Table 8 Comparison of multi-modal benchmark functions.

From: HIL co-simulation of an optimal hybrid fractional-order type-2 fuzzy PID regulator based on dSPACE for quadruple tank system

Algorithms	Parameters	Benchmark functions
Algorithms	Parameters	F8	F9	F10	F11	F12	F13
PSO⁵⁴	Mean	− 4840	46.7	0.28	0.01	0.01	0.01
PSO⁵⁴	STD	1150	11.6	0.51	0.01	0.03	0.01
GSO⁵⁵	Mean	− 2820	26	0.06	27.7	1.8	5.9
GSO⁵⁵	STD	493	7.47	0.24	5.04	0.95	7.13
DEO⁵⁶	Mean	− 11,100	69.2	0	0	0	0
DEO⁵⁶	STD	575	38.8	0	0	0	0
FFO⁵⁷	Mean	− 12,600	0.05	0.02	0.02	0	0
FFO⁵⁷	STD	52.6	0.01	0	0.02	0	0
ALA⁵⁸	Mean	− 1610	0	0	0.02	0	0
ALA⁵⁸	STD	314	0	0	0.01	0	0
SOSO⁵⁹	Mean	− 4.21	1.33	0	0.71	0.12	0.01
SOSO⁵⁹	STD	0	0.33	0	0.91	0.04	0
BO⁶⁰	Mean	− 1070	1.23	0.13	1.45	0.4	0.39
BO⁶⁰	STD	858	0.69	0.04	0.57	0.99	0.12
FPO^61,62	Mean	− 1840	0.27	0.01	0.09	0	0
FPO^61,62	STD	50.4	0.07	0.01	0.04	0	0
CS³²	Mean	− 2090	0.13	0	0.12	0	0
CS³²	STD	0.01	0	0	0.05	0	0
FO⁶³	Mean	− 1250	0.26	0.17	0.1	0.13	0
FO⁶³	STD	353	0.18	0.05	0.02	0.26	0
GO⁶⁴	Mean	− 2090	0.66	0.96	0.49	0.11	0.13
GO⁶⁴	STD	2.47	2.47	0.81	0.22	0.07	0.07
GOA⁶⁵	Mean	1	0	0.1	0	0	0
GOA⁶⁵	STD	0	0	0	0	0	0
MFO⁶⁶	Mean	− 5080	0	7.4	0	0.34	1.89
MFO⁶⁶	STD	696	0	9.9	0	0.22	0.27
MVA⁶⁷	Mean	− 11,700	118	4.07	0.94	2.46	0.22
MVA⁶⁷	STD	937	39.3	5.5	0.06	0.79	0.09
DO⁶⁸	Mean	− 2860	16	0.23	0.19	0.03	0
DO⁶⁸	STD	384	9.48	0.49	0.07	0.1	0
BBO⁶⁹	Mean	− 924	1.81	0.39	0.19	0.15	0.04
BBO⁶⁹	STD	65.7	1.05	0.57	0.11	0.45	0.06
BA⁷⁰	Mean	− 989	4.83	2.15	0.48	0.41	0.31
BA⁷⁰	STD	16.7	1.55	0.54	0.13	0.23	0.24
BGSO⁷¹	Mean	− 861	10.3	2.79	0.79	9.53	2220
BGSO⁷¹	STD	80.6	3.73	1.19	0.25	6.51	5660
SCO⁷²	Mean	1	0.01	0.38	0.01	0.01	0.01
SCO⁷²	STD	0.01	0.73	1	0.01	0.01	0.01
SSO⁷³	Mean	0.06	0.01	0.2	0.01	0.14	0.08
SSO⁷³	STD	0.81	0.01	0.15	0.07	0.56	0.71
WOA⁷⁴	Mean	− 8500	84.6	1.26	0.02	0.89	0.12
WOA⁷⁴	STD	726	16.2	0.73	0.02	0.88	0.19
BMFA1⁷⁵	Mean	− 3140.3	1.63	0.04	0	0	0
BMFA1⁷⁵	STD	290.75	0.96	0.21	0	0	0
BMFA2⁷⁵	Mean	− 3361.2	0.39	0	0	0	0
BMFA2⁷⁵	STD	287.32	0.72	0	0	0	0.01
E-GWO^76,77	Mean	− 5787.77	2.022602	0	0.002304	0.054082	0.655472
E-GWO^76,77	STD	1251.334	3.476486	0	0.005280	0.030135	0.237728
hAGWOCS	Mean	− 11,271.1	1.318777	0	0.006341	0.040212	0.627942
hAGWOCS	STD	12,224.55	2.103797	0	0.012583	0.02329	0.235043

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Table 8 Comparison of multi-modal benchmark functions.

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