## 1 Introduction  
Alternating current (AC) motors are widely used in industrial applications such as electric vehicles, hybrid electric vehicle, wind generation systems, mining machinery, marine propulsion, industrial robots, air conditioners, cranes, among other applications, as well as particular demand in residential and commercial areas [1, 2]. There are two main categories of AC motors: induction motors (IM) and synchronous motors (SM), the last one, are gaining great attention due to its self-excitation, high efficiency, low maintenance cost, high power density, fast dynamic response, high torque-to-inertia, and more economic maintenance compared to direct current and induction machines. [3, 4]. During recent decades, the permanent magnet synchronous motor (PMSM) has been used more and more due to its advantages; but at the same time, too many control structures have been developed to enhance the performance due to the nonlinears, and endogenous and exogenous disturbances [5]. Some of them include classical controllers, intelligent control, vector control, backstepping technique, linear quadratic regulator, fuzzy logic controller, artificial neural networks, genetic algorithm, predictive control, and robust control [6]. Vector control with classical controllers,Figure 1, is the most widely used controller in industries due its easy design and implementation procedure. Moreover, in the literature, proportional integral derivative (PID) controllers have been extensively studied to prove stability of PMSM dynamic performance and improve its operation under critical scenarios [7; 8]. In [9] shows that the PMSM is globally regulated around a desired equilibrium point, with a simple PI control around current errors, if the gains are suitably chosen. Therefore, a control scheme based on traditional PI controllers is a proper choice for velocity trajectory tracking. The advantages of a PID controller are its feasibility, simplicity, ease to understand, and implementation stage. However, a disadvantage of the traditional PID controller is the procedure to find the optimal PID gains for a particular system that is very difficult and tedious to perform [10; 11; 12; 13]. In the literature, there are several techniques to solve the problem of gains tuning of conventional PID and PI controllers when the system is non-linear, such as the PMSM. Traditional tuning techniques do not provide optimal adjustment parameters for the PID controllers because the transient response could produce oscillations and a large overshoot due to the system dynamics [14]. Therefore, new regulation strategies know as advanced and intelligent schemes have been applied to overcome the main drawbacks of PID controller in tuning stage, by the solution of an objective function to achieve optimal gains. A crucial problem in nonlinear systems control design and PID scheme is the definition of a set of controller gains that provide a system performance near-optimal. This process, which is usually carried out at the design stage as a compensation regarding various adverse factors, often produces restricted results. Thus, algorithms inspired in nature and population seems like an important alternative to synergistically operate with PID controller for improving the system behavior [10]. In recent literature, various metaheuristic algorithms have been used to carry out an inclusive search for optimal solutions, which cannot be solved by classical techniques. That the metaheuristic algorithms try to emulate the natural behavior of different class of animals, outcomes to have a high level of efficiency, overcome limitations of classical methodologies. Besides, search results depend on the system complexity or the problem to solve; because a metaheuristic algorithm can be more efficient than other to solve a particular problem, while the same metaheuristic algorithm could present a poor performance for another system [15].