Transient Response Improvement through Controller Design

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 ∓ j3.8
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%.