Table 4 The qualitative evaluation of error indices for test system 2.

Original system

0.1826

Proposed 3rd -order model

\(\frac{{0.09147{s^2} - 0.6571s}}{{{s^3}+4.3035{s^2}+0.334s+0.1801}}\)

4.6663

0.1678

3.1795

0.1728

EPSO-DV IMPPA [15]

\(\frac{{ - 0.15811s}}{{{s^2}+0.07078s+0.041276}}\)

5.5162

0.1666

3.2640

0.1766

BTM-PSO [29]

\(\frac{{ - 0.1432{s^2} - 0.06946s}}{{{s^3}+0.4514{s^2}+0.07559s+0.0162}}\)

5.6558

0.2103

3.2812

0.1779

BTM MM [35]

\(\frac{{ - 1.516{s^2} - 0.0162s}}{{{s^3}+0.4514{s^2}+0.07559s+0.0162}}\)

17.9155

54.9876

49.9909

4.8542

MBTM [36]

\(\frac{{ - 0.09222{s^2} - 0.07393s}}{{{s^3}+0.4514{s^2}+0.07559s+0.0162}}\)

5.4456

0.2983

3.5604

0.1704

BTM-MCCF [37]

\(\frac{{ - 0.1{s^2} - 0.0162s}}{{{s^3}+0.4514{s^2}+0.07559s+0.0162}}\)

5.0231

2.4677

10.2127

0.0284

LPC-IPAM [38]

\(\frac{{ - 0.1{s^2} - 0.0155s}}{{{s^3}+0.1554{s^2}+0.0518s+0.003}}\)

6.0791

1.1362

6.5795

0.0751

BSPA [40]

\(\frac{{ - 0.1132{s^3}+0.4623{s^2} - 2.189s+1.501 \times {{10}^{ - 16}}}}{{{s^3}+13.54{s^2}+1.002s+0.5691}}\)

25.0139

0.1720

3.3189

Inf

SEM-PSO [42]

\(\frac{{ - 0.20632{s^2} - 0.00039653s}}{{{s^3}+0.1837{s^2}+0.0375s+0.00134}}\)

5.3087

0.7032

6.5322

0.1439