Page 200 - 2024-Vol20-Issue2

196 | Al-mtory, Alnahwi & Ali

The open loop gain of the uncontrolled system is symbolized E. Simple mass – damper system:
as G(s), so the transfer function of closed loop system with
PID controller is as given below: G(S) = 1
S2+10S+20
Based on the fitness function in (16) and by selecting a to

L(S)G(S) be equal to 0.5 for each system, the unit step response after
Gcloseloop(S) = 1 + L(S)G(S)
(15) optimization results in the coefficient values given in Table

VI: To know the performance of each system before and After

In fact, the main goal of the PID controller is to obtain high ac- applying the PID controller, Table VII gives a comparison
curacy of data in addition to the level of quality in the transient
process [42]. Thus, the three main parameters that are used to of the time response before and after inserting the optimized
describe the quality of the transient response any system are
listed below [43]: controller, where the optimized system results in negligible
• Rising time tr: The time required for the response change to
reach 90% of its steady state value. values of rising time, settling time, and overshoot for all sys-
• Settling time ts: the time required for the response to be
within l2%l of the steady state value. tems. Fig. 5. illustrates the unit step time response for each
• Overshoot Mr (%) : the maximum value that the response
reaches above the steady state value. system before and after the optimization process.
In this work, the CDOA is used to optimize the PID coeffi-
cients KP, Ki, andKd to get the best transient response which B. Cantilever Beam
has as minimum values of tr,ts, andMr. The fitness function In mechanical engineering systems, there is an important
that is used to optimize the aforesaid three transient response element called the cantilever beam shown in. Fig. 6. From
parameters is as follows[41]: Table IX, it is noticed that the resulted best value of the fitness
function in the proposed algorithm is less than that of other
F = min (1 - e-a )(MP - ESS) + e-a (tr + ts) (16) algorithms. This indicates the success of the algorithm in
this application. The results can be seen in Table IX row No.
where a is the weight factor, and ESS is the steady state error 7, which lists the best values of the fitness functions for this
which denotes Laplace transform of the error signal e(t) at application.
the steady state (t ? 8) as given in (17)[41]. The main objective in this optimization process is to reduce the
weight of the side beam. As shown in the figure, the cantilever
beam contains five hallows in the form of square boxes. The
lengths of the five boxes in this process are variable. The
fitness function and its constraints for this problem are as in
(19) and (20), respectively [44].

f (x) = 0.6224(x1 + x2 + x3 + x4 + x5) (19)

Ess = lim S R(s) (17)

S?0 1 + Gcl osel oo p 61 27 19 7 1
x13 x23 x33 x43 x53
g(x) = + + + + - 1 = 0 (20)

where R(s) is the desired set of input in S-domain. If the input

is a unit step, its Laplace transform is equal to 1 . As a result, where x1, x2, x3x4, andx5 represent the dimensions of the can-
s tilever beam. The rang of these variables is 0.01 = xi = 100.
Table IX gives the results of the optimization for this prob-
the steady state error becomes as in (18): lem after applying the proposed CDOA algorithm, and then
comparing it with particle swarm optimization (PSO), ge-
1 (18) netic algorithm (GA), multi-verse optimizer (MVO), water
Ess = 1 + Gcloseloop(0) wave optimization (WWO), sine cosine algorithm (SCA), and
whole optimization algorithm (WOA) whose results are listed
The CDOA is applied to different standard systems, and the in [45].

time response is demonstrated before and after the optimiza-

tion process. The standard systems that were selected in this

work are as follows [41]:

A. Second order system: G(S) = 20
S2+0.5S+10
B. Forth order system: VI. CONCLUSION

G(S) = 25.2S2+21.2S+3 A new algorithm based on the diffusion of the pitting corro-
S4+16.4825S3+23.8021S2+14.8566S+10.2497 sion on the metal surface has successfully been presented in
C.Fifth order system this work. The oxidation-reduction chemical reactions and the
Gibbs free energy equation have been utilized to describe the
G(S) = 25.2S2+21.2S+3 corrosion diffusion that emulates the searching mechanism
S5+16.58S4+25.41S3+17.18S2+11.7S+1
D. Forth order system with time delay

G(S) = S4 +10S3 10 +50S+24 e-3S
+35S2