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Fig. 17 The non-dominated
solutions of the hybrid
method for the ZDT3 test
function
1
Pareto optimal front
Hybrid algorithm
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
f
1
(x)
number of members in the set of non-dominated solutions, di
i
is the Euclidean
distance between consecutive solutions in the gained non-dominated set, and
d
P
n
1
i¼1
d
i
n
¼
1
.
For the most extent spread set of non-dominated solutions
0
The performance of the hybrid algorithm comparing to NSGA-II (Deb et al.
2002
), SPEA (Zitzler and Theile
1999
), and PAES (Knowles and Corne
1999
)
algorithms is illustrated in Tables
14
,
15
,
16
,
17
and
18
.
Based on the results of Tables
14
,
15
,
16
,
17
and
18
, the hybrid algorithm has
very proper
D
¼
values for all test functions excluding ZDT2. While NSGA-II pre-
Δ
sents proper
results for all test functions except ZDT3, the approaches SPEA and
PAES do not illustrate proper performance in the diversity metric. The hybrid
algorithm presents acceptable results for the convergence metric in all test func-
tions. On the other hand, NSGA-II ZDT3 function, SPEA for FON function, and
PAES for FON and ZDT2 functions do not show proper performance.
The hybrid optimization algorithm is used to design the parameters of state
feedback control for linear systems. In the following section, state space repre-
sentation and the control input of state feedback control for linear systems will be
presented.
Δ
Table 14 The results of the comparison of multi-objective optimization algorithms for the SCH
test function
Metrics
NSGA-II
SPEA
PAES
The hybrid algorithm
10
−
1
10
0
10
0
10
−
1
Mean
4.77
×
1.02
×
1.06
×
6.00
×
Δ
Standard deviation 3.47
×
10
−
3
4.37
×
10
−
3
2.86
×
10
−
3
1.81
×
10
−
2
10
−
3
10
−
3
10
−
3
10
−
3
!
Mean
3.39
×
3.40
×
1.31
×
3.22
×
10
−
6
10
−
4
Standard deviation 0
0
3.00
×
1.35
×
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