Table 3 Optimal parameters of FOPID controllers
Parameters | PSO | WCA | HBA | AOS | COA |
|---|---|---|---|---|---|
\({\text{K}}_{\text{P}1}\) | −2.2754 | −3.0000 | −3.0000 | −2.8961 | −2.9923 |
\({\text{K}}_{\text{d}1}\) | −2.9885 | −1.0941 | −0.4974 | −2.5512 | −2.9951 |
\({\text{K}}_{\text{i}1}\) | −2.1337 | −2.6009 | −3.0000 | −2.0296 | −2.9963 |
\({{\uplambda }}_{1}\) | 1.1628 | 1.1499 | 0.6951 | 1.1039 | 0.8981 |
\({{\upmu }}_{1}\) | 0.5785 | 0.6957 | 1.5000 | 0.8235 | 0.6212 |
\({\text{K}}_{\text{P}2}\) | −2.0994 | −1.7360 | −0.4000 | −2.0270 | −1.5198 |
\({\text{K}}_{\text{d}2}\) | −2.2324 | −3.0000 | −3.0000 | −0.8984 | −2.0910 |
\({\text{K}}_{\text{i}2}\) | −1.1722 | −0.6284 | −0.4201 | −1.5313 | −1.5401 |
\({{\uplambda }}_{2}\) | 1.0357 | 1.0972 | 0.9776 | 1.0640 | 0.9851 |
\({{\upmu }}_{2}\) | 1.2783 | 1.3344 | 1.5000 | 0.8986 | 0.8889 |
\({\text{K}}_{\text{P}3}\) | −2.2041 | −2.7547 | −3.0000 | −2.6479 | −1.7242 |
\({\text{K}}_{\text{d}3}\) | −1.8897 | −3.0000 | −0.4017 | −0.4000 | −1.3896 |
\({\text{K}}_{\text{i}3}\) | −2.3044 | −2.2270 | −3.0000 | −2.4039 | −2.9688 |
\({{\uplambda }}_{3}\) | 0.9339 | 1.1775 | 0.7186 | 0.8567 | 0.7946 |
\({{\upmu }}_{3}\) | 0.9786 | 0.9378 | 1.5000 | 1.3548 | 1.0835 |
\({{\uptau }}_{\text{W}}\) | 18.7768 | 25.0082 | 30.0000 | 28.8367 | 14.7073 |
\({\text{K}}_{{\upbeta }}\) | 23.6464 | 33.2494 | 50.0000 | 46.8303 | 19.4407 |
\({\text{R}}_{\text{w}\text{t}}\) | 0.2149 | 0.1992 | 0.1000 | 0.1895 | 0.2110 |