Contents 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 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 flux-weakening control region with large direct axis current and the uncontrolled generator mode caused by the permanent-magnet 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
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 high-performance 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, 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.
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.
Torque versus current angle at various stator current magnitudes by using TMTPA control and MTPA control.
The voltage equations of the stator of PMASynRM in the 
\begin{align*} v_{d} &=R_{s} i_{d} +\frac {d}{dt}\lambda _{d} -\omega _{e} \lambda _{q} \tag{1}\\ v_{q} &=R_{s} i_{q} +\frac {d}{dt}\lambda _{q} +\omega _{e} \lambda _{d} \tag{2}\end{align*}
\begin{align*} \lambda _{q} &=L_{q} i_{q} \tag{3}\\ \lambda _{d} &=L_{d} i_{d} +\lambda _{m} \tag{4}\end{align*}
\begin{equation*} T_{e}=\frac {3}{2}\frac {P}{2}[\lambda _{m} i_{q} +(L_{d} -L_{q})i_{q} i_{d}] \tag{5}\end{equation*}
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 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,
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:\begin{equation*} T_{q} (s)=\frac {1}{L_{q} s+R_{s}},\quad T_{d} (s)=\frac {1}{L_{d} s+R_{s}} \tag{6}\end{equation*}
\begin{equation*} G_{cq} (s)=K_{p} +\frac {K_{i}}{s},\quad G_{cd} (s)=K_{p} +\frac {K_{i}}{s} \tag{7}\end{equation*}
The bode diagrams of the controlled plants are shown in Fig. 5 in blue. The design specification of the current controller is bandwidth (BW) \begin{align*} G_{cq} (s)=82.55+\frac {83407}{s},\quad G_{cd} (s)=18.69+\frac {19992.34}{s} \tag{8}\end{align*}
Bode diagrams of current controllers. Solid lines represent q- axis, and dashed lines represent d- axis.
Realization of MTPA Control
The output torque of the PMASynRM is composed of the electromagnetic torque of \begin{equation*} T_{e} =\frac {3}{2}\frac {P}{2}[\lambda _{m} I_{s} \cos (\beta)+(L_{d} -L_{q})I_{s}^{2} \sin (2\beta)] \tag{9}\end{equation*}
\begin{align*} \frac {\partial T_{e}}{\partial \theta _{i}}=2(L_{d} -L_{q})I_{s}^{2} \sin ^{2}(\beta)\lambda _{m} I_{s} \sin (\beta)-(L_{d} -L_{q})I_{s}^{2} \tag{10}\end{align*}
\begin{equation*} \beta _{MTPA} =\sin ^{-1}\left ({{\frac {-\lambda _{m} +\sqrt {\lambda _{m}^{2} +8(L_{d} -L_{q})^{2}I_{s}^{2}}}{-4(L_{d} -L_{q})I_{s}}} }\right) \tag{11}\end{equation*}
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 \begin{equation*} T_{e} =J\frac {d\omega _{r}}{dt}+B\omega _{r} +T_{L} \tag{12}\end{equation*}
\begin{align*} \dot {\omega }_{r} &=-\frac {\bar {B}}{\bar {J}}\omega _{r} +\frac {3P[\bar {\lambda }_{m} +\left ({{\bar {L}_{d} -\bar {L}_{q}} }\right)i_{d}^{\ast }]}{4\bar {J}}i_{q}^{\ast } -\frac {T_{L}}{\bar {J}} \\ &=A_{m} \omega _{r} +B_{m} i_{q}^{\ast } +C_{m} T_{L} \tag{13}\end{align*}
“–“ 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:\begin{align*} \dot {\omega }_{r} &=(A_{m} +\Delta A_{m})\omega _{r} +(B_{m} +\Delta B_{m})U+(C_{m} +\Delta C_{m})T_{L} \\ &=A_{m} \omega _{r} +B_{m} U+F \tag{14}\end{align*}
\begin{equation*} F=\Delta A_{m} \omega _{r} +\Delta B_{m} U+(C_{m} +\Delta C_{m})T_{L},\left |{ F }\right |\le F_{b} \tag{15}\end{equation*}
\begin{align*} e_{1} &=\omega _{r}^{\ast } (t)-\omega _{r}(t) \tag{16}\\ \dot {e}_{1} &=\dot {\omega }_{r}^{\ast } (t)-\dot {\omega }_{r}(t) \tag{17}\\ \lambda _{1} &=-c_{1} e_{1} -\dot {\omega }_{r}^{\ast }(t) \tag{18}\end{align*}
\begin{equation*} e_{2} =\dot {\omega }_{r} (t)+\lambda _{1} =\dot {\omega }_{r} (t)-c_{1} e_{1} -\dot {\omega }_{r}^{\ast }(t) \tag{19}\end{equation*}
\begin{equation*} V_{1} =\frac {1}{2}e_{1}^{2} +\frac {\left |{ {\tilde {F}} }\right |^{2}}{2a}=\frac {1}{2}e_{1}^{2} +\frac {\left |{ {F-\hat {F}} }\right |^{2}}{2a}>0 \tag{20}\end{equation*}
\begin{align*} \dot {V}_{1} &=e_{1} \dot {e}_{1} -\frac {1}{a}\tilde {F}\dot {{\hat {F}}}=e_{1} (\dot {\omega }_{r}^{\ast }-\dot {\omega }_{r})-\frac {1}{a}\tilde {F}\dot {{\hat {F}}} \\ &=e_{1} (-c_{1} e_{1} -e_{2})-\frac {1}{a}\tilde {F}\dot {{\hat {F}}} \\ &=-c_{1} e_{1}^{2}-(\omega _{r}^{\ast }-\omega _{r})e_{2} -\frac {1}{a}\tilde {F}\dot {{\hat {F}}} \\ &=-c_{1} e_{1}^{2}-\left[{\omega _{r}^{\ast }-\frac {1}{A_{m}}(\dot {\omega }_{r} -B_{m} U-F)}\right]e_{2} -\frac {1}{a}\tilde {F}\dot {{\hat {F}}} \\ &=-c_{1} e_{1}^{2}-\left({\omega _{r}^{\ast }-\frac {1}{A_{m}}\dot {\omega }_{r} +\frac {1}{A_{m}}B_{m} U+\frac {1}{A_{m}}\hat {F}}\right)e_{2} \\ &\quad -\,\frac {\tilde {F}}{a}\left({\frac {ae_{2}}{A_{m}}+\dot {{\hat {F}}}}\right) \tag{21}\end{align*}
\begin{align*} U_{CTC} &=i_{q}^{\ast } =B_{m}^{-1} (-A_{m} \omega _{r}^{\ast } +A_{m} c_{2} e_{2} +\dot {\omega }_{r}-\hat {F}) \tag{22}\\ \dot {{\hat {F}}}&=-\frac {ae_{2}}{A_{m}} \tag{23}\end{align*}
\begin{equation*} \dot {V}_{1} =-c_{1} e_{1}^{2} -c_{2} e_{2}^{2} \le 0 \tag{24}\end{equation*}
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:\begin{equation*} U=\hat {U}_{RLFNN} +\hat {U}_{c} \tag{25}\end{equation*}
\begin{align*} &U_{RLFNN} (e_{1},e_{2}, \boldsymbol {W}, \boldsymbol {W}_{ \boldsymbol {l}}, \boldsymbol {W}_{ \boldsymbol {MP}}, \boldsymbol {m}, \boldsymbol {\sigma })\equiv \boldsymbol {W} \boldsymbol {\Gamma } \\ & \boldsymbol {W}=[w_{1}^{6} w_{2}^{6} w_{3}^{6} w_{4}^{6} w_{5}^{6} w_{6}^{6} w_{7}^{6} w_{8}^{6} w_{9}^{6}]\in R^{1\times 9}; \\ & \boldsymbol {\Gamma } =[x_{1}^{6} x_{2}^{6} x_{3}^{6} x_{4}^{6} x_{5}^{6} x_{6}^{6} x_{7}^{6} x_{8}^{6} x_{9}^{6}]^{T}\in R^{9\times 1}; \\ & \boldsymbol {W}_{ \boldsymbol {l}} =[w_{1}^{4} w_{2}^{4} w_{3}^{4} w_{4}^{4} w_{5}^{4} w_{6}^{4} w_{7}^{4} w_{8}^{4} w_{9}^{4}]^{T}\in R^{9\times 1}; \\ & \boldsymbol {W}_{ \boldsymbol {MP}} =[w_{1}^{3} w_{2}^{3} \cdots w_{81}^{3}]^{T}\in R^{81\times 1}; \\ & \boldsymbol {m}=[m_{11} m_{12} m_{13} m_{24} m_{25} m_{26}]^{T}\in R^{6\times 1}; \\ & \boldsymbol {\sigma } =[\sigma _{11} \sigma _{12} \sigma _{13} \sigma _{24} \sigma _{25} \sigma _{26}]^{T}\in R^{6\times 1}; \tag{26}\end{align*}
\begin{align*} U_{CTC} & =U_{RLFNN}^{\ast } (e_{1},e_{2}, \boldsymbol {W}^{\ast }, \boldsymbol {W}_{ \boldsymbol {l}}^{\ast }, \boldsymbol {W}_{ \boldsymbol {MP}}^{\ast }, \boldsymbol {m}^{\ast }, \boldsymbol {\sigma } ^{\ast })+\varepsilon \\ &= \boldsymbol {W}^{\ast } \boldsymbol {\Gamma }^{\ast }+\varepsilon \tag{27}\end{align*}
\begin{align*} U\!=\!\hat {U}_{RLFNN} (e_{1},e_{2},\hat { \boldsymbol {W}},\hat { \boldsymbol {W}}_{ \boldsymbol {l}},\hat { \boldsymbol {W}}_{ \boldsymbol {MP}},\hat { \boldsymbol {m}},\hat { \boldsymbol {\sigma }})+\hat {U}_{c} \!=\!\hat { \boldsymbol {W}}\hat { \boldsymbol {\Gamma }}+\hat {U}_{c}\!\! \tag{28}\end{align*}
\begin{equation*} \tilde {U}=U_{CTC} -U=\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }^{\ast }+\hat { \boldsymbol {W}}\tilde { \boldsymbol {\Gamma }}+\varepsilon -\hat {U}_{c} \tag{29}\end{equation*}
\begin{equation*} \tilde { \boldsymbol {\Gamma }}= \boldsymbol {\Gamma }_{ \boldsymbol {m}} ^{T}\tilde { \boldsymbol {m}}+ \boldsymbol {\Gamma }_{ \boldsymbol {\sigma }} ^{T}\tilde { \boldsymbol {\sigma }}+ \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {l}}} ^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {l}} + \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {MP}}}^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}} +N_{h} \tag{30}\end{equation*}
\begin{align*} \boldsymbol {\Gamma }_{m}^{T}&=\left.{ {\left [{ {{\begin{array}{cccccccccccccccccccc} {\dfrac {\partial x_{1}^{6}}{\partial m_{11}}} & {\dfrac {\partial x_{1}^{6}}{\partial m_{12}}} & \cdots & {\dfrac {\partial x_{1}^{6}}{\partial m_{26}}} \\[0.8pc] {\dfrac {\partial x_{2}^{6}}{\partial m_{11}}} & {\dfrac {\partial x_{2}^{6}}{\partial m_{12}}} & \cdots & {\dfrac {\partial x_{2}^{6}}{\partial m_{26}}} \\[0.8pc] \vdots & \vdots & \ddots & \vdots \\[0.5pc] {\dfrac {\partial x_{9}^{6}}{\partial m_{11}}} & {\dfrac {\partial x_{9}^{6}}{\partial m_{12}}} & \cdots & {\dfrac {\partial x_{9}^{6}}{\partial m_{26}}} \\ \end{array}}} }\right]} }\right |R^{9\times 6}\\ \boldsymbol {\Gamma }_{\sigma }^{T}&=\left.{ {\left [{ {{\begin{array}{cccccccccccccccccccc} {\dfrac {\partial x_{1}^{6}}{\partial \sigma _{11}}} & {\dfrac {\partial x_{1}^{6}}{\partial \sigma _{12}}} & \cdots & {\dfrac {\partial x_{1}^{6}}{\partial \sigma _{26}}} \\[0.8pc] {\dfrac {\partial x_{2}^{6}}{\partial \sigma _{11}}} & {\dfrac {\partial x_{2}^{6}}{\partial \sigma _{12}}} & \cdots & {\dfrac {\partial x_{2}^{6}}{\partial \sigma _{26}}} \\[0.5pc] \vdots & \vdots & \ddots & \vdots \\[0.8pc] {\dfrac {\partial x_{9}^{6}}{\partial \sigma _{11}}} & {\dfrac {\partial x_{9}^{6}}{\partial \sigma _{12}}} & \cdots & {\dfrac {\partial x_{9}^{6}}{\partial \sigma _{26}}} \\ \end{array}}} }\right]} }\right |\in R^{9\times 6}\\ \boldsymbol {\Gamma }_{W_{l}}^{T}&=\left.{ {\left [{ {{\begin{array}{cccccccccccccccccccc} {\dfrac {\partial x_{1}^{6}}{\partial w_{1}^{4}}} & {\dfrac {\partial x_{1}^{6}}{\partial w_{2}^{4}}} & \cdots & {\dfrac {\partial x_{1}^{6}}{\partial w_{9}^{4}}} \\[0.8pc] {\dfrac {\partial x_{2}^{6}}{\partial w_{1}^{4}}} & {\dfrac {\partial x_{2}^{6}}{\partial w_{2}^{4}}} & \cdots & {\dfrac {\partial x_{2}^{6}}{\partial w_{9}^{4}}} \\[0.8pc] \vdots & \vdots & \ddots & \vdots \\[0.8pc] {\dfrac {\partial x_{9}^{6}}{\partial w_{1}^{4}}} & {\dfrac {\partial x_{9}^{6}}{\partial w_{2}^{4}}} & \cdots & {\dfrac {\partial x_{9}^{6}}{\partial w_{9}^{4}}} \\ \end{array}}} }\right]} }\right |\in R^{9\times 6}\\ \boldsymbol {\Gamma }_{W_{MP}}^{T}&=\left.{ {\left [{ {{\begin{array}{cccccccccccccccccccc} {\dfrac {\partial x_{1}^{6}}{\partial w_{11}^{3}}} & {\dfrac {\partial x_{1}^{6}}{\partial w_{12}^{3}}} & \cdots & {\dfrac {\partial x_{1}^{6}}{\partial w_{99}^{3}}} \\[0.8pc] {\dfrac {\partial x_{2}^{6}}{\partial w_{11}^{3}}} & {\dfrac {\partial x_{2}^{6}}{\partial w_{12}^{3}}} & \cdots & {\dfrac {\partial x_{2}^{6}}{\partial w_{99}^{3}}} \\[0.8pc] \vdots & \vdots & \ddots & \vdots \\[0.8pc] {\dfrac {\partial x_{9}^{6}}{\partial w_{11}^{3}}} & {\dfrac {\partial x_{9}^{6}}{\partial w_{12}^{3}}} & \cdots & {\dfrac {\partial x_{9}^{6}}{\partial w_{99}^{3}}} \\ \end{array}}} }\right]} }\right |\in R^{9\times 81}\end{align*}
\begin{align*} \boldsymbol {\Gamma }^{\ast }\!=\!\hat { \boldsymbol {\Gamma }}\!+\!{\tilde { \boldsymbol {\Gamma }}}\!=\!\hat { \boldsymbol {\Gamma }}\!+\! \boldsymbol {\Gamma }_{ \boldsymbol {m}} ^{T}\tilde { \boldsymbol {m}}\!+\! \boldsymbol {\Gamma }_{ \boldsymbol {\sigma }} ^{T}\tilde { \boldsymbol {\sigma }}\!+\! \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {l}}} ^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {l}} \!+\! \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {MP}}}^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}}\! +\!N_{h}\!\!\! \tag{31}\end{align*}
\begin{align*} \tilde {U} & =\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }^{\ast }+\hat { \boldsymbol {W}}\tilde { \boldsymbol {\Gamma }}+\varepsilon -\hat {U}_{c} \\ & =\tilde { \boldsymbol {W}}\hat { \boldsymbol {\Gamma }}+\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {m}} ^{T}\tilde { \boldsymbol {m}}+\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {\sigma }} ^{T}\tilde { \boldsymbol {\sigma }}+\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {l}}}^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {l}} \\ &\quad +\,\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {MP}}} ^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}} +H-\hat {U}_{c} \tag{32}\end{align*}
\begin{align*}&\hspace {-0.5pc}H=\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {m}} ^{T}\tilde { \boldsymbol {m}}+\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {\sigma }} ^{T}\tilde { \boldsymbol {\sigma }}+\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {l}}}^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {l}} +\tilde { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {MP}}} ^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}} \\&+\, \boldsymbol {W}^{\ast }N_{h} +\varepsilon \tag{33}\end{align*}
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)–(38); 3) The compensators equipped with an adaption law are created as depicted in (39) and (40).\begin{align*} \dot {\hat { \boldsymbol {W}}}^{T} & =\eta _{w} e_{2} {\hat { \boldsymbol {\Gamma }}} \tag{34}\\ \dot {\hat { \boldsymbol {m}}}^{T} & =\eta _{m} e_{2} \hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {m}} ^{T} \tag{35}\\ \dot {\hat { \boldsymbol {\sigma }}}^{T} & =\eta _{\sigma } e_{2}\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {\sigma }} ^{T} \tag{36}\\ \dot {\hat { \boldsymbol {W}}}_{l}^{T} & =\eta _{w_{l}} e_{2}\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {l}}} ^{T} \tag{37}\\ \dot {\hat { \boldsymbol {W}}}_{MP}^{T} & =\eta _{w_{MP}} e_{2}\hat { \boldsymbol {W}} \boldsymbol {\Gamma }_{ \boldsymbol {w}_{ \boldsymbol {MP}}} ^{T} \tag{38}\\ \hat {U}_{c} & =\hat {\psi } \tag{39}\\ \dot {{\hat {\psi }}} & =\gamma e_{2} \tag{40}\end{align*}
Proof:
Considering a Lyapunov function candidate as:\begin{align*} &\hspace {-2pc}V_{2} (e_{1} (t),\tilde {\psi }(t),\tilde { \boldsymbol {W}},\tilde { \boldsymbol {W}}_{ \boldsymbol {l}},\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}},\tilde { \boldsymbol {m}},\tilde { \boldsymbol {\sigma }}) \\ &=\frac {1}{2}e_{1}^{2} +\frac {\left |{ {\tilde {F}} }\right |^{2}}{2a}+\frac {B_{m}}{2A_{m} \eta _{w} }\tilde { \boldsymbol {W}}\tilde { \boldsymbol {W}}^{T} \\ &\quad +\,\frac {B_{m}}{2A_{m} \eta _{w_{l}} }\tilde { \boldsymbol {W}}_{ \boldsymbol {l}}^{T}\tilde { \boldsymbol {W}}_{ \boldsymbol {l}} +\frac {B_{m}}{2A_{m} \eta _{w_{MP}}}\tilde { \boldsymbol {W}}_{ \boldsymbol {MP}} ^{T} \\ &\quad \times \, \tilde { \boldsymbol {W}}_{ \boldsymbol {MP}} +\frac {B_{m}}{2A_{m} \eta _{m} }\tilde { \boldsymbol {m}}^{\textrm {T}}\tilde { \boldsymbol {m}} \\ &\quad +\, \frac {B_{m}}{2A_{m} \eta _{\sigma }}{\tilde { \boldsymbol {\sigma }}}^{T}\tilde { \boldsymbol {\sigma }}+\frac {B_{m}}{2A_{m} \gamma }\left \|{ {\tilde {\psi }}}\right \|^{2}>0 \tag{41}\end{align*}
\begin{equation*} \psi =H-\frac {\tilde {F}}{B_{m}}-\frac {1}{ae_{2} B_{m}}\tilde {F}\dot {{\hat {F}}} \tag{42}\end{equation*}
\begin{align*} & \dot{V}_{2}=-c_{1} e_{1}^{2}-c_{2} e_{2}^{2}+\left[\frac{e_{2} B_{m}}{A_{m}} \tilde{\boldsymbol{W}} \hat{\boldsymbol{\Gamma}}-\frac{B_{m}}{A_{m} \eta_{w}} \tilde{\boldsymbol{W}} \dot{\hat{\boldsymbol{W}}}^{T}\right] \\ & +\left[\frac{e_{2} B_{m}}{A_{m}} \hat{\boldsymbol{W}} \boldsymbol{\Gamma}_{\boldsymbol{m}}^{T} \tilde{\boldsymbol{m}}\right. \\ & \left.-\frac{B_{m}}{A_{m} \eta_{m}} \dot{\hat{\boldsymbol{m}}}^{T} \tilde{\boldsymbol{m}}\right]+\left[\frac{e_{2} B_{m}}{A_{m}} \hat{\boldsymbol{W}} \boldsymbol{\Gamma}_{\boldsymbol{\sigma}}^{T} \tilde{\boldsymbol{\sigma}}-\frac{B_{m}}{A_{m} \eta_{\sigma}} \dot{\hat{\sigma}}^{T} \tilde{\boldsymbol{\sigma}}\right] \\ & +\left[\frac{e_{2} B_{m}}{A_{m}} \hat{\boldsymbol{W}} \boldsymbol{\Gamma}_{w_{l}}^{T} \tilde{\boldsymbol{W}}_{\boldsymbol{l}}-\frac{B_{m}}{A_{m} \eta_{w_{l}}} \dot{\hat{\boldsymbol{W}}}_{\boldsymbol{l}}^{T} \tilde{\boldsymbol{W}}_{\boldsymbol{l}}\right] \\ & +\left[\frac{e_{2} B_{m}}{A_{m}} \hat{\boldsymbol{W}} \boldsymbol{\Gamma}_{w_{M P}}^{T} \tilde{\boldsymbol{W}}_{\boldsymbol{M P}}\right. \\ & \left.-\frac{B_{m}}{A_{m} \eta_{w_{M P}}} \dot{\hat{W}}_{M \boldsymbol{M}}^{T} \tilde{\boldsymbol{W}}_{\boldsymbol{M P}}\right]+\frac{e_{2} B_{m}}{A_{m}}\left(\hat{\psi}-\hat{U}_{c}\right) \\ & +\frac{B_{m}}{A_{m} \gamma} \tilde{\psi}\left(\gamma e_{2}-\dot{\hat{\psi}}\right) \tag{43}\end{align*}
\begin{align*} \dot {V}_{2} (e_{1} (t),\tilde {\psi }(t),\tilde { \boldsymbol {W}},\tilde { \boldsymbol {W}}_{ \boldsymbol {l}}, \tilde { \boldsymbol {W}}_{ \boldsymbol {MP}},\tilde { \boldsymbol {m}},\tilde { \boldsymbol {\sigma }})=-c_{1} e_{1}^{2} -c_{2} e_{2}^{2} \le 0 \tag{44}\end{align*}
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:
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
$\omega _{r}$ $e_{1}=\omega _{r}^{\ast } (t)-\omega _{r}(t)$ $e_{2} =\dot {\omega }_{r}(t)+\lambda _{1}$ RLFNN Input Layer: The input variables of the proposed FNN are
$x_{1}^{1}=e_{1} $ $x_{2}^{1} =e_{2}$ $y_{i}^{1}(N)$ RLFNN Membership Layer: The Gaussian functions are adopted to implement the fuzzification operation and the outputs are
$y_{ij}^{2}(N)$ $y_{ij}^{2}(N)$ RLFNN Legendre Layer: This layer uses the Legendre polynomial as the expansion function, and its input variable vector
$x=[x_{1}^{1},x_{2}^{1}]$ \begin{align*} \lambda & =[\lambda _{1}^{3},\lambda _{2}^{3},\ldots, \lambda _{9}^{3}]^{T} \\ & =[L_{0}, L_{1} (x_{1}^{1}), L_{2} (x_{1}^{1}), L_{3} (x_{1}^{1}), L_{4} (x_{1}^{1}), L_{1} (x_{2}^{1}), \\ &\qquad L_{2} (x_{2}^{1}), L_{3} (x_{2}^{1}), L_{4} (x_{2}^{1})]^{T}\end{align*} View Source\begin{align*} \lambda & =[\lambda _{1}^{3},\lambda _{2}^{3},\ldots, \lambda _{9}^{3}]^{T} \\ & =[L_{0}, L_{1} (x_{1}^{1}), L_{2} (x_{1}^{1}), L_{3} (x_{1}^{1}), L_{4} (x_{1}^{1}), L_{1} (x_{2}^{1}), \\ &\qquad L_{2} (x_{2}^{1}), L_{3} (x_{2}^{1}), L_{4} (x_{2}^{1})]^{T}\end{align*}
$L_{h}(x)$ ${h}$ $y_{p}^{3}(N)$ $w_{MP}^{3}$ $\lambda _{P}^{3}$ 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_{l}^{4}(N) $ 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
$y_{n}^{5}(N)$ RLFNN Output Layer: The node performs the summation and multiplying operation. Then, the output is given as
$y_{o}^{6}(N)$ $i_{q}^{\ast } $ Online Network Parameters Learnings: The online parameters learning are achieved by online tuning of the connective weights
$w_{o}^{6}$ $w_{l}^{4}$ $w_{MP}^{3}$ $m_{ij}$ $\sigma _{ij}$
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.
Experimentation
Fig. 10 depicts the experimental setup, which encompasses the PMASynRM drive, torque meter, the DSP TMS320F28075 control board, input/output (I/O) extension board, encoder interface board, IPMSM load, and the personal computer for the development system. A controlled DC power supply powers the DC link of the 4.5 kW SiC based VSI. During the experiment, the team sets an industrial drive of the 7.5 kW IPMSM at torque control mode and 5 Nm and 10 Nm for load torque. An incremental encoder operated through a QEP interface featuring a 1 ms sampling rate ascertains the position of the PMASynRM. 0.1 ms serves as the current control loop’s switching and sampling intervals. Subsequently, the PMASynRM drive is operated by the SVPWM’s switching commands being delivered to the VSI.
\begin{align*} T_{M} &=\max \limits _{N} (\left |{ {T_{error} (N)} }\right |) \tag{45}\\ T_{aver}&=\frac {\sum \limits _{N=1}^{h} {\left |{ {T_{error} (N)} }\right |}}{h} \tag{46}\\ T_{sd} &=\sqrt {\frac {\sum \limits _{N=1}^{h} {(T_{error} (N)-T_{aver} (N))^{2}}}{h}} \tag{47}\end{align*}
\begin{equation*} \frac {\omega _{n}^{2}}{s^{2}+2\xi \omega _{n} s+\omega _{n}^{2} }=\frac {30}{s^{2}+11s+30} \tag{48}\end{equation*}
\begin{align*} a& =1.5, \quad c_{1} =545, c_{2} =0.24 \\ c_{1} & =545, \quad \eta _{w} =0.01, \eta _{m} =0.175, \tag{49}\\ \eta _{\sigma } & =0.017, \quad \eta _{w_{l}} =0.22, \eta _{w_{MP}} =0.95, \gamma =18 \tag{50}\end{align*}
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 q-d 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. Figs. 12 and 13 depict the experimental results of the command tracking due to the periodical step and sinusoidal commands of CTC speed controller at case 1 and case 2. The rotor responses and tracking errors are shown in Figs. 12(a), 12(c), 13(a) and 13(c); the current commands are shown in Figs. 12(b), 12(d), 13(b) and 13(d). On the other hand, Figs. 14 and 15 depict the experimental results of the command tracking due to the periodical step and sinusoidal commands of the proposed ICTCRLFNN speed controller at case 1 and case 2. The rotor responses and tracking errors are shown in Figs. 14(a), 14(c), 15(a) and 15(c); the current commands are shown in Figs. 14(b), 14(d), 15(b) and 15(d). From the experimental results, the
Current responses of q-d axis current control by using PI controllers. (a) Step command and response of
Results of CTC speed controller for periodical step command. (a) Rotor response and tracking error at case 1. (b) Stator and
Results of CTC speed controller for periodical sinusoidal command. (a) Rotor response and tracking error at case 1. (b) Stator and
Results of ICTCRLFNN speed controller for periodical step command. (a) Rotor response and tracking error at case 1. (b) Stator and
Results of ICTCRLFNN speed controller for periodical sinusoidal command. (a) Rotor response and tracking error at case 1. (b) Stator and
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.
Maximum, average and standard deviation of tracking errors CTC and ICTCRLFNN. (a) Step command at case 1. (b) Sinusoidal command at case 1. (c) Step command at case 2. (d) Sinusoidal command at case 2.
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.