ME132 Interconnections Lec3
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Transcript of ME132 Interconnections Lec3
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ME 132 Dynamic Systems and Feedback
Lecture 3
Feedback Interconnections
Closed-Loop VS Open-Loop Control
Static and Linear Analysis
Summary• The structure of control systems• Basic control system (open-loop vs closed-loop)• Basic block-diagram algebra – static and linear
systems– Open-loop control– Closed-loop control
• Example: Cruise Control for a Car – Open-loop control– Closed-loop control
• Advantages and disadvantages of feedback
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The Structure of Control Systems3
Basic Control Systems• Open-Loop System
4
• Closed-Loop System
Open-loop system – static blocks5
controller plant
• r - reference input• d - disturbance input• y - output• e = r – y - error
Open-loop system – linear static blocks6
controller plant
y = Gm
m = u+ dy = G(u+ d)
u = Kr
y = G(u+ d) y = G(Kr + d)
Open-loop system – linear static blocks7
controller plant
y = G(Kr + d)
e = (1�GK)r �Gd e = r � y
Closed-loop system – static blocks8
controller plant
• Inputs– r - reference input– d - disturbance input
• Outputs– y - plant output– u - controller output– e = r – y - error
Closed-loop system – linear static blocks9
controller plant
Plant: y = G(u+ d)
Controller:u = Ke
e = r � yu = K(r � y)
Closed-loop system – static blocks10
controller plant
y = G(u+ d) u = K(r � y)
y = �GKy +GKr +Gd
y =GK
1 +GKr +
G
1 +GKd
Closed-loop system – static blocks11
controller plant
y =GK
1 +GKr +
G
1 +GKd
e =1
1 +GKr � G
1 +GKd e = r � y
We want to make small
Closed-loop VS Open-loop12
e =1
1 +GKr � G
1 +GKd
e = r � y
e = (1�GK)r �Gd
Making K large, results in e being small
Difficult to select K
K ⇡ r �Gd
Gr
Feedback system’s algebra13
controller plant
• Inputs– r - reference input– d - disturbance input
• Outputs– y - plant output– u - controller output– e = r – y - error
y =GK
1 +GKr +
G
1 +GKd
e =1
1 +GKr � G
1 +GKd
Closed-loop transfer functions
| {z }Gc(r!e)
| {z }Gc(d!e)
| {z }Gc(r!y)
| {z }Gc(d!y)
Feedback system’s algebra14
• Open-loop transfer function
Ge!f
= Go
= GK
Negative feedback system’s algebra15
• Closed-loop transfer function
Gclosed loop
=
Gforward path
1 +Gopen loop
Gc(r!y) =
Gfp(r!y)
1 +Go
=GK
1 +GK
Gc(d!y) =
Gfp(d!y)
1 +Go
=G
1 +GK
Negative feedback system’s algebra16
• Closed-loop transfer function
Gclosed loop
=
Gforward path
1 +Gopen loop
Gc(r!e) =
Gfp(r!e)
1 +Go
=1
1 +GK
Gc(d!e) =
Gfp(d!e)
1 +Go
=�G
1 +GK
Example: Cruise Control for a Car We want to regulate the linear speed of a car, y, to a
desired value r
Static car model:
G
H
Nominal car parameters
speed
control input
disturbance input
Cruise Open-Loop Control
G
HOpen-loop control:
u = Kol
r
Assume nominal system:
y = Go
Kol
r
G = Go, H = Ho, d = 0
Cruise Closed-Loop Control
Closed-loop control:
Closed-loop response:
| {z }Gc(r!y)
| {z }Gc(d!y)
y =GK↵
1 +GKfbr +
�H
1 +GKfbd
Cruise Closed-Loop Control Objective:
y =GK↵
1 +GKfbr +
�H
1 +GKfbd
Closed-loop response :
y ⇡ r
Assume nominal system:
| {z }⇡1
G = Go, H = Ho, d = 0
Cruise Closed-Loop Control Objective:
y =GK↵
1 +GKfbr +
�H
1 +GKfbd
Closed-loop response :
y ⇡ r
Assume nominal system: G = Go, H = Ho, d = 0
| {z }⇡�0.01
Ho
1 +GoKfb= 0.01 Kfb & Ho
0.01Go
Closed-Loop VS Open-Loop - Disturbance Rejection
Kfb & Ho
0.01Go
G
H
yol
= r �Hod ycl = r � 0.01d
Ho = 5 >> 0.01Closed-loop system exhibits superior external disturbance rejection
Open-Loop – Sensitivity to Plant Variations
G
H
Assume no disturbances (d = 0).
Question: • what is the variation of the output y around the nominal
output yo when there is a variation of the plant G around the nominal plant Go?
Open-Loop – Sensitivity to Plant Variations
G
H
S =
✓Go
yo
◆@y
@G
����G = G
o
y = y
o
=
✓Go
yo
◆@�
G
G
o
r�
@G
�����G = G
o
y = y
o
yo = r
S = 1
y =G
Go
r
1 percent deviation of the plant G from its nominal value Go will approximately result in a 1 percent deviation in the output y
Closed-Loop – Sensitivity to Plant Variations
S =
✓Go
yo
◆@y
@G
����G = G
o
y = y
o
y =GK↵
1 +GKfbr
yo =GoK↵
1 +GoKfbr
=
✓Go
yo
◆ @⇣
GKff1+GKfb
r⌘
@G
������G = G
o
y = y
o
Closed-Loop – Sensitivity to Plant Variations
y =GK↵
1 +GKfbr
yo =GoK↵
1 +GoKfbr
Ho
1 +GoKfb= 0.01
remember that we chose …
and Ho = 5
Scl =1
1 +GoKfb⇡ 0.002 << S
ol
Closed-Loop VS Open-Loop - Sensitivity
G
H
Closed-loop system is much less sensitive to plant variations
Scl =1
1 +GoKfb⇡ 0.002S
ol
= 1
Closed-Loop – Sensitivity to Noise
noise
| {z }�Gc(n!y)
Closed-Loop – Sensitivity to Noise
Gc(n!y) =GKfb
1 +GKfb
Ideally we require��Gc(n!y)
�� << 1
But this is in conflict with “high-gain” feedback,
which is necessary for disturbance rejection and robustness to plan variations. More on this later …
|GoKfb| >> 1
Summary• The structure of control systems
– Open-loop control– Closed-loop control
• Advantages of feedback :– Closed-loop systems exhibit superior external
disturbance rejection– Closed loop systems are significantly less
sensitive to plan parameter variations• Disadvantages of high-gain feedback
– High sensitivity to measurement noise– May destabilize feedback system (dynamic
models)
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