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

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

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

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

14

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