Control Systems and laplace transform systems on control.pdf

CONTROL SYSTEMS
Sisil Kumarawadu, PhD, SMIEEE
Senior Professor in Electrical Engineering
Head, Department of Electrical Engineering
University of Moratuwa, SRI LANKA
Distinguished Professor-Shanghai University of Electric Power
Chairman of the BoG-Arthur C. Clarke Institute for Modern Technologies

I have a RL circuit and a variable dc voltage source. How do I control the
current in the circuit to follow a predefined profile?
a] manually b] automatically
𝑖(𝑡)
𝑡

Feedback (closed-loop) Control System
Feedback
element
+
-
Reference
current profile
Actual current
in the circuit
Current
controller
Plant
Power supply + RL circuit
Current Sensor
Block diagram of a standard error feedback control system
𝑟(𝑡)
𝑡
𝑒(𝑡) 𝑣 (𝑡)
𝑖(𝑡)

Understanding the Plant
 By working with the plant sufficiently longer
 By mathematically modeling the plant
Eg.
 Plant is a dynamical system. Mathematical
models of dynamical systems are always
differential equations

Proportional (P) Controller
In the proportional control law, the control action is
proportional to the instantaneous value of the error
Here, , is a positive constant to be chosen by the
controller designer.

Linear Time-invariant Dynamical Systems
• Dynamical system is a system whose states vary with time
• If system dynamics or the equations do not vary with time,
such a system is called a time-invariant system
• Mathematical models of a linear system contain linear
differential equations
• Review “Linear Systems Theory”. Remember here we are
considering dynamical systems and not static systems

Laplace Transform
The Laplace transform of a time domain function or signal, f(t), is defined as



0
)
(
)
( dt
t
f
e
s
F st

Laplace Transform
Can analyse linear time-invariant systems
• Born23 March 1749
Beaumont-en-Auge, Normandy,
France
• Died5 March 1827(1827-03-05)
(aged 77) Paris, France
• Nationality French
• FieldsAstronomer and
Mathematician
• InstitutionsÉcole Militaire (1769–
1776)
• Academic advisorsJean d'Alembert
Christophe Gadbled
Pierre Le Canu
• Doctoral students Siméon Denis
Poisson
Pierre-Simon Marquis de Laplace
(1749–1827). Source: WIKIPEDIA

Laplace transform of some
common time functions in
engineering

Transfer Function
Recall the RL circuit
By taking Laplace transform, we get
Assuming initial conditions to be zero, i.e., and rearranging
the above equation we get

Feedback (closed-loop) Control System
Feedback
element
+
-
Reference I/P Measured O/P
Controller Plant
Is a signal or information
processing device
Controlled system
Sensors
Block diagram of a standard error feedback control system

Control System—Laplace Domain
)
(s
GP
)
(
)
(
)
(
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
s
H
s
G
s
G
s
G
s
G
s
R
s
I
s
M
s
I
s
E
s
G
s
G
s
I
s
H
s
R
s
E
P
C
P
C
P
C






+
-
Reference I/P O/P
)
(s
H
)
(s
R
)
(s
GP
)
(s
I
)
(s
GC
Closed-loop
transfer function (CL T/F)
)
(s
I
)
(s
GC
)
(s
GP
)
(s
H
)
(s
R
Controller T/F
Plant T/F
FB element T/F
Laplace transform
of the O/P
Laplace transform
of the reference I/P
)
(s
E

Example
• Assume unity feedback, i.e.,
• As the controller, we try
Here, , is a positive constant to be chosen by the controller designer. This
means
• The closed-loop transfer function is

Unit Step Response
Try K = 100
MATLAB® code is
L=0.2; R=40;
K=100;
M=tf([K],[L R+K]);
step(M), grid on
steady state error (off-set)
=
0 1 2 3 4 5 6 7 8 9
x 10
-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
System: M
Time (sec): 0.00833
Amplitude: 0.712
Step Response
Time (sec)
A
m
p
litu
d
e
𝐼 𝑠
𝑅 𝑠
=
𝐾
𝐿𝑠 + (𝑅 + 𝐾)

Steady state error with increasing gain
Steady state
error (%)
K
28.8
100
7.4
500
3.8
1000

0 1 2 3 4 5 6 7 8 9
x 10
-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
System: K = 1000
Time (sec): 0.00202
Amplitude: 0.962
System: K = 500
Time (sec): 0.00365
Amplitude: 0.926
System: K = 100
Time (sec): 0.00878
Amplitude: 0.713
Step Response
Time (sec)
Amplitude
K = 100
K = 500
K = 1000
MATLAB® code is
L=0.2; R=40;
for K=[100 500 1000], M=tf([K],[L R+K]);
step(M), hold on; end, grid on

Proportional (P) Controller
By taking the Laplace transform and zeroing the initial
conditions, we get
Here, , is a positive constant to be chosen by the controller
designer.

Integral (PI) Controller
Here, , are positive constants to be chosen by the
controller designer. By taking the Laplace transform and
zeroing the initial conditions, we get

Integral (I) Controller
0 1 2 3 4 5 6 7 8 9
x 10
-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Step Response
Time (sec)
Am
plitude

Let us try PI control
The closed-loop transfer function is
Let .
Check the step response:
R=40; L=0.2;
Kp=100; Ki=500;
M=tf([Kp Ki],[L R+Kp Ki]); step(M)

0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Step Response
Time (sec)
Amplitude

PI Controller
 P control action is used to obtain a faster
response
 I control action is used to zero the steady-
state error

Control Systems and laplace transform systems on control.pdf

More Related Content

Similar to Control Systems and laplace transform systems on control.pdf (20)

Recently uploaded (20)

Control Systems and laplace transform systems on control.pdf