1-1 回饋控制系統 1
Fig.1-2 P1-P2參數平面上的D分割圖 9
Fig.2-1 PID回饋控制系統 12
Fig.2-2 Gp(s)=(s^3+3s^2+s+7)/(s^5-8s^4+62s^2-97s+42)之kp對omega關係圖 19
Fig.2-3 多邊形區塊 20
Fig.2-4 kp(omega)對omega的關係圖 21
Fig.2-5 四條直線分割xy平面圖 23
Fig.2-6 四條直線分割區域R所形成之多邊形 24
Fig.2-7 Gp(s)=e^-0.4s / (s^2 +s+1)之kp(omega)對omega的關係圖 32
Fig.3-1 tau=1,kp對omega的關係圖 41
Fig.3-2 h-tau參數平面 45
Fig.3-3 (xi,h)=(0.5,2)之kp-kd平面D分割圖 47
Fig.3-4 (xi,h)=(0.5,4)之kp-kd平面D分割圖 47
Fig.3-5 (xi,h)=(0.5,6)之kp-kd平面D分割圖 48
Fig.3-6 (xi,h)=(0.5,8)之kp-kd平面D分割圖 48
Fig.3-7 (xi,h)=(0.5,10)之kp-kd平面D分割圖 49
Fig.3-8 Ks,PD存在判斷 50
Fig.3-9 邊界三角形之搜尋圖 57
Fig.3-10 轉軸程序示意圖 57
Fig.3-11 PD可穩定參數邊界圖 59
Fig.4-1 sigma=0.3284，無限定R-PID 72
Fig.4-2 sigma=0.3284，有限定R-PID 74
Fig.4-3 時延系統，根在複數平面分布圖 76

表目錄
表 2-1 線相交矩陣表 26
表 3-1 kp,m=-0.766696之kp,m-kd,m 穩定關係 54
表 3-2 kp,m=-6.685522之kp,m-kd,m 穩定關係 55
表 3-3 kp,m=-0.530691之kp,m-kd,m 穩定關係 55
表 4-1 無時延系統，無限定測試範圍，最大穩定度表 71
表 4-2 無時延系統，有限定測試範圍，最大穩定度表 73
表 4-3 時延系統，h=0.2，以Pade'近似求解，最大穩定度表 75
表 4-4 時延系統，有限定測試範圍，最大穩定度表 77
表 4-5 時延系統，有限定測試範圍，最大穩定度表 78

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