## 5 Dynamic Particle Swarm Optimization (PSO) Mechanism
### Implementation of the Dynamic PSO Mechanism  
In this proposed PSO-based optimization mechanism, the main goal is to identify and generate a global minimizer for the objective function (based on \(E_{k}\) and \(dE_{k}\)). Many of the optimization mechanisms fulfil this requirement through deterministic methods and probabilistic methods. Deterministic-method-based optimization mechanisms use heuristic approaches (learning by themselves) by taking into consideration the \(E_{k}\) and \(dE_{k}\) (the error and rate of change of the error). Therefore, the optimization process is controlled through the objective function threshold level, and this objective function is a function of the parameters \(E_{k}\) and \(dE_{k}\). Mostly, the deterministic optimization mechanisms apply penalties to the generated output. The PSO mechanism is a progressive computation technique based on swarm intelligence that gives a potential solution among the population members in the hyperspace. The optimized solutions are represented by adaptable position change (velocity) particles. The position of each particle in the hyperspace (search space) can be represented by n-dimensional axes. Figure 2 shows that these _coordinate positions_ are assigned to dynamically tune the fuzzification process, de-fuzzification process and the FIM. These optimum parameters are the final solution of the PSO controller, which is generated while taking into consideration the personal best \(P_{Best}\) of each particle (feasible solutions) until reaching the global best \(G_{Best}\) or the social best solution in the search space. These parameters need to be quickly re-optimized because of the dynamic behavior of the wheel slip. Figure 5 shows the updated positions of the particles that are determined by taking into consideration the updated velocity. When considering the original PSO mechanism [107], in hyperspace if the position of the particle \"\(i\)\" is denoted by \(x_{i}(t)\) at a discrete time \(t\) then the updated position \(x_{i}(t+1)\) and the updated velocity \(v_{i}(t+1)\) are determined according to (18), (19) and [107]:  
\[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1) \tag{18}\]  
\[v_{i}(t+1)=\omega v_{1}(t)+r_{1}a_{1}(P_{Best}(t)-x_{i}(t))+r_{2}a_{2}(G_{Best} (t)-x_{i}(t)) \tag{19}\]  
where \(\omega\) is the inertia weight; \(v_{i}(t)\) the velocity at a time step \(t\); \(r_{1}\), \(r_{2}\) are the randomly generated and uniformly distributed numbers (\(0\leq r_{1}\leq 1\) and \(0\leq r_{2}\leq 1\)), respectively; \(a_{1}\), \(a_{2}\) are the acceleration coefficients; \(P_{Best}\) is the personal best; and \(G_{Best}\) is the global best. In (18) and (19), the heuristic parameters critically affect the optimization mechanism. The inertia weight \(\omega\) critically affects the convergence behavior of the PSO mechanism. This fast convergence behavior occurs in the PSO mechanism because the inertia weight \(\omega\) highly influences the exploration process of the PSO mechanism. This takes place due to the inertia weight creating an impact on the current velocity based on the previous velocity of each particle. Therefore, a large \(\omega\) value toward the particles moves over the hyperspace rapidly, and the smaller \(\omega\) value towards particles moves around its neighborhood particles. In this case, the large \(\omega\) values allow global exploration, and smaller \(\omega\) values create a tendency for exploitation or local exploration (a fine-tuning process). Equation (19) shows that in the original PSO mechanism [107], the properly tuned values of parameters \(a_{1}\) and \(a_{2}\) allow for rapid convergence. The extended practical experiments show that this rapid convergence occurs with a large value of the cognitive parameter \(a_{1}\) and with a large value of the social parameter \(a_{2}\) when \(a_{1}+a_{2}\leq 4\). [108]. The values \(r_{1}\) and \(r_{2}\) control the diversity of the optimal solutions, and originally they are uniformly distributed between the range of \(0\) and \(1\). Figure 5: Geometric representation of the PSO model.