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Fig. 10. Multi step response Goriginal
Fig. 11. Step response Gapproximated
C. Fitted MATLAB tuner
As shown in Fig. 11 and Fig. 12, the approximated function
(Gapproximated) has a negative part and this problem affects
the control. To solve this problem curve fitting was applied.
The fitted transfer function (G fitted) was calculated using the to the food. Position and velocity update equations of the par-
system identification toolbox of MATLAB. The fitted transfer
function is as below:
9.56e-5s2 + 2.32e-4s + 7.3486e-6
G fitted(s) = s3 + 1.17s2 + 0.036s + 0.001122689 (7)
Fig. 13 shows the step response of the fitted transfer
function (G fitted) using MATLAB Tuner. Fig. 14 shows the
Multi-Step Response of the fitted transfer function (G fitted)
using MATLAB Tuner.
D. Fitted PSO
As shown in Fig. 13 and Fig. 14 The delay time was treated
using Pade approximation and the negative part was treated
using curve fitting but the performance of the fitted function
response must be improved. This can be done using particle
swarm optimization PSO algorithm tuning.
Al-Najari, Hen, Paw, & Marhoon
Fig. 12. Multi step response Gapproximated
Fig. 13. Step response G fitted
1) PSO Overview
The particle swarm optimization PSO algorithm developed
by Kennedy and Eberhart is a swarm based evolutionary al-
gorithm [17]. The algorithm is based on the movement of the
birds in the flock according to the location of the bird closest
ticles are used to model the movements of the flock. Equations
of velocity and position are given below, respectively.
Vik+1 = W kVik + c1r1(PBkest - Xik) + c2r2(GBk est - Xik) (8)
Xik+1 = Xik + Vik+1 (9)
Where k is the number of iteration, i, is the index of the
particle, W is the inertia weight that directly affects velocity,
c1 and c2 are the acceleration factors called cognition and
social constants respectively, r1 and r2 are random numbers
between 0 and 1. PBest is the best local solution; GBest is the
best global solution, Vi and Xi are the velocity and position
of particle i, respectively [18]. The particle swarm algorithm
PSO is to find the optimal solution in the set area to minimize
the value of the objective function. In this paper, integrated