A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 73

Table 15 The results of the comparison of multi-objective optimization algorithms for the FON
test function
Metrics NSGA-II SPEA PAES The hybrid algorithm

3.78 × 10 − 1 7.92 × 10 − 1 1.16 × 10 0 5.90 × 10 − 1 Δ Mean

10 − 4 10 − 3 10 − 3 10 − 2 Standard deviation 6.39 × 5.54 × 8.94 × 3.60 ×

10 − 3 10 − 1 10 − 1 10 − 3 ! Mean 1.93 × 1.25 × 1.51 × 1.56 ×

3.80 × 10 − 5 9.05 × 10 − 4 1.71 × 10 − 4 Standard deviation 0

Table 16 The results of the comparison of multi-objective optimization algorithms for the ZDT1
test function
Metrics NSGA-II SPEA PAES The hybrid algorithm

3.90 × 10 − 1 7.84 × 10 − 1 1.22 × 10 0 6.55 × 10 − 1 Δ Mean

10 − 3 10 − 3 10 − 3 10 − 2 Standard deviation 1.87 × 4.44 × 4.83 × 4.91 ×

10 − 2 10 − 3 10 − 2 10 − 3 ! Mean 3.34 × 1.79 × 8.20 × 8.16 ×

10 − 3 10 − 6 10 − 3 10 − 3 Standard deviation 4.75 × 1.00 × 8.67 × 2.73 ×

Table 17 The results of the comparison of multi-objective optimization algorithms for the ZDT2
test function
Metrics NSGA-II SPEA PAES The hybrid algorithm

10 − 1 10 − 1 10 0 10 − 1

Mean 4.30 × 7.55 × 1.16 × 9.57 ×

Δ

10 − 3 10 − 3 10 − 3 10 − 2 Standard deviation 4.72 × 4.52 × 7.68 × 3.20 ×

7.23 × 10 − 2 1.33 × 10 − 3 1.26 × 10 − 1 3.04 × 10 − 2

!

Mean

10 − 2 10 − 2 10 − 2 Standard deviation 3.16 × 0 3.68 × 1.84 ×

Table 18 The results of the comparison of multi-objective optimization algorithms for the ZDT3
test function
Metrics NSGA-II SPEA PAES The hybrid algorithm

10 − 1 10 − 1 10 − 1 10 − 1

Mean 7.38 × 6.72 × 7.89 × 6.28 ×

Δ

Standard deviation 1.97 × 10 − 2 3.58 × 10 − 3 1.65 × 10 − 3 5.30 × 10 − 2

10 − 1 10 − 2 10 − 2 10 − 3

! Mean 1.14 × 4.75 × 2.38 × 8.88 ×

10 − 3 10 − 5 10 − 5 10 − 3

Standard deviation 7.94 × 4.70 × 1.00 × 6.97 ×

6 State Feedback Control for Linear Systems

The vector state equation can be utilized for a continuous time system, as follows:

x ðÞ ¼ _ Ax ðÞþ Bu ðÞþ B v v ð t Þ ð 15 Þ

n stands for the state vector,

where x t ðÞ2 R x t _ ðÞ denotes the time derivative of state

m is the input vector. The disturbance v t

vector and u t ðÞ2 R ðÞ is assumed to be a