Comparative Study of Two Nonlinear Control Strategies of Induction Motors considering Heating and Magnetic Saturation

Feedback linearization technique (FLT) linearizes the model of induction machine (IM). Tis actuator sufers from the variation of its inductances due to saturation and resistances due to joule and skin efects. Sliding-mode control (SMC) is widely recognized as a robust technique against parametric variations of IM. Tis control strategy has the advantage of being simple to implement and requires only a simple fux observer. Tis explains the use of FLTand SMC in this work to control an IM while taking into account the magnetic saturation and heating of the IM. Simulation results, conducted in a MATLAB/Simulink environment, demonstrate the relevance and efciency of the proposed control schemes.


Introduction
Induction machine (IM), as the name dictates, is a strongly inductive machine that allows the transformation of electrical energy into mechanical energy and vice versa. By nature, an IM is a strongly nonlinear system due to its inductive properties. Field-orientation techniques (FOC) allow simplifying the machine by making the fux and the torque dependent only on the direct and quadrature components of the stator currents [1]. However, FOC is sensitive to parameters variations [2]. Rotor resistance variation is the main problem that causes the loss of the decoupling between fux and torque [3]. Several solutions have been proposed in the literature to estimate the electrical parameters of IM. Rotor time constant (L r /R r ) estimator has been proposed for an IM without saturation in [2], and identifcation of all IM parameters was presented in [4][5][6][7]. Magnetic saturation can also afect the inductances of IM, causing their variation. Several works have studied the efect of magnetic saturation in IM [8][9][10][11] to understand the behavior of the IM in the saturation area. Tese studies show that the best performances can be achieved when saturation is taken into account.
To extract maximum torque from IM, adaptive linear and nonlinear control laws have been proposed. To take into account the magnetic saturation of IM, the synthesis of nonlinear control laws is necessary. Several control laws are proposed and experimentally verifed in the literature. For example, a PID controller is used in [2,3,6] with adaptation of some parameters. Advanced control laws were also applied to IM such as fuzzy logic [12,13], backstepping technique [14][15][16][17], artifcial neural networks (ANNs) [18][19][20][21], sliding mode (SM) [22][23][24][25][26], feedback linearization technique (FLT) [27,28], and more. For sensorless control of IM without saturation phenomena, the authors in [12] suggest a model reference adaptive system (MRAS) speed estimator that uses type-1 and type-2 fuzzy logic controllers, with the type-2 fuzzy logic controller being proposed to handle higher degrees of uncertainty and improve performance under various operating conditions. In [13], the authors suggest a fuzzy adaptive PI-sliding-mode controller for controlling the speed of IM. However, it assumes that the parameters are constant and ignores the efects of magnetic saturation. In [14], the speed control of IM using a backstepping design is proposed and compared to conventional PID control, but without considering the saturation efect. An adaptive backstepping-based nonlinear controller incorporating the iron loss is developed under the parameter uncertainties [15], and with a recursive online estimation of the rotor time constant and load torque [16], nonetheless, the variation of inductances is ignored. In [17], a maximum torque per ampere (MTPA) method based on the backstepping controller is presented for IM drives, taking into account the efects of both iron loss and saturation; however, the saturation of the main fux path was defned as a function of magnetizing current without physical meaning. In [18][19][20][21], the authors introduce the artifcial neural networks (ANNs) for the control of IM using FOC principles, but they require data for training and consume a lot of memory resources. Te authors in [22] present a model-based lossminimization approach, which is combined with a backstepping direct torque control of the IM and describes a sliding-mode rotor fux observer to calculate the rotor speed, rotor time constant, and rotor fux space vector simultaneously. Te authors in [23] use the proportional integral type SM switching surfaces to control the stator fux and torque of IM. Te motor iron losses are modeled by a shunt rotor speed-dependent core resistance, but the saturation efect is not discussed. A sliding-mode controller is proposed to compensate for the uncertainties including parameter variations [24], but this strategy is applied to a linear induction motor (LIM) drive system without saturation efects. A suggested SMC-based model predictive control (MPC) scheme combines the advantages of SMC and MPC to develop a robust and fexible system that enhances tracking performance and torque ripple reductions [25] but the robustness concerning the variation of certain parameters is not discussed. Authors in [26] present the use of a higher-order sliding-mode scheme based on a supertwisting algorithm for sensorless control of IM, neglecting magnetic saturations.
When applying such nonlinear control laws, some assumptions are made which allow for obtaining interesting performances. Recently, in [27], the authors have developed a control law based on the feedback linearization technique (FLT) for saturated and unsaturated IM described in [10]. Te shape of the magnetizing curve of IM has suggested its representation as the sum of an exponential function with a linear one [27]. Tis nonlinear function has three parameters to be determined. Tese parameters have a physical meaning [27,29] which justifes the choice of this type of interpolation. However, the performance of the FLT controller depends strongly on the variation of the rotor and stator resistance (heating). In addition, estimation of the load torque is necessary to calculate the control voltages. Te robustness problem motivates us to propose and synthesize a sliding-mode control (SMC) law in this paper. Contrary to FLT, the load torque in SMC is not necessary to estimate. Te robustness of these two control laws against rotor resistance variation is discussed and compared.
Tis paper is organized as follows: Section 2 presents the mathematical model of the IM taking magnetic saturation into account. In Section 3, a new rotor fux model is developed that considers the machine's saturation, and a new inductance is defned. Section 4 describes the design of a feedback linearization technique (FLT) for controlling both the speed and rotor fux. In Section 5, a sliding-mode control (SMC) is proposed for the same purpose. Simulation results for both control strategies are presented in Section 6, with constant and variable rotor resistance. Section 7 provides the simulation confguration and nominal characteristics of the IM used in this study. Finally, Section 7 ofers conclusions based on the fndings.
For the symbols used in this paper, the reader can refer to Table 1.

Modeling of the Induction Machine considering the Magnetic Saturation
A dynamic model of IM taking into account the magnetic saturation of the iron core was introduced in [10] for direct torque control (DTC). Te state space form of this model is described by [27] for the synthesis of a control law which combines the principles of rotor fux orientation and feedback linearization. Tis linearization is derived from [30] for induction motors (IM) and from [28] for linear induction motors (LIM). Nonlinear observers of the stator and magnetizing currents are studied and experimentally verifed by [29] in the stationary reference frame. Based on the state model used in these papers, the robustness of the feedback linearization (FLT) control on the speed and torque performances, with respect to the variation of the rotor resistance, is compared to sliding-mode control (SMC). Starting from the IM state model given by [27] in a reference frame (x, y) (general reference frame) rotating at speed ω g , this model is given by Te magnetizing current vector magnitude |i mr | � �������� � i 2 mrx + i 2 mry will be oriented along the x-axis of the rotating reference frame. Te rotating speed ω g is equal to the case of unsaturated machine [27].
Te magnitude of the magnetizing current vector will be equal to |i mr | � i mrx . It is useful to note that the speed of rotation ω g is equal to the case of the unsaturated machine. Tis means that the saturation efects only change the magnitude or vector of the fux rotor, while its angle remains unchanged [27]. By substituting (6) in (1)-(4) and setting i mry (0) � 0, the state model (1)-(5) of IM, taking into account the magnetic saturation, in the rotor fux-oriented reference frame (x, y) can be written as follows: We add to this model the equation of the electromagnetic torque, expressed in the rotating reference frame (x, y): Te coefcients appearing in the IM model are defned by For simplifcation, these parameters are grouped in a vector ″ par ″ defned by Note: Te parameters with " * " vary as a function of the magnetizing current magnitude |i mr | and indirectly as a function of time.

Rotor Flux Model Taking into Account the Saturation of the Machine
Several works have addressed the IM saturation efect in the literature. In [31], nonlinear functions are proposed to model the magnetizing inductance and the leakage inductance of the rotor. From experimental data, the authors in [17] used an interpolation function consisting of three (3) parameters to model the magnetic fux. Tis function is the basis for calculating two inductances: dynamic and static inductances, which will be incorporated into the parameters of IM. In [32], magnetizing inductance and stator and rotor leakage inductances are introduced, but it requires twelve (12) parameters to be determined which makes these inductances very difcult to exploit. Te work published in [27,29,33] has allowed experimentally verifying the rotor fux profle as a function of the rotor magnetizing current (see Figure 1).
As can be seen in Figure 1, the rotor fux can be described as a sum of an exponential function and a linear function [27,29] as follows: Te coefcients α, β, and c were obtained by means of an optimal optimization (see [33]).
Te magnetizing inductance L m is defned by [10] L m � Ψ r i mr .
A new inductance has been introduced in [10], called modifed inductance, and is defned by From (11), the expressions of L m and L are obtained analytically as follows:

Feedback Linearization Technique (FLT)
From equations (7) and (8), the two linearizing and stabilizing control inputs u sx and u sy of the system are given by where v x and v y are additional control inputs that will be designed later.
As can be seen in model (7)-(10), the dynamics of torque and fux is not decoupled in each working condition. Indeed, the decoupling between torque and fux only works if the machine is operating at constant fux; otherwise, the speed dynamics has a nonlinearity with respect to the inputs, and another important coupling between torque and magnetizing current comes from the dependence of the model coefcients on the current i mr due to saturation efects. For this reason, two variables will be introduced in the IM model. Te input-output feedback law of the model will be applied to the new model [27].
Let us defne a new state variable "a" called angular acceleration, instead of i sy as follows: Journal of Electrical and Computer Engineering a � −a 33 ω r + f 3 i mrx i sy − f 4 T L . (20) Assuming that the change in load torque is sufciently slow, i.e., _ T L � 0 [27,34], then the derivative of a can be written as follows: By analyzing (21), it is easy to see that the feedback term that linearizes the speed dynamics can be defned as follows: Finally, we defne a new state variable v i mrx as follows: Ten, we derive v i mrx with respect to time, which gives Looking at (24), the feedback term that linearizes the dynamics of the magnetizing current can be defned as follows: Replacing (22) in (21) and (25) in (24), model (7)-(10) can fnally be written as follows: Model (26)- (29) are the linearized model of the induction motor, with a decoupled dynamic between the speed and the magnetizing current, taking into account the magnetic saturation [27].
In order that i mrx and ω r follow their respective references i ref mr and ω ref r , we introduce the following two errors: Te input signals v x ′ and v y ′ are then designed as follows: Coefcients k 1m , k 2m , k 1ω et k 2ω are parameters of the control. An appropriate choice of these parameters allows a good tracking of desired trajectories for speed and rotor fux. Now, if we consider the closed-loop stability of this strategy and replace (27) and (29) by the expressions of the control commands v x ′ and v y ′ , we obtain the following homogeneous diferential equations: If equations (34) and (35) are Hurwitz polynomials, the system is stable.
Te canonical form of a second-order system is as follows: where s is the Laplace variable, ξ is the damping factor, and ω n is the natural pulsation.
Knowing the profle of the desired references, we can determine the parameters of the state feedback using the following expressions: Notes: (i) We note the dependence of the machine parameters on the magnetizing current i mrx and therefore these parameters are time varying.
(ii) Te magnetizing current observer is designed from equation (10) by replacing i mrx by its estimate i mrx Figure 2 shows the complete block diagram of IM and feedback linearization (FLT) controller.

Sliding Mode Based on Exact Linearization (SMC)
In practice, all the parameters of IM vary, due to saturation (variation of inductances) and heating (variation of rotor and stator resistances). Variations in rotor and stator resistances will be considered as disturbances in each state variable. Models Terms d sx , d sy , and d mr are added to compensate uncertainties caused by rotor and stator resistances variation. Te term d L is used to model the disturbances coming from the load and friction torque: Synthesis of the sliding-mode control law is based on input-output linearization. To obtain the relation between ω r and u sy , we diferentiate ω r until the command appears: where Journal of Electrical and Computer Engineering 7 From (46), we deduce the control command u sy as follows: We defne the speed error: e 1 � ω r − ω ref r , and the sliding surface is selected as follows: With λ 1 > 0, v sy is the intermediate command that will be chosen and defned later.
We defne the Lyapunov function as follows: Te derivative with respect to time is given by Equation (52) becomes Te condition for that _  To fnd η 1 , we use the following reaching law: where τ 1 � 2π/ω c1 is the reaching time and ω c1 is chosen to be equal to bandwidth, and then D ωr ≥ |f 3 i mrx d sy + _ d L | � f 3 |i mrx d sy | because _ d L � 0 and f 3 is always positive.
Te same procedure will be applied for the synthesis of the current control law for i mrx .
To obtain the relationship between i mrx and u sx we diferentiate i mrx until the command u sx : with From (57), we deduce the following control command u sy : Defning the error e 2 � i mrx − i ref mr and selecting the new sliding surface as follows: with λ 2 > 0 v sx is the intermediate command that will be chosen later. Let us defne the Lyapunov function as follows: Te derivative with respect to time gives (63) Equation (63) becomes To fnd η 2 , we use the reaching law: Te condition for that _ V 2 ≤ − η 2 |S 2 | is to choose where τ 2 � 2π/ω c2 is the reaching time, ω c2 � 5.ω c1 , and D r ≥ |f 3 i mrx d sy + _ d L | � f 3 |i mrx d sy |. Notes: (i) Terms D ωr and D r are chosen large enough; in this work, we use the simulation tool to estimate their values.

Journal of Electrical and Computer Engineering
(ii) To reduce the chattering problem, we replace the function sgn(.) by tan h(.).
Te block diagram of the sliding-mode control (SMC) is given in Figure 3.

Simulation Result
Te simulations for this study were conducted using the MATLAB/Simulink environment, employing a sample time of T sc � 50 μs. Tis choice of sample time was made to account for the development of microcontrollers, such as DSP and FPGA, and to improve the accuracy of simulation results. It should be noted, however, that the physical system being studied is not simulated, and the digital computer only executes the control algorithm, requiring fewer calculations and allowing for lower sampling steps. Te simulations employed the SPWM (sinusoidal PWM) technique based on the concept of "efective time" [35] and utilized the ode8 (Dormand-Prince) solver with a fxed step T s � 5 μs. Te switching frequency was set at 2 kHz. Table 2 gives the nominal characteristics of the IM used in this paper.
At time t � 0.5 s, nominal load torque T L � 14 Nm is applied. Figure 4 shows the simulation result obtained by the FLT controller.
We notice that the FLT controller allows us to follow the rotor speed and its reference (Figure 4(a)). When the load torque T L is applied (Figure 4(b)), there is a slight decrease in the speed controlled by FLT (Figure 4(d)). We can improve the accuracy of the FLT controller by increasing adjustment parameters k 1ω and k 2ω . Note that FLT controller uses linear control techniques which allows to control the desired dynamics by specifying the parameters ξ ωn and ω ωn .
Electromagnetic torque developed by IM (Figure 4(b)) compensates the load torque applied at t 1 � 0.5 s. According to (11), the torque is a function of the magnetizing current i mrx and the current i sy . So, if the magnetizing current i mrx is maintained constant and if the torque response is dynamically faster (by using a larger natural pulsation ω cm ), the control of the electromagnetic torque will be done only by the quadrature component i sy (see Figure 5). Figure 4(c) shows the magnetizing current (i mr ) and its reference in the FLT control case. When the IM parameters are constant, especially the rotor resistance (R r ), the magnetizing current error obtained by the FLT controller is about 10 − 3 A (Figure 4(e)). During transient operation, a decrease in magnetizing current is observed since the magnetizing inductance has also decreased in the same manner (Figure 4(f )). Figure 5 gives the stator current i sx and i sy , in rotating reference frame (x, y) and the three-phase currents absorbed by the motor (i sa , i sb , andi sc ).
At steady state, i sx � i mrx � 3.5 A in the FLT controller case. Tis means that current i sx allows controlling the magnetizing current i mr while the quadrature current i sy allows controlling the electromagnetic torque (it is the principle of feld-oriented control (FOC)). Te profle of currents i sx and i sy demonstrate these results ( Figure 5). Te corresponding stator's current waveform is shown in Figures 5(b) and 5(c).
Te SMC controller can address the challenges of parameter variations in the feld-oriented control (FOC) system. In addition, one advantage of SMC is its relative simplicity of implementation. Figure 6 shows the simulation results when the rotor resistance is assumed to be constant and equal to its nominal value. Figure 6(a) shows that the SMC controllers give quasiequivalent results as in the FLT case, except in the transient state. In steady state, Figure 6(d) clearly shows that SMC is more accurate than FLT controllers in terms of rotor speed.
Te electromagnetic torque (Figure 6(b)) and magnetizing current (Figure 6(c)) are the same as those developed by the FLT controller ( Figure 6(b)). However, the magnetizing current error (Figure 6(e)) obtained by SMC is lower than that obtained by FLT (see Figure 4(e)).
Te direct i sx and quadrature i sy currents have the same values and profles as those of FLT (Figure 6(g)), and the waveform of the three-phase stator currents shows the good quality of stator currents.

Variable Rotor Resistance.
In practice, the stator and rotor resistance of IM varies according to the temperature (heat). Physically, the resistance varies at slow dynamics (large time constant compared to the rotor time constant). Now, assume that the rotor resistance varies according to the profle given in Figure 7. Te choice of this profle (Figure 7) is just to test the dynamics and the robustness of the FLT controller. Regarding inductances, it is useful to note that all the inductances (L m , L, L s , L r ) of the machine are variable.
Te reference speed and current are kept at ω ref r � 100 rad/s and i ref mr � 3.5 A, respectively. Similarly, at the time t � 0.5 s, the nominal load torque of 14 Nm is applied. Figure 8 shows the simulation results when the rotor resistance varies according to the profle in Figure 7.
In the frst step, the rotor resistance value has been increased to R r � R rn , before the application of the load torque. At no load, the FLT controller allows following accurately the rotor speed. At t 1 � 0.4 s, when the rotor resistance increases to 2R rn , the rotor speed obtained by FLT remains equal to 100 rad/s. At t 2 � 0.5 s, the load torque is applied, and rotor resistor is maintained equal to 2R rn , and the speed controlled by FLT decreases to 94 rad/s. At t 3 � 1s, the rotor resistance is decreased to 0.2 R rn , the speed increases to 104.5 rad/s. Furthermore, from t 4 � 1.5 s onwards, as the resistance is exponentially increased to 2.R rn , it is observed that the speed decreases exponentially in the case of FLT. Tis implies that a rapid increase in rotor resistance leads to a rapid decrease in rotor speed.
Te stator currents obtained by FLT controller are given in Figure 8(d). It can be seen that i sx current controls the magnetizing current, while i sy current controls the electromagnetic torque. Te variation of the rotor resistance afects i sy current. So, the three-phase stator currents absorbed by the machine are also afected (Figure 8(e)). In this paper, the rotor time constant T r is variable, unlike the model with constant parameters. Figure 8(f ) shows the variation of the magnetizing inductance L m which modifes the rotor inductance (L r � L rσ + L m ), and therefore, it allows for modifcation of the time constant T r .
Tese results show that the control FLT is not robust with respect to the variation of the IM machine parameters. Te nonrobustness is due, frst, to the orientation condition (6), which assumes that the rotor resistance is constant. In   reality, the rotor resistance is no longer constant; it varies according to temperature (heat). Second, because of this error in orientation, the model (7)-(10) is no longer valid, because the quadrature component of the magnetizing current will be nonzero and must be taken into account. Finally, the load torque is part of the FL control law (see equations (5) and (19)), so the accuracy of the torque estimation afects the FLT controller.    the (a, b, c) frame, and (c) zooming in on the three-phase currents. Now, the rotor resistance varied as shown in Figure 7. Figure 9 shows the simulation results when the machine is controlled by the SMC technique.
As shown in Figure 9(a), the rotor speed accurately follows its reference even when the rotor resistance varies and in the presence of the load torque. In the case of SMC control, the electromagnetic torque is insensitive to these variations (Figure 9(b)).
Te magnetizing current error is very low (on the order of 5.10 − 3 ), indicating the superiority of the SMC control compared to FLT (Figure 9(c)).
It can be seen that i sx current controls the magnetizing current while i sy current controls the electromagnetic torque (Figure 9(d)). Unlike the FLT controller, the variation of the rotor resistance does not afect the current i sy . Tus, the three-phase stator currents absorbed by the machine are not  afected either. Figure 9(f ) shows the variation of the magnetizing and modifed inductances (L m , L) which modifes the rotor inductance (L r � L rσ + L m ), and therefore it allows the modifcation of the time constant T r . Figure 10 illustrates the variation of speed, magnetizing current, and torque under an exponential variation of the rotor resistance.
When the resistance increases exponentially (Figure 10(a)), it is observed that the speed obtained by SMC accurately follows its reference with acceptable precision. However, in the FLT case, the speed tends to become negative with each speed change. As the resistance continues to increase, the precision of the speed improves in the FLT case ( Figure 10(b)). Te electromagnetic torque developed by the machine (Figure 10(c)) exhibits peaks during speed changes (FLT case). Tis phenomenon can be attributed to the fact that the speed, which tends to reverse during each speed change, adds an additional torque to the load torque. In contrast, the SMC control is unafected by these speed changes. Te stator currents show little diference, without any signifcant variations (Figures 10(d) and 10(e)).
As mentioned before, all inductances vary according to the magnetizing current. Figure 10(f ) illustrates the variation of the magnetizing current reference, consequently resulting in variable magnetizing inductance and the rotor time constant of the machine.
It should be noted that in the SMC control law, the rotor resistance, stator resistance, and load torque are considered as disturbances. On the other hand, the machine inductances are dependent on the magnetizing inductance, which is modeled by (16). Tese inductances vary only with respect to the magnetizing current. We can make a comparison between FL and SMC controller at several levels: (i) Application of linear control laws: in the case of FL is easier than that of SMC (ii) Number of parameters to set: this number is the same in both controllers. Te total is equal to four: two for the speed and two for the magnetizing current. (iii) Robustness: the control law by SMC is more robust to the variation of IM parameters and to the estimation of some state variables. Te FL controller is sensitive to these variations, and its accuracy depends on the accuracy of the fux and resistive torque observers. (iv) Te SMC controller is hindered by the requirement of signifcant tuning eforts, which may be perceived as a disadvantage due to their time-consuming nature. In contrast, the FL controller does not require tuning eforts, which can be considered as an advantage.

Conclusion
Magnetic saturation introduces variability in all parameters of the induction motor (IM), thereby increasing the nonlinearity of the machine model. Te variation of rotor resistance plays a crucial role in controlling and manipulating the rotor speed. Te proposed control schemes, FLT and SMC, prove to be efcient in controlling an IM while accounting for the efects of magnetic saturation and motor heating. Te feedback linearization technique (FLT) provides a linear model that allows the application of linear systems theory. However, this control approach requires a higher level of accuracy in the IM model and assumes that the parameters remain constant or are precisely observable. On the other hand, sliding-mode control (SMC) is a robust technique that can efectively handle parametric variations in the IM. It ofers the advantage of being simple to implement and only requiring a basic fux observer. However, the estimation of the maximum values of the disturbances occurring on the IM remains a challenge and increases the complexity of this control law.

Data Availability
No underlying data were collected or produced in this study.