Controller Design for Transient Response Improvement Report

This report presents an analysis of transient response improvement through controller design. It begins by calculating the values of k and dominant poles for a given open-loop transfer function. The report then demonstrates the location of poles on the root locus and discusses desired performance characteristics such as settling time, overshoot, and steady-state error. A lead-lag compensator is designed to meet these requirements, and a comparison is made between the compensated and uncompensated control systems, highlighting the improved performance achieved through compensation. The simulation data shows that the compensated system has smaller settling time, rise time, peak and overshoot compared to the uncompensated system.

TRANSIENT RESPONSE IMPROVEMENT THROUGH CONTROLLER DESIGN
1. Calculating the value of k and dominant poles.
Given the open loop transfer function G(s) the closed loop transfer function can be
obtained as:
G ( s ) = k (s+1)
(s+1)( s+2)(s+3)
T ( s )= G( s)
1+G(s) = k ( s+1)
( s+1)(s2 +5 s +k +6)
Comparing the characteristic equation with the general characteristic equation the value of k
can be obtained as for ζ =0.55
s2 +2 ζ ω0 s+ω0
2
2 ζ ω0 =5 ω0 =50
11 k +6=50
11
2
k=14.66
Substituting for k in the characteristic equation the poles are located at: -1,−2.5 ∓ j 3.8
therefore the dominant pole is located -1.
2. Demonstrating that the poles are located on the root locus.
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
0.94
0.64
0.76
0.160.340.50.64
0.76
0.86
0.94
0.985
0.160.340.5
0.86
0.985
2
4
6
8
10
2
4
6
8
10
Root Locus
Real Axis (seconds -1 )
Imaginary Axis (seconds-1 )
System: Gmod
Gain: 0
Pole: -2.5 + 3.8i
Damping: 0.55
Overshoot (%): 12.6
Frequency (rad/s): 4.55
System: Gmod
Gain: 0
Pole: -2.5 - 3.8i
Damping: 0.55
Overshoot (%): 12.6
Frequency (rad/s): 4.55
System: Gmod
Gain: 0
Pole: -1
Damping: 1
Overshoot (%): 0
Frequency (rad/s): 1
Figure 1: Root locus of the closed loop transfer function showing location of poles
3. For the best performance characteristics of the system the settling time of system should
be 1.5s, overshoot should be 5% and steady state error should be ∓0.5%.

0 0.5 1 1.5 2 2.5 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Step Response
Time (seconds)
Amplitude
System: G
Time (seconds): 0.829
Amplitude: 0.799 System: G
Time (seconds): 1.29
Amplitude: 0.723
Figure 2: Step response for the uncompensated system
4. Design of lead-lag compensator achieve requirements for the system. When we plotted
the uncompensated closed loop transfer function we obtained the following data: settling
time 1.2829, overshoot 12.6311 and steady state error 0.1538. We also plotted the bode
plot and obtain the phase margin as 80.7. For safe operation of the controller the phase
margin should be more than 120 degrees. The compensator should have a value of 39.3
degrees to achieve the compensation.
A. Compensating process: Compensating calculations at 140deg:
Φ =140 – (80.7 – Ac) = 140 – (80.7 – 10) =69.3
β = 1−SinΦ
1+ sinΦ = 1−sin 59.3
1+sin 59.3 =0.03335
Gm = 10 log 1
β =10 log 1
0.03335 = 14.768
The gain margin was then found on the bode plot of the uncompensated system. This
then gave the ωm value of 9.41, from which τ was found
τ = 1
ωm √ β = 1
9.41 √0.03335 = 0.5819
From this the Glead was found, τs+1
τβs+1 = 0.5819 s+1
0.01941 s+1
5. Comparison between compensated and uncompensated control system. From the simulation
the data can be summarized as below:
Compensated: settling time 0.2501, Rise time 1.2829, peak 0.7162, overshoot 0.9304
Uncompensated: settling time 1.2829, Rise time 0.383, peak 0.7992, overshoot 12.6311
From the data above its observed compensated system has smaller values compared to
uncompensated one. Therefore a compensated controller has a better performance compared
to uncompensated.