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
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
$IGD¯$ Mean 4.9289 $†$ 4.8479 $†$ 3.7097 $†$ 3.6284 $†$ 3.9999 $†$ 4.0811 $†$ 4.6840 $†$ 3.4995
$S¯$ Mean 1.8673 $†$ 1.6659 $†$ 0.1303 $†$ 3.6284 $†$ 0.0417 § 0.0639 $†$ 0.0491 $†$ 0.0451
$HV¯$ Mean 2.3793 $†$ 2.3823 $†$ 2.4038 $†$ 2.4050 $†$ 2.4005 $†$ 2.3891 $†$ 2.4089 $†$ 2.6319