w load disturbance
Controller Plant
r + e u + + y
Gc(s) Gp(s)
sensed
reference - control output
input, or error
set-point
+
+n
sensor
noise
Process Control and
Applications (Lect.3)
Dr. Nader A. Mansour
naderabdelwahab@gmail.com
Mechanical Engineering Department
Contents
Previous Lecture
PID Controller
P term controller
I term controller
D term controller
Previous Lecture
1st Order System Response
Previous Lecture
2nd Order System Response
Undamped step response
Previous Lecture
2nd Order System Response
Overdamped step response
Previous Lecture
2nd Order System Response
Critically damped step response
Previous Lecture
2nd Order System Response
Underdamped step response
PID Controller
PID Controller is the most widely used control
technique.
PID Controller is also called the Three Term
Controller.
PID = Proportional + Integral + Derivative
Each of these terms is a function in the error
signal e(t)
PID Controller
PID Controlled system
Proportional Controller
𝑈 𝑡 = 𝐾𝑝 𝑒 𝑡
is a function of the present value of the error.
The larger the error, the larger the control
action.
Simple to be implemented.
Proportional Controller
Disadvantages
It can’t eliminate the steady state error.
High values of proportional gain reduces the
stability of the system
Proportional Controller
w load disturbance
Controller Plant
+
r + e u + y
Gc(s) Gp(s)
reference - sensed control output
input, or error
set-point
Gc G p ( s ) G p (s)
Y (s) R( s) + W (s)
1 + Gc G p ( s ) 1 + Gc G p ( s )
Proportional Controller
For a proportional controller Gc ( s ) K p
K1
And for a first order system G p (s)
1 + Ts
Gc G p ( s ) G p (s)
Y (s) R( s) + W (s)
1 + Gc G p ( s ) 1 + Gc G p ( s )
K1 K1
Kp
Y (s) 1 + Ts R ( s ) + 1 + Ts W (s)
K1 K1
1+ K p 1+ K p
1 + Ts 1 + Ts
Proportional Controller
Case 1 : R ( s ) unit step, W ( s ) is zero
1 K p K1
y () lim sG ( s ) should be 1
s 0 s 1 + K p K1
Case 2 : W ( s ) unit step, R ( s ) is zero
1 K1
y () lim sG ( s ) should be 0
s 0 s 1 + K p K1
This happens only if K p otherwise steady state error is produced
Proportional Controller
Simulink Model
Proportional Controller
Increasing the controller gain
Fast process response
Steady state error reduced
Too large controller gain
Undesirable degree of oscillation or even
Unstable response
An intermediate value
Best control result
Integral Controller
𝑡
𝑈 𝑡 = 𝐾𝑖 𝑒 𝑡 𝑑𝑡
0
Accumulates the past values of the error signal.
If e(t) is non-zero for any length of time, the
control signal gets larger as time goes on.
control action will continue correcting the error
until it vanishes.
Eliminates steady state error.
Integral Controller
Disadvantages
More oscillatory response &
overshoot
For a very slow system the error
signal will accumulate fast and a
large integrator action will be
introduced
this can cause serious overshoots
the system response becomes
more oscillatory
Integral Controller
Disadvantages
Integral windup
refers to the situation where the integral action
continues to integrate (ramp) indefinitely
This usually occurs when the controller's output can
no longer affect the controlled variable, which in
turn can be caused by controller saturation
Integral Controller
Simulink Model
Derivative Controller
𝑑𝑒(𝑡)
𝑈 𝑡 = 𝐾𝑑
𝑑𝑡
It uses the present and past values to predict the
future error signal.
the control action is based on the rate of change of
the error.
Derivative Controller
Avoiding overshoot
if the error is decreasing too fast that means that the
current control action is very high so it needs to be
decreased substantially to avoid overshoots in the
system.
Braking system
derivative action is against other actions like the
proportional or the integral actions, so the derivative
action acts as a braking system for the response.
Derivative Controller
Advantages
Reduces the system oscillations which reduces the
settling time accordingly.
Quick response for abrupt changes
Derivative Controller
Disadvantages
Amplifies the high frequency noise
(d/dt(a sin(ωt)) = (a ω cos(ω t)))
thus needs filtering the error signal.
Derivative Controller
Disadvantages
Realization problem
ideal derivative with the transfer function = ‘s’ is
not practical so a system of the following form is
adopted
Ds
D s + 1
of where is very small (still has a derivative like behaviour)
Derivative Controller
Simulink Model
