IEEE Systems, Man and Cybernetics Magazine - January 2023 - 55

Next, the system parameter
Finally, the system parameter
values of Example 2 as given in
Appendix C (available at https://
doi .org /10.1109/MSMC. 2022 .
3208393) are considered. The
authors of [31] proposed a PID
controller for the considered system
based on the maximum peak
resonance specification. In [30],
the linear quadratic regulator
(LQR)-based controller was proposed
for the considered system.
For the comparative study, the
weighting matrices (Q and R) as
mentioned in [30] are considered here. In this case, the
controller gain values obtained by the proposed method
are
kk
29 668
Mve
p
-
12 6 244==.,
., and k3 =- ..20 731 The results
are presented through Table 2 and Figure 4(b). The value of
()
by the proposed method is .,0 0944 10 Hz3
-
#
which is the minimum in comparison with all of the
cited works. Significant improvements in ISE and ts are
also observed.
GSA is a natureinspired
metaheuristic
optimization
algorithm inspired
by Newtonian
laws of gravity
and interactions
of masses.
values of Example 3 as given in
Appendix D (available at https://doi.
org/10.1109/MSMC.2022.3208393)
are considered. A finite-dimensional
controller synthesis approach
was proposed in [32], and a PID and
a PID with acceleration controllers
were proposed in [33]. In this case,
the proposed controller gain values
are
k .,0 631
1
=
k .,4 961
2
=and
k3
=- ..99 887 The results are presented
through Table 2 and Figure
4(c). Significant improvements
in all performances are observed by the proposed method.
For instance, the ts value obtained by the proposed method
is 3.24 s compared to the ts values of 41.5 and 62.51 s by the
methods proposed in [25] and [32], respectively.
Summarizing the comparative analysis, it is concluded
that the obtained stability ranges in all three examples are
appropriate in maintaining system stability. Moreover, the
results presented through Table 2 and Figure 4 corroborate
Table 2. The Comparative Performances of the LFC System Considering
Three Different Sets of System Parameters.
System
Example 1
Authors and Methods
Tan (IMC-PID) [28]
Padhan and Majhi (LS-PID) [29]
Anwar and Pan (DS-PID) [25]
Singh et al. (LMI-PID) [5]
Ghosh et al. (ZN-PID-PSO) [4]
Singh et al. (PFSFC) [13]
Ghosh et al. (SFC-GSA) [11]
Proposed
Example 2
Paliwal et al. (LQR) [30]
Padhan and Majhi (LS-PID) [29]
Khodabakhshian and Edrisi (MPR-PID) [31]
Anwar and Pan (DS-PID) [25]
Proposed
Example 3
Gundes and Chow (FCS) [32]
Anwar and Pan (DS-PID) [25]
Kumar and Anwar (DS-PID) [33]
Kumar and Anwar (DS-PIDA) [33]
Proposed
Mp (Hz) (-ve) (#10-3)
13.2
9.8
8.9
7.5
7.04
1.9
0.8866
0.8498
0.7442
0.244
0.272
0.131
0.0944
1.7
1.5
0.55
0.32
0.147
ts (s)
4.35
5.1
2.68
2.274
2.165
-
-
2.843
-
4.37
24.2
5.2
5.264
62.51
41.5
4.1
3.4
3.24
FCS: finite-dimensional controller synthesis; LQR: linear quadratic regulator; MPR: maximum peak resonance; PIDA: PID with acceleration.
ISE (#10-5)
13.62
3.635
2.471
2.117
1.382
1.79
0.32
0.00949
0.71
0.002937
0.01067
0.000693
0.000516
1.395
0.9625
0.05
0.03
0.00009
January 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 55

https://doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393 https://doi.org/10.1109/MSMC.2022.3208393

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