Ch8 Root Locus Techniques
根軌跡法
§ 8.1 Introduction and § 8.2 Defining Root Locus
Figure 8.1
a. Closed-loop system;
• Open-loop transfer function: KG(s)H(s)
• Closed-loop transfer function: T(s) =
b. equivalent transfer function
又稱 loop gain
• Open-loop transfer function: KG(s)H(s)
又稱 loop gain
Open-loop poles → 滿足 KG(s)H(s) = 0 之 s 值
K的變化不影響 open-loop poles 的位置
容易求解
Closed-loop poles → 滿足 1 + KG(s)H(s) = 0 之 s 值
K的變化會影響 closed-loop poles 的位置
系統性能由closed-loop poles決定
不容易求解
Example to show Root Loci
Figure 8.4
a. CameraMan®
Presenter Camera System automatically follows a
subject who wears infrared sensors on their front
and back (the front sensor is also a microphone);
tracking commands and audio are relayed to
CameraMan via a radio frequency link from a unit
worn by the subject.
Courtesy of ParkerVision.
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Table 8.1 for the system of Figure 8.4
closed-loop pole location as a function of gain K
1 + KG(s)H(s) = 0 →
Q: How many systems ?
Figure 8.5
a. Pole plot from Table 8.1;
b. root locus
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• 根軌跡法:
1. 繪圖法 (graphic technique);
用於分析、設計系統的穩定度與暫態反應
(1) 說明系統性質 (qualitative description)
Fig. 8.5:
K＜25 overdamped system
K＞25 underdamped system
(2) 有力的量化工具 (quantitative tool)
說明於次頁
(3) 可執行高階系統的分析與設計
2. 由open-loop poles & zeros
決定所有closed-loop poles
之分佈 ( 當 k = 0 →∞)
說明根軌跡法是有力
的量化工具:
由Closed-loop poles
位置 可繪出系統的
明確反應(量化反應)
數學基礎: 向量的大小和方向
呈現方法
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Figure 8.2
s =  + j
(s + a)
(s + a)
(s + 7)|s5 + j2
數學基礎: 函數的大小和方向
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數學基礎: 範例
Ex.8.1 at s =-3+j4
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§ 8.3 Properties of Root Locus
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• 屬根軌跡上一點(Closed-loop poles )的判別式:
即滿足 1 + KG(s)H(s) = 0 之 s 值
§ 8.3 Properties of Root Locus
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• 根軌跡上一點(Closed-loop poles )的判別式:
即滿足 1 + KG(s)H(s) = 0 之 s 值
KG(s)H(s) = －1 = 1∠(2k+1)180o
k= 0, ±1, ±2, ±3, …
→ ∣KG(s)H(s)∣ = 1 大小判別式(求k值)
∠KG(s)H(s) = (2k+1)180o 角度判別式
(求是否根軌跡上一點)
Figure 8.6 1/2
∣KG(s)H(s)∣ = 1
大小判別式
(求該系統的k值)
∠KG(s)H(s) = (2k+1)180o 角度判別式
(求是否根軌跡上一點)
註: KG(s)H(s)
Open-loop transfer function; 又稱 loop gain
Figure 8.7 Vector representation of G(s) from Figure 8.6(a) at -2+ j 3
∠KG(s)H(s) = (2k+1)180o 角度判別式
S = -2+ j 3 不合
S = -2+ j (√3 / 2) 合 (角度判別)
K = 0.33 at S = -2+ j (√3 / 2)
重覆前述步驟完成根軌跡的繪製
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§ 8.4 Rules for sketching the Root Locus
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1. No. of branches
根軌跡條數 = Closed-loop poles 數目
2. Symmetry
根軌跡對稱於實軸
3. 在實軸上的根軌跡 (由角度判別式鑑定)
Figure 8.9
Real-axis segments of the root locus
實軸上、右方有奇數個 poles+zeros 所有點的集合 = 實軸上的根軌跡
§ 8.4 Rules for sketching the Root Locus
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§ 8.4 Rules for sketching the Root Locus
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Example 8.2
Figure 8.11
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Figure 8.12
Root locus and asymptotes for
the system of Figure 8.11
σ0 = - 4/3
θ = π/3； π； 5π/3
Skill-Assessment Exercise 8.3
自修
§ 8.5 Refining the Sketch 精確繪圖
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1. Real-Axis Breakaway & Break-in Points
質性說明:
Figure 8.13
Root locus example showing real- axis
breakaway (-1) and break-in points (2)
量化定義:
Figure 8.14
Variation of gain along the real axis for the
root locus of Figure 8.13
Breakaway point: (-1)
having max. k among root loci where it belong
break-in point: (2)
having min. k among root loci where it belong
§ 8.5 Refining the Sketch
Real-Axis Breakaway or Break-in Points 求法:
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§ 8.5 Refining the Sketch
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§ 8.5 Refining the Sketch
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§ 8.5 Refining the Sketch
3. Angles of Departure and Arrival
i.e. 不在實軸上的open-loop poles and zeros其根軌跡的畫法
Figure 8.15
a. angle of departure, θ1
根軌跡離開出發點角度
由角度判別式求θ1
∠KG(s)H(s) = (2k+1)180o
ε→ 0 ;
p1， → p1
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§ 8.5 Refining the Sketch
Figure 8.15
b. angle of arrival
根軌跡抵達終止點角度
由角度判別式求θ2
∠KG(s)H(s) = (2k+1)180o
ε→ 0 ;
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§ 8.5 Refining the Sketch
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§ 8.5 Refining the Sketch
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• HW
Skill-Assessment Exercise 8.4
1. 手繪求解
2. 利用 MATLAB 軟體求解
1.
2.
3.
4.
5.
6.
7.
8.
9.
根軌跡條數
根軌跡對稱於實軸
在實軸上的根軌跡
根軌跡的起點、終點
在無窮遠處之根軌跡(Behavior at Infinity)
Real-Axis Breakaway & Break-in Points
Jω-axis crossing
Angles of Departure and Arrival
求根軌跡關鍵點位置(Calibrating the root
locus)
§ 8.6 An Example
HW
Skill-Assessment Exercise 8.5
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1.手繪求解
2.利用 MATLAB 軟體求解
§ 8.6 An Example
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§ 8.6 An Example
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§ 8.7 Transient Response Design via Gain Adjustment
藉由調整k → 使高階系統的反應同2階系統
(i.e. 利用2階系統公式求系統反應, 如ts, tr, tp, %OS等)
Example 8.8
Figure 8.23 Second- and third-order responses for a. Case 2; b. Case 3
各與其2階系統(2nd-order dominant pair)比較暫態反應
Case 3 is better.
•若高階系統無 2nd-order dominant pair, 需以電腦模擬
求其暫態反應
• HW Skill-Assessment Exercise 8.6