EE 380 Linear Control Systems Lecture 25
Transcript of EE 380 Linear Control Systems Lecture 25
EE 380 Fall 2014Lecture 25.
EE 380
Linear Control Systems
Lecture 25
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 25.
Lecture 25 Topics
• Common Cascade Compensators– P, PI, PD, PID– Phase-lag, Phase-Lead
• Determining Compensator Parameters using Root Locus
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EE 380 Fall 2014Lecture 25.
– Three term compensator (Minorsky, 1922)
P, PI, PD, PID Compensators
3
R
Y cG s
UE pG s
Compensator Plant
0
( )t
P I Ddeu t K e t K e d K dt
EE 380 Fall 2014Lecture 25.
Guidelines for Selecting Terms• Proportional Control
– Always try first
• Proportional plus Integral (PI) Control– Improves steady-state accuracy– Bends root locus towards the right-half plane
• Proportional plus Derivative (PD) Control– Improves transient response– Bends the root locus towards the legft-half plane
• Proportional plus Integral plus Derivative (PID) Control– Improves steady-state accuracy and transient response
4
EE 380 Fall 2014Lecture 25.
PID Disadvantages
• PD Control– Gain increases with frequency– Accentuates noise
• PI Control– Infinite gain at DC
5
EE 380 Fall 2014Lecture 25.
Exercise 1
• Sketch the Bode magnitude and phase plots for– PI Compensator– PD Compensator
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EE 380 Fall 2014Lecture 25.
Exercise 1 Solution
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EE 380 Fall 2014Lecture 25.
Exercise 1 Solution
8
EE 380 Fall 2014Lecture 25.
Phase Lag Compensator
• Approximates PI controller– Low-frequency gain rolls-off to a finite value
• Transfer function representation
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1, 11o
co
s aG s K as
,oc o
o
s a s zKG s K p za s s p
EE 380 Fall 2014Lecture 25.
Exercise 2
• Sketch Bode magnitude and phase plot of the phase-lag compensator
10
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
11
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
12
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
13
EE 380 Fall 2014Lecture 25.
Phase Lead Compensator
• Approximates a PD controller– High-frequency gain rolls-off to a finite value
• Transfer function representation
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1 , 11o
co
sG s K as a
,oc o
o
s s zG s Ka K p zs a s p
EE 380 Fall 2014Lecture 25.
Exercise 3
• Sketch Bode magnitude and phase plot of the phase-lead compensator
15
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
16
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
17
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
18
EE 380 Fall 2014Lecture 25.
EE 380
Linear Control Systems
Lecture 25
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 25.
Lecture 25 Topics
• Common Cascade Compensators– P, PI, PD, PID– Phase-lag, Phase-Lead
• Determining Compensator Parameters using Root Locus
2
EE 380 Fall 2014Lecture 25.
– Three term compensator (Minorsky, 1922)
P, PI, PD, PID Compensators
3
EE 380 Fall 2014Lecture 25.
Guidelines for Selecting Terms• Proportional Control
– Always try first
• Proportional plus Integral (PI) Control– Improves steady-state accuracy– Bends root locus towards the right-half plane
• Proportional plus Derivative (PD) Control– Improves transient response– Bends the root locus towards the legft-half plane
• Proportional plus Integral plus Derivative (PID) Control– Improves steady-state accuracy and transient response
4
EE 380 Fall 2014Lecture 25.
PID Disadvantages
• PD Control– Gain increases with frequency– Accentuates noise
• PI Control– Infinite gain at DC
5
EE 380 Fall 2014Lecture 25.
Exercise 1
• Sketch the Bode magnitude and phase plots for– PI Compensator– PD Compensator
6
EE 380 Fall 2014Lecture 25.
Exercise 1 Solution
7
EE 380 Fall 2014Lecture 25.
Exercise 1 Solution
8
EE 380 Fall 2014Lecture 25.
Phase Lag Compensator
• Approximates PI controller– Low-frequency gain rolls-off to a finite value
• Transfer function representation
9
EE 380 Fall 2014Lecture 25.
Exercise 2
• Sketch Bode magnitude and phase plot of the phase-lag compensator
10
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
11
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
12
EE 380 Fall 2014Lecture 25.
Exercise 2 Solution
13
EE 380 Fall 2014Lecture 25.
Phase Lead Compensator
• Approximates a PD controller– High-frequency gain rolls-off to a finite value
• Transfer function representation
14
EE 380 Fall 2014Lecture 25.
Exercise 3
• Sketch Bode magnitude and phase plot of the phase-lead compensator
15
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
16
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
17
EE 380 Fall 2014Lecture 25.
Exercise 3 Solution
18