Table 1: Parameter Settings NSGA-II.
Representation Real numbers
Cross-Over Operator Uniform
Cross-Over Probability 0.9
Mutation Gaussian Perturbation
Mutation Probability 0.1
Stop Condition A1 300 generations
Selection Scheme A1 (200+200)
Stop Condition A2,A3 50 generations
Selection Scheme A2,A3 (100+100)
Table 2: Results for the coverage indicator.
C(A
i
, A
j
) A1 A2 A3
A1 − 0.26(0.31) 0.23(0.33)
A2 0.26(0.43) − 0.20(0.33)
A3 0.23(0.39) 0.56(0.35) −
Table 3: Results for indicators ESS, HV and DMAX.
ESS HV MD
A1 0.0452(0.05) 0.03(0.03) 0.03(0.03)
A2 0.14(0.08) 0.13(0.00) 0.15(0.09)
A3 0.11(0.07) 0.10(0.02) 0.11(0.07)
Table 4: Results for Mann-Whitney-Wilcoxon tests with a
signiﬁcance level α = 0.05.
A1 A2 A3
A1 − =, ⇑, ⇓, ⇓ =, ⇑, ⇓, ⇓
A2 =, ⇓, ⇑, ⇑ − ⇓, ⇓, ⇑, ⇓
A3 =, ⇓, ⇑, ⇑ ⇑, ⇑, ⇓, ⇑ −
5 CONCLUSIONS
In this paper a new design method for solving multi-
objective control problems was described. Results
show that, for the PID control problem, the proposed
method (A2) is able to ﬁnd better approximations than
the conventional method (A1) with respect to the HV
and MD indicators, given the same number of func-
tions evaluations. Moreover, when compared to A3
(the algorithm using the true stabilizing region), re-
sults show that A2 is indeed able to ﬁnd a good ap-
proximation of the feasible region.
In the future the authors will try to extend this
work, in order to consider more complex control
problems, with more design variables and objectives.
For that goal, a more sophisticated representation of
the search space is needed (stage 1), in order to make
the sampling process more efﬁcient.
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