Feedback Linearization technique for Speed and Torque control of Switched Reluctance Machine

. This paper proposes a feedback linearization controller design for torque ripple minimization and speed controller in switched reluctance motor (SRM) drive systems. A dynamic model system of equations in SRM is used to develop the proposed controller. The control strategy is formed by two loops. The first loop aims to control the machine's speed and generate the torque reference. In addition, the other loop is used to control the current to minimize torque ripple and the error between the desired speed and true speed. In order to validate our proposed control strategy, we used the MATLAB-Simulink toolbox. The proposed controller is being evaluated and compared to a conventional PI regulator to showcase its superior performance and advantages. Finally, the obtained results validate the proposed controller's robustness and efficiency in variable speed and load.


Introduction
The switched reluctance motor (SRM) is a type of electric motor that operates by the principle of reluctance torque [1].In addition, The SRM consists of a rotor with salient poles and a stator with multiple coils.The rotor is connected to a mechanical load, and the stator is supplied with electric current to generate a magnetic field [2].However, SRMs are commonly used in several applications, such as electric vehicles.They have some advantages over other types of motors, such as high torque density, high efficiency, and good performance at high speeds.However, they also have some disadvantages, such as high acoustic noise and vibration levels, which can limit their use in some applications [3].Numerous studies have been carried out to minimize the high torque ripple and speed control in SRMs.In general, reducing torque ripple is successfully accomplished by optimizing the SRM structural parameters.Numerous academics have suggested novel structures, such as the 18/15/6 pole [4], the design principles and performance optimization for segmented-rotor switched reluctance motors [5], and a switched reluctance machine (SRM) with the minimum possible stator core [6].Besides, many techniques have been proposed in the literature for torque ripple and speed control of SRM using artificial intelligence.Accordingly, fuzzy logic and neural networks for speed control [7].In [8], neural networks and PID are used for minimizing torque ripples in SRM.Further, a fuzzy adaptive controller optimizes torque ripples is discussed in [9].In another hand, the issue of decreasing torque ripples has been solved by a number of control-based techniques, such as the optimization of the excitation angles [10] and the control of the currents absorbed by the machine [11].In fact, an adaptive control technique is proposed for the best turn-on and turn-off angles of SRM over a wide range of speed control [12].However, another approach for torque optimization using a fuzzy adaptive controller is developed in [13].In [14], an off-line Transfer sharing function is proposed for copper loss reduction and torque ripple.In recent decades, various nonlinear controllers have been developed taking into account SRM saturation (inductance maximum).For example, predictive controllers, intelligent nonlinear controllers, sliding mode controllers, etc. Predictive control gives lower torque ripple compared to other methods [15].This work aims to develop a new control system for SRM.Therefore, the statement indicates that the primary objective of the ongoing work or research project is to create an innovative control system designed for the specific application of SRM.This implies an interest in improving or optimizing the control and performance of SRM, possibly for various industrial or technological applications.However, the proposed controller includes two block (loops) speed/current controllers.The first control loop within the system has a dual purpose: regulating the machine's speed and generating a torque reference.This primary loop plays a central role in ensuring that the machine operates at the desired speed and generates the required torque.Moreover, the second loop in the proposed controller is to control the converter feed SRM to minimize the torque ripple and minimal error between the desired speed and the true speed.In summary, the statement suggests that the second control loop within the system is specifically designed to regulate or control the converter, which, in turn, provides power or feeds the Switched Reluctance Motor (SRM).This control loop likely plays a crucial role in ensuring the proper operation and performance of the SRM in the context of the larger system or process.Finally, a specific SRM 8/6 MATLAB model, is adopted to evaluate by simulation the proposed controller.The test results prove the proposed controller's effectiveness and robustness and accuracy when the load torque and speed change.In addition, according to the simulation, the proposed controller can minimize the torque ripple and the error between the desired speed and the true speed.Finally, to test the efficiency and robustness of the proposed controller we are comparing its performances with those of a conventional PI regulator.

Modelling of SRMs mathematically
The switched reluctance machine (SRM) is one of the machines that can operate at high speed and with a wide constant power range.The SRM consists of the stator and rotor phases.Additionally, SRMs are characterized by prominent poles on both the stator and rotor, with coils wrapped around the stator poles and coupled in diametrically opposite pairs.(as shown in Figure 1) [16].Switched Reluctance Motors (SRMs) offer several advantages in various applications due to their unique design and operating characteristics.Here are some of the key advantages of SRM: • High Efficiency: SRMs can achieve high levels of efficiency, especially at partial loads.Their simple and robust construction contributes to this efficiency, making them suitable for applications where energy savings are critical.
• Wide Speed Range: SRMs are capable of operating at a wide range of speeds without the need for complex control systems or gearboxes.This versatility makes them suitable for applications with varying speed requirements.
E3S Web of Conferences 469, 00058 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900058 • Robust and Reliable: SRMs have a relatively simple construction, with no brushes or permanent magnets.This simplicity makes them more robust and less prone to wear and tear compared to other motor types.They are well-suited for harsh environments and high-temperature applications.
• High Power Density: SRMs can achieve high power density, providing a lot of power output for their size and weight.This makes them suitable for applications where space is limited.
• Cost-Effective: SRMs can be cost-effective to manufacture because they do not require expensive permanent magnets.Their simplified design can lead to lower production costs, making them an attractive option for certain applications.
• Precise Control: SRMs can offer precise control of torque and speed, making them suitable for applications where accurate positioning and control are necessary.
• Regenerative Braking: SRMs can easily implement regenerative braking, allowing them to recover energy during braking and feed it back into the power supply, improving overall efficiency.
• Flexibility in Power Supply: SRMs can operate on various power supplies, including AC or DC, making them versatile for different applications and grid configurations.
It's important to note that while SRMs offer these advantages, they also have some limitations and may not be suitable for all applications.The choice of motor type depends on the specific requirements and constraints of the application.By using the fundamental laws of dynamics and the usual electrical laws the dynamic system of equation in SRM is given by: The voltage equation of a phase winding is given by: Where R is the resistance per phase and φ is the flux linkage per phase given by: = 0. So, the parameters   given by the relation: The torque   is expressed as the relationship between the current i and the angle θ.Fig. 2 shows the static torque characteristics for SRM.

PI controller
A PI controller is a type of proportional-integral controller used in control systems engineering.It is a feedback control mechanism that uses a combination of proportional and integral actions to regulate a process variable, such as temperature, pressure, or flow rate, to a set point value [18].Using the PI controller approach, the speed controller design is established.In an interval proposed of  we have these equations: In summary, the statement indicates that equation (7) represents a system in which speed is influenced by   , and this system is considered linear.To control the speed and generate the required torque reference, the proposal is to implement a PI controller.The PI controller's role is to adjust the control inputs to ensure that the system operates as desired in response to changes in   , making it suitable for linear control systems.The speed tracking error: Where  * and  the speed reference and speed of SRM respectively.In equation (7): However,   is not a true control is as a virtual command.Accordingly, let us suppose   * the torque reference signal so.
The PI controller is given follows: Where () present the error between the desired speed and the true speed.So, in the equation (11) we are replaced () with the torque reference   * we get.
Finally, we will replace the equations ( 8) and ( 10) in ( 12) then we found the following formula: We are considered the load torque and the reference speed is constant.In the equation (13) we will remove the part of the integrator.

𝐽
The transfer function of closed loop system is given by: So, we are identified the system given by equation ( 15) with second order system.Finally, the   and   as given follows: Fig. 3, Describes the strategy used to control the first loop.

feedback Linearization
The linearization feedback method is a technique used to design controllers for nonlinear systems by linearizing the system's dynamics around a specific operating point [19].The method involves finding the linearized dynamics of the system by Taylor series expansion around the operating point, and then designing a linear controller for the linearized system.According to the statement, the following is described a control methodology that begins with the linearization of a dynamic system through a Taylor series expansion around an operating point.Once the linearized dynamics are determined, a linear controller is designed for the simplified linear system.This approach is often used in control system design to achieve precise control over systems with nonlinear characteristics, particularly within a limited operating range.Finally, the proposed controller based on feedback Linearization technique for SRM is shown in Fig. 4.

E3S
To define the conventional TSF, a mathematical expression must be used to develop the dynamic behavior of each phase torque of the machine.The commonly used TSF types are the nonlinear TFS.The nonlinear TSFs include the sinusoidal and the exponential TSFs.A typical nonlinear TSF based on sinusoidal formulation is presented in Fig. 5.

Fig. 5. Sinusoidal TSF
The torque of the i phase can be defined using the following formula:  Finally, used the TSF sinusoidal, we are obtained the torque for each phase.In fact, used the torque to-current inverse lookup table, we can obtain the current references  1 * ,  2 * ,  3 * and  4 * of each torque reference respectively.In the second loop, the linearization feedback is used to control the converter drive SRM (see fig. 4).As a conclusion, the second control loop within the system uses feedback linearization as a control technique to regulate the operation of the converter, which, in turn, drives the Switched Reluctance Motor (SRM).This control loop likely plays a crucial role in ensuring the proper and precise operation of the SRM within the larger control system.We proposed the state variable  =  so according to equation (1) we have: Let use chose: Where  is the intermediate input.We replace  1 in equation ( 20) so the result is shown below: In this case, the low of control is given by: The proposed controller can be extended or applied to other phases or situations because of their similarity or shared characteristics.It implies a desire for consistency or efficiency in managing these phases.The proposed controller, initially designed for a specific phase or situation, has qualities or features that make it adaptable and suitable, for application to other phases or situations that exhibit similarities or shared characteristics, with the original context.This implies a desire for efficiency and consistency, in control strategies across related scenarios.

results and simulation
To test the performance of the proposed controller, a simulation example is affected by MATLAB/Simulink.In fact, the performances of the proposed controller, are compared with conventional PI-type regulator.To demonstrate the performance of the proposed controller several tests are affected.

variable speed 200rpm to 300rpm
The SRM 8/6 simulator is used to compare the performance of two controllers.Fig. 6, illustrates the two controllers speed responses when speed reference is changed (from 200rpm to 300rpm).It can be seen that the proposed controller succeeds in tracking the reference speed, but the PI controller needs more time to stabilize, which justifies the presence of a significant overshot.In fact, the proposed controller presents fast convergent and also the best accuracy.In addition, in case of the speed is changed (at 0.5s) we are observed the torque is changed, but one moment torque follows the load torque.

Variable load 50Nm to 100Nm
To test the reliability of the proposed controller, we used a variable load torque while maintaining a constant speed of 200rad/s.Fig. 7 depicts the torque response when the load torque control law considered takes these values 50Nm and 100Nm.It can be seen that despite the load torque variations, the proposed controller always succeeds in forcing the speed to track its reference while guaranteeing minimum torque ripples.Despite these variations, the proposed controller is effective in ensuring that the system's speed closely follows its reference signal while simultaneously minimizing torque ripple.This suggests that the controller is well-suited for applications where precise speed control and reduced torque fluctuations are essential.Moreover, the proposed control gives good results at the level of accuracy and torque ripple.

Variable speed
This comparison is carried out by applying both controllers to the SRM 8/6 system, likely to evaluate how well each controller controls the motor or achieves specific objectives within the simulated environment.Fig. 8, illustrates the two controllers speed responses when speed reference is a signal evolving.It can be seen that the proposed controller succeeds in tracking the reference speed when speed changed, but the PI controller needs more time, to stabilize when the speed changes.In fact, the proposed controller presents fast convergent and also the best accuracy.The observation is that the proposed controller demonstrates better tracking performance and faster response to changes in the reference speed compared to the PI controller.The simulation results obtained show that the proposed controller can effectively regulate speed and reduce torque ripple, even with high torque values.Furthermore, the proposed controller stands out for its performance characteristics, specifically its ability to achieve fast convergence and deliver the highest level of accuracy, among the controllers being considered or discussed.This implies that the proposed controller is well-suited for applications where rapid and precise control is crucial.

Conclusions
In this study, torque and speed control of a switched reluctance motor using feedback linearization.Firstly, a PI controller is used to obtain the torque reference.After a TSF to obtain each torque of phase and used look table, we can determine the current correspond of each torque.Finally, linearization feedback is used to control the asymmetrical converter feed SRM.According to the results obtained, the proposed controller provided the best results in terms of torque ripples and speed tracking error.

Fig. 3 .
Fig. 3. Proposed controller based on PI for first loop.

Fig. 4 .
Fig. 4. Proposed controller based on Linearization feedback technique.After generating the torque reference (used first loop) as shown in Fig.4.The Transfer charring function (TSF) bloc generates from the torque reference (  * ) four torque references  1 * ,  2 * ,  3 * and  4 * .In addition, The Torque Sharing Function (TSF) approach is being considered as a potentially effective solution to share the torque to a set of torque.In this case we have the torque reference is the sum of four torque because we are working with SRM 4 phases.  * =  1 * +  4 * +  4 * +  4 * )) ,   <  <   +   0, ℎ

E3SFig. 6 .
Fig. 6.Torque waveform of controllers for a 50Nm fixed load when speed is raised from 200 to 300rad/s

Fig. 8 .
Fig. 8. Simulation of the proposed controller and PI controller for variable speed Where   presents the control law;   the phase current; ω the rotor speed;   the electromagnetic torque;   the load torque; J the moment of inertia; and f the viscous friction coefficient.The model parameters  1 ,   and   are given by: [17]e   is the machine phase winding resistance,  online parameters are calculated based on analytical expressions developed in[17].Taking into account the magnetic saturation (inductance maximum) we have So, we have in equation of second order.The Laplace transformation of this equation is given by, () and  * () are the Laplace transform successively of () and  * ().()( 2 + ( +   ) +   ) = * ()