## 4 A Fuzzy PSO-Based Control Design  
### Preliminaries on PSO  
The PSO algorithm is evolutionary, and it uses several candidate solutions to produce the optimal solution fitting a specific problem. PSO is a stochastic optimization technique, and the changing behavior of social animals in swarms influences its concept. The technique models such social behaviors using equations to help the \"particles\" while they move. There are three aspects the influence particle movement: cognitive, social, and inertia components [45; 46; 47; 48]. Each of these components reflects part of the equation. PSO has the advantage of being effective on a wide variety of problems without the user having to modify the basic structure algorithm. The literature specifies that PSO is among the highly efficient techniques employed for finding solutions to non-smooth global optimization problems. The primary benefits of the technique are specified below:  
1. The literature specifies that PSO is a highly efficient technique employed to find solutions to non-smooth global optimization problems. The main benefits of the technique are listed below. This technique does not rely on derivatives, unlike similar meta-heuristic optimization techniques. 2. The coding and theory corresponding to this method are straightforward, unlike other heuristic optimization techniques;
3. It has considerably less sensitive to the aspects of the objective function, unlike traditionally used analytical and heuristic techniques;
4. Unlike many heuristic optimization methods, PSO has relatively fewer parameters: inertia weight coefficient and acceleration coefficients. Furthermore, the parameters corresponding to the optimal solutions are relatively less sensitive compared to other heuristic methods;
5. PSO is less contingent on the initial set of points compared to numerous evolutionary techniques; also, this algorithm has robust convergence characteristics;
6. PSO-based techniques produce high-quality solutions at a lower time-cost but with better stability corresponding to the convergence aspects, as compared to other stochastic methods. The PSO algorithm may be classified as an iterative (progressive movement towards the solution) technique. In contrast, stochastic techniques attempt to optimize the present solution by transitioning partially to other points following pre-specified rules to reach the desired solution ultimately [49]. There are prerequisites corresponding to PSO application: there should be a research space comprising particles associated with a cost function (also referred to as the objective function) that needs to be optimized. The algorithm is based on the fundamental that these particles must be moved in order to arrive at the optimal value [50; 51; 52]. All such particles are associated with:  
* A set of defined coordinates specifying the position of the particle;
* Speed corresponding to the movement of the particle. There are several iterations when the particles have different positions. The characteristics evolve, considering the best position, neighbor position, and previous position. Such evolution of particles implies that several particles interact directly with a single particle, especially with the one having the optimal criteria. A neighborhood where several particles interact with this particle directly, especially with the one having the most-optimal criteria. An optimization problem is associated with site quality in the search space being a function of the cost function corresponding to the point [53]. The choice of parameters of the PSO algorithm is imposed arbitrarily in which the number of particles. The decision values corresponding to parameters required to be assessed expressed in the form of the position vector corresponding to the \(i^{\text{th}}\) particle, as specified below:  
\[X^{(i)}=\left[ \begin{array}{c}X^{(i)}_{1}\\ X^{(i)}_{2}\end{array} \right],\;i\in\{1,\ldots,p\} \tag{48}\]  
where \(p\) denotes the swarm size (count of particles).