1. Review Automatic Control
Instructor: Cailian Chen, Associate Professor
Department of Automation
27 December 2012

2. Structure of the course
Linear Time-invariant System (LTI)
Analysis
Time Domain
Complex Domain
Frequency Domain
System Model
Concepts
Performance
Compensation Design
各章概念融会贯通
解题方法灵活运用
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3. Time Constant Canonical Form (Bode Form)；
Root Locus Canonical Form (Evan’s Form) ；
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4. Stability Analysis Method
(1) Routh Formula
(2) Root Locus Method
(3) Nyquist Stability Criterion
Z＝N＋P j 
Stable Region
Unstable Region
[S plane]
Stability: All of the roots of characteristic equation of the closed-loop system locate on the left-hand half side of s plane.
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5. Summary of Chapter 5 Understand and remember of root locus equation
Rules for sketching of root locus
Calculate Kg and K by using magnitude equation
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6. 当k’ 从 变化时，S平面上系统特征根的变化形成轨迹。每一个k’ 值，按幅值条件对应于根轨迹上的n个点。
R(s)
G(s)
H(s) + -
C(s)
E(S)
Characteristic equation of the closed-loop system
当k’ 从 变化时，S平面上系统特征根的变化形成轨迹。每一个k’ 值，按幅值条件对应于根轨迹上的n个点。
根轨迹 上的点符合相角条件，且 符合相角条件的点一定在根轨迹上。故利用相角条件就可以绘制根轨迹，无需考虑幅值条件。
Magnitude equation
Phase equation
Argument equation
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7. Rules for Plotting Root Locus
Content
Rules 1
Continuity and Symmetry
Symmetry Rule 2
Starting and end points
Number of segments
n segments start from n open-loop poles, and end at m open-loop zeros and (n-m) zeros at infinity. 3
Segments on real axis
On the left of an odd number of poles or zeros 4
Asymptote
n-m segments： 5

8. Breakaway and break-in points6
Breakaway and break-in points 7
Angle of emergence and entry
Angle of emergence
Angle of entry 8
Cross on the imaginary axis
Substitute s = j to characteristic equation and solve
Routh’s formula

9. Important rules Rule 4: Segments of the real axis
Rule 5: Asymptotes of locus as s Approaches infinity
Rule 6: Breakaway and Break-in Points on the Real Axis
根据根轨迹的定义,当系统增益K值由0→∞ 时,根轨迹的起点必为K=0时的闭环特征根,而终点则为K→∞时的闭环特征根。
Rule 7: The point where the locus crosses the imaginary axis
substituting s=jω into the characteristic equation and solving for ω； Use Routh Formula
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10. Summary of Chapter 6 Complete Nyquist Diagram Bode Diagram
Nyquist Stability Criterion
Relative Stability
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11. Open-loop transfer function with integration elements
Type I system（ν＝1）
Type II system (ν = 2)

12. Type I open-loop system only with inertial elements
Type II open-loop system
with inertial elements
Intercept with real axis is most important, and can be determined by the following method:
A. Solve Im[G(jω)]=0 to get ω and then get Re[G(jω)]；
B. Solve ∠G(jω) = k·180°(k is an integer)

13. Draw the complete Nyquist Diagram
(ω) ：＋90ν°→ 0°→－90ν°

14. Bode Diagram: Always Use Asymptotes
Change the open-loop transfer function into the Bode Canonical form
The slope of lower frequency line is －20νdB/dec，where ν is the type of open-loop system. For ω＝1, L(1)=201gK
If there exist any break frequency less than 1, the point with ω＝1 and L(1)=201gK is on the extending line of lower frequency line.

15. Nyquist stability criterion
If N≠－P，the closed-loop system is unstable. The number of poles in the right-hand half s plane of closed-loop system is Z＝N＋P.
If the open-loop system is stable，i.e. P=0，then the condition for the stability of closed-loop system is: the complete Nyquist diagram does not encircle the point
(－1, j0), i.e. N=0.

16. Relative Stability
Phase margin γ
2. Gain margin γ

17. 相角裕度和增益裕度

18. Summary of Chapter 7 Phase Lead Compensation
Multiplying the transfer function by α
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19. Rules to design phase lead compensation
(1) Determine K to satisfy steady-state error constraint
(2) Determine the uncompensated phase margin γ0
(3) estimate the phase margin in order to satisfy the transient response performance constraint
(4) Determine
Extra margin:
5o~10o
(5) Calculate ωm
(6) Determine T
(7) Confirmation