Using the rooted tree optimization to increase the performance of the improved backstepping control used to control the induction machine

Abstract

Using backstepping control to control the operation of an induction motor often leads to unwanted torque and current fluctuations. Also, the value of total harmonic distortion (THD) of current increases if the machine parameters change. These drawbacks can be attributed to the gain values of the backstepping control approach. In this work, a synergetic-sliding mode control (SSMC) technique is used to compensate for the gains of the backstepping control. Using an SSMC controller is an effective and convenient solution. On the other hand, the rooted tree optimization (RTO) algorithm is used to calculate the parameter values of the SSMC controller to improve the performance and effectiveness of the backstepping control of the induction motor. Four SSMC-RTO controllers were used in addition to the pulse width modulation strategy to control the operation of the machine’s inverter. This proposed strategy is characterized by high performance, great robustness, and high efficiency in reducing torque ripples and reducing the value of THD of current. Also, the proposed approach improves the values of the integral of square error (ISE), integral of absolute error (IAE), integral of time multiplied square error (ITSE), and integral of time multiplied absolute error (ITAE) compared to the SSMC-BC approach. The proposed approach was implemented using MATLAB, and the performance was compared with the BC-SSMC approach. The effectiveness of the proposed approach is evaluated using several different tests. Simulation results show that the proposed SSMC-BC approach based on RTO algorithm has high performance and significantly improves current and torque quality compared to the BC-SSMC approach. The proposed approach significantly reduces the overshoot value of the flux compared to the SSMC-BC technique. The simulation results show that the proposed approach significantly outperforms the SSMC-BC technique and some related work in terms of the THD value of current, as the THD value was reduced by approximately 13.56%, 11.11%, and 34.51% compared to the SSMC-BC technique. These results make the proposed approach of interest in other industrial applications in the future.

Highlights

• ✓ Backstepping control combined with the synergetic-sliding mode control (SSMC) is developed to regulate the induction motor speed.

• ✓ The proposed control gain values were calculated using the rooted tree optimization (RTO) algorithm, as using this algorithm contributes to increasing the efficiency and performance of the machine compared to the traditional approach.

• ✓ The fluctuations of torque, current, and flow of the machine are reduced in all tests.

• ✓ The proposed approach is used in conjunction with the PWM approach to control the torque and improve the dynamic power response.

Introduction

During the recent decade, there has been a lot of interest in the control performance of AC motors, specifically induction motors (IMs). IMs have various industrial uses owing to their low maintenance and robustness. ¹ To attain maximum torque and effectiveness, the IMs speed control is more necessary.^2,3 In the past two decades, control strategy has gained wide popularity as a means of controlling and altering the performance of AC motors with very high efficiency. ⁴ Nonlinear control and power electronics have all advanced in the past few decades, allowing for intensive studies into improved IM control techniques. ⁵ In addition, IMs may be regulated by utilizing different techniques, for example, field-oriented control (FOC), ⁶ fractional-order control, ⁷ direct torque control (DTC), predictive torque control (PTC), ⁸ sliding mode control (SMC), ⁹ backstepping control (BC), super-twisting algorithm (STA), ¹⁰ synergetic control (SC), ¹¹ using processes intended to control the IM speed and flux.

Several nonlinear controllers, such as the SMC and SC techniques, have more resilience than linear solutions because of their nonlinear structure. However, these methods have a low current quality: they are hard to use on systems, especially ones that are complicated like SMC and BC techniques. In addition, these methods depend a lot on the parameters of the power system (PS), which is not good because it causes a lot of problems when the machine breaks down. Fuzzy logic (FL) and neural networks (NNs) control are two examples of intelligent control family members. These approaches are controls that don’t require in-depth information on the mathematical model (MM) of the PS; instead, they are affordable, easy, and simple to implement, depending on application experience. The dynamic speed is also good, which makes it a good choice for application. ¹² Table 1 presents some works that show improved performance compared to the proportional-integral (PI) applied to rotor speed control of IM.

Table 1.

Some works addressing the control of IM.

Published papers	Control methods	Summary	Limitations
Yang and Lu 13	- Torque control.	- This study contributes a novel torque control approach for IM drives, enhancing decoupling performance and torque response during variable flux operations by using electromagnetic and reactive torque as state variables, eliminating complex coordinate transformations, and improving overall dynamic performance.	- The paper’s limitations include a lack of comparison with other advanced control approaches like DTC and model predictive control (MPC) and potential challenges in practical implementation.
Quan and Long 14	- SMC;	- The novelty of this paper is the improvement of a sensorless SMC method for three-phase IM. It reduces ripples and eliminates the need for rotor speed sensors by using a NN-based observer, improving stability, robustness, and tracking performance.	- Complexity in implementing the NN-based observer, the computational burden of online updating algorithms, and the lack of comparison with other advanced sensorless control techniques under varying operating conditions.
	- Sensorless control;
	- NN approach.
Rachaputi et al. 15	- Novel SMC.	- The paper contributes a novel vectorial SMC (VSMC) for a single-stage IM drive in solar water pumping applications. It integrates improved particle swarm optimization (IPSO) for maximum energy extraction, even under partial shading conditions, achieving high accuracy and eliminating the need for Park and Clark transformations.	- One of the significant limitations of the designed PS is the usual approach of starting, which draws a large starting current. This can lead to potential issues in systems where current draw needs to be minimized, especially in solar applications where energy efficiency is crucial
Yi et al. 16	- Improved BC-SMCtechnique.	- The paper proposes a novel fault detection method that combines an analytical model with machine learning techniques. This approach	- The influence of model error on accuracy was not fully considered in the proposed methods. This oversight could affect the reliability of the fault detection system under varying operational conditions, as model inaccuracies can lead to incorrect fault diagnosis.
		addresses the challenges posed by nonlinear parts, unknown load disturbances, and noise disturbances in actual motor systems, enhancing the accuracy of fault detection. - Use of Extreme Learning Machine (ELM): The authors utilize ELM to adaptively approximate and estimate the nonlinear functions affecting the system’s tracking performance. This integration helps in reducing the influence of nonlinearities on fault detection.	- The residual error is sensitive to the running state of the motor due to the existence of model error. This sensitivity could lead to false positives or missed detections in fault diagnosis, particularly in environments with high external disturbances.
		- Backstepping sliding mode observer: The paper introduces an improved BC-sliding mode observer, which is designed to enhance the convergence speed and stability accuracy of the fault detection system. This observer is optimized using the Mayfly optimization algorithm, which contributes to better performance in diagnosing faults.	- Complexity of implementation.
Shahid et al. 17	- Model predictive torquecontrol (MPTC).	- The paper contributes by introducing a torque error-based auto-tuning approach for optimizing the weighting factors in model predictive torque control (MPTC) for IM drives, enhancing control performance and accuracy.	- Increased computational complexity due to the auto-tuning process, dependence on predefined threshold values, and the need for precise tuning of weighting factors to achieve optimal performance.
Dursun 18	- Fractional-order SMC (FOSMC)	- The paper contributes by proposing an FOSMC for IM vector control, improving robustness and stability. It introduces a vector diagram-based approach, enhancing performance under uncertainties and disturbances, and offers better tracking and reduced chattering compared to traditional controllers.	- The paper’s limitations include potential high control gains, finite time convergence issues, sensitivity to parameter tuning, and the need to address chattering and unwanted oscillations in the control system.

In Che et al., ¹⁹ the author believes that in the case of model reference adaptive IM speed sensorless system based on flux linkage, there are ripples at the level of rotational speed in the transient and permanent cases. In the case of IM speed estimation, the integral part of the voltage model affects the accuracy of the estimated speed with high-frequency and noisy signals. To solve this problem and improve the system’s anti-interference performance and speed estimation accuracy at low speeds, an improved speed estimation method combining fuzzy proportional integral control and SMC technique is proposed. The gain values of this proposed approach were calculated using a genetic algorithm. Adopting the proposed approach to this algorithm allows for significantly improving the effectiveness and performance of the control system. This proposed approach was implemented in a MATLAB environment, comparing the performance with the traditional method. The simulation results show the effectiveness of the proposed method in medium and low-speed regions with improved robustness against external disturbances and show the high accuracy of the estimated speed, the small amplitude and repeatability of the speed fluctuations, and the large dynamic performance indicators of the system compared to the traditional approach. In Kraiem, ²⁰ a new method for direct torque and flux control of an IM fed from a photovoltaic system is presented. This proposed strategy is based on static field guidance with torque and flux control. This proposed method differs from the DTC and FOC strategy. In this proposed approach, the inverter is supplied with a DC voltage extracted from a PV system to create a voltage and frequency profile for controlling the IM. Also, a typical adaptive reference system and Luenberger Observer are designed to determine the stator flux and rotor speed. This proposed approach has been implemented in MATLAB using different tests. Simulation results show that the proposed strategy can significantly reduce torque and flux undulations compared to the conventional strategy. Also, the simulation results show that the proposed approach works to maintain a constant switching frequency compared to the traditional approach. However, despite this performance, this approach has drawbacks that lie in its complexity and its reliance on the MM of the machine, which makes it affected if the machine’s parameters change, which is undesirable. A highly robust nonlinear strategy is proposed in Uyulan ²¹ to control an IM. This proposed strategy is a robust-adaptive linearizing scheme based on high-order sliding modes and robust differentiators. This proposed strategy has been used for sensorless position control. This proposed strategy has high durability, distinctive performance, great efficiency, and a high ability to reduce torque ripples and reduce the THD of current. MATLAB was used to implement the proposed approach, where several different tests were used and the results were compared to the traditional approach. The simulation results include both estimation and sensorless speed control of IMs over a wide operating range, especially at low and zero speed, all of which are promising and indicate significant superiority of the proposed approach over other existing strategies in terms of high-accuracy, direct drive, and sensorless control of speed/position of squirrel-cage IMs. In Chen et al., ²² an adaptive load torque observer based on the BC technique is designed to control the IM speed. Also, a smooth rotor flux switching strategy based on speed error is designed to reduce motor losses at low loads. In this proposed approach, the relationship between loss and rotor flux is determined by analyzing the loss model of the IM, and the optimal rotor flux is obtained. This proposed strategy has been experimentally verified in the LINKS-RT platform. The results show that the proposed control strategy has excellent performance in mitigating load disturbance and reducing energy loss compared with the traditional approach. In Ben Salem et al., ²³ the author proposed the use of an adaptive fractional-order sliding mode (FO-SM) control approach in order to replace traditional controls in the DTC strategy of IM. In this proposed approach, FO-SM-type controllers are used to control both flux and torque, where the outputs of these controllers are voltage reference values. The space vector modulation strategy is used to generate the necessary pulses based on the reference values generated by the FO-SM controllers. Accordingly, the DTC-SVM-FO-SM strategy is characterized by high performance and great durability. This strategy was implemented in the MATLAB environment using different tests and comparing the results with the traditional approach. All completed tests show the superiority of this approach over the traditional strategy in terms of reducing torque ripples and improving current quality. Compared to the traditional approach, this approach is characterized by a greater degree of complexity and the presence of a large number of gains, which makes it difficult to adjust the dynamic response. Also, using the SMC strategy in the proposed approach creates the phenomenon of chattering, which is an undesirable negative. In Araújo et al., ²⁴ it was proposed to use an auto-regressive neural network with exogenous inputs (NARX) to control the IM. This proposed strategy is characterized by high accuracy and fast dynamic response. The parameters of this network were determined through a process of selecting the best network by using the scanning method with multiple training and validation iterations with the introduction of new data. The NARX strategy was used in the early identification of faults through accurate diagnosis and classification of faults through analysis of current, temperature, and vibration signals. The results obtained showed the effectiveness and efficiency of using the NARX approach in improving the operational performance of the machine and identifying faults accurately. In Reyes-Malanche et al., ²⁵ it was proposed to use a fuzzy logic algorithm based on the Mamdani model to diagnose IM faults. This strategy is characterized by high durability and outstanding performance. This strategy allows for good control of the machine and early detection of faults that could occur to the machine during operation. This proposed strategy was implemented experimentally using a Digital Signal Processor (DSP) TMS320F28335, where the results demonstrated the effectiveness and effectiveness of the proposed approach in accurately detecting machine malfunctions. The negative of this strategy lies in the lack of a rule that allows for facilitating the use of fuzzy logic and determining the number of rules necessary to obtain good results. In Bacha et al., ²⁶ it was proposed to use a new BC technique with an SVM strategy to control the speed of an IM. This proposed approach has been proven stable using Lyapunov’s theory. This strategy was implemented in the MATLAB environment and used for experimental work, where the dSPACE 1104 real-time interface was used. The experimental results of the BC-SVM approach confirm the obtained simulation results. These obtained results highlight the effectiveness and strength of the BC-SVM approach in improving the dynamic response to speed and reducing the ripples of both current and torque. The BC-SVM approach has the disadvantages of being complex, expensive, and having a large number of gains which is undesirable. In Ali et al., ¹⁰ both the BC technique and super-twisting SMC strategy were combined to obtain a highly efficient and reliable strategy for controlling IM operation. The effectiveness and performance of this proposed strategy were compared with other strategies using MATLAB. The performance was compared in terms of the values of ISE (Integral Square error), IAE (Integral Absolute error), and ITAE (Integral Time Absolute error). The results were extracted in the following operational states: SSM (Start-Stop Mode), NOM (Normal Operation Mode), and DRM (Disturbance Rejection Mode). The simulation results showed the superiority of the proposed approach over other strategies in terms of improving the performance and effectiveness of the machine. Despite this high performance, the proposed approach has drawbacks. These drawbacks lie in the dependence of the proposed approach on the machine model, which creates several problems when the IM parameters are changed. Also, the proposed strategy is characterized by complexity and the presence of a large number of gains, which makes it difficult to adjust the dynamic response. In Zhang et al., ²⁷ a projection-based adaptive control filtered fuzzy nonsingular terminal sliding mode backstepping (PACFTB) control method was proposed to control the speed of an IM. The stability of the proposed approach was proven using Lyapunov’s theory. This proposed strategy has been verified using simulation and experimental work. This proposed approach was compared with the performance of conventional control filter backstepping and PI controller. The simulation results, along with the experimental results, confirm that the proposed control strategy has better control performance compared to other strategies. However, this proposed strategy has drawbacks despite the high performance, as it is characterized by the presence of a large number of gains and a greater degree of complexity.

In recent decades, meta-heuristic algorithms (MA) have grown prevalent in many practical fields due to their excellent performance, limited processing capacity, and time needed by deterministic algorithms in optimization issues. To get excellent results, you need easy concepts that are simple to use in different areas. Furthermore, deterministic algorithms are prone to sinking into the local optimum due to the absence of unpredictability at the last step. On the other hand, meta-heuristic algorithms may successfully avoid the local optimum by searching for all optimal solutions in the search space, thanks to the random components.

Metaheuristic methods use comparisons between things like plants, people, birds, the environment, water, gravity, and electromagnetic forces. Metaheuristic algorithms can be divided into two main kinds: nature and human as shown in Figure 1. Three subcategories emerge from the first category, which imitates physical or biological events. These three approaches are based on physics, evolution, and swarm dynamics. The primary source of inspiration for the second category is human phenomena. ²⁸

Figure 1.

Different kinds of meta-heuristic approaches.

Short types: An ant colony optimization (ACO) and a PSO, Artificial Colony of Bees (ACB), Artificial Fish Swarm Algorithm (AFSA), DE: Evolution with Differences, Central Force Optimization (CFO), Genetic Algorithm (GA), Evolution Strategy (ES), Gravitational Search Algorithm (GSA), Biogeography-Based Optimizer (BBO), Big Bang and Crunch (BBBC), Group Search Optimizer (GSO), Teaching-Learning-Based Optimization (TLBO), the Forensic-Based Investigation (FBI), and Harmony Search (HS). One of the meta-methods of improvement is through the roots of the tree.

Various works propose rooted tree optimization (RTO) to enhance the competencies of the control PSs. In Labbi et al., ²⁹ the authors propose the RTO to address the economic dispatch problem with valve-point effects, which introduce nonlinearities. The methodology draws inspiration from tree growth processes, optimizing power dispatch by efficiently exploring solution paths. The key contribution is the algorithm’s ability to handle complex, non-convex constraints, achieving better convergence, reduced computational time, and improved solution quality compared to conventional techniques such as PSO and GA, as validated by simulation results. The lateral growth rooted tree optimization (LGRTO), an enhancement of the RTO is presented in Edagbami et al. ³⁰ The main contribution of this study is eliminating premature convergence by allowing independent root growth and introducing hydrotropism and lateral growth equations. LGRTO was tested on benchmark functions and a Butterworth filter design problem, where it outperformed RTO and showed competitive results against advanced algorithms like teaching learning-based optimization (TLBO) in solving complex optimization tasks. In Bekakra et al., ³¹ an integral-proportional (IP) regulator is employed in the speed loop control of a doubly-fed IM (DFIM) using DTC, with the RTO algorithm used to tune the controller parameters by minimizing a multi-objective function. This approach addresses the limitations of usual tuning techniques, such as the pole placement and Ziegler-Nichols methods, which often result in high overshoot, undershoot, and slow settling times. The RTO algorithms enhance performance by significantly improving settling time, reducing overshoot and undershoot, and increasing robustness against system parameter variations, particularly fluctuations in stator resistance. Simulation results demonstrate that the RTO-tuned IP controller achieves superior dynamic performance compared to conventional methods, with faster response times and lower error metrics, including the integral square error (ISE), and integral absolute error (IAE), under varying load conditions.

In Edagbami et al., ³² the authors propose an advanced optimization technique designed to address the challenges posed by non-convex economic dispatch problems, particularly those involving cost functions with multiple local minima due to valve-point effects. The adaptive RTO algorithm enhances solution accuracy and convergence speed by dynamically adjusting parameters based on the search space. Simulation results show improved performance over classical optimization techniques, making it an effective solution for complex PS optimization tasks. In Bossoufi et al., ³³ an adaptive nonlinear BC approach is proposed for the doubly-fed induction generator in a wind turbine system. The method employs the RTO algorithm to optimize control parameters, enhancing system stability and robustness, particularly during parameter variations, and fluctuating wind conditions. Simulations and real-time tests using MATLAB/Simulink and a dSPACE board show notable improvements in power tracking, reduced electromagnetic torque oscillations, and lower total harmonic distortion. Another use of the RTO strategy was proposed in Sannigrahi et al. ³⁴ to improve the fuzzy logic properties of strategic incorporation of distributed generation (DG) and distributed static compensator (DSTATCOM) in radial distribution system (RDS). This proposed strategy has been used to improve the voltage profile, reduce losses, and maximize economic and environmental benefits. Fuzzy logic is used to keep the targets in the same range and the RTO strategy is used to determine the optimal location and size of the devices. This proposed strategy has been tested on 69-carrier and 118-carrier RDSs. The performance of the FL-RTO approach is compared with other existing techniques using MATLAB. The Wilcoxon signed rank test (WSRT) was applied to confirm the superiority of the proposed algorithm. All the results obtained prove the effectiveness and efficiency of using the FL-RTO strategy compared to other strategies. The FL-RTO approach is negative, as the negative of this strategy lies in choosing the number of FL rules to obtain good results. As is known, if a large number of rules are used, the control system becomes slow and requires a large memory, which is undesirable. In Meghni et al., ³⁵ the direct power control strategy based on the RTO-PI controller was used to improve the quality of energy produced by an energy system based on wind turbines. The RTO strategy was used to determine the gain values of the PI controller. The RTO-PI controller is used to determine the voltage reference values, and these reference values are used to determine the pulses needed to operate the machine inverter. The DPC-RTO-PI strategy is characterized by simplicity and ease of implementation. Also, it features a fast dynamic response. MATLAB was used to implement the proposed approach and compare it with the traditional approach. The results obtained showed the superiority of the proposed RTO-PI algorithm compared to the traditional strategy in terms of THD value, overshoot reduction, stabilization time, rise time, average active power value, total efficiency, and active power steady-state error. A new control strategy based on the RTO approach is proposed in Benamor et al. ³⁶ to reduce the phenomenon of chatter in the active and reactive powers, and to reduce the harmonic currents that appear mostly at the rotor side converter level of a double-fed induction generator. The RTO approach is used to adjust the parameters (K _i and K _p ) of the PI controller (RTO-PI), as the use of this strategy allows for improving the characteristics of the studied system in terms of power quality and reducing the THD value. The proposed approach is implemented using MATLAB, where several different tests were used to prove the effectiveness of the proposed technique compared to the traditional approach. Simulation results show that the approach based on the RTO-PI controller provides better dynamic and static performance than the traditional approach. Also, using this proposed approach increases the power quality and reduces the THD of current significantly compared to the traditional approach.

In this paper, a new controller is proposed to control IM operation. This new control depends on the use of the synergetic-SMC (SSMC) controller to improve the performance and effectiveness of the BC of IM strategy. Therefore, the SSMC-BC strategy is considered the first major contribution of this paper. This proposed strategy is characterized by high durability and outstanding performance in reducing torque and current ripples. In addition, the gains values of the proposed strategy were used to calculate the RTO algorithm. The use of this algorithm was relied upon due to the results presented in Meghni et al. ³⁵ The use of the RTO algorithm is the second major contribution of this paper. The use of the RTO algorithm allows for increasing the characteristics of the SSMC-BC strategy and increases the effectiveness of reducing torque ripples and reducing the value of THD of current.

The strategy SSMC-BC-RTO was applied to IM using MATLAB and compared the performance with the SSMC-BC approach. Several different tests were used to study the efficiency and effectiveness of the proposed approach, where the graphical and numerical results showed the superiority of the proposed approach over the SSMC-BC approach in terms of dynamic response and reducing the values of IAE, ITAE, ISE, and ITSE. The objectives achieved from the work completed can be summarized in the following points:

• Overcoming the problems and disadvantages of the BC of IM strategy

• Significantly improve the quality of torque and current.

• Increased robustness compared to the SSMC-BC approach.

• Underestimating the THD of current compared to the SSMC-BC approach and related works.

• Underestimating the overshoot of rotor flux.

• Improved dynamic response to speed, flux, and torque.

The following is the organization of the rest of the paper: the IM model is presented in Section 2. Section 3 presents the use of the BC technique in MI, applied to the FOC approach of MI. The designed SSMC method is presented in Section 4. The concept of RTO and its mathematical formulas are discussed in Section 5. The application of RTO to the minimization of a multi-objective function in the SSMC approach is explained in Section 6. Section 7 presents simulation results obtained on an MI. Finally, section 8 presents the conclusions. Section 6 presents the graphical and numerical simulation results obtained from the proposed tests. Finally, Section 7 presents the overall conclusions obtained from the performed work while mentioning limitations and future work.

IM model

IM is considered one of the most prominent machines that can be relied upon in the industrial field because of its positive features that make it the best compared to other types. These machines have a high return and low cost, as their use in industrial systems does not require periodic maintenance, which makes their lifespan longer. ³⁷ There are types of IMs, the most prominent types of which can be mentioned are the squirrel-cage IM ³⁸ and doubly-fed IM. ³⁹ Among the most prominent applications of these machines can be renewable energies, as electrical energy can be generated using these machines. ⁴⁰ Also, IMs are used in the field of pushing and traction. The electric car is one of the most prominent applications of IMs, as the use of IMs in the electric car ledes to high durability and great speeds. In this paper, a squirrel cage type IM was used, as this type was used because of its many advantages such as high durability and low cost. Also, this type is easy to control. ⁴¹ These machines have two main parts: the stator and the rotating part. The stator contains coils that are responsible for generating the electric field. ⁴² The rotating part is made of aluminum rods. To give the mathematical model for this machine under study, the Park transformation is used. The mathematical model of the machine depends on the relationships between flux, current, and voltage. The mathematical model of the IM can be divided into two parts: one section represents the electrical equations and another section represents the mechanical equations.

Equation (1) represents the mathematical model in the d-q axes of the studied machine using matrices. ⁴³ This equation makes it easier to understand the IM and implement it in MATLAB.

$$\begin{array}{c}\stackrel{·}{x}=A\left[x\right]+B\left[u\right]\hfill \\ \multicolumn{1}{c}{y=C\left[x\right]}\end{array}$$

(1)

With:

$$\begin{array}{c}\multicolumn{1}{c}{A=\left[\begin{array}{cccc}-(\frac{1}{\sigma T_s}+\frac{M^2}{\delta L_r{T}_r})\hfill & 0& \frac{M}{\delta L_r{T}_r}& \frac{M}{\delta L_r}\\ \multicolumn{1}{c}{0}& -(\frac{1}{\sigma T_s}+\frac{M^2}{\delta L_r{T}_r})& -\frac{M}{\delta L_r}& \frac{M}{\delta L_r{T}_r}\\ \multicolumn{1}{c}{\frac{M}{T_r}}& 0& -\frac{1}{T_r}& 0\\ \multicolumn{1}{c}{0}& \frac{M}{T_r}& 0& -\frac{1}{T_r}\end{array}\right];}\\ \multicolumn{1}{c}{\mathrm{B}=\left[\begin{array}{cc}\delta \hfill & 0\\ \multicolumn{1}{c}{0}& \delta \\ \multicolumn{1}{c}{0}& 0\\ \multicolumn{1}{c}{0}& 0\end{array}\right]}\end{array}$$

Using equation (2), the mechanical part of the machine can be expressed. ³⁷ This equation gives the development of speed and through this equation, the operation of a generator or engine can be controlled. The speed development is related to the difference between the torque of the machine and the torque of the load.

$$\frac{d\mathrm{\Omega }}{\begin{array}{c}\mathrm{dt}\hfill \end{array}}=\frac{1}{J}(T_e-T_L-f.\mathrm{\Omega })$$

(2)

Where, T _e is the electromagnetic torque, J is the inertia, TL is the load torque, and Ω is the speed.

The value of the electromagnetic torque of the machine is related to the flux and current. This torque can be changed by changing the current. Equation (3) expresses the torque of the machine used in Jose Ramon et al. ⁴¹

$$T_e=\frac{3M}{\begin{array}{c}2L_r\hfill \end{array}}.P({\varphi }_{\mathrm{dr}}{i}_{\mathrm{qs}}-{\varphi }_{\mathrm{qr}}{i}_{\mathrm{ds}})$$

(3)

In the field of control, several controllers have been proposed to control the operation of IM. The most prominent of these controllers are the PI, SMC, and SC controllers. As is known, the PI controller is characterized by simplicity, ease of implementation, and fast dynamic response. However, using a PI controller in the field of control gives unsatisfactory results if the system parameters change, which is a negative matter. On the other hand, using the SMC approach gives satisfactory results, but it is linked to the mathematical model of the system, which makes its application in complex systems difficult. Moreover, the SMC approach is associated with the chattering problem, as this negativity creates several problems in the control system. Accordingly, in the next part, a new, reliable controller will be discussed in the field of control. This controller is based on the combination of SMC and SC techniques.

SSMC approach design

In this section, a new method that relies on a hybrid of SMC and SC theory, wishes to enhance the approach and minimize the chattering phenomenon. ⁴⁴ This proposed strategy is a development of the SMC approach using the SC strategy. According to the work done in Nicola and Nicola, ⁴⁵ the SC approach is a different strategy from the SMC technique. Moreover, in Habib and Hamza, ⁴⁶ the SC approach gave a faster dynamic response and lower energy undulations than the SMC approach.

Using the work done in Yousfi et al., ⁴⁷ the SMC approach can be expressed by equation (4).

$$U(t)=U_n(t)+U_{\mathrm{eq}}(t)$$

(4)

With:

$$U_n(t)=-K\mathrm{sgn}(S(t))$$

(5)

Where, U _n is the switching control.

U _eq is the equivalent control, K is a positive switching gain, and S(t) is the sliding surface.

In the SMC approach, the equivalent control is responsible for the chattering problem. Therefore, it was proposed in Benbouhenni and Bizon ⁴⁸ to replace it with the SC strategy. In this way, the performance and effectiveness of the SMC approach are improved, as the use of the SC strategy increases the robustness and effectiveness of the SMC strategy.

According to the work done in Bounasla, ¹¹ the MM of the SC approach can be expressed by equation (6). According to this equation, the SC approach is characterized by simplicity, few gains, easy to implement, and can be applied in complex systems with ease. ⁴⁹ Using the SC strategy does not require knowledge of the MM of the carefully studied system, which makes it the most suitable for industrial applications. ⁵⁰

$$T\frac{\mathrm{ds}}{\mathrm{dt}}+s=0$$

(6)

Where, T is the convergence speed (T > 0).

The SC approach is used as an effective solution to increase the performance and efficiency of the SMC approach, where the equivalent control of the SMC technique is compensated by the SC strategy. The combination of the SC and SMC approach results in the SSMC approach. This proposed approach can be expressed by equation (7).

$$K(t)=T\frac{\mathrm{ds}}{\mathrm{dt}}-\lambda \mathrm{sgn}\left(s\right)+s$$

(7)

Where, λ is the positive gain.

Through equation (7), it is noted that the SSMC approach is characterized by simplicity, ease of implementation, and few gains, and does not require knowledge of the mathematical model of the system under study. The SSMC approach is easy to adjust, as gain values can be calculated using smart strategies such as genetic algorithm, or can rely on simulation and experimentation. In this work, the RTO algorithm was relied upon to calculate the gain values of the SSMC controller.

To prove the stability of the SSMC approach, Lyapunov’s theory can be relied upon. This theory is based on the use of equation (8). Also, the Budd curve can be relied upon to prove the stability of this approach.

$$s(x).\stackrel{·}{s}(x)\le \begin{array}{c}0\hfill \end{array}$$

(8)

The SSMC approach is used in this paper in order to improve the performance and effectiveness of the BC strategy of IM. Before applying the SSMC approach to the BC approach, the BC strategy must first be discussed in detail, mentioning its negatives, and positives. In the next section, the BC strategy used to control IM operations is discussed.

Application of BC technique to control the IM speed

The BC technique is considered one of the nonlinear strategies that can be relied upon in the field of control. Using the BC technique gives better results than several existing controls such as FOC and DTC. ⁵¹ This strategy was used to control the double-fed induction generator in Sara et al. ⁵² Using this strategy gave satisfactory results compared to the traditional approach. According to the work done in Mohamed et al., ⁵³ the BC technique provided satisfactory results compared to using PI control. Using the BC technique allows for significantly improved durability and increased performance compared to using PI control. In Abdessamad et al., ⁵⁴ the BC technique was used to improve the characteristics of the maximum power point tracking (MPPT) strategy of photovoltaic systems. This proposed strategy is characterized by high performance and great effectiveness in improving the characteristics of the studied system. The simulation results showed that using the BC technique leads to increasing the effectiveness and robustness of the MPPT strategy. The BC technique was compared with the SMC approach in Echiheb et al. ⁵⁵ This comparison was done in MATLAB. First, the characteristics of each approach were given, mentioning the differences and similarities. The simulation results showed that the BC technique has outstanding performance compared to the SMC approach, which makes it a promising solution. To use the BC approach requires knowledge of the mathematical model of IM. To apply this strategy, the current expressions in the Park system must first be determined. Also, the flux expression and the relationship between torque and speed must be determined. Therefore, equations (9)–(12) represent the mathematical model of the machine that is used in the BC approach.

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{i}_{\mathrm{sd}}}{\mathrm{dt}}=-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r}).i_{\mathrm{sd}}+{\omega }_s{i}_{\mathrm{sq}}}\\ \multicolumn{1}{c}{+\frac{M}{\delta .L_r.T_r}.{\varphi }_r+\delta .V_{\mathrm{sd}}}\end{array}$$

(9)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{i}_{\mathrm{sq}}}{\mathrm{dt}}=-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sq}}-{\omega }_s{i}_{\mathrm{sd}}}\\ \multicolumn{1}{c}{-\frac{M}{\delta .T_r}\omega {\varphi }_r+\delta V_{\mathrm{sq}}}\end{array}$$

(10)

$$\frac{d{\varphi }_r}{\mathrm{dt}}=\frac{M}{T_r}{i}_{\mathrm{sd}}-\frac{1}{T_r}{\varphi }_r$$

(11)

$$\frac{d\omega }{\mathrm{dt}}=\frac{3}{2}\frac{P.M}{J{L}_r}{\varphi }_r.i_{\mathrm{sq}}-\frac{T_L}{J}-\frac{f}{J}\omega$$

(12)

With:

$$T_s=\frac{L_s}{R_s},T_r=\frac{L_r}{R_r},\sigma =\frac{M}{L_s.L_r},\delta =\frac{1}{\sigma .L_s}$$

BC technique is a recursive design procedure that links the choice of a control Lyapunov function with the design of a feedback controller and guarantees global asymptotic stability of strict feedback systems.^56,57 The Lyapunov function ensures both performance and global stability. ⁵⁸

To apply the BC approach to control IM operation there are two stages.

Step 1

$$\begin{array}{c}{e}_1={\omega }^{*}-\omega \hfill \\ e_2={\varphi }_r^{*}-{\varphi }_r\hfill \end{array}$$

(13)

Where, e ₁ is the speed error and e ₂ is the rotor flux error.

Using the first Lyapunov function (V₀) represented in equation (14).

$$V_0=\frac{1}{2}(e_1^2+e_2^2)$$

(14)

Then, their derivative of V₀ is given by

$${\stackrel{·}{V}}_0=e_1{\stackrel{·}{e}}_1+e_2{\stackrel{·}{e}}_2$$

(15)

Then the error dynamical equations are:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_0=e_1({\stackrel{·}{\omega }}^{*}-\frac{3}{2}\frac{\mathrm{pM}}{J{L}_r}{\varphi }_r{i}_{\mathrm{sq}}-\frac{T_L}{J})}\\ \multicolumn{1}{c}{+e_2({\stackrel{·}{\varphi }}_r^{*}-\frac{M}{T_r}{i}_{\mathrm{sd}}+\frac{1}{T_r}{\varphi }_r)}\end{array}$$

(16)

Where, K₁ and K₂ are the positive gain.

Equation (17) represents reference values for currents components in the Park system:

$$\begin{array}{c}{i}_{\mathrm{sq}}^{*}=\frac{2}{3}\frac{J{L}_r}{\mathrm{pM}{\varphi }_r}({\stackrel{·}{\omega }}^{*}+\frac{T_L}{J}+K_1{e}_1)\hfill \\ i_{\mathrm{sd}}^{*}=\frac{T_r}{M}({\stackrel{·}{\varphi }}_r^{*}+\frac{1}{T_r}{\varphi }_r+K_2{e}_2)\hfill \end{array}$$

(17)

• Step 2

According to the principle of the BC strategy, equations (9) and (10) can be written by equations (18) and (19).

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{i}_{\mathrm{sd}}}{\mathrm{dt}}=-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r}).i_{\mathrm{sd}}+{\omega }_s{i}_{\mathrm{sq}}}\\ \multicolumn{1}{c}{+\frac{M}{\delta .L_r.T_r}.{\varphi }_r+\delta .V_{\mathrm{sd}}}\end{array}$$

(18)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{i}_{\mathrm{sq}}}{\mathrm{dt}}=-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sq}}-{\omega }_s{i}_{\mathrm{sd}}-\frac{M}{\delta .T_r}\omega {\varphi }_r}\\ \multicolumn{1}{c}{+\delta V_{\mathrm{sq}}}\end{array}$$

(19)

The error current (i* _sq and i* _sd ) of the stator in the Park system can be expressed by the equation (20).

$$\begin{array}{c}{e}_3=i_{\mathrm{sq}}^{*}-i_{\mathrm{sq}}\hfill \\ e_4=i_{\mathrm{sd}}^{*}-i_{\mathrm{sd}}\hfill \end{array}$$

(20)

Where, e₃ is the error of the quadrature stator current and e₄ is the error of the direct stator current.

Equation (21) is considered a function of Lyapunov’s theorem used to embody the BC approach.

$$V_T=V_0+\frac{1}{2}(e_3^2+e_4^2)$$

(21)

The derivation of equation (21) can be written according to equation (22).

$${\stackrel{·}{V}}_T={\stackrel{·}{V}}_0+e_3{\stackrel{·}{e}}_3+e_4{\stackrel{·}{e}}_4$$

(22)

The derivation of equation (20) can be written according to equation (23).

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{e}}_3=\frac{{\mathrm{di}}_{\mathrm{sq}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r}).i_{\mathrm{sd}}+{\omega }_s.i_{\mathrm{sq}}}\\ \multicolumn{1}{c}{-\frac{M}{\delta .T_r}.\omega .{\varphi }_r-\delta .V_{\mathrm{sq}}}\\ \multicolumn{1}{c}{{\stackrel{·}{e}}_4=\frac{{\mathrm{di}}_{\mathrm{sd}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r}).i_{\mathrm{sd}}-{\omega }_s.i_{\mathrm{sq}}}\\ \multicolumn{1}{c}{-\frac{M}{\delta .L_r.T_r}.{\varphi }_r-\delta .V_{\mathrm{sd}}}\end{array}$$

(23)

The reference values of the stator voltage in the Park system can be expressed by the equation (24).

$$\begin{array}{c}\multicolumn{1}{c}{V_{\mathrm{sq}}^{*}=\delta (\frac{{\mathrm{di}}_{\mathrm{sq}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sq}}+{\omega }_s{i}_{\mathrm{sd}}}\\ \multicolumn{1}{c}{-\frac{M}{\delta .T_r}\omega {\varphi }_r+K_3{e}_3)}\\ \multicolumn{1}{c}{V_{\mathrm{sd}}^{*}=\delta (\frac{{\mathrm{di}}_{\mathrm{sd}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sd}}}\\ \multicolumn{1}{c}{-{\omega }_s{i}_{\mathrm{sq}}-\frac{M}{\delta .L_r.T_r}{\varphi }_r+K_4{e}_4)}\end{array}$$

(24)

Where, $K_3\mathrm{and}{K}_4$ are positive design gains constants.

In order to ensure the stability of the BC approach, the condition listed in equation (25) must be met.

$${\stackrel{·}{V}}_T=k_1{e}_2^2+k_2{e}_2^2+k_3{e}_3^{2}e+k_4{e}_4^2\le 0$$

(25)

Despite the high performance and robustness of the BC approach, using this approach does not give good results in terms of ripples, especially if the machine parameters change. Several solutions were proposed in the field of control to increase the performance and effectiveness of the BC approach. The most prominent of these solutions was the use of neural networks, ⁵⁹ fractional calculus, ⁶⁰ FL approach, ⁶¹ and SMC technique. ⁶² Using these solutions does not give very satisfactory results. In the next part, it is proposed to use an SSMC controller based on the RTO algorithm to improve the performance and effectiveness of the BC approach.

BC approach based on SSMC-RTO controller

In this section, a new BC strategy is proposed to control IM operations. This is a new strategy based on the use of the SSMC controller based on the RTO algorithm. First, the errors are identified, and these errors used in the proposed approach are included in equation (26).

$$\begin{array}{c}{s}_{\omega }={\omega }^{*}-\omega \hfill \\ s_{{\varphi }_{\mathrm{rd}}}={\varphi }^{*}-{\varphi }_{\mathrm{rd}}\hfill \\ s_{\mathrm{isd}}=i_{\mathrm{sd}}^{*}-i_{\mathrm{sd}}\hfill \\ s_{\mathrm{isq}}=i_{\mathrm{sq}}^{*}-i_{\mathrm{sq}}\hfill \end{array}$$

(26)

Through equation (26), it is noted that there are four errors present. Therefore, four SSMC-RTO controllers can be used in the proposed strategy.

The proposed strategy is a development and change to the traditional BC approach. Relying on the use of SSMC-RTO type controls allows for increasing the effectiveness, efficiency, and durability of the BC of IM strategy.

The aim of the proposed approach based on SSMC-RTO controllers is to minimize errors in the flux rotor, speed rotor, direct load currents, and quadrature values between these sliding surfaces.

Figure 2 represents the SSMC-BC-RTO strategy proposed in this work to control the IM.

Figure 2.

The diagram of the SSMC-BC strategy based on RTO algorithm of the IM.

The proposed strategy relies on the use of the PWM strategy to generate the pulses necessary to control the operation of the machine inverter. Relying on the PWM strategy allows simplifying the control system and reducing its costs. Using the PWM strategy allows converting the reference voltage values generated by the SSMC-RTO controller into pulses. The most prominent features of the proposed approach are high durability, outstanding performance, and great efficiency.

In the proposed strategy, the SSMC controller is used to compensate for the gain values of the BC of IM approach controllers. The gain values of the BC approach are calculated using the SSMC controller according to equation (27).

$$\begin{array}{c}{K}_1=T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_1\mathrm{sgn}\left(s_{\omega }\right)+s_{\omega }\hfill \\ K_2=T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_2\mathrm{sgn}\left(s_{{\varphi }_{\mathrm{rd}}}\right)+s_{{\varphi }_{\mathrm{rd}}}\hfill \\ K_3=T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_3\mathrm{sgn}\left(s_{i_{i_{\mathrm{sq}}}}\right)+s_{i_{i_{\mathrm{sq}}}}\hfill \\ K_4=T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_4\mathrm{sgn}\left(s_{i_{\mathrm{sd}}}\right)+s_{i_{\mathrm{sd}}}\hfill \end{array}$$

(27)

The control gain values listed in equation (27) are calculated using the RTO algorithm. Figure 3 represents the SSMC-RTO control used to calculate the values of K₁, K₂, K₃, and K₄.

Figure 3.

Proposed SSMC controller: (a) error of rotor speed, (b) error rotor flux, (c) error load current quadrature, and (d) error load current direct.

Using the proposed approach, the stator currents (reference values) in the Parke system are written in the form of equation (28).

$$\begin{array}{c}{i}_{\mathrm{sq}}^{*}=\frac{2}{3}\frac{J{L}_r}{\mathrm{pM}{\varphi }_r}({\stackrel{·}{\omega }}^{*}+\frac{T_L}{J}+(T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_1\mathrm{sgn}\left(s_{\omega }\right)+s_{\omega }).e_1)\hfill \\ i_{\mathrm{sd}}^{*}=\frac{T_r}{M}({\stackrel{·}{\varphi }}_r^{*}+\frac{1}{T_r}{\varphi }_r+(T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_2\mathrm{sgn}\left(s_{{\varphi }_{\mathrm{rd}}}\right)+s_{{\varphi }_{\mathrm{rd}}}).e_2)\hfill \end{array}$$

(28)

Equation (29) represents the reference voltage values if the proposed approach is applied. This equation is a change and development of the equation (24).

$$\begin{array}{c}{V}_{\mathrm{sq}}^{*}=\delta (\frac{{\mathrm{di}}_{\mathrm{sq}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sq}}+{\omega }_s{i}_{\mathrm{sd}}\hfill \\ \multicolumn{1}{c}{-\frac{M}{\delta .T_r}\omega {\varphi }_r+(T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_3\mathrm{sgn}(s_{i_{i_{\mathrm{sq}}}})+s_{i_{i_{\mathrm{sq}}}}).e_3)}\\ \multicolumn{1}{c}{V_{\mathrm{sd}}^{*}=\delta (\frac{{\mathrm{di}}_{\mathrm{sd}}^{*}}{\mathrm{dt}}-(\frac{1}{\sigma .T_s}+\frac{M^2}{\delta .L_r.T_r})i_{\mathrm{sd}}-{\omega }_s{i}_{\mathrm{sq}}}\\ \multicolumn{1}{c}{-\frac{M}{\delta .L_r.T_r}{\varphi }_r+(T\frac{\mathrm{ds}}{\mathrm{dt}}-{\lambda }_4\mathrm{sgn}\left(s_{i_{\mathrm{sd}}}\right)+s_{i_{\mathrm{sd}}}).e_4)}\end{array}$$

(29)

The RTO algorithm was relied upon in this work because it is a new strategy and is characterized by high performance compared to other strategies. The RTO algorithm is a new approach that was introduced in 2016. Illustrates that the nearest roots to the aim may be used to get the grade of fitness and a new generation. In addition, the Roots that are far from the goal are eliminated.

Equation of RTO

The rooted tree algorithm for optimization starts, like other techniques, with the creation of an initial random population. A proposed solution is represented by one “root” in the RTO method, and a candidate’s evaluation, together with its fitness degree among the rest of the population, is represented by a “Wetness Degree (Dw).” ³¹

Rate of the nearest root to water (R_n)

This rate indicates the number of candidates, compared to the total population; need to assemble in the wetter location (the optimal solution). The novel population of the root closest to water is provided by:

$$\begin{array}{c}\multicolumn{1}{c}{x^{\mathrm{New}}(K,i_{\mathrm{ter}+1})=x^{\mathrm{Best}}(i_{\mathrm{ter}})+C_1.D_w(K)}\\ \multicolumn{1}{c}{.\mathrm{randn}.\mathrm{Upper}\frac{1}{N.i_{\mathrm{ter}}}}\end{array}$$

(30)

Where, x^New(k, i_ter+1) : new candidate for iteration (i_ter+1),

i _ter: iteration step,

Upper: upper limit of the parameter,

x ^Best(i _ter): The best solution of the preceding generation.

N: population scale,

k: candidate number,

randn: normal random number between [−1, 1].

Rate of the continuous root in its orientation (R_c)

The rate of members who have continued or advanced in the previous direction is shown by the number of points near the water. The novel random root population is provided as follows:

$$\begin{array}{c}\multicolumn{1}{c}{x^{\mathrm{New}}(K,i_{\mathrm{ter}+1})=x(k,i_{\mathrm{ter}})+C_2.D_w(K).\mathrm{rand}.}\\ \multicolumn{1}{c}{(x^{\mathrm{Best}}(i_{\mathrm{ter}})-x(k,i_{\mathrm{ter}}))}\end{array}$$

(31)

Where, rand is the number between 0,

x(k, i_ter) is the candidate for the previous iteration i_ter.

Rate of the random root (R_r)

It expresses the number of candidates in the total population, which they will randomly divide in the search space to obtain a global solution. It also replaces the low-moisture roots of the last generation. The new population of random roots is defined by:

$$x^{\mathrm{New}}(K,i_{\mathrm{ter}+1})=x_r(i_{\mathrm{ter}})+C_3.D_w(K).\mathrm{randn}.\mathrm{Upper}\frac{1}{i_{\mathrm{ter}}}$$

(32)

Where, x _r (i_ter) is the randomly selected individual from the previous generation,

C ₁, C₂, and C₃ are adjustable parameters.

The different steps of the RTO algorithm can be listed as follows:

• Step 1:

Create an initial generation, with N candidates, at random in the variable limits of the research space and calculate the values of the rates R _r , R _n , and R _c .

• Step 2:

Calculate the objective function (wetness degree (Dw)) for each member of the population in the following order:

$$D_w(k)=\{\begin{array}{c}\frac{f_k}{\mathrm{Max}(f_k)}\mathrm{for}\mathrm{Maximal}\mathrm{objective}\mathrm{function}\hfill \\ \multicolumn{1}{c}{1-\frac{f_k}{\mathrm{Max}(f_k)}\mathrm{for}\mathrm{Minimal}\mathrm{objective}\mathrm{function}}\end{array}$$

(33)

Where, $k=1,2,3.....,\mathrm{N}\mathrm{and}{f}_k\mathrm{degree}\mathrm{of}\mathrm{fitness}.$

• Step 3

Reproduce and be replaced by the novel population. For the replacement of the old population with the novel population based on R _r , R _n , and R _c , order these individuals by the wetness degree (Dw) using equations (30)–(32).

• Step 4

Go return to step 2 if the conditions for stopping are not completed.

The optimal solution is for the individual to function as the SSMC regulator and provide the optimal values for IM control (See Figure 4).

Figure 4.

Organizational diagram of the RTO algorithm implemented within the SSMC-BC approach.

Table 2 represents the parameters of the RTO algorithm used to calculate the values of the used controls.

Table 2.

RTO parameters.

Parameters	Values
Dim	8
Maximum iteration	100
Population size	25
R c  = R r	0.3
R n	1 − (R c  + R r ) = 0.4
C1 = C2	1.91
C3	1.25

Figure 5 represents the variation of the fitness function in the simulation.

Figure 5.

The variation of the fitness function in the simulation.

Table 3 represents fitness functions and optimal gains for each of the five iterations.

Table 3.

The presents the fitness functions and optimal gains for each of the five iterations.

No. of iterations	Fitness value	K 1		K 2		K 3		K 4
		λ1	T 1	λ2	T 2	λ3	T 3	λ4	T 4
1	0.00051	0.0945	0.0064	102.396	0.00040	86.43	0.00031	9.91e7	0.00296
40	0.000509	0.089	0.0042	102.763	0.00093	89.67e5	0.00063	9.99e7	0.000413
73	0.000486	0.105	0.0148	104.573	0.00036	87.21e5	0.00098	99979544.27	0.00498
77	0.000395	0.1058	0.0165	104.48	0.00043	85.85e5	0.00095	3e5	0.0001
90	0.000277	0.1091	0.0237	110	0.0001	87.73e5	0.00051	791056.46	0.0012

In Figures 6–9, the 2D and 3D concentrations of the roots (possible solutions) of the errors in rotor speed, error in flux, error in load current quadrature, and load error current direct are shown, respectively. A rapid convergence toward the optimal solution can be seen with the proposed method.

Figure 6.

(a) 2D concentration of the roots of error of rotor speed and (b) 3D concentration of the roots of error of rotor speed.

Figure 7.

(a) 2D concentration of the roots of error rotor flux and (b) 3D concentration of the roots of error rotor flux.

Figure 8.

(a) 2D concentration of the roots of error load current quadrature and (b) 3d concentration of the roots of error load current quadrature.

Figure 9.

(a) 2D concentration of the roots of error load current direct and (b) 3d concentration of the roots of error load current direct.

To study the stability of the proposed approach, both Lyapunov’s theory or the Bode curve can be used. Using Lyapunov’s theorem requires a derivation calculation. In most cases, using Lyapunov’s theorem requires complex calculations, which is not desirable. The Bode curve is a graphical method of proving stability, where both phase (°) and Magnitude (dB) are extracted. The Bode curve is a simple, easy method that does not require complex calculations. MATLAB is used to extract this curve. In this work, the Bode curve was used to prove the stability of the proposed approach. Figure 10 represents a Bode curve for two controls. According to Figure 10(a), the value of phase (°) for the SSMC-BC approach changes from 0 to −90°, and the value of Magnitude (dB) changes from −37 to −70 dB. Therefore, the value of both phase and Magnitude (dB) take negative values. Since both phase and Magnitude (dB) take negative values, the value of phase margin and gain margin is positive, and thus the SSMC-BC approach is stable.

Figure 10.

Bode curve: (a) SSMC-BC and (b) SSMC-BC-RTO.

Figure 10(b) represents the Bode curve for the proposed SSMC-BC-RTO approach. Through this figure and comparing it with Figure 10(a), it is noted that the change in the values of Magnitude (dB) and Phase (°) for the proposed SSMC-BC-RTO approach is the same as the change in the case of using the SSMC-BC approach. Since the values of Magnitude (dB) and Phase (°) take negative values, the values of phase margin and gain margin are positive, and therefore the proposed SSMC-BC-RTO approach is stable.

In the next part, the proposed approach is implemented using MATLAB, where several different tests are used to verify the proposed approach and compare it with the BC-SSMC approach.

Results

In this section, simulations are studied with an IM-operated motor that is used in this work is a 1.5 kW in SI units. The parameters of the IM as: p = 2, F = 50 Hz, L _m  = 0.556 H, R _r  = 4.05 Ω, R _s  = 5.35 Ω, L _s  = 576.3 mH, L _r  = 576.3 mH, fr = 0 Nm/s, and J = 0.0498 kg/m².

First test

In this test, the control speed stays at 157 rad/s, and a load torque of 3 at 6 and 3 N.m is provided at 1, 2, and 3s. Figures 11–16 show the simulation results for the first test.

Figure 11.

Speed (First test).

Figure 12.

Electromagnetic torque (First test).

Figure 13.

Rotor flux (First test).

Figure 14.

Stator current (First test).

Figure 15.

THD (SSMC-BC).

Figure 16.

THD (SSMC-BC-RTO).

Figure 11 represents the speed change of the machine as a function of time for the two controls. This speed changes linearly in the time domain from 0 to 0.51 s and takes a constant value estimated at 157 rad/s in the time domain from 0.51 to 4 s. The proposed approach offers a fast dynamic speed response compared to the SSMC-BC approach. Also, it is noted that the speed of the two controls is not affected by the change in load torque, which proves the robustness and performance of the two approaches. Figure 12 represents the torque for two controllers. This torque takes a maximum value at 0 s and then decreases after 0.51 s when the rotation speed reaches the nominal value. From Figure 12, it is noted that the torque follows the reference torque well, with undulations present.

Figure 13 shows the change in rotor flux as a function of time for the two controls. From this figure, it is observed that there is a fast dynamic response flux for the two controls. Also, the flux value remains constant and does not change with torque, as the flux value remains equal to the value 1 throughout the simulation period. It is also noted that there are ripples at the flux level and there is an exceedance of the reference value if the SSMC-BC approach is used.

Figure 14 represents the change in current as a function of time for the two controls. This current takes the form of a sinusoid for the two controls. The value of this current changes according to the change in torque. Also, it is noted that there are ripples at the current level, as the proposed approach gave a high quality of the current compared to the SSMC-BC approach.

Figures 15 and 16 represent the THD of current for the two controllers. From the two figures, it is noted that the THD value for the SSMC-BC approach was 0.59% and for the SSMC-BC-RTO technique it was equal to 0.51%. Therefore, the SSMC-BC-RTO technique significantly reduced the THD value compared to the SSMC-BC approach. This reduction was estimated at 13.56% compared to the SSMC-BC approach. Moreover, from Figures 15 and 16, the amplitude value of the fundamental signal (50 Hz) for the two controls was estimated at 1.469 A. So the two approaches offer the same goal. These results highlight that the quality of the current is high when using the proposed approach compared to the SSMC-BC approach.

Figures 17–20 represent Zoom in the graphical results of the first test. These figures show the superiority of the SSMC-BC-RTO technique over the SSMC-BC technique in terms of torque, flux, and current ripples. This proposed approach reduces these ripples significantly. Also, the proposed approach improves the dynamic response of both speed and flux. It is also noted that the proposed approach provided a lower value for both torque and flux overshoot than the SSMC-BC approach, which highlights the high performance of the proposed approach, as these results make this approach of interest for use in other applications.

Figure 17.

Zoom in speed response (First test).

Figure 18.

Zoom in torque response (First test).

Figure 19.

Zoom in rotor flux response (First test).

Figure 20.

Zoom in stator current response (First test).

Second test

To examine the robustness of the proposed control techniques, the nominal value of the R _r and R _s is multiplied by 2 at time 2.5 s. The results of this test are listed in Figures 21–26. Figure 21 represents the change in speed as a function of time for the two controls in the second test. Despite the change in resistance values, the speed remains the same as in the first test. The speed of the two controllers changes linearly in the time domain from 0 to 0.51 s, with the proposed approach having an advantage over the SSMC-BC approach in terms of response time. After the moment of time 0.51 s, the speed takes a constant value equal to 157 rad/s for two controls. At the moment in time, it is noted that the speed in the case of using the SSMC-BC approach was slightly affected by the change in load torque and that the speed was not affected by the change in load torque in the case of using the proposed approach.

Figure 21.

Speed response.

Figure 22.

Torque response.

Figure 23.

Rotor flux response.

Figure 24.

Current (Second test).

Figure 25.

Stator current THD (SSMC-BC).

Figure 26.

Stator current THD (SSMC-BC-RTO).

Figure 22 represents the torque change for the two controls in the second test. This torque follows the reference well for two controllers, despite changing resistance values. Also, it is observed that there are ripples at the level of this torque.

Figure 23 represents the flux change for the two controls in the second test. Despite changing the values of the resistances, the flux for the two controls follows the reference value well, with the presence of ripples. It is also noted that the flux has a fast dynamic response. If the SSMC-BC approach is used, the limit value is exceeded. From Figure 23, it is noted that the flux is not affected by the change in load torque, as the flux value remains equal to the value 1 throughout the simulation period.

Figure 24 represents the change in current as a function of time for the two controls during the second test. Despite changing the values of the resistances, the current remains sinusoidal for the two controllers. Also, the value of the current is related to the form of torque change, as the value of the current increases with an increase in load torque and decreases with a decrease in torque. It is also noted that the quality of the current is higher when using the proposed SSMC-BC-RTO approach compared to the SSMC-BC approach.

Figure 25 represents the THD of current if the SSMC-BC approach is used. Figure 26 represents the THD value if the proposed approach is used. Through these figures, the THD value of the current was estimated at 0.74% and 1.13% for both the proposed SSMC-BC-RTO approach and the SSMC-BC technique, respectively. These values show that the proposed approach significantly reduced the THD value despite the change in resistance values compared to the SSMC-BC technique. Therefore, the proposed SSMC-BC-RTO approach reduces the THD value in this test by an estimated percentage of 34.51% compared to the SSMC-BC technique. Figures 25 and 26 show that the amplitude value of the fundamental signal (50 Hz) for two controls is 1.468 A. Despite the change in resistance values, the two approaches provided the same amplitude, which is desirable for the proposed approach. These results highlight the effectiveness and efficiency of the proposed approach in improving the characteristics of the control system compared to the SSMC-BC approach.

Figures 27–30 show Zoom in graphical results for the second test for the two controls. From Figure 27, it can be seen that the speed was affected by the change in torque at the moment of 2.5 s, as the speed no longer followed the reference value well. This can be attributed to the effect of the two controls on changing the values of the resistances. Figure 28 shows that the torque ripples are lower when using the proposed approach compared to the SSMC-BC approach. Also, the proposed approach reduces the overshoot value of the torque compared to the SSMC-BC technique.

Figure 27.

Zoom in speed response (Second test).

Figure 28.

Zoom in torque response (Second test).

Figure 29.

Zoom in rotor flux response (Second test).

Figure 30.

Zoom in current response (Second test).

Figure 29 represents a Zoom of the rotor flux if both approaches are used. From this figure, it is noted that the proposed approach improves the response time to the rotor flux and reduces the overshoot value compared to the SSMC-BC technique. From Figure 30, the current takes a sinusoidal shape, with the proposed approach having an advantage in terms of quality compared to the SSMC-BC approach. These results highlight the high effectiveness and robustness of the proposed approach compared to the SSMC-BC approach. These obtained results make the proposed approach of great importance in the field of control.

Table 4 represents a study of the change in the value of both fundamental signal (50 Hz) amplitude (50 Hz) and current THD of both techniques. From this table, it is noted that the amplitude value was slightly affected in the second test compared to the first test, where its value increased. The difference in amplitude between the first and second tests was estimated at 0.001 A for the two approaches. Therefore, the two approaches present the same difference. On the other hand, the THD value was affected in the second test compared to the first test for the two controls. The THD value increased significantly in the second test for two of the controls. This increase was estimated at 49.4% and 31.08% for the BC-SSMC method and the BC-SSMC-RTO technique, respectively. Therefore, the proposed BC-SSMC-RTO approach provided a lower rate of change for THD than the BC-SSMC approach. This study highlights the robustness and effectiveness of the proposed approach in improving the THD value.

Table 4.

Effect study change of fundamental signal (50 Hz) amplitude and THD current in the second test compared to the first test.

THD and amplitude		SSMC-BC	SSMC-BC-RTO
Amplitude offundamentalsignal (50 Hz)	Test 1	1.468	1.468
	Test 2	1.469	1.469
	Test 2 − Test 1	0.001	0.001
	Ratios (%)	0.07	0.07
THD (%)	Test 1	0.59	0.51
	Test 2	1.13	0.74
	Test 2 − Test 1	0.54	0.23
	Ratios (%)	47.79	31.08

Third test

This test is different from previous tests. In this test, the effectiveness and robustness of the proposed approach are studied in terms of velocity direction change. This velocity is changed at moment 2 from 100 to 100 rad/s. The results obtained are shown in Figures 31–37.

Figure 31.

Rotor speed (Third test).

Figure 32.

Torque.

Figure 33.

Rotor flux.

Figure 34.

Stator current.

Figure 35.

Current THD (SSMC-BC).

Figure 36.

Current THD (SSMC-BC-RTO).

Figure 37.

Zoom in speed response.

Figure 31 represents the change in rotational speed of a motor if two controls are used. This speed follows the reference well, giving the proposed approach an advantage over the SSMC-BC approach in terms of response time.

Figure 32 represents the torque change as a function of time for the two controls. When starting, it is observed that the torque takes the largest value when the speed is zero, and then it decreases with an increase in the speed value. At the time point of 2 s, it is observed that the torque was affected by the change in the direction of the speed, as it took a negative value with the presence of ripples. Also, the torque follows the reference value well with two controls having quick dynamic response.

Figure 33 represents the flux change as a function of time for the two controls. Through this figure, it is observed that the flux is not affected by changing the direction of the speed, as it remains constant and equal to the value 1. When using the SSMC-BC technique, it is observed that the reference value is exceeded. Also, it is observed that there are ripples at the flux level if both controls are used.

Figure 34 represents the change in current as a function of time for the two controls. From this figure, it is noted that the value of the current is related to the change in torque. This current has a sinusoidal shape, which is the same as the observations in the previous tests. Also, it is noted that the current is affected by the change in the direction of the speed at the moment 2 s. When a load is applied at moment 3 s, the current increases, which is normal.

Figures 35 and 36 represent the value of THD and amplitude of a fundamental signal (50 Hz) for the two controls. According to Figure 35, the values of THD and amplitude for the SSMC-BC strategy were estimated at 0.63% and 1.468 A, respectively. In the case of the proposed approach, the values of THD were estimated at 0.56% and 1.468 A, respectively. Through these values, it is noted that the proposed approach and the SSMC-BC strategy provided the same amplitude value for the fundamental signal (50 Hz). According to Figures 35 and 36, the THD value is low if the proposed approach is used compared to the SSMC-BC strategy. This reduction was estimated at 11.11% compared to the SSMC-BC strategy. The results obtained in this test highlight the high performance and effectiveness of the proposed approach in improving the quality of current and torque compared to the SSMC-BC strategy.

Figures 37–40 represent Zoom in the results of the third test. According to Figure 37, the proposed approach has a fast dynamic response to speed compared to the SSMC-BC strategy. Figure 38 shows that the torque ripples are lower when using the proposed approach compared to the SSMC-BC strategy. Also, the proposed approach gave a better dynamic torque response than the SSMC-BC strategy. According to Figure 39, the flux ripples are lower when using the proposed approach compared to the SSMC-BC strategy. Also, it is noted that the flux in the case of using the SSMC-BC strategy is affected at the time point of 3 s by applying the load, where a decrease in the flux value is observed and then it returns to the reference value. However, the flux in the case of using the proposed approach was not affected by the application of the load in 3 s. According to Figure 40, the value of the current is affected by a change in torque, and the quality of the current is high if the proposed approach is used compared to the SSMC-BC strategy.

Figure 38.

Zoom in torque response.

Figure 39.

Zoom in rotor flux response.

Figure 40.

Zoom in stator current response.

Table 5 represents the change in the values of the fundamental signal amplitude (50 Hz) and current THD for two controls between the third test and the first test. From this table, it is noted that the amplitude value was not affected in the third test, as it remained equal to the value in the first test. So the amplitude value was not affected in the third test of the two controls. However, the THD value of the current increased in the third test compared to the first test for the two controls. This rise can be attributed to a change in the direction of rotation. This increase was estimated at 6.53% and 8.93% for both the SSMC-BC approach and the proposed SSMC-BC-RTO approach, respectively. Therefore, the SSMC-BC-RTO approach provided a greater THD effect than the SSMC-BC strategy. This study shows that the THD value of a current is affected by changing the direction of rotation and that its amplitude is not affected by changing the direction of speed.

Table 5.

Effect study change of fundamental signal (50 Hz) amplitude and current THD in the third test compared to the first test.

Amplitude and THD (%)		SSMC-BC	SSMC-BC-RTO
Amplitude offundamentalsignal (50 Hz)	Test 1	1.468	1.468
	Test 3	1.468	1.468
	Test 3 − Test 1	+0	+0
	Ratios (%)	00	00
THD (%)	Test 1	0.59	0.51
	Test 3	0.63	0.56
	Test 3 − Test 1	+0.04	+0.05
	Ratios (%)	6.53	8.93

Performance criterion

The transient error can be used to characterize the quality of a given slave system. The quality assessment of a given servo system is made by evaluating a quality criterion or performance index. Evaluation is mainly based on the index response of the servo system considered.

In practice, the SSE is never zero. Integrating a function of this error over an infinite interval gives the performance index used an infinite value. To overcome this drawback, the integration is often carried out over a finite interval [0, T], the upper limit of which T of the interval is chosen to be greater than the response time of the system.

Criteria ISE

This performance criterion is the most popular. It is defined by:

$$J_{\mathrm{ISE}}=\underset{0}{\overset{T}{\int }}{e}^2(t)\mathrm{dt}$$

(34)

The presence of (t) in the criterion allows the highlighting of high amplitude transient deviations.

Criteria IAE

It is sometimes advantageous to highlight the end of regime values transient. The ITAE performance benchmark allows this. Such a criterion is defined by:

$$J_{\mathrm{IAE}}=\underset{0}{\overset{T}{\int }}t|e(t)|\mathrm{dt}$$

(35)

Criteria ITAE

The ITAE performance criterion makes it possible to lightly weight the start of the transient regime and simultaneously strongly weight the error values at the end of the transient regime.

Such a criterion is defined by:

$$J_{\mathrm{ITAE}}=\underset{0}{\overset{T}{\int }}t|e(t)|\mathrm{dt}$$

(36)

Criterion ITSE

The ITSE performance criterion makes it possible to lightly weight the start of the transient regime and simultaneously strongly weight the error values at the end of the transient regime. Such a criterion is defined by:

$$J_{\mathrm{ITSE}}=\underset{0}{\overset{T}{\int }}t{e}^2(t)\mathrm{dt}$$

(37)

In this part, the performance criterion for the PS control in the absence and presence of the algorithm is calculated and compared in each test with the results obtained in MATLAB.

Test 1

Table 6 represents the results of the competence criteria are presented, and the algorithm is compared to the conventional approach of test 1. It is noted that the proposed approach gave values for JISE, JIAE, JITAE, and JITSE much better than the BC-SSMC approach, which is a good thing, highlighting the superiority of this approach and its great ability to improve the characteristics of the control system.

Table 6.

A comparison between the two strategies in terms of the values of the ITSE, ISE, IAE, and ITAE for the first test.

	Techniques	J ISE	J IAE	J ITAE	J ITSE
Speed	BC-SSMC	5772	52.6	11.46	922.8
	BC-SSMC-RTO	5101	47.92	9.813	760
Flux	BC-SSMC	0.01436	0.02308	0.000734	0.0001516
	BC-SSMC-RTO	93.4e–3	0.014	41.68e–4	6.24e–5
Current I d	BC-SSMC	0.01609	0.08404	0.1585	0.004877
	BC-SSMC-RTO	0.0559	0.090	0.1627	0.00498
Current I q	BC-SSMC	0.0249	0.2593	0.5455	0.048
	BC-SSMC-RTO	0.02573	0.2544	0.5353	0.4709

Test 2

Table 7 represents the results of the performance criteria are presented, and the algorithm is compared to the conventional technique of test 2. From this table it is noted that the proposed approach provided much better values than the traditional approach, which shows the extent of its effectiveness and efficiency in improving the characteristics of the control system.

Table 7.

A comparison between the two techniques in terms of the values of the ISE, ITSE, IAE, and ITAE for the second test.

Physical and electrical properties	Strategies	J ISE	J IAE	J ITAE	J ITSE
Speed	BC-SSMC	5772	53.13	13.22	923.6
	BC-SSMC-RTO	5488	51.23	12.48	856.1
Flux	BC-SSMC	0.01436	0.02671	0.01278	0.0001832
	BC-SSMC-RTO	93.56e–3	0.01813	0.1147	8.891e–5
Current I d	BC-SSMC	0.0765	0.09744	0.1862	0.006788
	BC-SSMC-RTO	0.05654	0.09678	0.1842	0.00637
Current I q	BC-SSMC	0.02506	0.2553	0.5343	0.04678
	BC-SSMC-RTO	0.023652	0.2544	0.5339	0.04378

Test 3

Table 8 represents the values of ISE, ITSE, IAE, and ITAE obtained in the third test for the two controls. From this table, it is noted that the proposed SSMC-BC-RTO approach significantly reduces the values of ISE, ITSE, IAE, and ITAE compared to the SSMC-BC approach. These results highlight the superiority of the proposed SSMC-BC-RTO approach and its ability to improve the characteristics of the control system. These results make the proposed SSMC-BC-RTO strategy interesting for application in other fields.

Table 8.

A comparison between the two approaches in terms of the values of the ISE, ITSE, IAE, and ITAE for the third test.

Physical and electrical properties	Approaches	J ISE	J IAE	J ITAE	J ITSE
Speed	BC-SSMC	1.31e4	95.45	165.12	2.23e4
	BC-SSMC-RTO	1.104e4	92.86	164.2	2.102e4
Flux	BC-SSMC	0.000956	0.01486	0.00054	0.0004926
	BC-SSMC-RTO	0.0093	0.01477	0.00047	6.23e–6
Current I d	BC-SSMC	0.0765	0.09744	0.1862	0.006788
	BC-SSMC-RTO	0.05673	0.09734	0.1796	0.0067
Current I q	BC-SSMC	0.0174	0.238	0.3566	0.0295
	BC-SSMC-RTO	0.01679	0.1654	0.3326	0.02706

Table 9 represents a comparison with related works in terms of the THD of current value. From this table, it is noted that the proposed approach gave a better THD value than several existing strategies. The proposed approach in the first test gave a THD value equal to 0.51%. This value was better than the work ⁶³ which dealt with the BC technique based on the third-order SMC technique (1.09% and 3.92%). The proposed approach also gave a better value than the seven-level neural DTC approach in the work, ⁶⁵ where the THD value was equal to 12.29%. This comparison highlights the effectiveness and efficiency of the approach in reducing the THD value of a current, which can be said that the proposed approach significantly improves the quality of the current compared to several strategies such as the predictive DTC strategy ⁶⁷ and FOC technique based on parallel PI controllers. ¹ This comparison makes the proposed approach of future interest in the field of control.

Table 9.

Comparison with related works in terms of THD of current value.

References	Techniques	THD (%)
Zellouma et al. 63	BC approach based on third-order SMC techniques	1.09
		3.92
Zellouma et al. 64	BC technique	2.56
		2.74
		2.70
Benbouhenni 65	Seven-level neural DTC approach	12.29
	Seven-level DTC approach	26.92
Habib 66	24 secteurs neural DTC approach	10.41
	24 secteurs DTC approach	22.70
Davis et al. 67	Predictive DTC strategy	13.40
Salem 68	Speed sensorless control using extended Kalman filter	1.65
Zellouma et al. 1	FOC technique	3.58
	FOC based on parallel PI controllers	1.86
Proposed technique (BC-SSMC-RTO)	Test 1	0.51
	Test 2	0.74

Conclusions

This study proposed a new strategy based on the combination of BC and SSMC to control the induction motor. The RTO algorithm was used to calculate the gain values of the proposed control method. The proposed approach was implemented in a MATLAB environment using different tests and compared to the SSMC-BC approach. Simulation results showed that the proposed control method improves the operational performance of the induction motor under different operating conditions. The proposed approach significantly improves the dynamic response of speed and flux. Also, the proposed approach reduces the THD of current compared to the SSMC-BC approach. The simulation results showed that the proposed method performs well when changing the parameters of the induction motor. The effectiveness of the proposed approach was studied using simulation only, as the study was limited to using only some tests, which limits the work done and does not lead to significantly exploring the extent of the effectiveness and strength of the proposed approach. In addition, the lack of data is one of the most prominent challenges facing this work carried out to compare the proposed strategy with other works, as the comparison was limited to the value of THD of current. The lack of time and necessary equipment are among the most prominent challenges facing the proposed approach and the failure to explore performance in real operational conditions.

In the future, the work will be attempted experimentally using real tools and comparing the results with other strategies. Also, suggest other strategies such as a backstepping-super twisting algorithm to control the induction motor.