## 4 Applications in Optimization
### Particle Swarm Optimization (PSO)
#### 4.1.2 Pid Controller Using PSO  
A few years after the initial use of chaotic attractor properties in PSO by Araujo and Coelho, a team led by Ivan Zelinka from the Technical University of Ostrava (VSB-TU, Faculty of Electrical Engineering and Computer Sciences) extensively explored a quasi-similar method in a series of articles. They used a slightly different velocity equation, shown in Equation (26), introducing an inertia weight modification to the PSO strategy based on two control parameters, \(w_{start}\) and \(w_{end}\):  
\[v_{i}(t+1)=w\cdot v_{i}(t)+c_{1}\cdot Rand\cdot(pBest-x_{i}(t))+c_{2}\cdot Rand \cdot(gBest-x_{i}(t)) \tag{31}\]  
with (32)  
\[w=w_{start}-\frac{(w_{start}-w_{end})\cdot t}{n} \tag{32}\]  
where \(t\) stands for the current generation number and \(n\) for the total number of generations.  
Within the Chaos PSO algorithm, the basic inbuilt computer random generator was replaced by a chaotic generator (in this case, using the Lozi map).  
First, Pluhacek et al. [62] considered a Partial Integral Derivative (PID) controller for a Direct Current (DC) motor system in order to obtain optimal settings. A DC motor is any of a class of rotary electrical motors that converts direct current electrical energy into mechanical energy. Proportional Integral Derivative (PID) control is the most common control algorithm used by industry, and has been universally accepted in industrial control.  
The popularity of PID controllers can be attributed to their robust performance in a wide range of operating conditions and their functional simplicity, which allows engineers to operate them in a simple and straightforward manner.  
All PID controllers contain three unique basic coefficients, namely, proportional, integral, and derivative, which are varied to achieve optimal response (see [63] for a detailed description). A simplified form in the Laplace domain is provided by  
\[G(s)=K\bigg{(}1+\frac{1}{sT_{i}}+sT_{d}\bigg{)}, \tag{33}\]  
where \(K\) is the controller gain, \(T_{i}\) is the adjustable integral time parameter, and \(T_{d}\) is the rate time. For suitable analytical calculations, the following form is often chosen:  
\[G(s)=k_{p}+\frac{k_{i}}{s}+k_{d}s, \tag{34}\]  
where the parameters are related to the standard form (33) through \(k_{p}=K\), \(k_{i}=K/T_{i}\), and \(k_{d}=K/T_{d}\).  
The transfer function for the DC motor considered in [62] is provided by  
\[G(s)=\frac{0.9}{0.00105s^{3}+0.2104s^{2}+0.8913s}. \tag{35}\]  
The optimization process involving the PSO algorithm was applied to minimize errors of the output transfer function, which can indicate the quality of regulation of the controller. Four criteria were considered:  
* The integral of the time multiplied by absolute error (ITAE): \[I_{ITAE}=\int_{0}^{T}t|e(t)|dt,\] (36) where \(e(t)\) is the control error (see [63]).
* The integral of the absolute magnitude of the error (IAE): \[I_{IAE}=\int_{0}^{T}|e(t)|\;dt.\] (37)
* The integral of the square of the error (ISE): \[I_{ISE}=\int_{0}^{T}(e(t))^{2}\;dt.\] (38)
* The mean of the square of the error (MSE): \[I_{MSE}=\frac{1}{n}\sum_{i=1}^{n}{(e(t))^{2}}.\] (39)  
The presented data and graphical simulation results of [62] lend weight to the argument that the implementation of a chaotic Lozi map into the PSO algorithm as a random number generator can significantly improve its performance over other non-heuristic and heuristic methods when solving the problem of optimal PID controller design for DC motor systems.  
In [64], the same authors addressed a fourth-order system provided by  
\[G(s)=\frac{1}{s^{4}+6s^{3}+11s^{2}+6s}. \tag{40}\]  
For statistical reasons, in both articles the optimization for each criterion was repeated thirty times. The results for the fourth-order system showed the same improvement of the Lozi map-based PSO method versus other non-heuristic and heuristic methods.