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Nội dung Text: Lecture Fundamentals of control systems: Chapter 6 - TS. Huỳnh Thái Hoàng
- Lecture Notes
Fundamentals of Control Systems
Instructor: Assoc. Prof. Dr. Huynh Thai Hoang
Department of Automatic Control
Faculty of Electrical & Electronics Engineering
Ho Chi Minh City University of Technology
Email: hthoang@hcmut.edu.vn
huynhthaihoang@yahoo.com
Homepage: www4.hcmut.edu.vn/~hthoang/
6 December 2013 © H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
- Chapter 6
DESIGN OF CONTINUOUS
CONTROL SYSTEMS
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 2
- Content
Introduction
Effect of controllers on system performance
Control systems design using the root locus method
Control systems design in the frequency domain
Design
g of PID controllers
Control systems design in state-space
Design of state estimators
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 3
- Introduction
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4
- Introduction to design process
Design is a process of adding/configuring hardware as well as
software in a system so that the new system satisfies the
d i d specifications.
desired ifi ti
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 5
- Series compensator
The controller is connected in series with the plant.
plant
R(s) Y(s)
+ GC(s) G(s)
Controllers: phase lead, phase lag, lead-lag compensator, P,
PD PI,
PD, PI PID,…
PID
Design method: root locus,
locus frequency response
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 6
- State feedback control
All the states of the system are fed back to calculate the control
rule.
r(t) u(t) x(t) y(t)
+
x (t ) Ax (t ) Bu (t ) C
K
State feedback controller: u (t ) r (t ) Kx (t )
K k1 k2 kn
Design method: pole placement,
placement LQR,…
LQR
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 7
- Effects of controller on system
performance
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 8
- Effects of the addition of poles
The addition of a pole (in the left
left-half
half ss-plane)
plane) to the open
open-
loop transfer function has the effect of pushing the root locus
to the right, tending to lower the system’s relative stability and
t slow
to l d
down th settling
the ttli off the
th response.
Im s Im s Im s
Re s Re s Re s
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 9
- Effects of the addition of zeros
The addition of a zero (in the left
left-half
half ss-plane)
plane) to the open
open-
loop transfer function has the effect of pulling the root locus to
the left, tending to make the system more stable and to speed
up the
th settling
ttli off the
th response.
Im s Im s Im s
Re s Re s Re s
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 10
- Effects of lead compensators
Transfer function:
1 Ts
GC ( s ) K C ( 1)
1 Ts
Frequency response:
1 Tj
GC ( j ) K C
1 Tj
Characteristics of the Bode plots:
1 1
max sin
1
1
max
T
L ( max ) 20 lg K C 10 lg
The lead compensators improve
the transient response (POT, ts,..)
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 11
- Lead compensator implementation
Lead compensator transfer function:
U ( s ) R2 R4 1 R1C1s 1 Ts
KC ( 1 R1C1 R2C2 )
E ( s ) R1 R3 1 R2C2 s 1 Ts
E(s) U(s)
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 12
- Effects of lag compensators
Transfer function:
1 Ts
GC ( s ) K C ( 1)
1 Ts
Frequency response:
1 Tj
GC ( j ) K C
1 Tj
Characteristics of the Bode plots:
1 1
min sin
1
1
min
T
L (min ) 20 lg K C 10 lg
The lag
g compensators
reduce the steady-state error.
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 13
- Lag compensator implementation
Lag compensator transfer function:
U ( s ) R2 R4 1 R1C1s 1 Ts
KC ( 1 R1C1 R2C2 )
E ( s ) R1 R3 1 R2C2 s 1 Ts
E(s) U(s)
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 14
- Effects of lead
lead--lag compensators
1 1T1s 1 2T2 s
Transfer function: GC ( s) KC (1 1, 2 1)
1 T1s 1 T2 s
Bode diagram
The lead-lag
lead lag compensators improve transient response and
reduces the steady-state error.
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 15
- Effects of proportional controller (P)
T
Transfer
f function:
f ti GC ( s ) K P
Increasing proportional gain leads to decreasing steady-state
error, however,
h the
h system become
b l
less stable,
bl andd the
h POT
increases.
y(t)
(t)
Ex: response of a
proportional control
system
t whose
h
plant has the
transfer function
below:
10
G ( s)
( s 2)( s 3)
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
- Effects of proportional derivative controller (PD)
Transfer function: Bode diagram
GC ( s ) K P K D s K P (1 TD s )
The PD controller is a
special case of phase lead
compensator,
t the
th
maximum phase lead is
max=900 at the frequency
q y
max=+.
The PD controller speed up
the response of the system,
however it also makes the
system more sensitive to
high frequency noise.
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 17
- Effects of proportional derivative controller (PD)
Note: The larger the derivative constant,
constant the faster the
response of the system.
y( )
y(t)
unompensated
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 18
- PD controller implementation
PD controller transfer function:
U ( s ) R2 R4
(1 R1C1s ) K P K D s
E ( s ) R1 R3
E(s)
U(s)
( )
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
- Effects of proportional integral controller (PI)
Transfer function: Bode diagram
KI 1
GC ( s ) K P K P (1 )
s TI s
The PI controller is a
special case of phase lag
compensator,
t theth minimum
i i
phase lag is min= 900 at
the frequency
q y min=+.
PI controllers eliminate
stead state error to step
steady
input, however it can
increase POT and settling
time.
6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 20
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