Direct Tourism Contribution to Employment
Chapter 4: Methodology
4.7 Phase 2: Resident perceptions
4.6. Investigation of the DO Process under random delays with varying controller
4.6.1 Simulation results of case 1: Kpis varied from 2.2 to 5.2 while Ti is kept constant at 8000.
Figure 4.18: Case 1, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days, Ti= 8000, Kp= 2.2
Figure 4.19: Case 1, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Ti= 8000, Kp= 2.7
3 4 5 6 7 8 9 10
x 10-3 1.95
2 2.05 2.1
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08 2.1
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO random delay
Figure 4.20: Case 1, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Ti= 8000, Kp= 3.2
Figure 4.21: Case 1, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Ti= 8000, Kp= 3.7
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Figure 4.22: Case 1, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Ti= 8000, Kp= 5.24.6.2
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Simulation results of case 2: Kpis constant at 3.2 while Tivaries from 2000 to 14000
Figure 4.23: Case 2, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.2, Ti= 2000
Figure 4.24: Case 2, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.2, Ti= 5000
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Figure 4.25: Case 2, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp=3.2, Ti= 8000
Figure 4.26: Case 2, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.2, Ti= 11000
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO random delay
Figure 4.27: Case 2, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.2, Ti= 14000
4.6.3 Simulation results of case 3: Both Kpand Ti are increased from a minimum value to maximum (from 2.2 to 5.2 for Kpand from 2000 to 14000 for Ti).
Figure 4.28: Case 3, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 2.2, Ti= 2000
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Figure 4.29: Case 3, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 2.7, Ti= 5000
Figure 4.30: Case 3, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.2, Ti= 8000
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Figure 4.31: Case 3, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 3.7, Ti= 11000
Figure 4.32: Case 3, ca= 0.000014 days, sc= 0.000013 days, tot= 0.000027 days,
Kp= 5.2, Ti= 14000
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
3 4 5 6 7 8 9 10
x 10-3 1.96
1.98 2 2.02 2.04 2.06 2.08
Discrete Time (days)
DO (mg/l)
DO under random delay (
tot only)
DO
random delay
Table 4.3: Performance indices of the closed loop DO process under random time delays with varying controller parameters (KpandTi) Measurement
Conditions
Time delay Values
(days)
Kp Ti Rise Time (days)
Settling Time (days)
Percentage Overshoot (%)
Steady State Error (mg/l)
Oscillation
Amplitude (mg/l) Min; max.
Delay in Response (days) Case 1
0.000027 2.2 8000 0.002014 0.003400 4.01 -0. 0801 2.0738, 2.0801 0.000014
0.000027 2.7 8000 0.002014 0.003414 3.35 -0.0627 2.0583, 2.0671 0.000014
tot=c 0.000027 3.2 8000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
0.000027 3.7 8000 0.002014 0.003414 2.57 -0.0449 2.038, 2.0515 0.000014
0.000027 5.2 8000 0.002014 0.003414 1.97 -0.0296 2.0202, 2.0394 0.000014
Case 2
0.000027 3.2 2000 0.002014 0.003400 2.90 -0.0525 2.047, 2.058 0.000014
0.000027 3.2 5000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
tot= c 0.000027 3.2 8000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
0.000027 3.2 11000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
0.000027 3.2 14000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
Case 3
0.000027 2.2 2000 0.002014 0.003400 4.01 -0. 0801 2.0738, 2.0801 0.000014
0.000027 2.7 5000 0.002014 0.003414 3.35 -0.0627 2.0583, 2.0671 0.000014
tot= c 0.000027 3.2 8000 0.002014 0.003414 2.90 -0.0525 2.047, 2.058 0.000014
0.000027 3.7 11000 0.002014 0.003414 2.58 -0.0449 2.0383, 2.0515 0.000014
0.000027 5.2 14000 0.002014 0.003414 1.95 -0.0296 2.0203, 2.0389 0.000014
4.6.4 Analysis of the results of the DO Process under random delays with varying controller parameters (Kpand Ti)
In case 1, Kpwas increased to be (2.2, 2.7, 3.2, 3.7, 5.2) while Ti was kept constant at 8000. It could be observed that with increase in Kp , the DO process experienced decrease in percentage overshoot, steady state error and oscillation amplitude. However the delay in system response and the rise time remained constant. This is shown in Figures 4.18 to 4.22 and Table 4.3. It could also be observed that with the values of the less than 3.2, the DO process did not experience a sustained oscillation. This suggests that the critical delay for the values of Kp less than 3.2 must be more than 0.000027 days that was obtained in Table 4.2.
In case 2, Kpwas kept constant at 3.2 while Ti was increased to be (2000, 5000, 8000, 11000, 14000). Despite the increase inTi, the system response remained the same as shown in Figures 4.23 to 4.27 and Table 4.3. This suggests that the constant Kp must have made the DO process response the same (constant) despite the variations in the values of Ti .
In case 3, both Kp and Ti were varied. Kpwas increased to be (2.2, 2.7, 3.2, 3.7, 5.2) while Ti was also increased to be (2000, 5000, 8000, 11000, 14000). The DO process system performance in case 3 was found to be consistent with case 1 as shown in Figures 4.18 to 4.22, Figures 4.23 to 4.27 and Table 4.3 in terms of percentage overshoot, steady state error, delayed system response and rise time. The only noticeable difference could be found in the oscillation amplitude and steady state error. In case 3, the system experienced 1.95% percentage overshoot as compared to 1.97% in case 1. This occurred when Ti in case 3 was 14000 as compared to 8000 in case 1.
Possible explanation could be that the higher value of Ti of 14000 has helped to improve the system performance by reducing percentage overshoot in case 3 when compared to case 1 with a Ti lower value of 8000. Just like the cases 1 and 2, the system’s delayed response and rise time were not affected with the increased Kp and Ti.
From this investigation, one could therefore conclude that the value of the critical delay is not only dependent on the magnitude of the random delays introduced into the control system, it is also dependent on the choice of the controller parameters Kp and Ti as seen in Table 3, case 1. This dependency is found to be more with Kp thanTi. Increasing the value of Kp for the DO process from 2.2 to 5.2 showed some initial improvement in system performance as seen in cases 1 and 3 by reducing percentage overshoot, steady
state error and oscillation amplitude but at 3.2, a sustained oscillation was experienced from where the DO process control system became unstable. There would therefore be the need to design a new controller that would take cognizance of the random time delays in its design in order to be able to provide robustness and stability for the DO process under the influence of random time delays.