On integrable and approximate solutions for Hadamard fractional quadratic integral equations

1.   Introduction

Induction motors (IMs), which are used in industrial processes for variable speed and position control, have become more attractive in recent years because of their simplicity, lower cost, affordability, and easy maintenance needs. However, IM control has been a conceptually tough topic due to its complex dynamic characteristics, which are highly nonlinear and coupled as flux and electromagnetic torque ^[1,2]. Recent developments in power electronics with sophisticated control techniques have enabled the employment of electrical drives, particularly induction motors, in high-performance, variable-speed applications. In the literature, several methods for IM control have been studied, the most widely used being field-oriented control (FOC). In FOC, the rotor flux and torque that generate current components are decoupled to provide reaction properties akin to those of a direct current motor ^[3,4,5,6,7]. However, conventional strategies of FOC may struggle with parametric variations and uncertainty and be unable to provide the desired performance ^[8].

As a result, great effort has been expended on developing the performance and robustness of an induction motor's control approaches. An intriguing technique for controlling nonlinear systems is the sliding mode control, which is distinguished by its extreme simplicity and resilience to disturbances and parameter changes. This method drives the system state's trajectory toward a switching surface in the state space via discontinuous control ^[9]. The sliding mode control (SMC) has been used in combination with other methods as a model reference adaptive system (MRAS) ^[10] and with fuzzy logic technique and indirect field-oriented control (IFOC) techniques ^[11].

In spite of its advantages, there is a drawback associated with this method—the chattering phenomenon. Chattering can induce torque pulsation and current harmonics among other undesired consequences by exciting the system's high-frequency unmodeled dynamics. To overcome this drawback, many advanced methods have been developed, and several techniques for robust chattering reduction are dependent on modifying the switching functions ^[12,13,14]. While high-frequency chattering may be easily reduced or eliminated by using the saturation function, this will inevitably result in a loss of some robustness against load disturbances and uncertainties. However, the majority of steady-state errors will still be present.

To minimize the steady-state error and specify the integral sliding mode controller, an integral sliding surface is a recommended technique ^{[14,15,16,17,18,19,20]}. Steady-state error (SSE) in pseudo-sliding modes can be avoided by using a pseudo-sliding mode control, as introduced in ^[20], in which smooth functions replace the switching function with the inclusion of integral terms. This turns the system into pseudo-sliding modes, resulting in SSEs. On the other hand, by adding an integral term to the surface, the system can be vulnerable to windup. This causes the system response to significantly overshoot and exhibit undesired oscillations. Using a switching sigmoid function, authors in ^[21] suggested an integrated sliding mode controller with a novel anti-windup technique that has low overshoot and no steady-state inaccuracy.

Another class of SMC approaches to attenuate the chattering and achieve robust control is called high-order SMC ^{[22,23,24,25,26,27]}. Several suggested approaches are used to construct a variety of novel second-order sliding mode controllers (SOSMC), including the twisting algorithm ^[28,29] and the super twisting algorithm ^[30,31,32]. It should be mentioned that SOSM algorithms require bounded uncertainties in sliding mode dynamics because state variables are constantly related to uncertainties. This means that state variables should also be bounded, even though determining the domain of attraction is usually not easy ^[25,33].

This work aims to compare two sliding mode control techniques applied to IM—a classical SMC and a second-order SMC—using the twisting algorithm, whose aim is to reduce chattering. The findings of the experiment are examined and presented.

The structure of the paper is as follows: Section 2 presents the induction motor model in the synchronous reference frame, and the field-oriented model is explained. In Section 3, the classical sliding mode control theory for a nonlinear system is briefly reviewed. The two proposed controls are described and developed in Section 4. In Section 5, experimental results of the implemented controls are presented to examine the effectiveness and compare their performance. Section 6 provides a summary of the study's main conclusions.

2.   Induction motor model of field-oriented flux

Accurate mathematical modeling is essential to effectively control the various operating modes of IMs. The following expressions represent the mathematical model of the IM in terms of stator current and rotor flux in the synchronous reference frame ^[2]:

The mechanical modeling part of the system is given by:

Where the electromagnetic torque ${T}_{e}$ is given by:

With:

Where ${\phi }_{rd}$, ${\phi }_{rq}$ are the rotor flux components, and ${i}_{sd}$, ${i}_{sq}$ are the stator currents. ${u}_{s} = {\left[{u}_{sd}, {u}_{sq}\right]}^{T}$ is the control vector. ${T}_{L}$ is the load torque, ω is the mechanical frequency of rotor speed, ωs is the electrical stator frequency speed, σ is a total leakage factor, and ${T}_{L}$ is the rotor time constant. ${R}_{s}$ and ${R}_{r}$ denote stator and rotor resistance, ${L}_{s}$ and ${L}_{r}$ are the stator and rotor self-inductance, M is mutual inductance, p is the number of pole pairs, j is the moment of inertia, and F is the friction coefficient.

A rotor field orientation in the synchronous reference frame (d, q) is achieved if we let ${\phi }_{rq} = 0$ and ${\phi }_{rd} = {\phi }_{r}$, ^[3]. The IM model, after field-oriented transformation, becomes:

With:

The components of the function f(x) are nonlinear functions, $x = [{i}_{sd}~~ {i}_{sq}~~ {\phi }_{r}~~ \text{}\mathrm{\varOmega }{]}^{T}$ is the state space vector, and the control vector ${u}_{s}$ is defined by ${\left[{u}_{sd}~~ {u}_{sq}\right]}^{T}$. The objective of vector control is to enable decoupling control of torque and rotor flux. After orientation of the flux, the electromagnetic torque becomes:

Additionally, the direct stator current component may be used to reconstruct the rotor flux using the third equation in Eq (4):

Where s is the Laplace operator. According to (5) and (6), we obtain two independent actions: the stator component ${i}_{sd}$ controls the flux, and ${i}_{sq}$ controls the torque, achieving the classical direct FOC control objective ^[3].

The synchronous rotating frame angle ${\mathrm{\theta }}_{s}$ can be determined by integrating the synchronous angular frequency ${\omega }_{s}$ as below:

The slip angular frequency ${\omega }_{r}$ is given by:

In the following section we will apply two techniques, the first-order and the second-order sliding mode, to control the IM.

3.   Sliding mode control strategy

3.1.   Problem formulation

Although a sliding mode method is known to be robust to disturbances and parametric uncertainties, one particular drawback in classical sliding mode techniques (first-order SM) is the chattering phenomena, which is essentially motion that oscillates around the sliding surface. To overcome this undesirable phenomenon, the higher-order sliding mode concept has been introduced by Levant et al. ^[24].

Consider a dynamic system of the form:

Where $x\in {R}^{n}, u\in R$ are the state and control input of system, a(x), b(x) are smooth functions, S is the smooth output function, and n is the dimension of the system.

Let the system be closed by some dynamical discontinuous feedback; we may construct consecutive total time derivatives, and the closed-system state-space variables' continuous functions are $S = \dot{S} = \ddot{S} = \dots = {S}^{(r-1)}$. The fundamental objective is to establish a limited time convergence onto the so-called r-order sliding mode, which is composed locally of Filippov trajectories and is represented as the non-empty surface $S = \dot{S} = \ddot{S} = \dots = {S}^{(r-1)} = 0$. Additionally, the regulation of the rth-order sliding mode achieves greater accuracy in terms of chattering ^[25].

However, because the chosen outputs have a relative degree equal to two with regard to S, a finite time convergence to S = 0 can only be accomplished using higher-order sliding modes. As it occurs, a first-order sliding mode controller would yield an asymptotic convergence. This finding serves as further justification for using a second-order sliding mode in addition to the chattering reduction. The twisting method is a popular example of a second-order algorithm. In general, the second-order sliding controller needs the sliding surface and its derivative to be made available and is determined by the equalities $S = \dot{S} = 0$. The purpose is to produce $\dot{S} = 0$ in finite time while keeping S on zero by discontinuous feedback control. For every limited input, system trajectories are meant to be indefinitely extendible in time, and the system is understood in the Filippov sense ^[34]. The dynamics of sliding function S may be characterized as follows:

Where S and $\dot{S}$ are assumed to be measurable. This is accomplished by computing the second total time derivative of S along the trajectories of (9) under these conditions ^[25]: $A\left(x\right) = {\left.\ddot{S}\right|}_{u = 0}$ and $B\left(x\right) = \frac{\partial \ddot{S}}{\partial u}\ne 0$, where the functions $A\left(x\right)$ and $B\left(x\right)$ are some unknown smooth functions satisfying the following conditions:

Where ${C}_{0}, {K}_{m}, {K}_{M}$ are three positive coefficients. The uncertainties of the sliding mode dynamics of (10) are clearly bound by positive constants, implying that the state variables may also be confined. Then, (11) and (12) imply the following differential inclusion:

We suggest the twisting algorithm, created by A. Levant, which has an easy-to-use design and simple shape. This algorithm relies on appropriately switching control between two distinct values to ensure that the trajectories in the phase plane converge to the origin within a finite time ^[29]. The convergence of the twisting algorithm is ensured by a geometric progression that takes the shape of a spiral movement around the origin. The control law of the twisting algorithm is provided by:

Where ${\lambda }_{m}, {\lambda }_{M}$ are positive constants, with $0 < {\lambda }_{m} < {\lambda }_{M}, \mathrm{ }, {K}_{m}{\lambda }_{M}-{C}_{0} > {K}_{M}{\lambda }_{m}+{C}_{0}$.

The twisting algorithm is well-suited for controlling systems with second-order dynamics or those requiring precision in higher-order derivatives, such as trajectory tracking in different applications. Introducing a twisting algorithm into sliding mode control (SMC) is primarily aimed at addressing limitations of traditional sliding mode control, such as chattering, while improving control performance and robustness.

The twisting algorithm offers finite-time convergence to the sliding surface compared to conventional SMC techniques that could only guarantee asymptotic convergence. This property is crucial in time-critical systems or applications requiring fast stabilization. The twisting algorithm achieves continuous convergence; it gets rid of abrupt control actions with smooth control updates and, by mitigating chattering, it decreases mechanical stress on actuators. By producing smoother control signals, the twisting algorithm improves overall system performance and lowers control input energy, making it desirable to several applications ^[28,33]. In practice, implementing the twisting algorithm is easy, but we need to obtain the first derivative of the switching surface.

4.   Induction motor control by sliding mode method

4.1.   First-order SM control of IM

The chosen sliding surface is defined by:

Such as:

Where e is the error between the measured rotor speed Ω, the estimated value of the square of the flux modulus φ, and their references ${\mathrm{\varOmega }}_{ref}$ and ${\phi }_{ref}$.

For good tracking of speed and flux, it is important to make the surface invariant $\left(\dot{\mathrm{S}}\left(\mathrm{x}\right) = 0\right)$ and attractive $\left({S}^{T}\dot{\mathrm{S}} < 0\right)$. The derivative of the surface gives:

The surface $\mathrm{S} = 0$ is invariant if $\dot{\mathrm{S}} = 0,$ where $Q\left(x\right)+R\left(x\right){u}_{eq} = 0$, so we can therefore deduce the equivalent component as follows:

The surface S becomes attractive if we choose ${u}_{glis}$ in the following form:

λ_Ω, λ_φ, k_Ω, and k_φ are positive constants chosen such that:

In sliding mode, Equation (15) becomes:

From (19), the tracking errors in speed ${\mathrm{e}}_{\mathrm{\varOmega }}$ and flux ${\mathrm{e}}_{\mathrm{\varphi }}$ converge exponentially toward 0.

The sliding mode control law ${u}_{s} = {u}_{eq}+{u}_{glis}$ is given by:

In practice, we encountered vibration problems hindering the proper functioning of the motor due to the chattering. For this, we replaced the sign function [sng(S)] of the discontinuous part of the control law of (20) with the saturation function (Sat), to avoid or at least lessen it. We define the following saturation function ^[12,24]:

Where $\varepsilon$ is the boundary layer thickness.

Mitigating vibration problems caused by chattering by replacing the sign function Sng(S) with a saturation function Sat(S) is an effective strategy in control systems, particularly in sliding mode control (SMC). The continuous nature of Sat(S) within the boundary layer ε reduces high-frequency switching, minimizing mechanical vibrations and noise and improving actuator lifetime; this means that we can get softened control action near the switching surface.

The introduction of a boundary layer ε compromises the perfect tracking behavior of the sliding mode. If ε is too large, the system may exhibit steady-state error. Selecting an appropriate ε is critical; it should be large enough to avoid chattering but small enough to ensure accurate tracking within the desired response time. In practice, by judiciously adjusting the parameter ε, a compromise can be found between robustness, precision, and vibration reduction. This concept is widely used in industrial applications requiring precise and reliable control of motors ^[12,35]. There are several types of functions with different boundary layers, such as $\mathrm{tanh}\left(\frac{S}{\varepsilon }\right), \frac{S}{S+\mathrm{\varepsilon }}, \frac{2}{\pi }\mathrm{atan}\left(\frac{S}{\mathit{\varepsilon }}\right)$ , ... etc.

4.1.1.   Stability analysis

The suggested sliding mode controllers must meet the requirements of Lyapunov's stability theory in order to ensure the stability of the system. The selection of the positive Lyapunov's function is as follows:

The time derivative of V is that:

Substituting (16) and (20) into (23), the derivative of Lyapunov function V becomes:

With: $\lambda = \left[\begin{array}{cc}{\lambda }_{\mathrm{\varOmega }} & 0\\ 0 & {\lambda }_{\varphi }\end{array}\right], k = \left[\begin{array}{c}{k}_{\mathrm{\varOmega }}\\ {k}_{\varphi }\end{array}\right]$

The stabilized system will occur when the attaining Lyapunov function criterion $\dot{V} < 0$ is met by the designed nonlinear sliding mode controller, as indicated by (20).

4.2.   Second-order SM control of IM

We take into consideration the uncertain model of the asynchronous motor provided by the form in order to construct a control by second-order sliding modes that guarantees robust performance in the face of parametric changes and disturbances:

Where $f\left(x\right)$ and $g\left(x\right)$ constitute the nominal parts, and ∆f, ∆g represents uncertainties such that:

The control vector is ${\mathrm{u}}_{\mathrm{s}}$ so that ${u}_{s}\le {\left|u\right|}_{max}$.

The relative degree of a system is the derivative number of the output to explicitly see the input (the command). To develop the control law according to the twisting algorithm, we choose the sliding surface S given by (27), such that the relative degree equals 2.

Substituting (26) in the second derivative of S, we obtain:

with:

It is assumed that $\left|\Delta {H}_{1}\right|, \left|\Delta {H}_{2}\right|, \left|\Delta {\mathrm{G}}_{1}\right|and\left|\Delta {\mathrm{G}}_{2}\right|$ are limited. The load torque ${T}_{L}$ must also be bound as well as its first derivative.

By using the twisting method, the system's trajectories are forced to develop on the surface S after a finite amount of time, creating a second-order sliding regime that results in $S = \dot{S} = 0$.

The proposed final control using state space feedback is given by:

$\widehat{\mathrm{G}}\left(x\right)$ is invertible and $v = {\left[{v}_{1}{v}_{2}\right]}^{T}$ is considered the new discontinuous feedback control.

By replacing ${\mathrm{u}}_{\mathrm{s}}$ in (28), the dynamics of $\mathrm{\varOmega }$ and ${\phi }_{r}$ take the following form:

Assume that the following functions are bounded, ∀ $v$, such as:

with K_mi, K_Mi, and C_0i being positive constants.

According to (30) and (31), it is possible to apply the previously presented twisting algorithm. The control $v$ is then defined by:

Where λ_mi, λ_Mi are positive constants checking the following conditions:

For its implementation, we need the sign of the derivative of the surface $\left(\dot{S}\right)$, which can either be obtained by the Matlab function (du/dt) or estimated in a time interval by the sign of the expression $S\left(t\right)-S(t-\tau)$ where τ is the sampling period.

5.   Experimental setup

The presented sliding mode controls are implemented using an experimental configuration that includes an ADC interface board (CP1104) and a Dspace card (DS1104 controller board with TMS320F240 slave CPU). The DC bus voltage is linked to a three-phase VSI inverter with a 10 kHz switching frequency. PWM output channels, encoder inputs, and analog inputs are all included in the experimental setup. The control program is written in Matlab/Simulink real-time interface. The IM used is a three-phase Y-connected four-pole 1.5 kW, 50 Hz, 3.5 A rating. Detailed IM parameters are given in Table 1.

5.1.   Presentation of the benchmark

The overall configuration of the control system for IM is shown in Figure 2. All parameters of IM are taken to be known and constant, with the exception of the rotor time constant T_r, which will change while the motor is running.

The two proposed sliding mode vector controls use the measured velocity Ω, estimated flux ${\widehat{\phi }}_{r}$ given by (6), corresponding position angle ${\widehat{\theta }}_{s}$ given by (7), and stator current measurements (${i}_{sd}, {i}_{sq}$). These control inputs control the three-phase inverter, making it possible to impose the monitoring of the flux and speed trajectories. The load torque is imposed on the asynchronous machine by a powder brake powered by a direct current of 0.15 A; the sampling period is ${T}_{e} = {10}^{-4}s$. The reference speed is applied as soon as the rotor flux is established. 5 Hz is utilized as the low pass filter's cutoff frequency for the measured speed.

The sliding mode control block diagram of the first-order SMC and second-order SMC are shown in Figures 3 and 4, respectively.

To make a more relevant comparison of the proposed control laws, the speed profiles were designed, allowing us to evaluate their performance at variable speeds (low speed, nominal speed, inversed speed), in the presence of the load torque and the variation of the rotor time constant Tr, and to highlight the tracking capabilities of control algorithms at different regimes. Figure 5 shows the profiles used during the experimental tests. The criteria for comparing control laws include the quality of the transition regime response and the ability to track various speed references under conditions of load torque and parameter variations.

5.2.   Experimental results of the implemented controls

To compare and confirm the efficacy of the suggested first- and second-order sliding mode controls under the same conditions, several experimental findings are given. The response of each method is observed under different operating conditions such as a step change in the load torque, speed variation, and variation of the rotor time constant. The regulation parameters of the second-order SMC used in experimental tests are as follows: speed gain constant, ${\lambda }_{M1} = 3200$, ${\lambda }_{m1} = 1000$, rotor flux gain constant, ${\lambda }_{M2} = 8500$, ${\lambda }_{m2} = 4000$.

Test 1: Tracking performance

We present the disturbance rejection and evaluate the IM's speed progression. The test is related to the performances of the drive system at 150 rad/s reference speed shown in Figure 5; a load torque of 7.8 Nm is applied from t = 4 s to t = 10 s. Figure 6 shows that speed tracking is very quietly maintained while a proper rejection of the load torque is achieved. Speed and flux responses as well as disturbance rejection are improved by second-order SMC.

The stator current and the stator voltage responses in the synchronous reference frame (d, q) given by Figure 7 show a perfect decoupling between the torque and the rotor flux, with a good reduction of the chattering in the second-order SM control by the twisting algorithm. Stator current components ${i}_{sq}$ vary proportionally to the applied load torque, presenting the response of the electromagnetic torque.

Test 2: Speed variation

We take into account the speed tracking performances for a broad variety of reference speeds in this test. First, the induction motor is accelerated to 50 rad/s from a standstill; then, it is accelerated again to 140 rad/s and decelerated to a low speed of 10 rad/s. The twisting control algorithm, as shown in Figure 8, illustrates the strong speed tracking and proper rejection of the load torque, presenting extremely weak oscillations at low speeds compared to the control carried out by the first-order sliding mode.

As seen in Figure 9, the reference speed has been modified to the trapezoidal shape in order to validate speed tracking. It is evident that even in reverse speed operation, the speed converges to the reference trajectory and rejects the load torque rather effectively. Stator voltages $({u}_{s\alpha }, {u}_{s\beta })$ recorded in the stationary reference frame, shown by Figure 9, present a satisfactory response, and the SOSMC control with twisting algorithm improved chattering reduction.

Test 3: Robustness against parametric variations.

This test was performed to validate and evaluate system performance and robustness against parameter variation. During this test, 100% variation of 1/Tr is taken into account from t = 8.2 s to t = 12.2 s, and the load torque is applied at t = 4 s, as shown in Figure 10. The rotor flux is perfectly oriented on the d-axis, and the load torque is rejected with great efficiency, resulting in superb speed control, no noticeable changes when the torque is increased, and no significant chattering. Results obtained by the second-order SM control are improved compared to those of the first order.

6.   Conclusions

Two sliding mode techniques have been proposed and implemented for speed and rotor flux control of induction motors. Through the results obtained experimentally, the two sliding mode techniques allowed us to synthesize a robust control of the induction motor. Under load and rotor time constant variations, acceleration, deceleration, and speed reversal modes were investigated for speed convergence and speed tracking in various speed ranges. By comparing the first-order and second-order sliding mode control, we discovered that the twisting algorithm achieved smoother speed and rotor flux forms than the first-order sliding mode control. Also, the reduction in ripple in voltage and current responses revealed an improved dynamic performance of the induction motor. We conclude that the second-order sliding mode control reliably decreases chattering under all conditions while still maintaining the advantages of the classical sliding mode control.

Author contributions

Hadda Benderradji: Conceptualization, Software, Investigation, Writing - original draft, Writing - review and editing; Samira Benaicha: Writing - original draft; Larbi Chrifi Alaoui: Software, Investigation. All authors have read and agreed to the published version of the manuscript.

Use of AI tools declaration

The authors declared they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

All authors declare no conflicts of interest in this paper.