Intelligent Computed Torque Control With Recurrent Legendre Fuzzy Neural Network for Permanent-Magnet Assisted Synchronous Reluctance Motor

The goal of this research is to develop an intelligent controlled permanent-magnet assisted synchronous reluctance motor (PMASynRM) drive system by utilizing an intelligent computed torque control with recurrent Legendre fuzzy neural network (ICTCRLFNN), in order to adjust the nonlinear and time-varying control specifications of the motor. The team first proposes an ANSYS Maxwell-2D dynamic model that contains a maximum torque per ampere (MTPA) control PMASynRM drive. A lookup table (LUT) is composed of the finite element analysis (FEA) results, which bring about the current angle of command within the MTPA. Subsequently, the team designs a computed torque control (CTC) system to control the speed reference command. Creating a working CTC for practical applications is quite complex because the detailed system dynamics, which includes the unpredictability of the PMASynRM drive system, is not available beforehand. Thus, this study suggests that a recurrent Legendre fuzzy neural network (RLFNN) can act as a close substitute for the CTC to resolve its existing complications. Furthermore, the team modifies an adaptive compensator to proactively adjust for the potential calculated deviance of the RLFNN. Asymptotical stability is assured by using the Lyapunov stability method, which generates the RLFNN’s online learning algorithms. This study concludes that certain experimental results verify the effective and robust qualities of the suggested ICTCRLFNN controlled PMASynRM drive.


I. INTRODUCTION
Owing to rare-earth permanent magnets (PMs) being mounted on the rotor, the Interior permanent magnet synchronous motor (IPMSM) have a high torque and power density, wide speed operating range, and high efficiency. Therefore, they are widely used in electric vehicles (EVs) and vehicular applications [1], [2], [3], [4], such as starters/alternators, traction motor, power steering, and The associate editor coordinating the review of this manuscript and approving it for publication was Feifei Bu . air conditioning motors. However, the rare-earth elements such as NdFeB are more expensive, and the monopoly of supply chain and trade war are two big concerns. Moreover, the disadvantages such as operated at the fluxweakening control region with large direct axis current and the uncontrolled generator mode caused by the permanentmagnet produced flux linkages are the restricting factors of high performance applications. Thus, a rather new machine called permanent-magnet assisted synchronous reluctance motor (PMASynRM) has been designed by reducing the amount of rare-earth PMs or using ferrite magnets in the rotor to alleviate such difficulties [5]. Since the amount of rare-earth PMs and the resulted magnet flux linkages of the PMASynRM are rather small comparing with the conventional IPMSM, the reluctance torque becomes dominant in the developed torque. Furthermore, the PMASynRM can offer better power factor, torque capability and efficiency with respect to the synchronous reluctance motor (SynRM) [6], [7], [8].
Since the maximum torque per ampere (MTPA) control can improve the torque output in the constant torque region of the IPMSMs, some methods of MTPA have been published in recent years [9], [10], [11], [12], [13]. The MTPA is a control technique to reduce the copper losses by producing the required torque using the minimum current magnitude, which helps. A MTPA control using a saliency back-electromotive force-based intelligent torque observer was proposed in [9] for the improvement of speed estimating performance of a sensorless IPMSM drive system. In [10], a fuzzy control system, which adopted high-frequency mechanical power variation information, was developed to obtain the advance angle for a MTPA controlled IPMSM. Moreover, a small virtual current angle signal was injected in [11] to produce the d-axis current demand as well as to follow the MTPA operation point. In [12], a novel MTPA algorithm was proposed without using any motor dynamics and parameters, in which the current phase is varied continuously to minimize the magnitude of stator current vector at the given load torque. Furthermore, a machine learning approach was used to design the MTPA and flux-weakening control for an IPMSM in [13]. In addition, there are already some research for the MTPA controlled PMASynRM drive to improve its control characteristics in the constant torque region [14], [15]. However, the stator resistance and PM flux linkage are varying with motor temperature [15] and the d-q axis inductances are much influenced by the operating conditions of the motor drives.
The proportion-integral (PI) controller has been adopted in many control applications due to its simplicity. However, the disadvantages, such as sensitive to parameter variations and external disturbances, of the PI controller is well known. On the other hand, the computed torque control (CTC), which is designed by using Lyapunov stability theory [16], [17] has attracted great amount of attention for the nonlinear feedback control. The CTC is utilized to linearize the nonlinear equation of controlled plant by cancellation of some nonlinear terms [17]. Nevertheless, the objection to the real-time use of such control scheme is the lack of knowledge of the detailed system dynamics of the controlled plant. The intelligent control system by using fuzzy systems or neural networks can solve the above difficulty. Moreover, in various neural networks, the Legendre neural network (LNN), which adopts Legendre orthogonal polynomials, can effectively expand the input vector with nonlinear transformations [18], [19], [20]. When compares with the other orthogonal polynomials, the Legendre polynomial offers much less computational complexity than the functional link neural network using the trigonometric function [20]. Hence, LNN can effectively increase the dimensionality of the input vector with lower computational burden and faster convergence rate. Furthermore, by using varied functions to construct the consequent part in the fuzzy neural networks (FNNs) [21] can improve system performance [22], [23]. In [23], since the nonlinear and time-varying control characteristics of a synchronous reluctance motor (SynRM) limited the high-performance applications of this motor, an intelligent backstepping control using recurrent feature selection fuzzy neural network was proposed to construct a robust position controller for a SynRM servo drive system. Nevertheless, an LNN is added in the consequent part of the recurrent fuzzy neural network (RFNN) [24] to form a recurrent Legendre fuzzy neural network (RLFNN) to enhance the control and approximation performance by using the nonlinear combination of input variables in this study. Comparing with a fuzzy neural network (FNN), a recurrent Legendre fuzzy neural network (RLFNN) has several advantages. Firstly, RLFNN is capable of capturing the temporal characteristics of the system, making it well-suited for time-series prediction and control tasks. Secondly, RLFNN has a more flexible architecture, allowing for the use of multiple inputs and outputs, as well as the incorporation of recurrent connections that enable memory and feedback. This makes RLFNN more powerful in handling complex and dynamic systems with nonlinearities and uncertainties.
The main objective of this study is to build a high performance PMASynRM drive to achieve high energy efficiency and robust speed control simultaneously by using the MTPA and intelligent controls. In order to find the optimal current angle to maximize the output torque for a given stator current, the MTPA control has been widely adopted for the control of IPMSMs and SynRMs. Therefore, the copper loss can be minimized. On the other hand, the highperformance uses of the PMASynRM are constrained by the nonlinear and time-varying control elements of this motor even though it is of resilient composition, highly efficient, and inexpensive. Therefore, a Maxwell 2D simulation tool was adopted to assist the design of the PMASynRM to reach the necessary function in [25]. Moreover, this study aims to create an intelligent computed torque control (CTC) by operating the speed of a PMASynRM drive through the utilization of a recurrent Legendre fuzzy neural network (ICTCRLFNN) to achieve the robust control. In order to achieve optimal MTPA control, knowledge of the motor parameters is crucial. To address this issue, the team proposes an MTPA operated PMASynRM model that utilizes ANSYS Maxwell-2D. The current angle command for MTPA is generated by a lookup table (LUT), which is derived from the results of the finite element analysis (FEA). However, the process used to create the PMASynRM drive involves the introduction of various factors such as parameter variations, external disturbance, and lumped uncertainty, which can impact its properties. To mitigate these effects, the team employs a speed control method using a computed torque control (CTC). However, 54018 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
applying CTC in practical situations can be challenging due to the unpredictable system dynamics inherent in the PMASynRM drive system. To address this challenge, the team proposes the use of a recurrent Legendre fuzzy neural network (RLFNN) as an approximation of the CTC. In addition, the team augments an adaptive compensator to adjust for the potential approximated deviance of the RLFNN. The Lyapunov stability method generates the RLFNN's online learning algorithms, which guarantees the robust control performance. In the end, a digital signal processor (DSP) TMS320F28075 with 32-bit floating point powers the PMASynRM drive's vector mechanism and suggested intelligent control system. Additionally, the major contributions of this study are: 1) The successful implementation of a high-performance PMASynRM speed drive with FEA-based MTPA in a 32-bit floating point DSP.
2) The successful development of an ICTCRLFNN and an adaptive compensator for the high-performance PMASynRM speed drive. 3) The successful derivation of a learning algorithm for the online training of the RLFNN using the Lyapunov stability theorem.

II. MODELLING OF PMASYNRM DRIVE SYSTEM
Owing to the high magnetic saturation of PMASynRMs under heavy loads, which will induce nonlinear torque generation, the FEA software such as Ansys Maxwell or JMAG is usually adopted for the design and analysis of PMASynRMs. Moreover, through the analysis of the motor mechanical design tool of ANSYS Maxwell-2D by using FEA, the current angle between the stator current and the MTPA can be obtained. Furthermore, the resulted minimum currents under various load torque conditions by using FEA are made into lookup table (LUT) to generate the current angle command of the MTPA. The cutaway of the adopted PMASynRM is shown in the Fig. 1. In addition, the designed parameters shown in Table 1 are used in developing the 2D FEA model of PMASynRM. To highlight the importance of considering magnetic saturation by using the parameters of PMASynRM shown in Table 2, Fig. 2 shows the relationship between the torque and current angle at various stator current magnitudes, where the red dotted line is the traditional maximum torque per ampere (TMTPA) trajectory without considering the magnetic saturation phenomenon; the black dotted line is the MTPA trajectory considering the magnetic saturation, which is the simulated results using Maxwell-2D. It can be found that the required stator current considering the magnetic saturation by using the MTPA is lower under the same torque.
The voltage equations of the stator of PMASynRM in the d-q reference frame can be expressed as follows:   In (1) and (2), R s is the stator resistance; v d and v q are the d-axis and q-axis voltage; i d and i q are the d-axis and q-axis stator currents; λ d and λ q are the d-axis and q-axis flux linkages; ω e is the rotor electrical angular velocity. Moreover, the flux linkage equations in the d-q reference frame are: In the above equations, L d and L q are the d-axis and q-axis inductances; λ m is the ferrite PM flux of the rotor. The produced electromagnetic torque of the PMASynRM without considering magnetic saturation expressed in the d-q reference frame can be expressed as follows: and the number of pole is represented by P.
The control block diagram of PMASynRM drive system including the proposed CTC speed controller and ICTCRLFNN speed controller, current angle LUT, d-axis and q-axis PI current controllers and coordinate VOLUME 11, 2023 54019 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.   transformation is shown in Fig. 3. The adopted PMASynRM is a 4 poles, 36 slots, 4.5 kW, 214 V, 9.4 A, 1500 rpm, 25 Nm type motor. In Fig. 3, ω * r is the motor speed command; ω r is the speed response; e r is the speed error; i * q is the q-axis current command; i * d is the d-axis current command; i a , i b , and i c are the three phase are obtained by using the PI current controllers with voltage compensation terms and coordinate transformations as shown in Fig. 3. The three phase currents and are measured by using the Hall current sensors with 1 V/6.67 A transformation ratio and via the analog-to-digital converter (ADC). The space vector pulse width modulation (SVPWM) technology is implemented to control a voltage source inverter (VSI) also shown in Fig. 3. The switching frequency of the VSI is 20 kHz. In addition, silicon carbide (SiC) power MOSFET C3M0065090D manufactured by CREE with 900 V/36 A are adopted in the VSI. The current controllers in this article are implemented by the PI controllers. The small signal models of the controlled plants for the design of q-d axis current controllers, which are shown in Figs. 4(a) and 4(b) respectively, can be represented by: where represents small signal; s is the Laplace operator. Moreover, the PI current controllers are in the following forms: The   controllers according to the parameters in Table 2. The resulted current controller is: The resulted bode diagrams of the loop gain are also shown in Fig. 5 in red. In addition, in the control block diagram shown in Fig. 3, the feedforward terms derived in (1) and (2), ω e (L d i d + λ m ) and −ω e L q i q , are used for the decoupling control to let the current controllers be independent on speed ideally.

III. REALIZATION OF MTPA CONTROL
The output torque of the PMASynRM is composed of the electromagnetic torque of λ m i q and the reluctance torque of (L d − L q )i q i d , respectively, and the reluctance torque is determined by the d-axis inductance and the q-axis inductance. Fig. 6 is the relationship between the stator current vector ⃗ I s and the d-q axis, in which represented by the resultant current vector I s , then: where β is the current angle. The current vector should be found for its magnitude to be as low as possible at a specific torque in order to operate the PMASynRM at the MTPA condition. Thus, the differentiation of the torque with respect to the current angle β in the synchronous reference frame depicts as follows [26]: Furthermore, the equation below provides the MTPA operation point and the resulted current angle β [15]: Based on (11), the optimal MTPA control only can be reached when there are clear motor parameters of L d , L q and λ m . Despite this, the motor parameters still contain time-varying properties due to the motor's age, magnetic saturation, and operating temperature. Thus, solely utilizing (11) in realworld circumstances makes reaching MTPA control very complex. The team brings ANSYS Maxwell-2D into play in order to examine the MTPA's minimum current solution while taking in account for the PMASynRM's magnetic saturation properties. Finally, Fig. 3 depicts the MTPA's current angle command, which is generated by an LUT made from the results of the finite element analysis (FEA).

IV. DESIGN OF CTC AND ICTCRLFNN
Three challenges of this study are: (1) the design of an optimal MTPA control, which has been proposed in the previous section by using a FEA-based LUT; (2) the design of a robust speed control using intelligent control; (3) the development of a reliable online learning algorithm for the intelligent control to guarantee the stability. The last two challenges will be overcome by the proposed ICTCRLFNN in this section.

A. DESIGN OF CTC
The mechanical dynamic equation of the PMASynRM is The ideal dynamic equation can use (5) and (12) to rewrite as follow:ω where ; C m = − 1 J ; ''−'' represents the nominal value of the motor parameter.   With the consideration of the uncertainties, which includes parameter variations and external disturbance, the dynamic equation (13) can be rewritten as follow: where U = i * q is the torque current command. C m = − 1 J ; A m , B m and C m denote the time-varying parameter variations; F is named the lumped uncertainty and defined by (15) and F b represents the lumped uncertainty bound. Define the speed tracking error and its derivative as follows: and c 1 is a positive constant. Additionally, a virtual control error is defined: Moreover, a Lyapunov function candidate is considered as: with V 1 being a positive definite function;F = F −F,F is the estimated value of F; a is a positive constant. Then, the derivative of V 1 can be derived as: According toV 1 , an CTC law U and adaptive lawḞ [16], [17] are designed as follows: Substituting (22), (23) into (21), the following equation can be obtained:V indicating that e 1 and e 2 will converge to zero as t → ∞.
As the result, the CTC speed control is asymptotically stable. Though the CTC system can guarantee the asymptotically stability of PMASynRM drive system, the lumped uncertainty F is unknown in practical applications. However, the CTC system requires the actual parameters of the PMASynRM drive system in advance. In addition, the slow adaptation of the lumped uncertainty shown in (23) resulted in sluggish dynamical response. Thus, it is unfeasible to design a CTC law for practical applications.

B. DESIGN OF ICTCRLFNN
A RLFNN controller is proposed to approximate the CTC law shown in (22) to overcome the above shortcomings of the CTC as shown in Fig. 7. Moreover, the control law for the ICTCRLFNN system is designed as follow: 54022 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
In the above equation,Û RLFNN is designed to learn the CTC law andÛ c is used to compensate the approximated error of the RLFNN controller. The six-layer RLFNN shown in Fig. 8 can be represented in the following form: In (26), m and σ are the vector of mean and the standard deviation of the Gaussian functions in Layer 2. There is an optimal U * RLFNN for any nonlinear function according to the universal approximation property. Thus, an optimal U * RLFNN is designed to learn the CTC law U CTC such that In (27), ε is the minimum reconstructed error; W * , W * l , W * MP , m * , σ * and Γ * are the optimal values of W , W l , W MP , m, σ and Γ respectively. Furthermore, the control law shown in (25) can be represented as: andW = W * −Ŵ ,Γ = Γ * −Γ . In order to transform the nonlinear RLFNN into partially linear form, a linearization technique is applied to obtain the expansion ofΓ in Taylor series as follows: Rewriting (30), Γ * can be obtained that Substituting (30) and (31) into (29), the estimated error in (29) can now be written as and H represents the uncertain term and can be written as: Theorem 1: Taking account for the PMASynRM drive system, which is depicted in (14), the suggested ICTCRLFNN will acquire absolute asymptotical stability once the parameters below are achieved: 1) The ICTCRLFNN control is created as depicted in (25); 2) The RLFNN's adaptation law is created as depicted in (34) where η w , η m , η σ , η w l and η w MP are learning rates parameters and are all sets as positive constants;ψ is the online estimated value of approximated error; and γ is a positive constant. Proof: Considering a Lyapunov function candidate as: where V 2 is a function of positive definite. Additionally, ψ = ψ −ψ is the approximated error and defined as: Then, differentiate V 2 with respect to time the following equation can be obtained: Moreover, substitute (34)-(40) into (43), it can be concluded thaṫ SinceV 2 (e 1 (t),ψ(t),W ,W l ,W MP ,m,σ ) ≤ 0 is negative semidefinite, all signals contained in (41) are bounded. Furthermore, by using Barbalat's Lemma, it can be inferred that e 1 , e 2 will converge to zero as t → ∞. This complete the proof that the proposed ICTCRLFNN control system is asymptotically stable. Detailed derivation can be referred to [23]. The flowchart for the proposed RLFNN controller is shown in Fig. 9. The operating mechanism in the proposed RLFNN are described in detail as follows: 1. Sampling: An incremental encoder with resolution 2500 counts/rev is adopted to measure the rotor position by using the eQEP module in the DSP. Then, the derivative of rotor position ω r is calculated. Following that, the tracking error e 1 = ω * r (t) − ω r (t) or e 2 =ω r (t) + λ 1 is generated and sent to the proposed RLFNN controller.

RLFNN Input Layer:
The input variables of the proposed FNN are x 1 1 = e 1 and x 1 2 = e 2 . In the input layer, the node outputs are y 1 i (N ) and sent to the membership layer.

RLFNN Membership Layer:
The Gaussian functions are adopted to implement the fuzzification operation and the outputs are y 2 ij (N ). Then, the outputs y 2 ij (N ) are sent to the rule and recurrent layer.
4. RLFNN Legendre Layer: This layer uses the Legendre polynomial as the expansion function, and its input variable vector x = [x 1 1 , x 1 2 ] can be divided into: where h is the order of the exponent. In this layer, the node outputs y 3 p (N ) are obtained by performing the summation and multiplying operations with w 3 MP and λ 3 P .

RLFNN Rule and Recurrent Layer:
Using the recurrent technique, the previous system state can obtained through time delay, which can effectively obtain internal feedback. Signals and approximate information allow the system to have better dynamic mapping capabilities. Then, the outputs y 4 l (N ) are sent to the rule and recurrent layer.

RLFNN Consequent Layer:
The nodes of consequent layer multiply the output signals from rule and recurrent layer, Legendre layer and recurrent property, and output the result of product for dynamic mapping. The outputs are y 5 n (N ) and sent to the output layer.
7. RLFNN Output Layer: The node performs the summation and multiplying operation. Then, the output is given as y 6 o (N ), which add adaptive compensator to get the i * q current command.
8. Online Network Parameters Learnings: The online parameters learning are achieved by online tuning of the connective weights w 6 o in the output layer, the connective weights w 4 l in the rule and recurrent layer, the connective weights w 3 MP in the Legendre layer, and the mean m ij and standard deviations σ ij of the membership functions in the membership layer.
Though all the learning rate parameters and set as positive constants, too large positive constants will result in the divergence of the RLFNN, and too small positive constants will result in the slow convergence of the RLFNN. Therefore, the learning rate parameters are tuned by trial and error in the experimentation.   T M , T aver , and T sd , which are the maximum tracking error, the average tracking error, and the standard deviation of the tracking error [17], respectively, are adopted to verify the control performance within the suggested speed control system:

V. EXPERIMENTATION
where T error (N ) = ω * r (N ) − ω r (N ); h is the total number of iterations. The responses of periodical step and sinusoidal reference trajectories are measured to show the control system's control performance. The following second-order transfer function with rise time 0.6 s serves as the reference VOLUME 11, 2023 54025 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. In (48), ξ and ω n denote the damping ratio and undamped natural frequency, respectively. Moreover, the experimental results of CTC and proposed ICTCRLFNN speed controllers are provided and analyzed for the comparison of the control performance. The parameters of the CTC speed controller is designed using the motor parameters shown in Table 2. The parameters of the CTC and the proposed ICTCRLFNN speed controller are given in (49)  The parameters are tuned by trial and error to achieve the best transient control performance with the consideration of the stability requirement. Though optimization algorithms, such as genetic algorithm and particle swarm optimization, can be used to search for the optimal set of parameter values that achieve the desired performance criteria, the disadvantage of the optimization algorithms is time consuming. Furthermore, to compromise between the computation amount and the control performance, two, six, nine, nine, nine and one neurons have been selected in the input, membership, Legendre, rule, consequent and output layer for the network structure of the RLFNN.
The control objective in the experimentation is to control the rotor speed of the PMASynRM to track the periodically speed commands with minimum tracking error including step (100 rpm) and sinusoidal (±100 rpm) commands. Some experimental results are provided to demonstrate the effectiveness of the proposed PMASynRM drive. First, the control performance of the current control is shown in Fig. 11. Figs. 11 (a) and 11(b) show the current responses of the qd axis current control by using the designed PI controllers shown in (10) with step command where 13 A is the rated value of the phase current. Owing to the specification PM 52 • , the overshoot of current step responses are both 13% as shown in Figs. 11(a) and 11(b). Then, two different operating points 500 rpm (case 1) and 1000 rpm (case 2) are presented where the load torque transiting from 5 Nm to 10 Nm at 10 s for the speed control. In order to test the parameter sensitivity and robustness of the proposed  controllers at different operating conditions, the test scenarios in the experimentation have been shown in Table 3 From the experimental results, the d-axis current command can be effectively generated by using the FEA-based LUT for the MTPA control. Additionally, comparing with CTC speed controller, the rotor responses of the PMASynRM using the proposed ICTCRLFNN speed controller are much enhanced and the tracking errors are reduced at different reference commands owing to the RLFNN with the capabilities of parallel processing and online learning of the network parameters. That is to say, the robustness of the speed control is improved by using the proposed ICTCRFNN speed controller. The results of all the experiments shown in Figs. 12 to 15 have been quantified and compared in Table 4.   The performance measurements of CTC and proposed ICTCRLFNN speed controllers at two operating cases with step and sinusoidal reference commands are compared in Fig. 16. The proposed ICTCRLFNN speed controller has lower values of maximum, average and standard deviation tracking errors due to its faster convergence rate and improved generalization performance. Moreover, the execution or compute time of the ''C'' program in the TMS320F28075 32-bit floating point DSP with 120 MHz can be obtained by the clock tool of Texas Instruments Code Composer Studio v6 program editing interface. The total operation cycles and total execution time for the CTC are 401 and 0.00334ms; the proposed ICTCRLFNN are 7437 and 0.0619 ms, respectively. As a result, the total execution time of the proposed ICTCRLFNN is still less than 1 ms, which is the sampling interval of the speed control loop.

VI. CONCLUSION
During this study, an ICTCRLFNN is suggested to serve for a high-performance PMASynRM drive system. First, the team brings the dynamic model into play, which features an MTPA operated PMASynRM drive with ANSYS Maxwell-2D capabilities. FEA results derive an LUT, which the team uses to create the MTPA's current angle command. Subsequently, the team fashions a CTC speed tracking system. Moreover, this study suggests that RLFNN can act as a stand in for the CTC law to resolve issues with the CTC's necessary motor specifications within the PMASynRM drive. Furthermore, the Lyapunov stability method generates the RLFNN's online learning algorithms, which guarantee asymptotical stability. In the end, the experiment concludes that the suggested ICTCRLFNN has more than adequate control performance in terms of speed tracking control.