臺灣博碩士論文加值系統

傳統PID（Proportional-Integral-Derivative）控制器至今仍被廣泛應用於各個工業領域。PID自問世以來已近百年歷史，因其結構簡單、穩定性佳、調整方便等優點，而成為工業控制中極其重要的一環。在面對無法完全掌握的被控對象或無法獲得精確數學模型的情況下，其他控制技術可能難以應用，而PID控制技術則能派上用場。
PID控制器可細分為PI與PD兩種控制功能，其中PI控制器因能有效地消除穩態誤差而受到廣泛青睞。與PD控制器相比，PI控制器在面對持續性輸入時具有明顯優勢，因為PD控制器無法完全消除穩態誤差，這使得PI控制器在要求高穩態性能的應用中更具吸引力。例如，在製造業中，PI控制器常用於溫度控制、液位控制和壓力控制等系統，以確保系統在穩態下能精確達到目標值。這些優勢使得PI控制器成為許多自動化生產線及過程控制中的首選，特別是在需要精確跟蹤和穩定性的場合。因此，本文將深入探討PI控制器在不同系統中的穩定性分析，與魯棒性能探討。
雖然傳統PID控制器在工業中使用已久但仍無法有效處理具有時延的系統，這對系統的穩定性造成了挑戰。為解決這一問題，我們引入了帕德近似(Padé approximant method)來處理具有時延的系統。然而帕德近似法存在精度不足的問題，為此我們進一步採用牛頓迭代法(Newton-Raphson method)來進行精確化改進，同時，為了確保穩定性的全面性，我們使用了幅角定理(Argument principle)進行檢驗，確認無遺漏的根位於右半平面。通過D-Partition方法，我們找出了控制器參數的穩定區域，並利用根軌跡(Root Locus)分析了參數對控制器根的影響。但考慮到D分割法僅能在特定時延下確定穩定區域，這可能導致在時延增大時原本穩定的參數變為不穩定，所以我們進一步探討了系統的最大時延。此外我們還利用H_∞-norm原理和靈敏度函數(sensitivity function) 與D分割結果結合，檢驗其同時滿足穩定性和魯棒性的特性。本研究通過6個不同的Cases來驗證上述方法，分別應用於一階與二階系統，並探討了不同零點與極點對系統穩定性與魯棒性的影響。這些案例的結果表明，所提出的方法能夠有效提升含時延系統的控制性能，並為相關領域的研究提供了新的思路和方法。

Traditional PID (Proportional-Integral-Derivative) controllers are still widely used in various industrial fields today. Since their inception nearly a century ago, PID controllers have become an essential component in industrial control due to their simplicity, stability, and ease of tuning. In scenarios where the controlled object cannot be fully understood or where an accurate mathematical model is not obtainable, other control techniques may be difficult to apply, whereas PID control techniques can be effectively utilized.
PID controllers can be further divided into PI and PD control functions. Among these, PI controllers are widely favored for their ability to effectively eliminate steady-state errors. Compared to PD controllers, PI controllers have a significant advantage when dealing with continuous inputs, as PD controllers cannot completely eliminate steady-state errors. This makes PI controllers more attractive in applications that demand high steady-state performance. For example, in the manufacturing industry, PI controllers are commonly used in temperature control, level control, and pressure control systems to ensure that the system can accurately reach the target value in a steady state. These advantages make PI controllers the first choice in many automated production lines and process control applications, especially in scenarios requiring precise tracking and stability. Therefore, this paper will delve into the stability analysis and robustness performance of PI controllers in different systems.
Although traditional PID controllers have been used in industry for a long time, they are still ineffective in handling systems with time delays, posing a challenge to system stability. To address this issue, we introduced the Padé approximant method to handle systems with time delays. However, the Padé approximant method has accuracy limitations, so we further adopted the Newton-Raphson method for precise improvement. Additionally, to ensure comprehensive stability, we used the phase angle theorem to verify that there are no missing roots in the right half-plane. Through the D-Partition method, we identified the stability regions of the controller parameters and analyzed the impact of the parameters on the controller roots using Root Locus analysis. Considering that the D-Partition method can only determine the stability region at a specific time delay, which may cause originally stable parameters to become unstable when the time delay increases, we further explored the system's maximum time delay. Moreover, we combined the H_∞ norm principle with the sensitivity function and D-Partition results to examine their characteristics in simultaneously meeting stability and robustness. This study validated the above methods through six different cases applied to first-order and second-order systems and explored the impact of different zeros and poles on system stability and robustness. The results of these cases demonstrate that the proposed methods can effectively enhance the control performance of time-delay systems and provide new ideas and methods for research in related fields.

目錄
緒論
1.1研究動機 ………………………………………………………………… 1
1.2文獻回顧 ………………………………………………………………… 2
1.3章節與組織 ………………………………………………………………. 3
控制器穩定性分析
2.1 前言 ………………………………………………………………….. 4
2.2 D-Partition …………………………………………………………. 4
2.2-1 Root Tendency ……………………………………………………. 6
2.3 Padé approximant method …………………………………………. 7
2.3-1 牛頓迭代法 ……………………………………………………….. 9
2.3-2 幅角定理 ………………………………………………………….. 10
2.4 Root Locus Analysis …………………………………………………… 12
2.5 範例 …………………………………………………………………… 13
第三章 控制器之最大時延分析
3.1 前言 ………………………………………………………………….. 22
3.2 範例 ………………………………………………………………….. 22
第四章 H_∞-norm 分析
前言 ………………………………………………………………….. 39
4.2 H_∞-norm最適參數區域構建 ………………………………….. 39
4.3 範例 ………………………………………………………………….. 41
第五章 結論與未來展望
5.1 結論 ………………………………………………………………….. 59
5.2 未來展望 ……………………………………………………………. 59
參考文獻 61



圖目錄
圖2.1 牛頓迭代法 ………………………………………………………… 10
圖2.2 幅角定理分割圖 …………………………………………………… 11
圖2.3 D分割穩定區域 ……………………………………………………… 18
圖2.4在D分割圖中隨機取點 …………………………………………… 18
圖2.5 a點根的分布 ………………………………………………… 18
圖2.6 a點幅角定理判定右半平面根數 ………………………………… 18
圖2.7 b點根的分布 ………………………………………………… 19
圖2.8幅角定理判定b點右半平面根數 ………………………………… 19
圖2.9 c點根的分布 ………………………………………………… 19
圖2.10幅角定理判定c點右半平面根數 ………………………………… 19
圖2.11 f點根的分布 ………………………………………………… 20
圖2.12幅角定理判定f點右半平面根數 ………………………………… 20
圖2.13 e點根的分布 ………………………………………………… 20
圖2.14幅角定理判定e點右半平面根數 ………………………………… 20
圖2.15將圖2.5中的b(13,2)點平移至b^*(14.8,2)點.......... 21
圖2.16 c點平移至c^*點之根軌跡 …………………………………………… 21
圖 3.2 的分布 ……………………………………………………………… 25
圖 3.1 CaseI之最大時延 ………………………………………………… 34
圖 3.3 CaseII之最大時延( ) …………………………… 35
圖3.4 CaseII 之最大時延( ) …………………………… 35
圖3.5 CaseII 之最大時延( , ) …………………………… 35
圖3.6 CaseIII 之最大時延 ………………………………………………… 36
圖3.7 CaseIV 之最大時延(z=1.5) ……………………………………… 36
圖3.8 CaseIV 之最大時延(z=3) ……………………………………… 37
圖3.9 CaseV 之最大時延 …………………………………………… 37
圖3.10 CaseVI 之最大時延(z=0.5) ……………………………………… 38
圖3.11 CaseVI 之最大時延(z=1) …………………………………………… 38
圖3.12 CaseVI 之最大時延(z= -1) ……………………………………… 38
圖4.2(a) CaseI D分割結合 , γ=2,τ_p=0.1,b(3.5,1.7) ……… 44
圖4.2(b)CaseI 利用Sensitivity Function計算 γ 峰值…………………… 44
圖4.2(c) CaseI D分割結合 , γ=1.25,τ_p=0.1,b(2,0.06187)… 45
圖4.2(d)CaseI 利用Sensitivity Function計算 γ 峰值 (2,0.06187) ……… 45
圖4.3(a) CaseII D分割結合 , γ=3,τ_p=0.1, ,b(3,2)… 46
圖4.3(b) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 46
圖4.3(c) CaseII D分割結合 , γ=1.5,τ_p=0.1,,b(2.084,0.058)………………47
圖4.3(d) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 47
圖4.4(a) CaseII D分割結合 , γ=3,τ_p=0.1, ,b(3.2,2)… 48
圖4.4(b) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 48
圖4.4(c) CaseII D分割結合 , γ=2,τ_p=0.1, ,b(2.9,0.08) 49
圖4.4(d) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 49
圖4.5(a) CaseII D分割結合 , γ=4,τ_p=0.1, ,b(3.798,1) 50
圖4.5(b) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 50
圖4.5(c) CaseII D分割結合 , γ=2.8,τ_p=0.1, ,b(3.978,0.13)……………………… 51
圖4.5(d) CaseII 利用Sensitivity Function計算 γ 峰值 ………………… 51
圖4.6 CaseIII D分割結合 , γ=5,τ_p=0.1, ,z=0 …… 52
圖4.7(a) CaseIV D分割結合 , γ=5,τ_p=0.1, ,z=1.5,b(3.2,2.3) 53
圖4.7(b) CaseIV 利用Sensitivity Function計算 γ 峰值 ………………… 53

圖4.7(c)CaseIV D分割結合 , γ=1.5,τ_p=0.1, z=1.5,b(2,0.1)… 54
圖4.7(d) CaseIV 利用Sensitivity Function計算 γ 峰值 ………………… 54
圖4.8(a) CaseIV D分割結合 , γ=1.35,τ_p=0.1,z=3,b(0.9,0.05)…………………………… 55
圖4.8(b) CaseVI 利用Sensitivity Function計算 γ 峰值 ………………… 55
圖4.9 CaseV D分割結合 , γ=5,τ_p=0.1, ,z=0 …………… 56
圖4.10(a) CaseVI D分割結合, γ=8,τ_p=0.1,z=0.5,b(2.2,0.1)………………………………… 57
圖4.10(b) CaseVI 利用Sensitivity Function計算 γ 峰值………………… 57
圖4.10(c) CaseVI D分割結合, γ=5.15,τ_p=0.1,z=0.5,b(1.8,0.04)…………………… 58
圖4.10(d) CaseVI 利用Sensitivity Function計算 γ 峰值………………… 58



表目錄
表2.1 Padé近似法之階數對照表……………………………………………… 9
表2.2 經牛頓迭代法修正前的參數點a之根值……………………………… 15
表2.3 經牛頓迭代法修正後的參數點a之根值……………………………… 15
表2.4 參數點b之根植……………………………………………………… 16
表2.5 參數點c之根植………………………………………………………… 16
表2.6 參數點f之根植………………………………………………………… 17
表2.7 參數點e之根植………………………………………………………… 17
表3.1 CaseI 最大時延驗證…………………………………………………… 30
表3.2 CaseII 最大時延驗證…………………………………………………… 30
表3.3 CaseII 最大時延驗證…………………………………………………… 31
表3.4 CaseII 最大時延驗證…………………………………………………… 31
表3.5 CaseIV 最大時延驗證………………………………………………… 32
表3.6 CaseIV 最大時延驗證………………………………………………… 32
表3.7 CaseVI 最大時延驗證………………………………………………… 33
表3.8 CaseVI 最大時延驗證………………………………………………… 33
表3.9各Case的最大時延…………………………………………………… 33
表4.1 各Case的γ值………………………………………………………… 43

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