IEEE Systems, Man and Cybernetics Magazine - January 2023 - 51

where
P1 =
TT TT TT
t
tt sg
TT ksg ps
ps++
sg ps
and it is a function of only system parameters. Similarly,
(8) with the help of (6) leads to
kP2
where
P2 =
RT Tk kRT++ +
kk Tt
()() ps
sg ps
1
t
sg
ps
1
which is a function of the controller component k1 and the
system parameters. Finally, (8) with the help of (7) leads to
kP3
where
P3 =
Rk k1
kk k -+1
ps
sg ps
k
1 k
t
◆ Finally, considering the inequality of (20) and calculating
further, the following can be obtained:
d 2 a
a
1
3
11
10
kP1 =- d+
kP2=-d
kP3=-d
22
33
where 00
12
22
dd and d3 0 .2
,,
11
(12)
(13)
(14)
It
to define the values or ranges of d1 , d and 3
2 ,
Now, the objective is set
d as a function
of system parameters. In [2] and [11], the controllers
were initialized from a random range, and the stability was
ensured by checking the conditions given in (8) for each
agent in each iteration. The optimization of variables
d and 3
d
2 ,
1 ,
d from (12), (13), and (14), respectively, can be a
choice in line with those works. However, the optimization
process will still have three decision variables. In this article,
first, a range of stable operations depending upon system
parameters is presented. Next, a process to reduce the
decision variables is presented for efficient computation.
In the proposed analysis, we have considered 1
independent variable, whereas 2d and 3
d to be an
d are considered
dependent variables (dependent on 1d and the system
parameters). This leads to one independent variable, and,
by optimizing the concerned variable through GSA, the
system performances can be influenced. A step-by-step
process to orchestrate the variables 2
d and 3
dent variables is given:
◆ Considering (5) and (12) and calculating further,
A11 1
=ad .
(15)
◆ To make P2 dependent on 1d and thereby considering
(12), P2 is given by
PP .
22 12 13=- +
ad aa
(16)
a ,
=
da d=
d =
22 2
1
2aa da da11 110
-+ .
112 1
2
a9
i 1176f are all dependent on the system parameis
noteworthy to mention that
i ,,
ters only. These are given in Appendix A (available at
https://doi.org/10.1109/MSMC.2022.3208393). The value of
1
d satisfying (21) ensures system stability. The parameters
2d and 3
d are obtained from equality conditions as
per (22) and (23), respectively, using 1d and the system
parameters. In an effort to improve the steady state and
the transient performances along with ensuring system
stability, the GSA-based approach is used and discussed in
the next section.
d as depenOptimization
of the System
Performances Based on GSA
GSA was first introduced by the authors of [15]. It is a
nature-inspired metaheuristic optimization algorithm
inspired by Newtonian laws of gravity and interactions of
masses. In GSA, the decision variables of a problem are
encoded as the positions of the agents. Uses of GSA can be
found in various medical and engineering applications [2],
[16], [17], [18], [19], [20], [21]. Furthermore, different variants
of GSA were also introduced and successfully implemented
in different problems [22].
Inspired by earlier works [1], [2], [11], the paradigm of
GSA is followed to modulate the controller gains for performance
improvements in the output response. In GSA,
the position of an agent i is defined as
Dd dd iN ,1== (24)
i ii i
12 l
[, , ,]
f6 f,
January 2023 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 51
(21)
(22)
(23)
the parameters
()+
t
2
aad da dand
is a function of the controller components k1 and k2
and the system parameters. Now, adding a slack variable
in (9) and surplus variables in (10) and (11), the following
equations are obtained:
1 412 9320 .
(20)
1 3
(11)
◆ Considering (12), (13), and (14) in (7), A3 can be written
as
A39 3
=ad .
(19)
◆ Putting the values of A1, A2, and A3 from (15), (17),
and (19), respectively, in A4 [refer to (8)], the following
is obtained:
1 2
(10)
◆ Considering (12) and (13) in (6) and calculating further,
.
A24 2
=ad
◆ Considering (12) and (13) in P3, it can be written as
()
P35 61 72 8
=- ++aa da da .
(17)
(18)
https://www.doi.org/10.1109/MSMC.2022.3208393

IEEE Systems, Man and Cybernetics Magazine - January 2023

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