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Recent metaheuristics on control parameter determination
Table 6. Dynamic performance characteristics for balance angle

M ess
No Method T r T s T p
1 CO 0.0100 2.0400 0.4500 0.1521 1.20 e-03
2 PO 0 9.8800 0.1000 0.0817 7.47 e-04
3 AO 0 12.2000 0.0900 0.0766 7.47 e-04
4 GOA 0 11.0100 0.1000 0.0823 7.47 e-04
5 POA 0 12.8900 0.1000 0.0824 7.47 e-04
6 GJO 0 11.0400 0.1000 0.0827 7.47 e-04
7 FDA 0.0100 10.0900 0.1000 0.0820 7.47 e-04
8 EO 0 10.9800 0.1000 0.0821 7.47 e-04
9 ARO 0.0100 10.3500 0.1000 0.0823 7.47 e-04
10 CSA 0 11.0300 0.1000 0.0826 7.47 e-04
11 WOA 0.0100 19.8500 0.5900 0.2033 1.78 e-02
12 GWO 0 11.2200 0.1000 0.0824 7.47 e-04
13 FPA 0 10.0200 0.1000 0.0820 7.47 e-04
14 CMA-ES 0.01 4.38000 0.1600 0.3075 7.47 e-04

Table 7. Dynamic performance characteristics for speed responses
M ess
No Method T r T s T p
1 CO 0.4500 3.8270 0.7100 4.1331 0.0865
2 PO 0.6300 16.8500 1.7700 0.3397 2.5455e-05
3 AO 0.7000 14.6900 1.9000 0.3467 1.6699e-05
4 GOA 0.6300 16.8300 1.7500 0.3410 2.3981e-05
5 POA 0.6400 16.4900 1.7600 0.3399 2.5923e-05
6 GJO 0.6200 16.0000 1.7200 0.3511 1.1546e-05
7 FDA 0.6500 15.9600 1.7600 0.3420 2.2301e-05
8 EO 0.6400 15.1000 1.7600 0.3406 2.4552e-05
9 ARO 0.6200 12.9400 1.7500 0.3412 2.3618e-05
10 CSA 0.6400 15.0900 1.7600 0.3386 2.9485e-05
11 WOA 0.5800 15.8600 0.9000 7.0768 1.1954
12 GWO 0.6300 16.0600 1.7200 0.3518 1.0643e-05
13 FPA 0.6400 18.5100 1.7700 0.3378 2.8406e-05
14 CMA-ES 0.1600 4.95000 0.4800 6.7859 4.8849e-15

is determined to capture the reference in a very considered methods. M and ess denote the maxi-
short time and with a very small steady-state er- mum overshoots and the steady-state error of the
ror, are found to be 16.5358 and 3.8075. Although signals, respectively.
the Kp value of the CMA-ES method does not dif- When the dynamic performance characteristics of
fer significantly, the unnecessary increase in the
Ki value by approximately ten times causes ex- the system, whose details are more clearly under-
stood in Table 7, are examined, the lowest error is
cessive oscillation and delay in the system’s speed
found with the CMA-ES method, while the over-
response. Kp and Ki values of the CO algorithm
shoot is also at the highest value in this method.
are 78.5239 and 1.0038, respectively.
While WOA has the highest steady-state error,
the lowest steady-state error after CMA-ES is ob-
Figure 5 shows that the CO method produces an
tained with the GJO method. When the balance
excessive oscillatory response due to its higher Kp
of overshoot and steady-state error of the GJO
value when compared to other methods. WOA
and GWO methods is considered, it can be said
finds the optimal parameter values for the two-
that they are the methods that find the optimal
wheeled vehicle as 27.6981 and 1.0696 for Kp and solution with the most precision among the com-
Ki, respectively. Similarly, the WOA method pared methods. However, the AO, FDA, ARO,
produces an oscillatory and delayed response as
GOA, EO, PO, POA, FPA, and CSA methods
a result of the slightly higher Kp and lower
follow the GJO and GWO methods, respectively.
Ki. The dynamic performance characteristics ob-
tained from the system response for speed control The convergence curves for all algorithms are pre-
are presented in Table 7. In the table, the terms sented in Figure 6. For a fair comparison, all algo-
of the Tr, Ts, and Tp represent the rise time, set- rithms are performed under the same conditions
tling time, and peak time of the signals for all and on the same platform.
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