The document discusses proportional (P), integral (I), and derivative (D) controllers which are commonly used in closed-loop control systems. It provides examples of how each controller type affects characteristics like rise time, overshoot, settling time, and steady-state error. A P controller reduces rise time but cannot eliminate steady-state error. An I controller eliminates steady-state error but can increase overshoot and settling time. A D controller reduces overshoot and improves transient response. The document recommends using PID control to achieve the best response by reducing rise time, overshoot, steady-state error, and improving stability.

1. Shri G. S. Institute of Technology and
Science, Indore Madhya Pradesh
Department of Electrical Engineering

2. Proportional Controller
A proportional controller (P-controller) is a type of feedback control system that
adjusts the control signal of a system u(t) in proportion to the error e(t), calculated
between the setpoint and the actual output y(t). This type of controller is
commonly used in closed-loop control systems to ensure that the output of the
system remains as close as possible to the setpoint.

3. Proportional Control - Example
The proportional controller (Kp) reduces the rise time, increases the
overshoot, and reduces the steady-state error.
Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
From: U(1)
To:
Y(1)
T s
( )
Kp
s
2
10 s

 20 Kp

( )

Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
From: U(1)
To:
Y(1)
K=300 K=100

4. The Characteristics of P, I, and D controllers
A proportional controller (Kp) will have the effect of reducing the
rise time and will reduce, but never eliminate, the steady-state
error.
An integral control (Ki) will have the effect of eliminating the
steady-state error, but it may make the transient response worse.
A derivative control (Kd) will have the effect of increasing the
stability of the system, reducing the overshoot, and improving the
transient response.

5. Proportional Control
By only employing proportional control, a steady state error
occurs.
Proportional and Integral Control
The response becomes more oscillatory and needs longer to
settle, the error disappears.
Proportional, Integral and Derivative Control
All design specifications can be reached.

6. CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR
Kp Decrease Increase Small Change Decrease
Ki Decrease Increase Increase Eliminate
Kd Small Change Decrease Decrease Small Change
The Characteristics of P, I, and D controllers

7. Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
From: U(1)
To:
Y(1)
Proportional - Derivative -
The derivative controller (Kd) reduces both the overshoot and the
settling time.
T s
( )
Kd s
 Kp

s
2
10 Kd

( ) s

 20 Kp

( )

Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
From: U(1)
To:
Y(1)
Kd=10
Kd=20

8. Proportional - Integral - Example
The integral controller (Ki) decreases the rise time, increases both
the overshoot and the settling time, and eliminates the steady-state
error
MATLAB Example
Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
From: U(1)
To:
Y(1)
T s
( )
Kp s
 Ki

s
3
10 s
2

 20 Kp

( ) s

 Ki

Time (sec.)
Amplitude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
From: U(1)
To:
Y(1)
Ki=70
Ki=100

9. Designing a PID Controller
1. Obtain an open-loop response and determine what needs to be improved
2. Add a proportional control to improve the rise time
3. Add a derivative control to improve the overshoot
4. Add an integral control to eliminate the steady-state error
5. Adjust each of Kp, Ki, and Kd until you obtain a desired overall
response.
Lastly, please keep in mind that you do not need to implement all three
controllers (proportional, derivative, and integral) into a single system, if not
necessary. For example, if a PI controller gives a good enough response (like the
above example), then you don't need to implement derivative controller to the
system. Keep the controller as simple as possible.

10. Proportional+Integral+Derivative Control
Although PD control deals neatly with the overshoot and problems
associated with proportional control it does not cure the problem with the
steady-state error. Fortunately it is possible to eliminate this while using
relatively low gain by adding an integral term to the control function
which becomes

11. Thank you