Modeling and Speed Tuning of a PMSM with Presence of Fissure Using Dragonﬂy Algorithm

: This paper presents a robust trajectory tracking control for a Permanent Magnet Synchronous Motor (PMSM) with consideration a fault, parametric uncertainties and external disturbances by e ﬀ ectively integrating robust optimal linear quadratic control. One kind of fault is considered in the machine, particularly the presence of ﬁssure rotor. The dynamic model of the PMSM with the presence of ﬁssure presents highly non-linear behaviors, which means that tuning is quite complicated, which the tuning was chosen through swarm intelligence optimization (Dragonﬂy Algorithm). A sensitivity analysis is carried out, in order to limit the search range to minimize the evaluation time. This methodology was used to diminish these defects during motor operation. Simulation results show that the optimal linear quadratic control method has a robust fault-tolerant performance.


Introduction
The Permanent Magnet Synchronous Motors (PMSM), in addition to providing high performance in applications where it is necessary to correct the power factor, provide high torque and constant speed under variable loads, which makes them increasingly studied and used in applications that, until a few years ago, were restricted to induction motors [1]. One of the failures that causes more interest, especially in electric motors of considerable sizes, is due to vibratory problems, caused by imbalance, which, in turn, are generated by degradation in the rotor shaft, that is, fatigue phenomena which, finally, causes fracture in the rotor shaft [2]. The behavior of the propagation of fissures in solid materials is a subject of great interest in the field of engineering, thereby helping to preserve the life of mechanical devices [3]. A contribution to the fault-monitoring approach and input-output feedback linearization control of the induction motor (IM) in the closed-loop drive is presented in [4]. Two kinds of faults are considered in the machine, particularly the broken rotor bars and stator inter-turn short circuit faults. Therefore, the neural network (NN) technique is applied in order to identify the faults and distinguish them. However, the NN requires a relevant database to achieve satisfactory results. Hence, the stator current analysis based on the HFFT combination of the Hilbert transform and fast Fourier transform is applied to extract the amplitude of the harmonics and used them as an input dataset for NN.
Rotor faults have drawn more attention from the Artificial Intelligent (AI) research community in terms of utilizing fault-specific characteristics in its feature engineering. In [5], a review and definition gains in the dynamic model of the PMSM using the optimization dragonfly algorithm (DA) technique, a sensitivity analysis of the same is carried out, in order to limit the definition domain, minimize the evaluation time and help the convergence of the dynamic system. The control scheme of the PMSM is simple and robust and can operate within a very wide speed range.
The rest of the document is organized as follows: Section 2 develops the dynamic model of PMSM with the presence of a rotor fissure. In Section 3, a presentation of the control scheme of the dynamic PMSM model is given. In Section 4, the control tuning procedure through the Dragonfly algorithm is presented. Section 5 describes the procedure for conducting a sensitivity analysis. In Section 6, the simulation of the dynamic system of the PMSM is carried out. Finally, Sections 7 and 8 present the results of the implementation and a discussion of the results, respectively.

Dynamic PMSM Model with Presence of Rotor Fissure
Considering the voltage balance equations, the dynamic model of the PMSM in the reference system dq is obtained in a similar way to the modeling of a synchronous machine with field winding.
To obtain the PMSM model, the flow links equations are eliminated and currents are defined as equal to zero by damping windings. Replacing the field current with a constant parameter due to the permanent magnet flow link, the model obtained is characterized by [13].
The dynamics of the PMSM rotor regarding angular velocity and angular position is defined by where T e = P 2

Fracture Dynamics in the Rotor Shaft
The fissure behavior of the rotor shaft is proposed as a result of the variation in the stress on it, caused by the external load. The rotational effects of the rotor shaft generate a dynamic of opening and closing the fissure, which will generate, through work cycles, fatigue fracture that has a behavior similar to the fragile fracture. The fissure behavior takes the structure of the Paris equation, as in [14,15].
The stress intensity factor is defined as From the expression defined by (7), the variation in torsional stress is proposed based on the behavior of a hollow circular section, which will increase in size due to degradation dynamics, starting from the concentration point of effort, where the presence of the fissure exists [16,17]. The smaller diameter of the rotor shaft d is proposed from the variation in the crack size, (d = ga), where g takes the breathing behavior of the axis proposed by Mayes and Davis, and can be defined as [18] g = l + cos(ω r t) 2 (9) because of the presence of the fissure in the rotor shaft, the degradation in the rotational inertia of the rotor shaft has an effect proposed as

Dynamic Coupled Model
The dynamic study model with the presence of degradation in the rotational inertia of the rotor shaft can be formulated as

Reference Model
The control scheme of the dynamic model of the PMSM with the characteristics defined by Equations (1)-(5) is carried out based on its linearization, without the presence of fissures, at the single point of global stable equilibrium of the dynamic system, defined as [19] The first-order linear differential equations for the PMSM under which the simulation will be carried out, and, subsequently the development of the linear quadratic speed regulator, are

Desired Behavior of the Error
The error behavior regarding the change in speed between the desired angular speed and the rotor speed delivered by the PMSM is established as [20] e ωr = ω rd − ω r (20) It is thought that the angular velocity error decreases exponentially in a limited time, which implies the relationship e ωr = exp(−c ωr t) The behavior of the current i qd desired for the system based on the angular velocity error ω r takes the form The load torque can be controlled directly by the current component of the axis q, therefore, the angular speed of the rotor can be controlled by the change in the current of the axis q, whereby the change in the current of the axis d is established at zero (i dd = 0) to minimize current and resistance losses [21].

Optimal Linear Quadratic Control for States ω r , i q
For the control of the PMSM, the feedback of the states ω r , i q from the defined linear reference model is proposed, where the optimal gains are determined from the energy function E Q as in [22] The positive Hermitian matrices Q, R are defined as The vector X takes the form The control function U q takes the structure The linearized system, for the variables i q and ω r under the action of the control, takes the form

PID Controller for i d Current
The structure of the proposed PID controller takes the form [23] To regulate the current i d , it is based on the form of the decoupled linear equation defined in (17); when defining the control action, it is established as The block diagram for regulating the PMSM i d is shown in Figure 1. To regulate the current id, it is based on the form of the decoupled linear equation defined in (17); when defining the control action, it is established as The block diagram for regulating the PMSM is shown in Figure 1.  Using the geometric place of the roots, the controller gains k i , k p , k d are calculated as

Tuning of Controller Using the Dragonfly Algorithms
The inspiration for the DA [24] is taken from the social behavior of the dragonflies when hunting their food (static swarm) and when they migrate (dynamic swarm). Considering these two behaviors, there are five factors involved in determining the individual dragonfly position: (a) separation; (b) alignment motion; (c) cohesion motion; (d) food Attraction; (e) predator distraction. There are two ways of updating the individual dragonfly position depending on the neighborhood position. If there is no dragonfly in the neighborhood radius, the individual position is updated considering the Levy flight equation and given as follows where dn is the number of decision variables. The Lévy flight function is given by lévy(dn)= 0.01 r 1 ρ where r 1 and r 2 are two random numbers in [0, 1]; β is a constant and ρ is computed as where Γ(x) = (x − 1)!. Otherwise, the new position is calculated as follows where ∆X t+1 is the step vector and can be obtained as where s shows the separation weight; a is the alignment weight; c is the cohesion weight; f is a food actor; e is the enemy factor; w is the inertia weight, S i indicates the separation of the i − th individual, A i is the alignment of i − th individual, C i is the cohesion of the i − th individual, F i is the food source of the i − th individual, E i is the position of enemy of the i − th individual and t is the iteration number. The optimization process of DA is further explained by the pseudo code below [25]. The optimization process of DA is further explained by the pseudo code below (Algorithm 1): Update position vector using the Lévy flight function end if end for i Sort the population/dragonflies from best to and find the current best end while Therefore, in this work, the dragonfly algorithm is used to calculate the optimization of the speed controller gains for the PMSM with the presence of degradation.

Sensitivity Analysis
The proposed tuning algorithm, using the dragonfly algorithm, searches for gains and the subsequent simulation of them in a defined domain. In order to limit the search domain, a sensitivity analysis is proposed for the PMSM speed control system. The sensitivity analysis will determine the variations in each of the gains involved, defined as the k j parameters [26], where In addition, the study aims to ensure that this domain is optimal in terms of the consumption of the runtime of the algorithm. An energy function is defined, which aims to analyze the sensitivity of the PMSM model, defined as where E T is the total energy, E C is the kinetic energy and E P potential energy. The previous two energy functions can be defined as follows [1] E C = T e ω r dt (40) The total energy is calculated using (1) and (2) Appl. Sci. 2020, 10, 8823 Equation (42) can be expressed as a differential equation as The energy cost function G, for the temporary evaluation period, can be defined as The speed of change in the cost function in differential form, in the analysis time interval, can be described as The set of differential equations that determines the behavior of the sensitivity of the parameters of the PMSM system is determined by combining the mathematical model of the machine and the energy cost function, and is expressed as From the system of Equations (46)-(50), the behavior of the cost function is obtained for the defined simulation time interval. The sensitivity of the system S j , for each parameter of interest k j , is as in

Simulation
The cost function to evaluate the performance of the dragonfly algorithm of profit search, considered as optimal, is that which minimizes the desired trajectories with those obtained in the interactions of the proposed system, which is defined from the state of interest ω r as To carry out the simulation of the dynamic PMSM system, the parameters and simulation coefficients are defined, Table 1 illustrates the numerical values that are taken into consideration during the model analysis.
The gains in the linear comparison system are determined from the linear analysis of speed regulation and take the following values The process of searching for the gains of the speed control scheme for the non-linear dynamic system of PMSM is presented in the diagram shown in Figure 2. The step of searching and testing the parameters of the controller is carried out by means of dragonfly algorithm. The gains in the linear comparison system are determined from the linear analysis of speed regulation and take the following values The process of searching for the gains of the speed control scheme for the non-linear dynamic system of PMSM is presented in the diagram shown in Figure 2. The step of searching and testing the parameters of the controller is carried out by means of dragonfly algorithm.

Results
Using the proposed model of the PMSM with the presence of fissure in Section 3 and the linear comparison reference defined in Section 4, a convergence analysis is performed between both models subjected to the disturbance defined by the system of equation [

Results
Using the proposed model of the PMSM with the presence of fissure in Section 3 and the linear comparison reference defined in Section 4, a convergence analysis is performed between both models subjected to the disturbance defined by the system of equation [27] as The variation in the displacement angle of the rotor δ is determined from In Figure 3 it can be seen that, under the disturbance given to both models of the PMSM under study, the simulation convergence time interval, for the i d current, is less than 0.01 s; subsequently, the i d current diverges in the reference linear and the nonlinear proposal. For the states i q and ω r , it is observed that the convergence is for the entire simulation time interval.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 19 the current diverges in the reference linear and the nonlinear proposal. For the states and , it is observed that the convergence is for the entire simulation time interval.  From the comparison between the proposed model with the presence of degradation in rotational inertia and the linear model of the PMSM, it is observed that they only converge under a small region of time (0.01 s). This time interval is insufficient for a proper degradation analysis, where simulation times will be greater than 2000 s, and where the controller must show its efficiency to overcome the modeling limitations and ensure that the variables of interest converge to the desired reference values (angular velocity and current consumption).
The delimitation of the search domain of each of gains involved in controller action will help achieve this goal. Figures 4-8 show the behavior of the cost function proposed in the analysis of sensitivity for each of the gains involved. simulation times will be greater than 2000 s, and where the controller must show its efficiency to overcome the modeling limitations and ensure that the variables of interest converge to the desired reference values (angular velocity and current consumption).
The delimitation of the search domain of each of gains involved in controller action will help achieve this goal. Figures 4-8 show the behavior of the cost function proposed in the analysis of sensitivity for each of the gains involved.  It is observed that sensitivity of the dynamic system and the energy consumption are exponentially increased, with gains of k1 less than 200. Similarly, the numerical instability of algorithm grows exponentially, achieving the non-convergence of the solution of the mathematical model, therefore, those values should be avoided. For the gain k2, a concave behavior is observed in the range of values near to 120; where the minimum sensitivity point is located, energy consumption will be minor as well as the time of algorithm computation, therefore, an interval close to that value of sensitivity must be chosen.
It is observed that, within the most representative variations in energy consumption, there is the gain kp, which is very sensitive to values greater than −185. Where the sensitivity is reflected in the energy consumption, which increases exponentially, the time of computation will be similar, increasing the numerical instability of the algorithm; therefore, these values should be avoided. It is observed that sensitivity of the dynamic system and the energy consumption are exponentially increased, with gains of k1 less than 200. Similarly, the numerical instability of algorithm grows exponentially, achieving the non-convergence of the solution of the mathematical model, therefore, those values should be avoided. For the gain k2, a concave behavior is observed in the range of values near to 120; where the minimum sensitivity point is located, energy consumption will be minor as well as the time of algorithm computation, therefore, an interval close to that value of sensitivity must be chosen.
It is observed that, within the most representative variations in energy consumption, there is the gain kp, which is very sensitive to values greater than −185. Where the sensitivity is reflected in the energy consumption, which increases exponentially, the time of computation will be similar, increasing the numerical instability of the algorithm; therefore, these values should be avoided.   For the gains ki and kd the behavior of the sensitivity in the face of gain variation is indifferent, therefore, any interval chosen as the search function will not affect the time spent on computing resources. Using the previous sensitivity analysis, intervals of search for gains were chosen close to trajectories that minimize the proposed energy consumption, which are defined as   For the gains ki and kd the behavior of the sensitivity in the face of gain variation is indifferent, therefore, any interval chosen as the search function will not affect the time spent on computing resources. Using the previous sensitivity analysis, intervals of search for gains were chosen close to trajectories that minimize the proposed energy consumption, which are defined as The response of behavior of the angular velocity ωr and current consumption corresponding to iq of linear comparison model of the PMSM in the desired angular velocity input ωrd are shown in Figure 9. The parameters for proper convergence of search-tuning of the DA after numerical simulation tests are shown in Table 2. It is observed that sensitivity of the dynamic system and the energy consumption are exponentially increased, with gains of k 1 less than 200. Similarly, the numerical instability of algorithm grows exponentially, achieving the non-convergence of the solution of the mathematical model, therefore, those values should be avoided. For the gain k 2 , a concave behavior is observed in the range of values near to 120; where the minimum sensitivity point is located, energy consumption will be minor as well as the time of algorithm computation, therefore, an interval close to that value of sensitivity must be chosen.
It is observed that, within the most representative variations in energy consumption, there is the gain k p , which is very sensitive to values greater than −185. Where the sensitivity is reflected in the energy consumption, which increases exponentially, the time of computation will be similar, increasing the numerical instability of the algorithm; therefore, these values should be avoided.
For the gains k i and k d the behavior of the sensitivity in the face of gain variation is indifferent, therefore, any interval chosen as the search function will not affect the time spent on computing resources. Using the previous sensitivity analysis, intervals of search for gains were chosen close to trajectories that minimize the proposed energy consumption, which are defined as For the convergence test between answers to the control action of the linear test model and the proposed model of nonlinear PMSM, the desirable behavior of angular velocity is proposed through a defined step function where ω rd = 188.5 rad s i f t < 0.0005 s 100.0 rad s i f t > 0.0005 s The response of behavior of the angular velocity ω r and current consumption corresponding to i q of linear comparison model of the PMSM in the desired angular velocity input ω rd are shown in Figure 9. The parameters for proper convergence of search-tuning of the DA after numerical simulation tests are shown in Table 2.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19 Figure 9. Desired behavior of angular velocity (lower) and current consumption iq (higher) in the linear comparison system. In Figure 10, the behavior of the DA regarding the pursuit of gains for tune the PMSM model is illustrated, with the presence of degradation in rotational inertia, under search conditions defined by the sensitivity analysis and the desired model of behavior of linear PMSM, as well as angular speed's desired behavior under a first search iteration. Figures 10 and 11 show, in different colors, the dynamic response of the closed-loop system (controller-PMSM), with all gains calculated in the first iteration and in the tenth search evolution, respectively.  In Figure 10, the behavior of the DA regarding the pursuit of gains for tune the PMSM model is illustrated, with the presence of degradation in rotational inertia, under search conditions defined by the sensitivity analysis and the desired model of behavior of linear PMSM, as well as angular speed's desired behavior under a first search iteration. Figures 10 and 11 show, in different colors, the dynamic response of the closed-loop system (controller-PMSM), with all gains calculated in the first iteration and in the tenth search evolution, respectively.
In Figure 10, the behavior of the DA regarding the pursuit of gains for tune the PMSM model is illustrated, with the presence of degradation in rotational inertia, under search conditions defined by the sensitivity analysis and the desired model of behavior of linear PMSM, as well as angular speed's desired behavior under a first search iteration. Figures 10 and 11 show, in different colors, the dynamic response of the closed-loop system (controller-PMSM), with all gains calculated in the first iteration and in the tenth search evolution, respectively.  In Figure 11, the behavior of the collection of gains generated by the DA is illustrated, and applied to a proposed model of the PMSM with the presence of degradation in rotational inertia. In the tenth search evolution, the convergence of the behaviors of all these gains obtained by the DA is obtained for the desired model of linear reference to the PMSM. In Figure 12, the best response obtained in the ωr angular velocity tuning process is illustrated. The consumption of current iq by the controller action defined by the dragonfly algorithm, in the case of the analysis, is It is observed that the obtained trajectory has a performance with behavior close to the desired angular velocity, ωrd, therefore, the gains obtained are considered optimal. The behavior of the proposed nonlinear model and linear model PMSM reference for times greater than 0.001 s is achieved with these gains, so the robustness of the tuning algorithm is checked under the presence Figure 11. Behavior of the DA for the dynamic system PMSM with crack presence, better control gains in the tenth search evolution.
In Figure 11, the behavior of the collection of gains generated by the DA is illustrated, and applied to a proposed model of the PMSM with the presence of degradation in rotational inertia. In the tenth search evolution, the convergence of the behaviors of all these gains obtained by the DA is obtained for the desired model of linear reference to the PMSM. In Figure 12, the best response obtained in the ω r angular velocity tuning process is illustrated. The consumption of current i q by the controller action defined by the dragonfly algorithm, in the case of the analysis, is Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 19 Figure 11. Behavior of the DA for the dynamic system PMSM with crack presence, better control gains in the tenth search evolution.
In Figure 11, the behavior of the collection of gains generated by the DA is illustrated, and applied to a proposed model of the PMSM with the presence of degradation in rotational inertia. In the tenth search evolution, the convergence of the behaviors of all these gains obtained by the DA is obtained for the desired model of linear reference to the PMSM. In Figure 12, the best response obtained in the ωr angular velocity tuning process is illustrated. The consumption of current iq by the controller action defined by the dragonfly algorithm, in the case of the analysis, is It is observed that the obtained trajectory has a performance with behavior close to the desired angular velocity, ωrd, therefore, the gains obtained are considered optimal. The behavior of the proposed nonlinear model and linear model PMSM reference for times greater than 0.001 s is achieved with these gains, so the robustness of the tuning algorithm is checked under the presence of the initial fissure given. Figure 13 illustrates the ωr angular velocity behavior and consumption of It is observed that the obtained trajectory has a performance with behavior close to the desired angular velocity, ω rd , therefore, the gains obtained are considered optimal. The behavior of the proposed nonlinear model and linear model PMSM reference for times greater than 0.001 s is achieved with these gains, so the robustness of the tuning algorithm is checked under the presence of the initial fissure given. Figure 13 illustrates the ω r angular velocity behavior and consumption of current i q under the action of the gains obtained through the simulation done for an evaluation time of 0.03 s, which shows that the system keeps responding to the action of control.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 19 Figure 13. Comparison, best gains found, nonlinear system, linear system for times greater than the attractor of the equilibrium point.
In Figure 14, the speed behavior angular ωr and current consumption iq of the nonlinear model of the PMSM, proposed with degradation in rotational inertia for a simulation time of 450 s, are illustrated. The time of evaluation is large so that the effects of steady state be seen, and it is observed that the behavior of the desired angular velocity ωrd is maintained under the action of control. For the given simulation, an angular velocity given by a Bézier polynomial ¥ is desired to provide a sufficiently smooth transfer between the actual and desired speed reference values, within a specific time interval. Then, the reference trajectory profile is as follows * = with r1 = 252, r2 = −1050, r3 = 1800, r4 = −1575, r5 = 700 and r6 = −126. Figure 13. Comparison, best gains found, nonlinear system, linear system for times greater than the attractor of the equilibrium point.
In Figure 14, the speed behavior angular ω r and current consumption i q of the nonlinear model of the PMSM, proposed with degradation in rotational inertia for a simulation time of 450 s, are illustrated. The time of evaluation is large so that the effects of steady state be seen, and it is observed that the behavior of the desired angular velocity ω rd is maintained under the action of control. For the given simulation, an angular velocity given by a Bézier polynomial ¥ is desired to provide a sufficiently smooth transfer between the actual and desired speed reference values, within a specific time interval. Then, the reference trajectory profile is as follows where ω 1 = 0 rpm, ω 2 = 800 rpm, ω 3 = 1600 rpm, ω 4 = 600 rpm, T 1 = 0 s, T 2 = 18 s, T 3 = 135 s, T 4 = 150 s, T 5 = 290 s, T 6 = 305 s, T 7 = 420 s, T 8 = 450 s, and Y = is the Bézier interpolation polynomial with r 1 = 252, r 2 = −1050, r 3 = 1800, r 4 = −1575, r 5 = 700 and r 6 = −126. ¥ = 5 [ 1 − 2 + 3 2 − 4 3 + ⋯ − 6 5 ] (56) with r1 = 252, r2 = −1050, r3 = 1800, r4 = −1575, r5 = 700 and r6 = −126.  The behavior of the fissure inside the PMSM is shown in Figure 15. It is observed that the fissure grows suddenly with the PMSM startup and, because of inertial effects, as the working time of the PMSM continues, the growth of the fissure is gradual and in an exponential form, which will exhibit a progressive degradation in the rotational inertia of the rotor shaft, thereby validating the proposed dynamic behavior. The tracking error for optimal linear state feedback controller is shown in Figure 16. The behavior of the fissure inside the PMSM is shown in Figure 15. It is observed that the fissure grows suddenly with the PMSM startup and, because of inertial effects, as the working time of the PMSM continues, the growth of the fissure is gradual and in an exponential form, which will exhibit a progressive degradation in the rotational inertia of the rotor shaft, thereby validating the proposed dynamic behavior. The tracking error for optimal linear state feedback controller is shown in Figure  16.

Conclusions
This paper presents an application of one metaheuristic (dragonfly algorithm) for tuning a PID, and optimal linear quadratic controllers for a PMSM with the presence of rotor fissure. The optimization procedure was employed considering simulation with the non-linear system model. This strategy contains the speed and current control loops. In order to define the search range for DA, a sensitivity analysis of gains of the PMSM controller was carried out. The sensitivity analysis of gains of the PMSM controller with the presence of degradation in rotational inertia can find search intervals that help to minimize the tuning time of the DA. The dragonfly algorithms confirm the feasibility and The behavior of the fissure inside the PMSM is shown in Figure 15. It is observed that the fissure grows suddenly with the PMSM startup and, because of inertial effects, as the working time of the PMSM continues, the growth of the fissure is gradual and in an exponential form, which will exhibit a progressive degradation in the rotational inertia of the rotor shaft, thereby validating the proposed dynamic behavior. The tracking error for optimal linear state feedback controller is shown in Figure  16.

Conclusions
This paper presents an application of one metaheuristic (dragonfly algorithm) for tuning a PID, and optimal linear quadratic controllers for a PMSM with the presence of rotor fissure. The optimization procedure was employed considering simulation with the non-linear system model. This strategy contains the speed and current control loops. In order to define the search range for DA, a sensitivity analysis of gains of the PMSM controller was carried out. The sensitivity analysis of gains of the PMSM controller with the presence of degradation in rotational inertia can find search intervals that help to minimize the tuning time of the DA. The dragonfly algorithms confirm the feasibility and effectiveness of the parameter optimization for the optimal linear quadratic and PID controller. The

Conclusions
This paper presents an application of one metaheuristic (dragonfly algorithm) for tuning a PID, and optimal linear quadratic controllers for a PMSM with the presence of rotor fissure. The optimization procedure was employed considering simulation with the non-linear system model. This strategy contains the speed and current control loops. In order to define the search range for DA, a sensitivity analysis of gains of the PMSM controller was carried out. The sensitivity analysis of gains of the PMSM controller with the presence of degradation in rotational inertia can find search intervals that help to minimize the tuning time of the DA. The dragonfly algorithms confirm the feasibility and effectiveness of the parameter optimization for the optimal linear quadratic and PID controller. The results of the simulation show that the controller has a good performance and fast tracking speed under presence of fissure in the rotor shaft and external perturbation. The controller aims to ensure speed tracking tasks while significantly reducing the speed overshoot. Note that there was no need to retune the controllers for different kinds of operations; therefore, the designed controller is convenient to be realized. This shows the advantages of advanced controller tuning. Since the controller exhibits an excellent performance, it is ideal for application in process industries.
The PMSM model with the rotational inertia degradation coupling in the shaft allows for applications in the field of preventive maintenance, failure control or determination of failure intervals of rotating machines, to name a few fields of application. It is observed that, under the given working conditions, the dynamic model of the PMSM with the presence of degradation in the rotational inertia of the rotor axis is adequately tuned to the requested references by means of the DA used for any analysis time. The proposed PMSM model with the presence of rotational inertia degradation is considered congruent with the expected degradation behavior typical of rotating machines: the size of the internal fissure continues to grow in an exponential form and gradual manner until the fracture finally occurs.

Conflicts of Interest:
The authors declare that there is no conflict of interest.

Nomenclature
The variables involved in the modeling of the PMSM with the presence of degradation in rotational inertia are defined in Table 1 Cost localization function of optimal earnings k j Optimal earnings vector ω rj Value obtained from the jth search for speed gain