ReviewAutomatic Control
Instructor: Cailian Chen, Associate ProfessorDepartment of Automation
27 December 2012
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Structure of the courseLinear Time-invariant System (LTI)
各章概念融会贯通解题方法灵活运用
Concepts System Model
PerformanceTime DomainComplex DomainFrequency Domain
Analysis
Compensation Design
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1 2
1 2
1 1
2
1 1
2
]12)[()1(
]12)[()1(
)(
)(n
j
n
l pnpnj
m
i
m
k znkznki
ll
ssss
sssK
sR
sC
Time Constant Canonical Form (Bode Form)
;
; Root Locus Canonical Form (Evan’s Form)
1 2
1 2
( )( ) ( )( )( )
( ) ( )( ) ( )r m
n n
K s z s z s zN sG s
D s a s p s p s p
LL
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j
0
Stable RegionUnstable Region
[S plane]
(1) Routh Formula (2) Root Locus Method (3) Nyquist Stability Criterion
Stability Analysis Method
Stability: All of the roots of characteristic equation of the closed-loop system locate on the left-hand half side of s plane.
Z = N + P
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Understand and remember of root locus equation
Rules for sketching of root locus Calculate Kg and K by using magnitude
equation
Summary of Chapter 5
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Characteristic equation of the closed-loop system
1 ( ) 0G s H s ( ) 1G s H s
R(s)
G(s)
H(s)
+
-
C(s)E(S)
...2,1,0,)12()()(
1|)(|
|)(|
11
1
1
kkpszs
ps
zsK
n
jj
m
ii
n
jj
m
ii
g
Phase equationArgument equation
Magnitude equation
7
Content Rules1 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 Asymptote
mn
zpn
j
m
iij
1 1
)()(
,2,1,0,)12(
kmn
k =
Rules for Plotting Root Locus
8
7Angle 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
n
j
m
zii
ijz k1 1
)12(
m
i
n
pjj
jip k1 1
)12(
6Breakaway
and break-in points
0
][
ds
sFd 0 sZKsPsF g
0 sZsPsZsP
m
i
n
j ii pz1 1
11
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180 (2 1)( 0,1,2, )
kk
n m
1 1
n m
j ij i
p z
n m
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
0 sZsPsZsP ji pszs
11
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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Summary of Chapter 6
Complete Nyquist Diagram Bode Diagram Nyquist Stability Criterion Relative Stability
Open-loop transfer function with integration elements
Type I system ( ν = 1 )
Type II system (ν = 2)
0
(0)A (0) 90
( ) 0A ( ) ( ) 90n m
1 2
1 2
(1 )(1 )...(1 )( )
( ) (1 )(1 )...(1 )m
n
K j j jG j
j j T j T j T
n m
0
(0)A (0) 180
( ) 0A ( ) ( ) 90n m
Type I open-loop system only with inertial elements
Type II open-loop systemwith 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)
0 s
s平面
ε
C1
C2
R
jj
j
(ω) :+ 90ν°→ 0°→ - 90ν°
: 0< <<1, =[-90 ,+90 ]je
1
1
1
1j
m ji jvi
v vns e j jjj v
K e KG s H s e
e T e
Draw the complete Nyquist Diagram
Bode Diagram: Always Use Asymptotes
① Change the open-loop transfer function into the Bode
Canonical form
② The slope of lower frequency line is
- 20νdB/dec , where ν is the type of open-loop
system. For ω = 1, L(1)=201gK
③ 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.
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.
Relative Stability
1. Phase margin γ
( ) ( 180 ) 180 ( )c c
1 ( )g gK A
20lg ( ) ( )g gGM A L
2. Gain margin γ
相角裕度和增益裕度
0
-90°
-180°
-270°
0
-90°
-180°
-270°稳定系统
(c)
正相角裕度
正增益裕度
负增益裕度
负相角裕度
不稳定系统(d)
0
A
正相角裕度
-1
1
0
A
-1
1
稳定系统(a)
不稳定系统(b)
负增益裕度
负相角裕度
正增益裕度
( )(dB)L
( )( )
( )(dB)L
( )( )
( )jY
( )X
( )jY
( )X
1
gK
1
gK
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Phase Lead Compensation
Summary of Chapter 7
1m
T
)(L
decdB /20
alog20
alog20
aT/1 m
)(
T/1
m
( ) 10logc mL
0 20
Multiplying the transfer function by α
1 sin
1 sinm
m
1
1c
TsG s
Ts
(4) Determine
(2) Determine the uncompensated phase margin γ0
(7) Confirmation
(1) Determine K to satisfy steady-state error
constraint
(5) Calculate ωm
(6) Determine T
(3) estimate the phase margin in order to satisfy the transient response performance constraint
Rules to design phase lead compensation
m
Extra margin:5o~10o