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EE 380 Fall 2014Lecture 19.
EE 380
Linear Control Systems
Lecture 19
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 19.
Lecture 19 Topics
• PID Controller Design – Examples
2
EE 380 Fall 2014Lecture 19.
Exercise 1 • Consider the effect of P control on a second-order plant
• Compare the open-loop and closed-loop natural frequencies and dimensionless damping ratios
• Is it possible to arbitrarily place the poles of the closed-loop system to achieve a desired transient response?
• Determine the closed-loop steady-state error for a unit-step input
3
( )R s ( )Y sPK
( )E s 2,
2 2, ,2n ol
ol n ol n ol
Ks s
Controller Plant
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
4
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
5
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
6
EE 380 Fall 2014Lecture 19.
Exercise 2 • Consider the effect of PI control on a second-order plant
• Assuming that the controller gains are chosen so that the closed-loop system is BIBO stable, determine the closed-loop steady-state error for a unit-step input
• Compared to Exercise 1, why is it more challenging to choose controller gains to achieve desired transient response characteristics?
7
( )R s ( )Y sI
PKKs
( )E s
Controller Plant2
2 22n
n n
Ks s
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
8
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
9
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
10
EE 380 Fall 2014Lecture 19.
Exercise 3• Consider the effect of PD control on a second-order plant
assuming that for frequencies of interest sτ « 1
• Compare the open-loop and closed-loop natural frequencies and dimensionless damping ratios
• Is it possible to arbitrarily place the poles of the closed-loop system to achieve a desired transient response?
• Determine the closed-loop steady-state error for a unit-step input
11
( )R s ( )Y s
1D
PK sKs
( )E sController Plant
2,
2 2, ,2n ol
ol n ol n ol
Ks s
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
12
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
13
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
14
EE 380 Fall 2014Lecture 19.
Exercise 4 • Consider the closed-loop system below
• Is the open-loop plant BIBO stable?
• Can we form a BIBO stable closed-loop system using – P control? – PI control? – PD control?
15
( )R s ( )Y s cG s
( )E s2
14s
Controller PlantU( )s
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
16
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
17
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
18
EE 380 Fall 2014Lecture 19.
EE 380
Linear Control Systems
Lecture 19
Professor Jeffrey SchianoDepartment of Electrical Engineering
1
EE 380 Fall 2014Lecture 19.
Lecture 19 Topics
• PID Controller Design – Examples
2
EE 380 Fall 2014Lecture 19.
Exercise 1 • Consider the effect of P control on a second-order plant
• Compare the open-loop and closed-loop natural frequencies and dimensionless damping ratios
• Is it possible to arbitrarily place the poles of the closed-loop system to achieve a desired transient response?
• Determine the closed-loop steady-state error for a unit-step input
3
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
4
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
5
EE 380 Fall 2014Lecture 19.
Exercise 1 Solution
6
EE 380 Fall 2014Lecture 19.
Exercise 2 • Consider the effect of PI control on a second-order plant
• Assuming that the controller gains are chosen so that the closed-loop system is BIBO stable, determine the closed-loop steady-state error for a unit-step input
• Compared to Exercise 1, why is it more challenging to choose controller gains to achieve desired transient response characteristics?
7
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
8
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
9
EE 380 Fall 2014Lecture 19.
Exercise 2 Solution
10
EE 380 Fall 2014Lecture 19.
Exercise 3• Consider the effect of PD control on a second-order plant
assuming that for frequencies of interest sτ « 1
• Compare the open-loop and closed-loop natural frequencies and dimensionless damping ratios
• Is it possible to arbitrarily place the poles of the closed-loop system to achieve a desired transient response?
• Determine the closed-loop steady-state error for a unit-step input
11
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
12
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
13
EE 380 Fall 2014Lecture 19.
Exercise 3 Solution
14
EE 380 Fall 2014Lecture 19.
Exercise 4 • Consider the closed-loop system below
• Is the open-loop plant BIBO stable?
• Can we form a BIBO stable closed-loop system using – P control? – PI control? – PD control?
15
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
16
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
17
EE 380 Fall 2014Lecture 19.
Exercise 4 Solution
18
