Table 1 Statistical analysis of different agents for GOA, PSO and GA optimized PDF plus (1+ PI) controllers

F1

Shifted Sphere Function

\( {f}_1(x)=\sum \limits_{i=1}^n{x}_i^2 \)

Average

1.6219X10⁵⁶

8.931X10⁶⁶

9.2656 X10²¹

Standard Deviation

1.6266X10⁵⁷

7.0111X10⁶⁸

4.1437 X10²²

Minimum

0.1088X10³

0.0059X10³

0.0021 X10³

Maximum

1.8291X10⁶⁸

9.468X10⁶⁶

1.8531 X10²³

F2

Shifted Schwefel’s Function

\( {f}_2(x)=\sum \limits_{i=1}^n\left|{x}_i\right|+{\prod}_{i=\frac{1}{2}}^n\left|{x}_i\right| \)

Average

3.7673 X10⁶⁷

13.045 X10¹¹

1.4177 X10²⁴

Standard Deviation

3.7688 X10⁶⁸

11.366X10¹²

1.428 X10²⁵

Minimum

0.0062 X10³

0.0379 X10³

0.0023 X10³

Maximum

3.8265 X10⁶⁹

13.415 X10¹³

1.4319 X10²⁴

F3

Shifted Rosenbrock’s

Function

\( {f}_3(x)=\sum \limits_{i=1}^n{\left(\left[{x}_i+0.5\right]\right)}^2 \)

Average

2.7399 X10³²

12.121 X10⁹

4.3317 X10¹⁹

Standard Deviation

1.0523 X10³³

11.598 X10¹¹

3.9854 X10²⁰

Minimum

0.0161 X10³

4.686 X10²

0.0082 X10³

Maximum

4.0777 X10³³

10.032 X10¹³

4.3987 X10¹⁹

F4

Shifted Rastrigin’s Function

\( {f}_4(x)=\sum \limits_{i=1}^n\left[{x}_i^2+10\cos \left(2\pi {x}_i\right)+10\right] \)

Average

5.2988 X10⁴⁴

6.8166 X10³⁸

2.2347 X10²⁷

Standard Deviation

2.9021 X10⁴⁵

7.0025 X10³⁹

1.4921 X10²⁸

Minimum

0.0063 X10³

1.792 X10²

0.0076 X10³

Maximum

1.5896 X10⁴⁶

9.568 X10⁴¹

2.4682 X10²⁷

F5

Shifted Schwefel’s Problem with Noise in Fitness

f₅(x) = max {|x_i|, 1 ≤ i ≤ n}

Average

2.2974 X10³⁴

5.6982 X10³⁴

3.8751 X10²²

Standard Deviation

2.2798 X10³⁵

6.2784 X10³⁸

2.9852 X10²³

Minimum

0.0009 X10³

0.0561 X10³

0.0023 X10³

Maximum

2.2799 X10³⁶

5.5862 X10³⁶

4.0012X10²²

F6

Schwefel’s Problem with Global Optimum on Bounds

\( {f}_6(x)=\sum \limits_{i=1}^n\left[100{\left({x}_{i+1}-{x}_i^2\right)}^2+{\left({x}_i-1\right)}^2\right] \)

Average

3.0414 X10³⁰

2.1187 X10⁴²

1.8955 X10⁴³

Standard Deviation

2.5276 X10³¹

1.5182 X10⁴⁴

1.9865 X10⁴⁴

Minimum

0.0015 X10³

0.172 X10²

0.0089 X10³

Maximum

2.4682 X10³²

2.930 X10⁴⁵

2.2219 X10⁴³

F7

Shifted Rotated Griewank’s Function

\( {f}_7(x)=\sum \limits_{i=1}^n{ix}_i^4+ random\left[0,1\right] \)

Average

3.4099 X10³³

11.530 X10³⁵

7.6482 X10³⁴

Standard Deviation

2.4053 X10³⁴

13.028 X10³⁶

5.6214 X10³⁵

Minimum

0.0008 X10³

1.406 X10²

0.0019 X10³

Maximum

1.8920 X10³⁵

11.022 X10³⁸

4.5875 X10³⁴

F8

Shifted Rotated Ackley’s Function with Global Optimum on Bounds

\( {f}_8(x)=\sum \limits_{i=1}^n-{x}_i\sin \left(\sqrt{\left|{x}_i\right|}\right) \)

Average

5.0129 X10³²

6.321 X10³¹

8.8626 X10²⁹

Standard Deviation

2.2418 X10³³

5.4213X10³²

4.5897 X10³⁰

Minimum

0.0018 X10³

2.1235X10³

0.0062 X10³

Maximum

1.0026 X10³⁴

8.082X10³³

6.9871 X10²⁹