Enhanced Control Technique for Induction Motor Drives in Electric Vehicles: A Fractional-Order Sliding Mode Approach with DTC-SVM

Abstract

The present paper proposes the use of fractional derivatives in the definition of sliding function, giving a new mode control applied to induction motor drives in electric vehicle (EV) applications. The proposed Fractional-Order Sliding Mode Direct Torque Control-Space Vector Modulation (FOSM-DTC-SVM) strategy aims to address the limitations of conventional control techniques and mitigate torque and flux ripples in induction motor systems. The paper first introduces the motivation for using fractional-order control methods to handle the nonlinear and fractional characteristics inherent in induction motor systems. The core describes the proposed FOSM-DTC-SVM control strategy, which leverages a fractional sliding function and the associated Lyapunov stability analysis. The efficiency of the proposed strategy is validated via three scenarios. (i) The first scenario, where the acceleration of the desired speed is defined by pulses, leading to Dirac impulses in its second derivative, demonstrates the advantage of the proposed control approach in tracking the desired speed while minimizing flux ripples and generating pulses in the rotor pulsation. (ii) The second scenario demonstrates the effectiveness of filtering the desired speed to eliminate Dirac impulses, resulting in smoother rotor pulsation variations and a slightly slower speed response while maintaining similar flux ripples and stator current characteristics. (iii) The third scenario consists of eliminating the fractional derivatives of the pulses existing in the expression of the control, leading to the elimination of Dirac impulses. These results demonstrate the potential of the FOSM-DTC-SVM to revolutionize the performance and efficiency of EVs. By incorporating fractional control in the control scheme for PV-powered EVs, the paper showcases a promising avenue for sustainable transportation.

1. Introduction

An overwhelming number of systems can be characterized by nonlinear partial differential equations. In addition, these systems exhibit fractional characteristics, which give rise to the idea of fractional-order or noninteger-order systems [1]. It stands to reason that fractional-order or non-integer controllers would be better suited than integer-order controllers in these instances. In fact, fractional models, which capture the dynamics of non-integer order systems, have found applications across diverse domains, ranging from physics and engineering to biology and finance. This prevalence of fractional models in the natural and artificial worlds has sparked curiosity and exploration of fractional control strategies in the context of electric vehicle (EV) motors [1,2]. Induction motors receive a lot of attention as potential EV applications based on their (i) low cost, (ii) low maintenance needs, (iii) robustness, and (iv) ruggedness. Several approaches are available to control the torque of an induction machine. DTC and FOC are the two most widely used methods for regulating induction motor torque [3].

Conventional control techniques often struggle to manage the torque and flux ripples effectively, leading to reduced efficiency, increased wear and tear on motor components, and undesirable noise and vibrations. These limitations hinder the overall performance and driving experience of EVs. In order to lessen flux and torque fluctuations, ref. [4] built a chip based on fuzzy DTC. An adaptive SMC was employed instead of a PID controller to improve anti-interference capability [5]. With fuzzy logic, we can fine-tune the torque hysteresis controller’s bandwidth. To enhance the performance of the SVM-DTC, high-order sliding mode controllers were employed [6]. To eliminate parametric uncertainties, ref. [7] proposed a speed and rotor flux control method that combines sliding mode and nonlinear control techniques. To accomplish directly modulated torque, the authors of [8] suggested using space vector modulation to account for flux ripples and torque. In [9], a fractional-order SM control method has been presented for modifying a DC–DC converter’s output voltage. A dependable nonlinear control methodology for handling modeling uncertainties and disturbances in nonlinear control methods is the sliding mode control approach [10]. It makes use of control over variable structure. Compared to traditional DTC schemes, the direct torque control approach using the space vector modulation (DTC-SVM) technique provides better responses (transient and steady-state), presenting good dynamic torque response, fewer ripples, and superior speed control [11,12,13].

Furthermore, fractional characteristics are frequently seen in electric car induction machine systems, which results in differential equations of non-integer order. These rationales have led scholars to propose the concept of fractional-order derivatives or non-integer-order systems. Fractional-order controllers might be more suitable in these situations than integer-order controllers, notwithstanding the rapid advancement in research.

This paper proposes a novel Fractional-Order Sliding Mode Direct Torque Control-Space Vector Modulation (FOSM-DTC-SVM) strategy to address the previously mentioned limitations. The proposed approach aims to achieve smoother torque and flux control, improve transient response, and enhance overall system performance, paving the way for more efficient and reliable EV propulsion.

More recently, [2] has applied fractional-order derivatives using sliding mode control (FO-SMC) to induction motors under DTC-SVM. Their work proposed fractional-order derivatives with sliding surfaces. This yielded a FO-SMC for a class of nonlinear systems. However:

• The stability analysis is not studied.
• Authors approximate the differential of a signal u by ${\displaystyle \frac{\Delta u}{\Delta t}}$. They compute the fraction derivative $D_t^{\sigma }u$ by ${\left[{\displaystyle \frac{\Delta u}{\Delta t}}\right]}^{\sigma }$, which is not suitable; moreover, signal u can be negative.

References [14,15] have applied the PI fractional-order derivatives to a fractional-order linear system dedicated to a linearized model of an induction machine.

Building upon the foundational work in [2,14,15], our proposed FO-SM strategy for induction motors under DTC-SVM introduces the use of fractional derivatives on the definition of the sliding function. This controller’s main goals are to achieve a smoother and faster transient response and more effective responsiveness operation solutions for the induction motor system. Moreover, the stability has been verified through Lyapunov functions.

Thus, this work proposes a fractional order approach based on sliding mode control applied to SVM-based DTC of induction motors dedicated to EV applications. Consequently, the proposed Fractional Order approach based on Sliding Mode Direct Torque Control-Space Vector Modulation (FOSM-DTC-SVM) presents an intriguing approach to revolutionizing the performance and efficiency of EVs. By leveraging the control strategy’s inherent fractional order nature, the proposed FOSM-DTC-SVM algorithm not only aims to address the limitations of conventional control strategies and mitigate torque and flux ripples, but it also stands out from the existing literature by providing a more straightforward and simplified implementation algorithm. Incorporating fractional control in the control scheme for PV-powered EVs showcases a promising avenue for sustainable transportation.

Below is the paper’s organizational structure.

Section 1 introduces the topic of fractional-order calculus based on sliding mode control (FO-SMC) dedicated to induction motor drives used in electric vehicle propulsion. It discusses the motivation for using fractional-order control techniques to address induction motor systems’ nonlinear and fractional characteristics. The section also provides an overview of the state-of-the-art induction motor control techniques based on DTC (direct torque control) and FOC (field-oriented control).

Section 2 presents the preliminaries, including the dynamic model of the induction machine-based DTC-SVM control.

The paper’s core is then presented in the subsequent sections, where the proposed Fractional-Order SMC-DTC-SVM (FO-SMC-DTC-SVM) control strategy is detailed. This includes the design of the two-term fractional sliding function and the associated Lyapunov stability analysis.

The efficiency of the proposed FO-SMC-DTC-SVM approach is then validated through simulation results in the later sections of the paper.

Finally, the paper concludes with a summary of the key contributions and findings.

2. Preliminary Steps

2.1. Modeling Induction Machines with DTC-SVM

The dynamic model of induction motors can be described using space variables as the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\displaystyle \frac{d}{dt}}{\varphi }_{\alpha s}& =& -R_s{i}_{\alpha s}+v_{\alpha s}\hfill \\ {\displaystyle \frac{d}{dt}}{\varphi }_{\beta s}& =& -R_s{i}_{\beta s}+v_{\beta s}\hfill \\ {\displaystyle \frac{d}{dt}}{\varphi }_{\alpha r}& =& -N_p{\Omega }_m{\varphi }_{\beta r}-R_r{i}_{\alpha r}\hfill \\ {\displaystyle \frac{d}{dt}}{\varphi }_{\beta r}& =& N_p{\Omega }_m{\varphi }_{\alpha r}-R_r{i}_{\beta r}\hfill \end{array}\right.\end{array}$$

(1)

where the subscripts r and s denote the rotor and the stator, respectively, and the subscripts $\alpha$ and $\beta$ refer to the ($\alpha , \beta$) frame. The variables $\varphi$, i, and v represent flux, current, and voltage, respectively, while $R_r$ and $R_s$ are the rotor and stator resistances, and ${\Omega }_m$ is the machine speed, which is related to the slip speed (${\omega }_s-{\omega }_r=N_p{\Omega }_m$) and the pole pair number ($N_p$).

The relationships between currents and fluxes are:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{\varphi }_{\alpha s}& =& {\mathcal{l}}_s{i}_{\alpha s}+{\mathcal{l}}_{rs}{i}_{\alpha r}\\ {\varphi }_{\alpha r}& =& {\mathcal{l}}_{sr}{i}_{\alpha s}+{\mathcal{l}}_r{i}_{\alpha r}\\ {\varphi }_{\beta r}& =& {\mathcal{l}}_{sr}{i}_{\beta s}+{\mathcal{l}}_r{i}_{\beta r}\\ {\varphi }_{\beta s}& =& {\mathcal{l}}_s{i}_{\beta s}+{\mathcal{l}}_{rs}{i}_{\beta r}\end{array}\right.\end{array}$$

(2)

The machine’s mechanical dynamics can be represented by the equation:

$$\begin{array}{ccc}\hfill J{\displaystyle \frac{d}{dt}}{\Omega }_m& =& T_e-T_L\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill T_e& =& N_p({\varphi }_{\alpha s}{i}_{\beta s}-{\varphi }_{\beta s}{i}_{\alpha s})\hfill \end{array}$$

(4)

The block diagram of DTC-SVM maintains the main benefits of the traditional DTC method, including eliminating coordinate transformations and achieving rapid torque and flux control (Figure 1). A Space Vector Modulation (SVM) block generates the pulses for the inverter, guaranteeing a consistent commutation frequency [16].

A sophisticated control system that manages an induction motor is a key component in many electric vehicles. The system utilizes a Fractional-Order Sliding Mode (FO-SM) control strategy, a highly precise method for ensuring smooth and efficient motor operation. A speed sensor acts as the system’s “speedometer”, providing real-time feedback on the motor’s rotation. The speed reference sets the desired speed, much like cruise control in a car. The speed controller, employing FO-SM, compares the actual speed to the desired speed and makes adjustments, acting like the driver’s foot on the gas pedal to maintain the set speed. The torque controller, also using FO-SM, ensures the motor has the necessary power to achieve that speed, akin to the engine providing the force to move the vehicle. A PV system ensures that the DC bus $V_{d}c$ provides the necessary power for the inverter. This system is designed to be highly responsive and accurate, ensuring smooth and efficient motor operation, which is a critical factor in maximizing battery life and performance in electric vehicles.

2.2. Tractive Systems

The force that propels the car forward and is transmitted to the ground via the drive wheels is known as the tractive effort. Consider a vehicle traveling at a speed of $V_v$ and mass $M_v$. The following forces must be resisted by the tractive effort or force that propels the vehicle forward [16]:

-

Rolling resistance, the force opposing a vehicle’s motion, is calculated as $F_{rrol}=\zeta M_{v}g$. This equation shows that rolling resistance is directly proportional to the vehicle’s mass and the rolling resistance coefficient, typically around 0.005 for electric vehicle tires. Minimizing rolling resistance is crucial for maximizing energy efficiency in electric vehicles.

-

Aerodynamic drag, the force opposing a vehicle’s motion through the air, is calculated as $F_{ad}={\displaystyle \frac{1}{2}}\rho A{C}_d{V}_v^2$. This equation shows that drag is directly proportional to air density, frontal area, drag coefficient, and the square of the vehicle’s speed. Minimizing drag is crucial for fuel efficiency, especially at higher speeds.

-

The force needed to climb a hill, $F_{hc}$, is calculated as $F_{hc}=M_{v}gsin\gamma$, where $\gamma$ is the slope angle. This means steeper hills and heavier vehicles require more force to climb.

We focus on a car moving over a straight, level route. When a vehicle travels on a flat surface, the hill-climbing force is zero. In this case, the tractive effort needed to maintain motion is simply the sum of rolling resistance and aerodynamic drag: $F_{te}=F_{rrol}+F_{ad}$. The load torque can be expressed as: $T_L=R_{g}r\left(F_{rrol}+F_{ad}\right)=$, with $R_g={\displaystyle \frac{1}{G}}$, and with G the gear ratio.

After development, the load torque is:

$$\begin{array}{ccc}{T}_L\hfill & =\hfill & k_0+k_2{V}_v^2\hfill \end{array}$$

(5)

with $k_0=R_{g}r\zeta M_{v}g$ and $k_2={\displaystyle \frac{1}{2}}{R}_{g}r\rho A{C}_d{V}_v^2$.

The following equation describes the link between the motor speed and the vehicle speed:

$$V_v=R_{g}r{\Omega }_m.$$

(6)

With reference to Equations (5) and (6), simulation tests were conducted under both healthy and defective situations. The motor drove a load torque, which may be stated as follows:

$$T_L=k_0+k_2{\left(R_{g}r\right)}^2{\Omega }_m^2=f_0+f_2{\Omega }_m^2.$$

(7)

2.3. Background on Fractional Calculus

Several well-known mathematical definitions for fractional-order derivatives have been cited in many references, such as (i) the Riemann–Liouville, (ii) Grünwald–Letnikov, and (iii) Caputo approaches. They are commonly discussed in the literature [17,18,19]. Additionally, the Oustaloup and refined Oustaloup methods are widely used for approximating fractional-order derivatives in the frequency domain [20,21].

In this work, we take into consideration the following Caputo definitions of fractional differentiation and integration of a function $g\left(t\right)$ for order $\sigma \in ]0 ,1[$:

$$\begin{array}{ccc}\hfill D_t^{-\sigma }g\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_0^t{(t-v)}^{\sigma -1}g\left(v\right)dv\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill D_t^{\sigma }g\left(t\right)& =& D_t^1\left(D_t^{-(1-\sigma )}g\left(t\right)\right)\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& =& {\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\int }_0^t{(t-v)}^{-\sigma }g\left(v\right)dv\right]={\displaystyle \frac{1}{\Gamma (1-\sigma )}}{\int }_0^t{(t-v)}^{-\sigma }{\displaystyle \frac{dg}{dv}}dv\hfill \end{array}$$

(10)

with the $\Gamma$ function:

$$\Gamma \left(\eta \right)={\int }_0^{\infty }{t}^{\eta -1}{e}^{-t}dt.$$

(11)

Consider a fractional-order Caputo system [17]:

$$D_t^{\sigma }X\left(t\right)=F(t,X\left(t\right))$$

(12)

where $F:[0,\infty [\times \mathcal{V}\to {\mathbb{R}}^n$ is piecewise continuous in t, $\mathcal{V}\subset {\mathbb{R}}^n$ is a domain containing the equilibrium point $X=0$, $\left[F\right(t,0)=0]$, and $\sigma \in ]0, 1]$. The initial state is $X\left(0\right)$.

Using Lyapunov theory, the asymptotic stability for system (12) is provided as follows:

Theorem 1

([22]). For the nonautonomous fractional-order system described by Equation (12), where $X=0$ represents the equilibrium point, we assume the existence of a Lyapunov function $V(t,X(t\left)\right)$. This Lyapunov function, along with two positive scalars ${\gamma }_1$ and ${\gamma }_2$, must satisfy the following conditions:

$$\begin{array}{ccc}\hfill {\gamma }_1\Vert X\Vert & \le & V(t,X(t\left)\right)\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill D_t^{\sigma }V(t,X\left(t\right))& \le & -{\gamma }_2\Vert X\Vert \hfill \end{array}$$

(14)

where $\sigma \in [0, 1]$. Then, system (12) is locally asymptotically stable around $X=0$.

Theorem 2

([18]). Let $X=0$ the nonautonomous fractional-order system’s equilibrium point (12). Let us assume that there is a scalar ${\gamma }_1>0$ and a continuous Lyapunov function $V(t,X(t\left)\right)$ such that $\forall X\ne 0$.

$$\begin{array}{ccc}\hfill {\gamma }_1\Vert X\left(t\right)\Vert & \le & V(t,X(t\left)\right)\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill D_t^{\sigma }V(t,X\left(t\right))& \le & 0,\sigma \in [0, 1].\hfill \end{array}$$

(16)

Consequently, system (12) is locally stable around $X=0$.

Lemma 1

([19]). Let $Ff\left(t\right)\in {\mathbb{R}}^n$ be a continuous function and Q a positive matrix of order n. Then for all $t\ge 0$:

$$D_t^{\sigma }{F}^T\left(t\right)QF\left(t\right)\le 2F{\left(t\right)}^{T}Q{D}_t^{\sigma }F\left(t\right),\forall \sigma \in [0, 1].$$

(17)

Now, let us consider a nonlinear system characterized by an nth order differential equation:

$$y^{\left(n\right)}=f(y, y^{\prime }, \dots , y^{(n-1)})+g(y, y^{\prime }, \dots , y^{(n-1)})u$$

(18)

where $g(y, y^{\prime }, \dots , y^{(n-1)})$ is a strictly positive function. $y\in \mathbb{R}$ is the output of the system and $u\in \mathbb{R}$ is its control. The case of multi-input and multi-output can be extended easily.

Let us define the desired trajectory defined by $\left(y_d, y_d^{\prime }, \dots , y_d^{(n-1)}\right)$, and the error $e=y-y_d$.

Let us define the generalized error w:

$$w=e^{(n-1)}+\sum _{i=0}^{n-2}{\alpha }_i{e}^{\left(i\right)}=e^{(n-1)}+{\alpha }_{n-2}{e}^{(n-2)}+\dots +{\alpha }_1{e}^{\prime }+{\alpha }_{0}e$$

(19)

such as the following polynomial:

$$Q\left(p\right)=p^{n-1}+{\alpha }_{n-2}{p}^{n-2}+\dots +{\alpha }_{1}p+{\alpha }_0=0$$

(20)

has roots all with negative real parts.

Let us define the following sliding function:

$$s=w+\eta D_t^{-\sigma }w$$

(21)

where $\sigma \in ]0, 1[$.

Assuming that:

$$\dot{s}=-{\zeta }_1\mathrm{sign}s-{\zeta }_{2}s.$$

(22)

The control law can be expressed as:

$$u={\displaystyle \frac{1}{g(y, y^{\prime }, \dots , y^{(n-1)})}}\left[y_d^{\left(n\right)}-f(y, y^{\prime }, \dots , y^{(n-1)})-\sum _{i=1}^{n-2}{\alpha }_i{e}^{(i+1)}-\eta D_t^{1-\sigma }w-{\zeta }_1\mathrm{sign}s-{\zeta }_{2}s\right].$$

(23)

The stability study of the system is done in two steps. The first one shows that the system’s state reaches the sliding surface $s=0$. For that, the norm of s should strictly decrease in time. Then, the following function is an adequate choice of a Lyapunov function:

$$V\left(s\right)={\displaystyle \frac{1}{2}}{s}^{T}s.$$

(24)

Differentiating it with respect to times gives:

$${\dot{V}}_1=s^T\dot{s}=-{\zeta }_1\left|s\right|-{\zeta }_2{s}^{T}s<0$$

(25)

which is negative. Then, the system should attain the sliding surface $s=0$. On this surface, we have:

$$w+\eta D_t^{-\sigma }w=0.$$

(26)

Applying $D_t^{\sigma }$ to this equation gives:

$$D_t^{\sigma }w+\eta w=0.$$

(27)

Now, consider the following Lyapunov function:

$$V_2\left(w\right)={\displaystyle \frac{1}{2}}{w}^{T}w.$$

(28)

Applying $D_t^{\sigma }$ to $V_2\left(w\right)$ gives:

$$D_t^{\sigma }{V}_2=w^T{D}_t^{\sigma }w=-\eta w^{T}w<0.$$

(29)

This shows that w goes to zero, and consequently e goes to zero. This ensures that the system will converge to the desired trajectory.

3. Suggested Control Approach

This work presents a novel FO-SMC legislation. Significant modifications have been incorporated with the goal of ensuring improved control system efficiency based on the control law defined in [23]. Indeed, to provide a smooth transient, fast convergence, and complete cancellation of the static error, the definition of the sliding function is described above. An important contribution is the proof of the asymptotic convergence of the suggested guidance law.

3.1. Speed Control

The speed and the torque are represented by two differential equations controlled by the rotor pulsation $u={\omega }_r$ [3,24]:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill J{\displaystyle \frac{d{\Omega }_m}{dt}}& =& T_e-T_L\hfill \\ \hfill \tau {\displaystyle \frac{d{T}_e}{dt}}+T_e& =& {\omega }_r\hfill \end{array}\right.\end{array}$$

(30)

with:

$$T_L=f\left({\Omega }_m\right)=(f_0+f_2{\Omega }_m^2)\mathrm{sign}{\Omega }_m+f_1{\Omega }_m$$

(31)

(As described above, we have $f_1=0$). This gives:

$$J{\displaystyle \frac{d^2{\Omega }_m}{d{t}^2}}+{\displaystyle \frac{df\left({\Omega }_m\right)}{dt}}{\displaystyle \frac{d{\Omega }_m}{dt}}+{\displaystyle \frac{1}{\tau }}\left[J{\displaystyle \frac{d{\Omega }_m}{dt}}+f\left({\Omega }_m\right)\right]={\displaystyle \frac{A}{\tau }}{\omega }_r$$

(32)

and then:

$${\displaystyle \frac{d^2{\Omega }_m}{d{t}^2}}+\left[{\displaystyle \frac{f_1+2f_2\left|{\Omega }_m\right|}{J}}+{\displaystyle \frac{1}{\tau }}\right]{\displaystyle \frac{d{\Omega }_m}{dt}}+{\displaystyle \frac{(f_0+f_2{\Omega }_m^2)\mathrm{sign}{\Omega }_m+f_1{\Omega }_m}{\tau }}={\displaystyle \frac{A}{J\tau }}{\omega }_r.$$

(33)

The sliding function used in this study is defined as:

$$s_{\Omega }=\dot{e}+{\eta }_{1\Omega }e+{\eta }_{2\Omega }{D}_t^{-\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)$$

(34)

where $e={\Omega }_m-{\Omega }_m^{★}$, $\dot{e}={\dot{\Omega }}_m-{\dot{\Omega }}_m^{★}$, and $\ddot{e}={\ddot{\Omega }}_m-{\ddot{\Omega }}_m^{★}$. $\sigma$ belongs in $]0, 1[$. (${\Omega }_m^{★}$, ${\dot{\Omega }}_m^{★}$, ${\ddot{\Omega }}_m^{★}$) represent the desired trajectory. ${\eta }_{1\Omega }>0$ and ${\eta }_{2\Omega }>0$. “sign” represents the signum function.

Assume that:

$${\dot{s}}_{\Omega }=-({\zeta }_{1\Omega }\mathrm{sign}{s}_{\Omega }+{\zeta }_{2\Omega }{s}_{\Omega })$$

(35)

${\zeta }_{1\Omega }>0$ and ${\zeta }_{2\Omega }>0$. This results in:

$$\ddot{e}+{\eta }_{1\Omega }\dot{e}+{\eta }_{2\Omega }{D}_t^{1-\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)=-({\zeta }_{1\Omega }\mathrm{sign}{s}_{\Omega }+{\zeta }_{2\Omega }{s}_{\Omega }).$$

(36)

Then:

$$\begin{array}{ccc}\hfill u& =& {\displaystyle \frac{J\tau }{A}}[\left({\displaystyle \frac{f_1+2f_2\left|{\Omega }_m\right|}{J}}+{\displaystyle \frac{1}{\tau }}\right){\displaystyle \frac{d{\Omega }_m}{dt}}+{\displaystyle \frac{(f_0+f_2{\Omega }_m^2)\mathrm{sign}{\Omega }_m+f_1{\Omega }_m}{\tau }}\hfill \\ & & + {\ddot{\Omega }}_m^{★}-{\eta }_{1\Omega }\dot{e}-{\eta }_{2\Omega }{D}_t^{1-\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)-{\zeta }_{1\Omega }\mathrm{sign}{s}_{\Omega }-{\zeta }_{2\Omega }{s}_{\Omega }].\hfill \end{array}$$

(37)

This control law ensures that the system’s state will reach the sliding surface defined by $s_{\Omega }=0$.

Theorem 3.

Examine the system denoted by (30) and the suggested sliding function (34). The control law guarantees the closed-loop system’s asymptotic stability (53).

Proof. 

First of all, we should demonstrate that the system state should reach the sliding surface $s_{\Omega }=0$. For this, we should demonstrate that the norm of $s_{\Omega }$ is a strictly decreasing function of time. Then, we consider the subsequent Lyapunov function:

$$V_1\left(s_{\Omega }\right)={\displaystyle \frac{1}{2}}{s}_{\Omega }^2.$$

(38)

   □

Its derivative is:

$${\dot{V}}_1\left(s_{\Omega }\right)=s_{\Omega }{\dot{s}}_{\Omega }=-{\zeta }_{1\Omega }\left|s_{\Omega }\right|-{\zeta }_{2\Omega }{s}_{\Omega }^2<0.$$

(39)

Next, when the system remains on the surface $s_{\Omega }=0$, this results in $D_t^{\sigma }{s}_{\Omega }=0$:

$$D_t^{\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)+{\eta }_{2\Omega }\left(\dot{e}+{\eta }_{1\Omega }e\right)=0.$$

(40)

This gives:

$$D_t^{\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)=-{\eta }_{2\Omega }\left(\dot{e}+{\eta }_{1\Omega }e\right).$$

(41)

In order to ensure that the trajectory error goes to zero, we consider the subsequent Lyapunov function:

$$V_2(e,\dot{e})={\displaystyle \frac{1}{2}}(\dot{e}+{\eta }_{1\Omega }e{)}^2\ge 0.$$

(42)

Its fractional derivatives with respect to time, noted $D_t^{\sigma }{V}_2$, is expressed by (Lemma 1):

$$\begin{array}{ccc}\hfill D_t^{\sigma }{V}_2(e,\dot{e})& \le & (\dot{e}+{\eta }_{1\Omega }e)D_t^{\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)=-{\eta }_{2\Omega }{\left(\dot{e}+{\eta }_{1\Omega }e\right)}^2\le 0.\hfill \end{array}$$

(43)

The choice of this Lyapunov is not totally adequate. But, in this case, we have shown that $s_{\Omega }$ goes to zero and $(\dot{e}+{\eta }_{1\Omega }e)$ converges to zero, and consequently, e converges to zero, and the system attains its desired trajectory.

We can use a third Lyapunov function, which is definite positive:

$$V_3(e,\dot{e})={\displaystyle \frac{1}{2}}(\dot{e}+{\eta }_{1\Omega }e{)}^2+{\eta }_{3\Omega }{D}_t^{1-\sigma }{e}^2$$

(44)

${\eta }_{3\Omega }>0$.

Its fractional derivatives with respect to time, noted $D_t^{\sigma }{V}_3$, is given by (Lemma 1):

$$\begin{array}{ccc}\hfill D_t^{\sigma }{V}_3(e,\dot{e})& \le & (\dot{e}+{\eta }_{1\Omega }e)D_t^{\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)+{\eta }_{3\Omega }e\dot{e}\hfill \\ & \le & -{\eta }_{2\Omega }{\dot{e}}^2-(2{\eta }_{1\Omega }{\eta }_{2\Omega }-{\eta }_{3\Omega })e\dot{e}-{\eta }_{2\Omega }{\eta }_{1\Omega }^2{e}^2<0\hfill \end{array}$$

(45)

if and only if:

$${(2{\eta }_{1\Omega }{\eta }_{2\Omega }-{\eta }_{3\Omega })}^2-4{\eta }_{1\Omega }^2{\eta }_{2\Omega }^2<0.$$

(46)

That is if:

$$0<{\eta }_{3\Omega }<4{\eta }_{1\Omega }{\eta }_{2\Omega }.$$

(47)

The case ${\eta }_{3\Omega }=2{\eta }_{1\Omega }{\eta }_{2\Omega }$ is adequate, giving:

$$\begin{array}{ccc}\hfill D_t^{\sigma }{V}_3(e,\dot{e})& \le & -{\eta }_{2\Omega }{\dot{e}}^2-{\eta }_{2\Omega }{\eta }_{1\Omega }^2{e}^2<0.\hfill \end{array}$$

(48)

3.2. Flux Control

The vector of the flux $\Phi$ controlled by the stator voltage vector $V_s$ is described by the differential equation:

$${\displaystyle \frac{d\Phi }{dt}}=V_s-R_s{I}_s.$$

(49)

Let us define the sliding function:

$$s_{\varphi }=e+{\eta }_{\varphi }{D}_t^{-\sigma }e$$

(50)

where ${\eta }_{\varphi }>0$, $e=\Phi -{\Phi }_d$, and $\dot{e}=\dot{\Phi }-{\dot{\Phi }}_d$. $\sigma$ belongs in $]0, 1[$. (${\Phi }_d$, ${\dot{\Phi }}_d$) is the desired trajectory.

The proposal is to determine the adequate control assuming that:

$${\dot{s}}_{\varphi }=-({\zeta }_{1\varphi }\mathrm{sign}{s}_{\varphi }+{\zeta }_{2\varphi }{s}_{\varphi })$$

(51)

${\zeta }_{1\varphi }>0$ and ${\zeta }_{2\varphi }>0$. This results in:

$$\dot{e}+{\eta }_{\varphi }{D}_t^{1-\sigma }e=-({\zeta }_{1\varphi }\mathrm{sign}{s}_{\varphi }+{\zeta }_{2\varphi }{s}_{\varphi }).$$

(52)

Then:

$$V_s=R_s{I}_s+{\dot{\Phi }}_d-{\eta }_{\varphi }{D}_t^{1-\sigma }e-({\zeta }_{1\varphi }\mathrm{sign}{s}_{\varphi }+{\zeta }_{2\varphi }{s}_{\varphi }).$$

(53)

This result guarantees that the system state reaches the sliding surface $s_{\varphi }=0$.

Theorem 4.

Consider the system denoted by (49) and the suggested sliding function (50). The control law guarantees the closed-loop system’s asymptotic stability (53).

Proof. 

First, we should demonstrate that the trajectory of the systems should reach the sliding surface $s_{\varphi }=0$. For this, we should demonstrate that the norm of $s_{\varphi }$ is a strictly decreasing function of time. Then, we consider the subsequent Lyapunov function:

$$V_1\left(s_{\varphi }\right)={\displaystyle \frac{1}{2}}{s}_{\varphi }^T{s}_{\varphi }.$$

(54)

   □

Its time differential is:

$${\dot{V}}_1\left(s_{\varphi }\right)=s_{\varphi }^T{\dot{s}}_{\varphi }=-{\zeta }_{1\varphi }\left|s_{\varphi }\right|-{\zeta }_{2\varphi }{s}_{\varphi }^T{s}_{\varphi }<0.$$

(55)

When the system is remaining at the surface $s_{\varphi }=0$, we can write $D_t^{\sigma }{s}_{\varphi }=0$, then:

$$D_t^{\sigma }e=-{\eta }_{\varphi }e.$$

(56)

To ensure that the trajectory error goes to zero, we consider the subsequent Lyapunov function:

$$V_2\left(e\right)={\displaystyle \frac{1}{2}}{e}^{T}e.$$

(57)

Its fractional derivatives with respect to time, noted $D_t^{\sigma }{V}_2$, is given by (Lemma 1):

$$\begin{array}{ccc}\hfill D_t^{\sigma }{V}_2\left(e\right)& \le & -e^T{D}_t^{\sigma }e=-{\eta }_{\varphi }{e}^{T}e<0.\hfill \end{array}$$

(58)

Then, the system converges to its intended trajectory, ensuring its asymptotic stability.

4. Evaluation of the FO-SM-DTC-SVM Method Based on Simulations Results

Table 1. Parameters of an IM.

$R_s=0.29$  $\Omega$	$R_r=0.38$  $\Omega$	$M=47.3$ mH
${\mathcal{l}}_s={\mathcal{l}}_r=50$ mH	$N_p=2$	$J=0.5$ kg/m2

contains the parameters of the three-phase induction motor. The motor is rated at 220 V, 10 kW, and 1470 rpm at 50 Hz.

$R_s$ and $R_r$ refer to stator and rotor resistances, $l_s$ and $l_r$ refer to stator and rotor inductance, M is the mutual one, $N_p$ is the pole pair number, and J is the motor inertia.

Table 2. Settings for the electric vehicle model.

$r=0.3$ m	$A=1$ m2
$M_v=400$ kg	$C_d=0.19$
G = 0.9	$\rho =1.2$ kg/m3

presents the parameters of the electric vehicle model.

The following simulation works represent a continuation of the author’s previous research efforts [25,26,27], all focused on developing robust and advanced control systems for induction motors. The proposed FO-SM-DTC-SVM strategy builds upon this foundation, leveraging fractional-order control to achieve enhanced performance.

4.1. Evaluation Metrics

Two evaluation matrics were selected to evaluate the effectiveness of three DTC-SVM techniques, considering the identical simulation settings and various reference speed levels. Initially, let us examine the $i_{as}$ current expression around a steady-state operating point, which is as follows:

$$i_{as}\left(t\right)=\Re e\left(\sum _{N=1}^{\infty }{I}_{N}exp\left(jN{\omega }_{s}t\right)\right).$$

(59)

The amplitude of the fundamental is $|I_1|$, while the amplitude of the harmonic N is $|I_N|$.

The stator current’s average total harmonic distortion (THD), which can be defined as follows, is the first criterion.

$$\begin{array}{ccc}\hfill \mathtt{THD}& =& {\displaystyle \frac{1}{|I_1|}}\sqrt{\sum _{N=2}^{\infty }{\left|I_N\right|}^2}\hfill \end{array}$$

(60)

The frequency spectrum of the stator current $i_{as}$ has been examined in this context. It entails monitoring each harmonic’s amplitude in relation to its harmonic frequency. A depiction of current amplitudes $I_N$ as functions of their harmonic ranks N is the $i_{as}$ frequency spectrum.

For varying speed levels, the torque waves around its steady-state value $T_{e,mean}$ are translated using the second comparison criterion. It can be represented as the average of the torque ripple ratio divided by the torque mean over a certain time:

$$\begin{array}{ccc}\hfill T_{RIP}& =& ∥{\displaystyle \frac{T_e\left(t\right)-T_{e,mean}}{T_{e,mean}}}∥=∥{\displaystyle \frac{T_e\left(t\right)}{T_{e,mean}}}-1∥\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill T_{RIP,1}& =& {\displaystyle \frac{1}{T}}{\int }_{t_0}^{t_0+T}\left|{\displaystyle \frac{T_e\left(t\right)-T_{e,mean}}{T_{e,mean}}}\right|dt\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill T_{RIP,2}& =& \sqrt{{\displaystyle \frac{1}{T}}{\int }_{t_0}^{t_0+T}{\left[{\displaystyle \frac{T_e\left(t\right)-T_{e,mean}}{T_{e,mean}}}\right]}^{2}dt}\hfill \end{array}$$

(63)

$$\begin{array}{ccc}\hfill T_{RIP,\infty }& =& \underset{t_0\le t<t_0+T}{max}\left|{\displaystyle \frac{T_e\left(t\right)-T_{e,mean}}{T_{e,mean}}}\right|\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill {\Phi }_{RIP,1}& =& {\displaystyle \frac{1}{{\Phi }_s}}{\int }_{t_0}^{t_0+T}\left|{\displaystyle \frac{{\Phi }_s\left(t\right)-{\Phi }_{s,mean}}{{\Phi }_{s,mean}}}\right|dt\hfill \end{array}$$

(65)

$$\begin{array}{ccc}\hfill {\Phi }_{RIP,2}& =& \sqrt{{\displaystyle \frac{1}{T}}{\int }_{t_0}^{t_0+T}{\left[{\displaystyle \frac{{\Phi }_s\left(t\right)-{\Phi }_{s,mean}}{{\Phi }_{s,mean}}}\right]}^{2}dt}\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill {\Phi }_{RIP,\infty }& =& \underset{t_0\le t<t_0+T}{max}\left|{\displaystyle \frac{{\Phi }_s\left(t\right)-{\Phi }_{s,mean}}{{\Phi }_{s,mean}}}\right|\hfill \end{array}$$

(67)

with ${\Phi }_{s,mean}=1$ Wb.

The value of T was chosen to be equal to the stator period. It is crucial to select the time $t_0$ so that the system reaches its steady-state regime for t greater than $t_0$. It is worth mentioning that these three metrics can be calculated from $t_0$ to ∞ without any noticeable variation in the obtained results.

We have considered a speed shape divided into three time zones:

• Time zone 1: $t\in [0,1]$ s.
• Time zone 2: $t\in [1,2]$ s.
• Time zone 3: $t\in [2,3]$ s.

Each zone presents a transient and a steady-state behavior of the speed. In the steady-state part, the speed is constant, and the electromagnetic torque has a certain mean value and a current having its spectrum. The spectrum of the current, its total harmonic distortion, and the torque ripples are calculated in each zone. However, the ripple of the flux is calculated on the overall interval of time (eliminating only its transient phase).

4.2. First Scenario

We have considered the derivative of the desired speed as pulses giving the desired speed. In this case, the second derivative of the desired speed presents Dirac implusions. This is illustrated in Figure 2 which represents the desired trajectory of the speed (${\Omega }_m^{★}$, ${\dot{\Omega }}_m^{★}$, ${\ddot{\Omega }}_m^{★}$).

Figure 3a represents the evolution of the speed. The speed tracking closely follows the desired behavior. Figure 3b provides a zoomed-in view highlighting the diminishing difference between the desired and actual speed, approaching zero. Figure 3c–f depict the behavior of the electromagnetic torque, rotor pulsation, stator voltage amplitude, and flux. It is obvious that these results are adequate, presenting several pulses on the rotor pulsation caused by the derivative of the desired pulses of the speed. Moreover, the ripples of the flux are sufficiently low.

Table 3. Flux ripples.

${\Phi }_{\mathit{RIP},1}$	${\Phi }_{\mathit{RIP},2}$	${\Phi }_{\mathit{RIP},3}$
0.0016	0.0021	0.0066

shows that the flux ripples are less than 1% of the flux.

Table 4. Torque ripples in each time zone.

	${\mathit{T}}_{\mathit{RIP},1}$	${\mathit{T}}_{\mathit{RIP},2}$	${\mathit{T}}_{\mathit{RIP},3}$
Time Zone 1	0.0482	0.0613	0.2267
Time Zone 2	0.0490	0.0611	0.2492
Time Zone 3	0.0832	0.1020	0.2631

represents the torque ripples calculated in each zone.

The torque ripples are sufficiently low, showing good performances of the proposed controller.

Figure 4a presents the evolution of the stator current $i_{as}$ and a period of this current in each time zone (Figure 4b–d). The spectrum of this current has been calculated in each zone based on these periods.

Figure 5 presents the reduced spectrum to the fundamental of the current $i_{as}$ in each zone. It is clear that its harmonics have sufficiently low components, and consequently, the total harmonic distortion THD of the current is sufficiently low in each time zone. To observe the harmonics of the current, in Figure 5a,c,e, the y-axis has been limited in the range [0, 0.1] despite the reduced fundamental being equal 1. However, in order to have the best observation of these harmonics, Figure 5b,d,f present these harmonics without drawing the fundamental. This is well confirmed by

Table 5. THD of Stator Current ($i_{as}$).

	Time Zone 1	Time Zone 2	Time Zone 3
THD	0.0141	0.0151	0.0047

(the THD is around $1\%$).

4.3. Second Scenario

In order to eliminate the Dirac impulsions for the derivative of the desired speed, we have considered a filter on the pulses. This is essential to smooth rotor pulsation variations (see Figure 6). This figure represents the evolution of the desired trajectory, which has been filtered by a first-order system with a constant equal to 0.01 s.

Figure 7 illustrates the evolution of speed, stator current, voltage current, flux, electromagnetic torque, and rotor pulsation. In fact, the evolution of the speed becomes slightly slow (Figure 7a,b), and the evolution of the rotor pulsation becomes more smooth, eliminating its sudden variations, and its amplitude becomes lower (Figure 7d). A similar observation applies to the evolution of the electromagnetic torque (Figure 7c). However, the amplitude of the stator voltage and flux remain consistent (Figure 7e,f). Moreover, the stator currents keep the same profile and the same spectrum. Compared to the first scenario, these figures show that the obtained results present smooth evolutions of these variables.

The total harmonic distortion and the ripples in the electromagnetic torque remain consistent because they are computed in the steady-state phase. Moreover, the flux ripples are the same.

4.4. Third Scenario

In this case, we have considered the non-filtered profile of the speed described in the first scenario and Figure 2. However, it is clear that the second differential of the desired speed is a sequence of Dirac pulses. This signal is equal to zero for all time t instead of $t={\displaystyle \frac{1}{2}}k$ s (where k is an integer: $k=1, 2, 3 \dots$). These instants correspond to sudden variations of ${\Omega }_m^{★}$. For this reason, and in order to eliminate these pulses, replace the control law defined in Equation (37) with:

$$\begin{array}{ccc}\hfill u& =& {\displaystyle \frac{J\tau }{A}}[\left({\displaystyle \frac{f_1+2f_2\left|{\Omega }_m\right|}{J}}+{\displaystyle \frac{1}{\tau }}\right){\displaystyle \frac{d{\Omega }_m}{dt}}+{\displaystyle \frac{(f_0+f_2{\Omega }_m^2)\mathrm{sign}{\Omega }_m+f_1{\Omega }_m}{\tau }}\hfill \\ & & + {\ddot{\Omega }}_m^{★}-{\eta }_{1\Omega }\dot{e}-{\eta }_{2\Omega }{D}_t^{1-\sigma }\left({\dot{\Omega }}_m+{\eta }_{1\Omega }e\right)-{\zeta }_{1\Omega }\mathrm{sign}{s}_{\Omega }-{\zeta }_{2\Omega }{s}_{\Omega }].\hfill \end{array}$$

(68)

This consists of replacing the term $D_t^{1-\sigma }\left(\dot{e}+{\eta }_{1\Omega }e\right)$ with $D_t^{1-\sigma }\left({\dot{\Omega }}_m+{\eta }_{1\Omega }e\right)$. Simulation outcomes are presented in Figure 8.

The pulses in the electromagnetic torque and control ${\omega }_r$ are effectively eliminated, resulting in a smoother response.

5. Conclusions

This research presents a novel fractional-order sliding mode control (FO-SMC) strategy, specifically the FOSM-DTC-SVM, for IM drives in electric vehicle (EV) applications. Three distinct simulation scenarios were conducted to evaluate the performance of this proposed control approach.

The first scenario examined the system’s response when the desired speed acceleration was defined by pulses, introducing Dirac impulses in its second derivative. The results demonstrated the FOSM-DTC-SVM’s ability to track the desired speed effectively while minimizing flux ripples and generating pulses in the rotor pulsation.

The second scenario investigated the impact of filtering the desired speed to eliminate Dirac impulses. This resulted in smoother rotor pulsation variations and a slightly slower speed response while maintaining similar flux ripple and stator current characteristics.

The third scenario focused on the effect of eliminating fractional derivatives in the control law. This led to the elimination of Dirac impulses and further improved the system’s performance.

Overall, simulation results confirmed the robustness and effectiveness of the FOSM-DTC-SVM strategy, highlighting its potential as a promising solution for enhancing the performance and efficiency of EV propulsion systems. This research lays the groundwork for future advancements in fractional-order control techniques within the electric vehicle sector.

Fractional-order sliding mode control offers greater design development flexibility than non-fractional sliding mode control due to its fractional differentiation. In fact, the ordinary differentiation is a particular case of fractional derivatives. However, this flexibility also increases the complexity of the tuning process, especially in the presence of unknown parameters, parametric fluctuations, and external disturbances. An adaptive strategy is needed to address these challenges, and the asymptotic stability of the adaptive control system presents a promising area for future research.