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Table 8 shows the statistical results of IGD¯, S¯ and HV¯ obtained by MOEA-OSD/SRS and other seven compared algorithms. Note, the PF corresponding to the nondominant solutions in the PS obtained by all eight algorithms makes up the real PF of each environment. It is obvious that MOEA-OSD/SRSA is of better performance, that is to say, the control effect of the PID controller optimized by MOEA-OSD/SRS is much better. Therefore, MOEA-OSD/SRS is promising to be applied to tune the parameters of the PID controller of the dynamic system.

Table 8:

The statistical results of IGD¯, S¯, and HV¯ for different algorithms.

FPSPPSMOEA
DNSGADNSGA-RM-RMImmune-OSD
MetricsStatistic-II-A-II-B-MEDA-MEDASGEAMOEA/DGDE3/SRS
IGD¯ Mean 4.9289  4.8479  3.7097  3.6284  3.9999  4.0811  4.6840  3.4995 
 Std 0.0579 0.1245 0.1196 0.0373 0.0815 0.1001 0.0968 0.0128 
S¯ Mean 1.8673  1.6659  0.1303  3.6284  0.0417 § 0.0639  0.0491  0.0451 
 Std 0.3479 0.5166 0.1156 0.0373 0.0019 0.0108 0.0030 0.0028 
HV¯ Mean 2.3793  2.3823  2.4038  2.4050  2.4005  2.3891  2.4089  2.6319 
 Std 0.0093 0.0164 0.0033 0.0011 0.0030 0.0032 0.0125 1.15E-04 
FPSPPSMOEA
DNSGADNSGA-RM-RMImmune-OSD
MetricsStatistic-II-A-II-B-MEDA-MEDASGEAMOEA/DGDE3/SRS
IGD¯ Mean 4.9289  4.8479  3.7097  3.6284  3.9999  4.0811  4.6840  3.4995 
 Std 0.0579 0.1245 0.1196 0.0373 0.0815 0.1001 0.0968 0.0128 
S¯ Mean 1.8673  1.6659  0.1303  3.6284  0.0417 § 0.0639  0.0491  0.0451 
 Std 0.3479 0.5166 0.1156 0.0373 0.0019 0.0108 0.0030 0.0028 
HV¯ Mean 2.3793  2.3823  2.4038  2.4050  2.4005  2.3891  2.4089  2.6319 
 Std 0.0093 0.0164 0.0033 0.0011 0.0030 0.0032 0.0125 1.15E-04 

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