Electromagnetic torque control for synchronous reluctance motor servo-drive system applied in continuously variable transmission system

Abstract

Compared with the classical linear controller, the nonlinear controller can result better control performance for the nonlinear uncertainties of the continuously variable transmission (CVT) system which is driven by the synchronous reluctance motor (SynRM). The better control performance can be shown in the nonlinear uncertainties behavior of CVT system by using the proposed novel admixed improved recurrent Hermite polynomial neural network (AIRHPNN) control system. The novel AIRRHPNN control system can carry out overlooker control, improved recurrent Hermite polynomial neural network control (IRHPNN) with an adaptive law, and reimbursed control with an appraised law. Additionally, according to the Lyapunov stability theorem, the adaptive law in the IRHPNN and the appraised law of the eimbursed controller are established. Furthermore, two weights with two varied learning rates according to increment Lyapunov function are derived to help improving convergence. Finally, comparative examples are illustrated by the experimental results to confirm that the proposed control system could be obtained better control performance.

Keywords

Continuously variable transmission Lyapunov stability recurrent Hermite polynomial neural network synchronous reluctance motor

1. Introduction

Compared with the others in servomotors, the synchronous reluctance motor (SynRM) through the optimal design methods [1,2] is one of highest efficiency and lowest cost motors. Many researchers [3–6] were dedicated to the dive and control of the SynRM because of advancement of optimal design methods and power electronics technologies. However, the SynRM servo-drive continuously variable transmission (CVT) system [7,8] is yet not proposed in studying of dynamic control. The CVT speed control has been reported in many reports [9,10]. The dynamic responses of CVT system thus are studied in this paper.

Owing to neural network’s inherent parallel structure and good learning ability [11,12], they have better approximation capability in modeling nonlinear systems. They thus yield computationally expensive and large number of iterations for their trainings. To bring down the computational complexity, a functional-type neural network (NN) [13–16] with much less computational cost has been introduced. The functional-type NN has lesser computational complexity and faster convergence than the NN in the performance execution phase. Ma et al. [17,18] proposed a computationally efficient Hermite polynomial NN. The constructing Hermite polynomial expansions were adopted by using the structure and function levels adaptation methodologies. This Hermite polynomial NN can catch the underlying input-output map effectively. Rigatos et al. [19] proposed the Hermite polynomial NN that can be used in nonparametric estimation. Siniscalchi et al. [20] proposed a Hermite polynomial NN to apply in connectionist speech recognition systems with speaker adaptation. However, these Hermite polynomial NNs were most applied for system modelling and image processing. Moreover, the weight updates of these NNs were not utilize the internal information of the NNs and were sensitive to the function approximation in the training procedure. Owing to more precision approximation in the modelling nonlinear system and dynamic control [13–16], many researchers have been fascinated with the recurrent neural network’s (RNN’s) studies. These RNNs are able to carry out identification and control of complex dynamics system, but they need higher computational costs. The proposed improved recurrent Hermite polynomial neural network (IRHPNN) thus has better dynamic mapping performance and less computational time when the uncertainties appeared. The proposed novel admixed improved recurrent Hermite polynomial neural network (AIRHPNN) control system for controlling SynRM servo-drive CVT system with nonlinear dynamics is presented in order to reduce computational complexity and raise system robustness in this study.

Table 1
Performance comparison of control systems

Performance Control system and three test cases

Well-known PI controller

157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u

Maximum error of e 9.5 rad/s 22.8 rad/s 42.7 rad/s

(91 rpm) (218 rpm) (408 rpm)

RMS error of e 5.0 rad/s 6.5 rad/s 5.5 rad/s

(48 rpm) (62 rpm) (53 rpm)

Performance Control system and three test cases

Three-layer feedforward NN control system

157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u

Maximum error of e 7.5 rad/s 9.5 rad/s 21.3 rad/s

(72 rpm) (91 rpm) (204 rpm)

RMS error of e 3.5 rad/s 3.5 rad/s 3.0 rad/s

(33 rpm) (33 rpm) (29 rpm)

Performance Control system and three test cases

Novel AIRHPNN control system

157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$ 314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u

Maximum error of e 4.5 rad/s 5.2 rad/s 7.6 rad/s

(43 rpm) (50 rpm) (73 rpm)

RMS error of e 2.0 rad/s 2.5 rad/s 2.0 rad/s

(19 rpm) (24 rpm) (19 rpm)

The novel AIRHPNN control system can carry out overlooker control, IRPHNN control with an adaptive law, and reimbursed control with an appraised law. According to Lyapunov stability, the adaptive law for training parameters online in the IRPHNN can be derived. The IRPHNN with learning procedure online thus could respond to nonlinear uncertainty of system. In addition, the two optimal values in the learning rates of connective and recurrent weights for the IRPHNN can be also derived and attain better convergence by using increment Lyapunov function. Finally, comparisons of the experimental results are demonstrated to confirm that the novel AIRHPNN control system can achieve good control performance.

This paper is composed in the following parts: Section 2 is indicated by the structure of the SynRM servo-drive CVT system, Section 3 is presented by a design method of novel AIRHPNN control system, Section 4 is denoted by experimental results and Section 5 is given by some conclusions.

2. Structure of the SynRM servo-drive CVT system

2.1. Configuration of CVT system with uncertainties

The geometric constitution for the simplified kinematics of the CVT system with negligible belt flexural effects and slip losses is illustrated in Fig. 1. The torque dynamic equations [7–10] shown in Fig. 1(a) and Fig. 1(b) with the front driving shaft and the rear driven shaft by using law of conservation could be simplified as $\begin{eqnarray}\displaystyle \text{} & \text{}T_{e}=J_{a}\dot{{\omega}}_{r}+B_{a}{\omega}_{r}+T_{1} & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{}T_{2}=J_{2}\dot{{\omega}}_{2}+B_{2}{\omega}_{2}+T_{2}^{l}(f_{l2}(v_{a2},{\tau}_{a2},F_{2l},B_{2},{\omega}_{2}^{2})), & \displaystyle\end{eqnarray}$ (2) where $T_{1}={\sigma}_{2}({\varsigma}_{1},{\varsigma}_{2}){\omega}_{2}T_{2}/{\omega}_{r}=T_{1}^{l}(T_{a},{\rm\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})+J_{1}∼\dot{{\omega}}_{r}+B_{1}{\omega}_{r}$ is the equivalent driven torque at the front pulley shaft. σ₂(𝜍₁, 𝜍₂) is the conversion ratio between the front pulley shaft and the rear pulley shaft. (𝜍₁, 𝜍₂) is the sliding active arcs to contribute torque transmission at low speed. (𝛼₁, 𝛼₂) is the wrap angles of the belt-pulley contacting arcs. T ₂ is the driven torque of the rear pulley shaft. $T_{2}^{l}(f_{l2}(v_{a2},{\tau}_{a2},F_{2l},B_{2},{\omega}_{2}^{2}))$ is the lumped nonlinear external disturbances including the rolling resistance v _a2, the wind resistance τ_a2 and the braking force F _2l. J ₂, B ₂, J _a and B _a are the mechanical angular velocity of the rotor represent the moment of inertia of the wheel, the viscous friction coefficients of the wheel, the moment of inertia of the front pulley side, the viscous friction coefficients of the front pulley side, respectively. ω₂ and ω_r and the velocity of the rear pulley shaft and the mechanical angular velocity of the rotor, respectively. Then the torque equation can be transformed from the rear pulley side to the front pulley side by using speed ratio and sliding ratio [7–10]. It is assumed that the belt flexural effects, power and slip losses are neglected. Consequently, the combined dynamic equation [7–10] of the SynRM servo-drive CVT system from Eq. (1) to Eq. (2) can be presented by $\begin{eqnarray}\displaystyle T_{e}=T_{1}^{l}(T_{a},{\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})+J_{r}\dot{{\omega}}_{r}+B_{r}{\omega}_{r}, & & \displaystyle\end{eqnarray}$ (3) where $T_{1}^{l}(T_{a},{\rm\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})=T_{a}+{\rm\Delta}T+T_{u}$ [7–10] is the combined lumped nonlinear external disturbances and parameter variations. T _a, ${\rm\Delta}T={\rm\Delta}J_{r}∼\dot{{\omega}}_{r}+{\rm\Delta}B_{r}{\omega}_{r}$ and $T_{u}=F_{l}(B_{2})+{\upsilon}_{a}(v_{2},B_{2})+{\tau}_{a}(v_{2}){\omega}_{r}^{2}$ are the fixed load torque, the combined parameter variation and the combined unknown nonlinear load torque, respectively. 𝜐_a(v ₂, B ₂), τ_a(v ₂) and F _l(B ₂) are the combined rolling resistance, the combined wind resistance and the combined braking force, respectively. B _r = B ₁ + B _a and J _r = J ₁ + J _a are the combined viscous friction coefficient and the combined moment of inertia, respectively. J ₁ and B ₁ represent the moment of inertia of the SynRM and the viscous friction coefficients of the SynRM, respectively.

Fig. 1.

Geometric constitution of the CVT system driven by the SynRM: (a) sketch graph of the SynRM-wheel set via the CVT connection, and (b) geometric graph of the CVT system.

2.2. Configuration of the SynRM servo-drive system

The synchronously rotating reference frame of the SynRM with d–q axis frame can be offered as [3–6]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{Q}=r_{s}i_{Q}+L_{Q}\dot{i}_{Q}+p_{n}{\omega}_{r}L_{D}i_{D}/2\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{D}=r_{s}i_{D}+L_{D}\dot{i}_{D}-p_{n}{\omega}_{r}L_{Q}∼i_{Q}/2,\end{eqnarray}$ (5) where u _D, i _D and L _D are the d-axis stator voltage, current and inductance, respectively. u _Q, i _Q and L _Q are the q-axis stator voltage, current and inductance, respectively. r _s is the stator resistance. p _n is the number of pole. The electromagnetic torque can be denoted as $\begin{eqnarray}\displaystyle T_{e}=3p_{n}[(L_{D}-L_{Q})∼i_{Q}i_{D}]/4, & & \displaystyle\end{eqnarray}$ (6) and the motor dynamics equation is $\begin{eqnarray}\displaystyle T_{e}=T_{l}+B_{1}{\omega}_{r}+J_{1}\dot{{\omega}}_{r}, & & \displaystyle\end{eqnarray}$ (7) where $T_{l}=T_{1}^{l}(T_{a},{\rm\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})+J_{a}∼\dot{{\omega}}_{r}+B_{a}{\omega}_{r}$ stands the load torque and T _e stands for the electromagnetic torque.

2.3. Structure of the SynRM servo-drive system

The Sketch graph of the SynRM servo-drive CVT system is shown in Fig. 2.

Fig. 2.

Sketch graph of the SynRM servo-drive CVT system.

The entire system of the SynRM servo-drive CVT system can be denoted as: a speed/torque control system, a field-oriented control, the interlock and isolated circuits, the voltage source inverter (VSI) with three-sets of insulated-gate bipolar transistor (IGBT) power modules inverter, a SynRM servo-drive system and a CVT sytem. The field-oriented control consists of a sinusoidal pulse width modulation (PWM) control modulator, a proportional-integral (PI) current control, a coordinate translation system including inverse coordinate translation, $sin \theta_f / cos\theta_f$ generation and lookup table generation. The mix signal field-programmable-gate-array (FPGA) system using Xilinx Spartan-XCS05XL chip was used to perform field-oriented control. The digital-signal-processor (DSP) control system using Texas Instrument TMS320F28335 chip was used to perform speed/torque control. Two gains for the PI current controller are given as follows: k _ic = k _pc∕T _ic = 6.22, k _pc = 12.5 through some heuristic knowledge [21–23] to obtain better dynamic response. The SynRM servo-drive CVT system was operated under lumped external disturbances and nonlinear uncertainties.

3. Novel AIRHPNN control system

The nonlinear uncertainties of the SynRM servo-drive CVT system, such as the nonlinear friction of the transmission V-belt and clutch, rolling resistance, wind resistance, and braking force, lead to degenerate tracking responses in the command current and speed of the SynRM servo-drive CVT system. These nonlinear uncertainties cause the rotor inertia and friction of the CVT system to vary. For simplifying the novel AIRHPNN control system design, the dynamic model from Eq. (3) can be represented as $\begin{eqnarray}\displaystyle \dot{{\omega}}_{r} & = & \displaystyle T_{e}/J_{r}-T_{1}^{l}(T_{a},{\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})/J_{r}-B_{r}{\omega}_{r}/J_{r}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle A_{e}u_{e}+B_{e}T_{1}^{l}(T_{a},{\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})/J_{r}+C_{e}{\omega}_{r},\end{eqnarray}$ (8) where u _e = T _e is the control effort, i.e., the command torque of the SynRM. A _e = 1∕J _r, B _e = −1∕J _r and C _e = −B _r∕J _r are three known constants. When the uncertainties occur, the parameters can be assumed to be bounded, i.e., $|B_{e}T_{1}^{l}(T_{a},{\rm\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})|\leq D_{2}$ , D ₁ ≤ A _e and |C _eω_r| ≤ D ₃(ω_r). D ₃(ω_r) is a known continuous function. D ₁ and D ₂ are known constants. Then, the tracking error can be defined as $\begin{eqnarray}\displaystyle e={\omega}-{\omega}_{r}, & & \displaystyle\end{eqnarray}$ (9) where e is the track error between the desired rotor speed ω and actual rotor speed ω_r. If all parameters of the SynRM servo-drive CVT system including the lumped nonlinear external disturbances and parameter variations are well known, the ideal control law can be designed as $\begin{eqnarray}\displaystyle u_{e}^{\ast }=[\dot{{\omega}}+k_{e}e-C_{e}{\omega}_{r}-B_{e}T_{1}^{l}(T_{a},{\Delta}T,F_{l}(B_{2}),{\upsilon}_{a}(v_{2},B_{2}),{\tau}_{a}(v_{2}),{\omega}_{r}^{2})]/A_{e}, & & \displaystyle\end{eqnarray}$ (10) where k _e is a positive constant. Equation (10) with $u_{e}^{\ast }=u_{e}$ can be substituted into Eq. (8), for obtaining the following error dynamic equation $\begin{eqnarray}\displaystyle {\dot{e}}+k_{e}e=0. & & \displaystyle\end{eqnarray}$ (11) If e (t) → 0 as in $t\rightarrow\infty$ in Eq. (11), then the state of the system can be used to asymptotically track the desired output trajectory. For improved tracking of the trajectory, the novel AIRHPNN control system was developed for controlling the SynRM servo-drive CVT system under uncertain perturbations.

The configuration of the proposed novel AIRHPNN control system is shown in Fig. 3. The proposed control system consists of an overlooker control, an IRPHNN control with an adaptive law, and a reimbursed control with an appraised law. The control law is designed as $\begin{eqnarray}\displaystyle u_{e}=u_{1}+u_{2}+u_{3}, & & \displaystyle\end{eqnarray}$ (12) where u ₁ is the overlooker control that is capable of stabilizing the states of the controlled system around a predetermined bound area, u ₂ is the IRHPNN control which is the major tracking controller. The control law is designed to imitate an ideal control law. The reimbursed control, u ₃ is designed to reimburse the difference between the ideal control law and the IRHPNN control. Because the overlooker control causes the overdone and chattering effort, the IRHPNN control and reimbursed control are proposed to reduce and smooth the control effort when the system states are within the predetermined bound area. When the IRHPNN control approximation properties cannot be guaranteed, the overlooker control is effective.

Fig. 3.

Configuration of the proposed novel AIRHPNN control system.

For the condition of divergence of states, the design of novel AIRHPNN control system is essential to stretch the divergent states back to the predestinated bound area. The novel AIRHPNN control system can uniformly approximate the ideal control law inside the bound area. Then stability of the novel AIRHPNN control system can be guaranteed. An error dynamic equation from Eq. (8) to Eq. (12) can be obtained $\begin{eqnarray}\displaystyle {\dot{e}}=-k_{e}e+[u_{e}^{\ast }-u_{1}-u_{2}-u_{3}]A_{e}. & & \displaystyle\end{eqnarray}$ (13)

3.1. Overlooker control

Firstly, the overlooker control u ₁ can be designed as $\begin{eqnarray}\displaystyle u_{1}=f_{1}\text{sgn}(eA_{e})[D_{3}({\omega}_{r})+D_{2}+|\dot{{\omega}}|+|k_{e}e|]/A_{e}, & & \displaystyle\end{eqnarray}$ (14) where $\mathrm{sgn} (\cdot)$ is a sign function. When the IRHPNN approximation properties cannot be guaranteed, the overseer control law is effective; in other words, f ₁ = 1. Because of the inadequate bound values, such as D ₁, D ₂, D ₃(ω_r) and the sign function, the overlooker control can produce in overdone and chattering effort. Therefore, the IRHPNN control and the reimbursed control can be designed to overcome the aforementioned drawback. The IRHPNN control raised to imitate the ideal control $u_{2}^{\ast }$ . Then the reimbursed control posed to reimburse the difference between the ideal control and the IRHPNN control.

3.2. IRPHNN control

Secondly, the construction of the proposed three-layer IRHPNN, which is composed of an input layer, a hidden layer and an output layer, is shown in Fig. 4.

Fig. 4.

Construction of the three-layer IRHPNN.

The activation functions and signal actions of nodes in each layer of the IRPHNN are listed in the following segments:

(1) In the input layer, input and output signals for each node i are expressed as $\begin{eqnarray}\displaystyle o_{i}^{1}=l_{i}^{1}\left(\mathop{\prod }_{k}a_{i}^{1}(N)u_{ik}^{1}∼o_{k}^{3}(N-1)\right),\quad i=1,2. & & \displaystyle\end{eqnarray}$ (15) In the input layer, each node i is indicated by using $\Pi$ , which multiplies by all input signals. $a_{1}^{1}=({\omega}-{\omega}_{r})$ is the tracking error. $a_{2}^{1}=e(1-z^{-1})={\rm\Delta}e$ is the tracking error change. $u_{ik}^{1}$ is the recurrent weight from output layer to input layer. $l_{i}^{1}$ is the activation function which is selected as a linear function. N is the number of iterations and $o_{k}^{3}$ is the output value in the output layer.

(2) In the hidden layer, input and output signals for each node j are expressed as $\begin{eqnarray}\displaystyle o_{j}^{2}=H_{j}\displaystyle \left(\mathop{\sum }_{i=1}^{2}o_{i}^{1}(N)+{\tau}∼o_{j}^{2}(N-1)\right),\quad j=0,1,\ldots ,∼m-1. & & \displaystyle\end{eqnarray}$ (16) In the hidden layer, each node j is indicated by using $\Sigma$ and summation, which summates all input signals. Hermite polynomials [17–20] are selected for activation function in the hidden layer. Hermite polynomials [17–20] H _n(x) is the argument of the polynomials with −1 < x < 1; n is the order of expansion. m is the number of nodes. The zero, the first and the second order Hermite polynomials are given by H ₀(x) = 1, H ₁(x) = 2x and H ₂(x) = 4x ² −2, respectively. The higher order Hermite polynomials may be generated by the recursive formula H _n+1(x) = 2xH _n(x) − 2nH _n−1(x). τ is the self-connecting feedback gain of the hidden layer which is selected between 0 and 1.

(3) In the output layer, input and output signals for each node k are expressed as $\begin{eqnarray}\displaystyle o_{k}^{3}=l_{k}^{3}\displaystyle \left(\mathop{\sum }_{j=0}^{m-1}u_{kj}^{2}o_{j}^{2}(N)\right),\quad k=1. & & \displaystyle\end{eqnarray}$ (17) Each node k is presented using $\Sigma$ , which denotes the summation of all input signals. The parameter $u_{kj}^{2}$ is the connective weight from the hidden layer to the output layer. $l_{k}^{3}$ is the activation function which is selected as a linear function. The signals of the jth input to a node of the output layer can be represented as $a_{j}^{3}(N)=o_{j}^{2}(N)$ . Output value of the output layer can be indicated as $o_{k}^{3}(N)=u_{2}$ . Then output value of the IRPHNN can be expressed as $\begin{eqnarray}\displaystyle u_{2}={\Phi}^{T}{\Omega}, & & \displaystyle\end{eqnarray}$ (18) where ${\Phi}=[\begin{array}{@{}ccc@{}}u_{10}^{2} & \cdots ∼ & u_{1∼(m-1)}^{2}\end{array}]^{T}$ is a vector of the weight parameters from the hidden layer to the output layer in the IRPHNN. ${\Omega}=[\begin{array}{@{}ccc@{}}a_{0}^{3} & \cdots ∼ & a_{(m-1)}^{3}\end{array}]^{T}$ is an inputs vector in the output layer of the IRPHNN, where $a_{j}^{3}$ presents selected Hermite polynomials.

3.3. Reimbursed control

Thirdly, to implement reimbursed control u ₃, a minimum affinity error 𝜑 can be defined as $\begin{eqnarray}\displaystyle {\varphi}=u_{e}^{\ast }-u_{2}^{\ast }=u_{e}^{\ast }-({\Phi}^{\ast })^{T}{\Omega}, & & \displaystyle\end{eqnarray}$ (19) where 𝛷^∗ is an ideal weight vector to achieve minimum affinity error. It is assumed that a small positive value 𝜌 is greater than the absolute value of 𝜑, that is, 𝜌 > |𝜑|. Then, the error dynamic equation (13) can be rewritten as $\begin{eqnarray}\displaystyle {\dot{e}} & = & \displaystyle -k_{e}e+[u_{e}^{\ast }-u_{1}-u_{2}-u_{3}]A_{e}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -k_{e}e+[(u_{e}^{\ast }-u_{2}^{\ast })+(u_{2}^{\ast }-u_{2})-u_{3}-u_{1}]A_{e}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -k_{e}e+[(u_{e}^{\ast }-({\Phi}^{\ast })^{T})+(({\Phi}^{\ast })^{T}-({\Phi})^{T}){\Omega}-u_{3}-u_{1}]A_{e}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -k_{e}e+[({\varphi}+({\Phi}^{\ast }-{\Phi})^{T}){\Omega}-u_{3}-u_{1}]A_{e}.\end{eqnarray}$ (20) The following Lyapunov function is selected as $\begin{eqnarray}\displaystyle P(t)=e^{2}/2+[({\Phi}^{\ast }-{\Phi})^{T}({\Phi}^{\ast }-{\Phi})/(2{\eta}_{1})+\tilde{{\delta}}^{2}/(2{\alpha})], & & \displaystyle\end{eqnarray}$ (21) where 𝜂₁ is the learning rate, 𝛼 is the adaptation gain, and $\tilde{{\delta}}=\hat{{\delta}}-{\delta}$ is the bound estimation error. Equation (20) can be represented as follows by differentiating the Lyapunov function and by using Eq. (19) $\begin{eqnarray}\displaystyle {\dot{P}}(t) & = & \displaystyle e∼{\dot{e}}-({\Phi}^{\ast }-{\Phi})^{T}\dot{{\Phi}}/{\eta}_{1}+\tilde{{\delta}}\hat{{\delta}}/{\alpha}\nonumber\\ \displaystyle & = & \displaystyle -k_{e}e^{2}+[{\varphi}+({\Phi}^{\ast }-{\Phi})^{T}{\Omega}-u_{3}-u_{1}]eA_{e}-({\Phi}^{\ast }-{\Phi})^{T}\dot{{\Phi}}/{\eta}_{1}+\tilde{{\delta}}\dot{\hat{{\delta}}}/{\alpha}.\end{eqnarray}$ (22) For achieving ${\dot{P}}(t)\leq 0$ , the adaptive law $\dot{{\Phi}}$ , the reimbursed controller u ₃, and the appraised law $\dot{\hat{{\delta}}}$ can be designed $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \dot{{\Phi}}={\eta}_{1}{\Omega}∼e∼A_{e}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{3}=\hat{{\delta}}\text{sgn}(e∼A_{e})\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \dot{\hat{{\delta}}}={\alpha}|e∼A_{e}|.\end{eqnarray}$ (25) If Eqs (14), (19) and (24) are substituted into Eq. (22), and if Eq. (14) with f ₁ = 1 is used, then Eq. (22) can be represented as $\begin{eqnarray}\displaystyle {\dot{P}}(t)=-k_{e}e^{2}+({\varphi}-\hat{{\delta}}\text{sgn}(eA_{e}))A_{e}e+\tilde{{\delta}}\dot{\hat{{\delta}}}/{\alpha}. & & \displaystyle\end{eqnarray}$ (26) The substitution of Eq. (25) into Eq. (26) gives $\begin{eqnarray}\displaystyle {\dot{P}}(t)\leq -k_{e}e^{2}+({\varphi}-{\delta})|e∼A_{e}|\leq -k_{e}e^{2}\leq 0. & & \displaystyle\end{eqnarray}$ (27) Equation (27) shows ${\dot{P}}_{1}(t)$ to be negative semi-definite (i.e., P (t) ≤ P (0)), implying that $e$ and (𝛷^∗−𝛷) are bounded. Additionally, the function is defined as $\begin{eqnarray}\displaystyle Q_{1}(t)=-{\dot{P}}(t)=k_{e}e^{2}. & & \displaystyle\end{eqnarray}$ (28) Integration Eq. (27) gives $\begin{eqnarray}\displaystyle \int _{0}^{t}Q_{1}({\varsigma})d{\varsigma}=\int _{0}^{t}-[P({\varsigma})]d{\varsigma}=P(0)-P(t). & & \displaystyle\end{eqnarray}$ (29) Because P (0) is bounded and P (t) is a non-increasing and bounded, then $\begin{eqnarray}\displaystyle \lim _{t\rightarrow \infty }\int _{0}^{t}Q_{1}({\varsigma})d{\varsigma}\lt \infty . & & \displaystyle\end{eqnarray}$ (30) Differentiating Eq. (28) gives $\begin{eqnarray}\displaystyle \dot{Q}_{1}(t)=2k_{e}e{\dot{e}}. & & \displaystyle\end{eqnarray}$ (31) All the variables on the right-hand side of (24) are bounded. This implies that ${\dot{e}}$ is also bounded. Then, Q ₁(t) is a uniformly continuous function [24,25]. It is denoted that lim_t→∞ Q ₁(t) = 0 by using Barbalat’s lemma [24,25]. Therefore e (t) → 0 as t →∞. Furthermore, sgn (e A _e) can be replaced by the equation (e A _e)∕(|e A _e| + 𝜅) in order to avoid chattering phenomenon of reimbursed controller u ₃, in which 𝜅 is a positive constant.

3.4. Training methodology of the IRHPNN

An online parameters training methodology of adaptive law $\dot{{\Phi}}$ in the IRHPNN can be derived and computed according to Lyapunov stability theorem and the gradient descent method. For achieving better convergence of tracking error, two optimal learning rates are used for training parameters of the IRHPNN. First, the parameter of adaptive law $\dot{{\Phi}}$ shown in Eq. (23) can be represented as $\begin{eqnarray}\displaystyle \dot{u}_{kj}^{2}={\eta}_{1}e∼A_{e}o_{j}^{2}. & & \displaystyle\end{eqnarray}$ (32) For describing online training algorithm of the IRHPNN, a cost function is defined as [13] $\begin{eqnarray}\displaystyle P_{1}=e^{2}/2. & & \displaystyle\end{eqnarray}$ (33) In accordance with the gradient descent method and chain rule, the adaptive law of the weight also can be represented as $\begin{eqnarray}\displaystyle \dot{u}_{kj}^{2}=-{\eta}_{1}{\displaystyle \frac{\partial P_{1}}{\partial u_{kj}^{2}}}=-{\eta}_{1}{\displaystyle \frac{\partial P_{1}}{\partial u_{2}}}{\displaystyle \frac{\partial u_{2}}{\partial o_{k}^{3}}}{\displaystyle \frac{\partial o_{k}^{3}}{\partial u_{kj}^{2}}}=-{\eta}_{1}{\displaystyle \frac{\partial P_{1}}{\partial u_{2}}}o_{j}^{2}. & & \displaystyle\end{eqnarray}$ (34) Comparing Eq. (32) with Eq. (34), yields ∂P ₁∕∂u ₂ = −eA _e. Then, the adaptive law of recurrent weight $u_{ik}^{1}$ using the gradient descent method and chain rule can be represented as $\begin{eqnarray}\displaystyle \dot{u}_{ik}^{1} & = & \displaystyle -{\eta}_{2}{\displaystyle \frac{\partial P_{1}}{\partial u_{ik}^{1}}}=-{\eta}_{2}{\displaystyle \frac{\partial P_{1}}{\partial u_{2}}}{\displaystyle \frac{\partial u_{2}}{\partial o_{k}^{3}}}{\displaystyle \frac{\partial o_{k}^{3}}{\partial o_{j}^{2}}}{\displaystyle \frac{\partial o_{j}^{2}}{\partial o_{i}^{1}}}{\displaystyle \frac{\partial o_{i}^{1}}{\partial u_{ik}^{1}}}\nonumber\\ \displaystyle & = & \displaystyle {\eta}_{2}eA_{e}u_{kj}^{2}H_{j}(\cdot )a_{i}^{1}(N)O_{k}^{3}(N-1)\end{eqnarray}$ (35) in which 𝜂₂ is a learning rate.

3.5. Derivation of two optimal learning rates in the IRHPNN and convergence analysis

To obtain better convergence speed two learning rates of the weights in the IRHPNN thus is proposed. Two optimal learning rates are thus derived to assure convergence of the output tracking error, and the convergence analysis is provided in the following two theorems.

TheoremA .
Let’s assume that 𝜂₁ is the learning rate of connective weight between the hidden layer and the output layer in the IRPHNN. Meanwhile, let R _1 max be defined as R _1 max ≡ max _N∥R ₁(N)∥, in which $R_{1}(N)=\partial ∼o_{k}^{3}/\partial ∼u_{kj}^{2}$ and ∥ ⋅ ∥ is the Euclidean norm in $\Re$ ⁿ. If 𝜂₁ is chosen as [26] $\begin{eqnarray}\displaystyle 0\lt {\eta}_{1}\lt 2/(A_{e}R_{1\max })^{2}. & & \displaystyle\end{eqnarray}$ (36) The convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate, which achieves the fast convergence, can be obtained $\begin{eqnarray}\displaystyle {\eta}_{1}^{\ast }=1/(A_{e}R_{1\max })^{2}. & & \displaystyle\end{eqnarray}$ (37)

TheoremB .
Since $\begin{eqnarray}\displaystyle R_{1}(N)={\displaystyle \frac{\partial o_{k}^{3}}{\partial u_{kj}^{2}}}=o_{j}^{2}. & & \displaystyle\end{eqnarray}$ (38) A Lyapunov function can be selected as $\begin{eqnarray}\displaystyle Y_{4}(N)={\displaystyle \frac{1}{2}}e^{2}(N). & & \displaystyle\end{eqnarray}$ (39) The increment in the Lyapunov function can be depicted as $\begin{eqnarray}\displaystyle {\Delta}Y_{4}(N)=Y_{4}(N+1)-Y_{4}(N)={\displaystyle \frac{1}{2}}[e^{2}(N+1)-e^{2}(N)]. & & \displaystyle\end{eqnarray}$ (40) Then, the error difference can be represented by $\begin{eqnarray}\displaystyle e(N+1)=e(N)+{\Delta}e(N)=e(N)+\left[{\displaystyle \frac{\partial ∼e(N)}{\partial ∼u_{kj}^{2}}}\right]^{T}{\Delta}u_{kj}^{2}, & & \displaystyle\end{eqnarray}$ (41) in which Δe (N) is the output error increment, and ${\rm\Delta}u_{kj}^{2}$ represents increment of the weight. The Eq. (41) using Eqs (32), (33), (34) and (38) can be obtained as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \frac{\partial ∼e(N)}{\partial ∼u_{kj}^{2}}={\displaystyle \frac{\partial ∼e(N)}{\partial ∼P_{1}}}{\displaystyle \frac{\partial P_{1}}{\partial u_{2}}}{\displaystyle \frac{\partial u_{2}}{\partial o_{k}^{3}}}{\displaystyle \frac{\partial ∼o_{k}^{3}}{\partial ∼u_{kj}^{2}}}=-{\displaystyle \frac{eA_{e}}{e(N)}}R_{1}(N)\end{eqnarray}$ (42) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle e(N+1)=e(N)-\left[{\displaystyle \frac{eA_{e}}{e(N)}}R_{1}(N)\right]^{T}{\eta}_{1}eA_{e}R_{1}(N).\end{eqnarray}$ (43) Thereby $\begin{eqnarray}\displaystyle \Vert e(N+1)\Vert & = & \displaystyle \Vert e(N)[1-{\eta}_{1}(eA_{e}/e(N))^{2}R_{1}^{T}(N)R_{1}(N)]\Vert \nonumber\\ \displaystyle & \leq & \displaystyle \Vert e(N)\Vert ∼\Vert 1-{\eta}_{1}(eA_{e}/e(N))^{2}R_{1}^{T}(N)R_{1}(N)\Vert .\end{eqnarray}$ (44) Then ΔY ₄(N) using Eqs (41), (42), (43) and (44) can be depicted as $\begin{eqnarray}\displaystyle {\Delta}Y_{4}(N) & = & \displaystyle {\displaystyle \frac{1}{2}}{\eta}_{1}[eA_{e}]^{2}R_{1}^{T}(N)R_{1}(N)\cdot \{{\eta}_{1}[eA_{e}/e(N)]^{2}R_{1}^{T}(N)R_{1}(N)-2\}\nonumber\\ \displaystyle & \leq & \displaystyle {\displaystyle \frac{1}{2}}{\eta}_{1}[eA_{e}]^{2}(R_{1\max }(N))^{2}\{{\eta}_{1}[eA_{e}/e(N)]^{2}(R_{1\max }(N))^{2}-2\}\end{eqnarray}$ (45) If 𝜂₁ is chosen as $0<{\eta}_{1}<∼2/\{(R_{1\max })^{2}[eA_{e}/e(N)]^{2}\}$ , the Lyapunov stability of Y ₄(N) > 0 and ΔY ₄ < 0 is guaranteed. Then, the output tracking error will converge to zero as t → 0. This completes the proof of the theorem. Furthermore, the optimal learning rate, which achieves the fast convergence, is corresponding to [26] $\begin{eqnarray}\displaystyle 2{\eta}_{1}^{\ast }\{(R_{1\max })^{2}[eA_{e}/e(N)]^{2}\}-2=0, & & \displaystyle\end{eqnarray}$ (46) i.e., $\begin{eqnarray}\displaystyle {\eta}_{1}^{\ast }=1/\{(R_{1\max })^{2}[eA_{e}/e(N)]^{2}\}=1/(A_{e}R_{1\max })^{2}, & & \displaystyle\end{eqnarray}$ (47) which comes from the derivative of (47) with respect to 𝜂₁ and equals to zero. This shows the interesting results for the optimal learning rate which can be on-line tuned at each instant.

TheoremC .
Let’s assume that 𝜂₂ is the learning rate of recurrent weight between the output layer and the input layer in the IRHPNN. Meanwhile, let R _{2 max} be defined as R _{2 max} ≡ max _N∥R ₂(N)∥, where $R_{2}(N)=\partial ∼o_{k}^{3}/\partial ∼u_{ik}^{1}$ and ∥ ⋅ ∥ is the Euclidean norm in $\Re$ ⁿ. If 𝜂₂ is chosen as [26] $\begin{eqnarray}\displaystyle 0\lt {\eta}_{2}\lt 2/(A_{e}R_{2\max })^{2}. & & \displaystyle\end{eqnarray}$ (48) Then, the convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate which achieves the fast convergence can be obtained as $\begin{eqnarray}\displaystyle {\eta}_{2}^{\ast }=1/[(A_{e}R_{2\max })^{2}]. & & \displaystyle\end{eqnarray}$ (49)

TheoremD .
Since $\begin{eqnarray}\displaystyle R_{2}(N)={\displaystyle \frac{\partial ∼o_{k}^{3}}{\partial u_{ik}^{1}}}=u_{kj}^{2}H_{j}(\cdot )a_{i}^{1}(N)o_{k}^{3}(N-1). & & \displaystyle\end{eqnarray}$ (50) The Lyapunov function can be selected as (39) and the increment in the Lyapunov function can be depicted as (40). Then, the error difference can be depicted as $\begin{eqnarray}\displaystyle e(N+1)=e(N)+{\Delta}e(N)=e(N)+\left[{\displaystyle \frac{\partial ∼e(N)}{\partial u_{ik}^{1}}}\right]^{T}{\Delta}u_{ik}^{1}, & & \displaystyle\end{eqnarray}$ (51) where ${\rm\Delta}u_{ik}^{1}$ represents increment of the weight. The Eq. (51) using Eqs (32), (33), (35) and (50) can be depicted as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial e(N)}{\partial ∼u_{ik}^{1}}}={\displaystyle \frac{\partial e(N)}{\partial ∼P_{1}}}{\displaystyle \frac{\partial ∼P_{1}}{\partial o_{k}^{3}}}{\displaystyle \frac{\partial ∼o_{k}^{3}}{\partial ∼u_{ik}^{1}}}=-{\displaystyle \frac{eA_{e}}{e(N)}}R_{2}(N)\end{eqnarray}$ (52) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle e(N+1)=e(N)-\left[{\displaystyle \frac{eA_{e}}{e(N)}}R_{2}(N)\right]^{T}{\eta}_{2}eA_{e}∼R_{2}(N).\end{eqnarray}$ (53) Thereby $\begin{eqnarray}\displaystyle \Vert e(N+1)\Vert & = & \displaystyle \Vert e(N)[1-{\eta}_{2}(eA_{e}/e(N))^{2}R_{2}^{T}(N)R_{2}(N)]\Vert \nonumber\\ \displaystyle \text{} & \leq \text{} & \displaystyle \Vert e(N)\Vert ∼\Vert 1-{\eta}_{2}(eA_{e}/e(N))^{2}R_{2}^{T}(N)R_{2}(N)\Vert .\end{eqnarray}$ (54) Then ΔY ₄(N) using (51), (52), (53) and (54) can be rewritten as $\begin{eqnarray}\displaystyle {\Delta}Y_{4}(N) & = & \displaystyle {\displaystyle \frac{1}{2}}{\eta}_{2}[eA_{e}]^{2}R_{2}^{T}(N)R_{2}(N)\{{\eta}_{2}[eA_{e}/e(N)]^{2}R_{2}^{T}(N)R_{2}(N)-2\}\nonumber\\ \displaystyle \text{} & \leq \text{} & \displaystyle {\displaystyle \frac{1}{2}}{\eta}_{2}[eA_{e}]^{2}(R_{2\max }(N))^{2}\{{\eta}_{2}[eA_{e}/e(N)]^{2}(R_{2\max }(N))^{2}-2\}.\end{eqnarray}$ (55) If 𝜂₂ is chosen as $0<{\eta}_{2}<∼2/\{(R_{2\max })^{2}[eA_{e}/e(N)]^{2}\}$ , then the Lyapunov stability of Y ₄(N) > 0 and ΔY ₄ < 0 is guaranteed, so that the output tracking error will converge to zero as t → 0. This completes the proof of the theorem. Moreover, the optimal learning rate, which achieves the fast convergence, is corresponding to [26] $\begin{eqnarray}\displaystyle 2{\eta}_{2}^{\ast }\{(R_{2\max })^{2}[eA_{e}/e(N)]^{2}\}-2=0, & & \displaystyle\end{eqnarray}$ (56) i.e., $\begin{eqnarray}\displaystyle {\eta}_{2}^{\ast }=1/\{(R_{2\max })^{2}[eA_{e}/e(N)]^{2}\}=1/\{(A_{e}R_{2\max })^{2}\}, & & \displaystyle\end{eqnarray}$ (57) which comes from the derivative of (57) with respect to 𝜂₂ and equals to zero. This shows the interesting results for the optimal learning rate which can be on-line tuned at each instant.

On the other hand, the guaranteed convergence of tracking error to be zero does not imply convergence of the estimated value of the lumped uncertainty to it real values. The persistent excitation condition [25] should be satisfied for the estimated value to converge to its theoretic value.

In summary, the online tuning algorithm of the IRHPNN control is based on two adaptive laws (34) and (35) for the connective weight adjustment and the recurrent weights adjustment with two optimal learning rates in (47) and (57), respectively. Moreover, the IRHPNN weight appraised errors are fundamentally bounded [26]. As long as the IRHPNN weight appraised errors are bounded, which is required to ensure that the control signal is bounded.

TheoremE .
The key point of the proposed design is to utilize the Lyapunov function for constructing the novel AIRHPNN control system in Eq. (16), which reduces the input dimensions of the IRHPNN controller.

TheoremF .
A block diagram of novel AIRHPNN control system is depicted in Fig. 3.

TheoremG .
IRHPNN approximation holds only in a compact set. Thus, the obtained result is semi-global in the sense that they hold for the compact sets, there exists a controller with a sufficiently large number of reformed IRHPNN nodes such that all the closed-loop signals are bounded.

4. Experimental results

The whole structure of the SynRM servo-drive CVT system by using the mixed signal FPGA system and DSP control system is shown in Fig. 2. The control algorithm was executed by using the DSP control system including 16 channels analog-digital converters with 12 bits, 18 programmable PWM ports, 6 high-resolution PWM ports and 2 quadrature encoder interfaces. The flowchart of executed control methodologies with real-time implementation by using the mixed signal FPGA system and DSP control system consists of a main program and an interrupt service routine (ISR) as shown in Fig. 5. In the main program, parameters and input/output (I/O) initialization are processed first. Then, the interrupt interval for the ISR is set. After enabling the interrupt, the main program is used to monitor control data. The ISR with 2 ms sampling interval is used for reading the rotor position of the SynRM servo-drive CVT system from encoder interface and three-phase currents from analog-to-digital (A/D) converter, calculating rotor position and speed, executing lookup table and coordinate translation, executing PI current control, executing novel AIRHPNN control system, and outputting three-phase current commands to sinusoidal PWM circuit for switching the three-sets of IGBT power modules inverter via interlock and isolated circuits. The three-sets of IGBT power modules voltage source inverter is switched by current-controlled sinusoidal PWM with a switching frequency of 15 kHz. The identifier s1 is the executed number of the field-oriented control. The identifier s2 is the executed number of the AIRHPNN control system. The identifier s1_max is the maxized number of the executed field-oriented control and it is set to 2. The specification of the SynRM is the three-phase, two-pole 48 V, 1.5 kW, 3600 rpm. All parameters of the SynRM are given as r _s = 0.88 Ω, L _Q = 22.15 mH, L _D = 146.68 mH J ₁ = 1.02 ×10⁻³ N ms², B ₁ = 3.28 ×10⁻³ N ms/rad. The specifications of the CVT system are given as 646.8 mm of V-belt length, 72.6 mm of front pulley diameter, 31.3 mm of rear pulley diameter, 9 of rings per set and 4.5 of conversion ratio. Owing to inherent uncertainty in the CVT system (e.g. the lumped nonlinear external disturbances and parameter variations) and output current limitation of battery capacity, the SynRM can only operate at 314 rad/s (3000 rpm) to avoid burning IGBT module for the CVT system at high speed perturbation.

Fig. 5.

Flowchart of the executing program by using mix signal FPGA system and DSP control system.

To show the control performance of the proposed novel AIRHPNN control system, two experimental tests are provided. One is the 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ , and the other is the 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ . In addition, for comparison control performances, three control systems are adopted as the well-known PI controller, the three-layer feedforward NN control system and the proposed novel AIRHPNN control system in this paper.

First, all of the gains of the well-known PI controller via some heuristic knowledge [21–23] on the tuning of the PI controller are given as k _is = k _ps∕T _is = 19.2, k _ps = 8.15 at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with the parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ for the speed tracking to achieve good transient and steady-state control performance.

Second, three-layer feedforward NN has 2-4-1 nodes in the input-hidden-output layer, and adopts the sigmoid activation function in the input and hidden layers. The control gains of the three-layer feedforward NN control system are chosen to achieve the best transient control performance in some experiments considering the requirement of stability. Moreover, the connective weights between the input layer and the hidden layer, and the connective weight between the hidden layer and the output layer in the three-layer feedforward NN are initialized with random number. Two learning rates of the connective weight between the hidden layer and the connective weight between the hidden layer and the output layer are chosen as 0.12 and 0.15, respectively. The parameter adjustment process remains continually active for the duration of the experimentation. Third, the control gains of the proposed novel AIRHPNN control system are given as 𝛼 = 0.22, τ = 0.12 to achieve the best transient control performance in some experiments under considering the requirement of stability. The parameter adjustment process remains continually active for the duration of the experiment. The structure of the adopted IRHPNN has 2-4-1 nodes in the input-hidden-output layer.

Firstly, the experimental results of the well-known PI controller for the SynRM servo-drive CVT system at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ and at 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ are shown in Fig. 6 and Fig. 7, respectively. The tracking responses of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r are shown in Figs 6(a) and 7(a); tracking responses of the speed error e are shown in Figs 6(b) and 7(b); responses of the command electromagnetic torque T _e are shown in Figs 6(c) and 7(c). Because the low speed operation is the same as the nominal case which is $T_{e}\cong T_{1}^{l}$ operation because of smaller disturbance, the speed tracking response shown in Fig. 6(a) has better tracking performance. Moreover, the tracking response of speed shown in Fig. 7(a) leads to degenerate tracking at bigger nonlinear disturbances (e.g. rolling resistance, wind resistance, and parameter variation) at high speed operation. Meanwhile, the tracking responses of the speed error e shown in Figs 6(b) and 7(b) appear larger tracking errors. Furthermore, the electromagnetic torque T _e emerges larger torque ripple shown in Figs 6(c) and 7(c).

Fig. 6.

Experimental results for the SynRM servo-drive CVT system obtained using the well-known PI controller at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

Fig. 7.

Experimental results for the SynRM servo-drive CVT system obtained using the well-known PI controller at 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

From the experimental results, sluggish tracking responses of the speed and current are obtained for the SynRM servo-drive CVT system using the well-known PI controller. The linear controller has the weak robustness under bigger nonlinear disturbances because of no appropriately gains tuning or no degenerate nonlinear effect. In addition, the dynamic response of command electromagnetic torque T _e brings about great torque ripple due to the electric scooter with CVT system’s nonlinear disturbance such as V-belt shaking friction, action friction between the front pulley and the rear pulley.

Secondly, the experimental results of the three-layer feedforward NN control system for the SynRM servo-drive CVT system at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ and at 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ are shown in Figs 8 and Fig. 9, respectively.

Fig. 8.

Experimental results for the SynRM servo-drive CVT system obtained using the three-layer feedforward NN control system at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

The tracking responses of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r are shown in Figs 8(a) and 9(a); tracking responses of the speed error e are shown in Figs 8(b) and 9(b); responses of the command electromagnetic torque T _e are shown in Figs 8(c) and 9(c). The low-speed operation was the same as the nominal case which is $T_{e}\cong T_{1}^{l}$ operation because of smaller disturbances; hence, the response of the speed shown in Fig. 8(a) shows higher tracking performance. The better speed tracking response is shown in Fig. 9(a) under the occurrence of the bigger lumped nonlinear external disturbances and parameter variations. Because the low speed operation is the same as the nominal case which is $T_{e}\cong T_{1}^{l}$ operation because of smaller disturbance, the speed tracking response shown in Fig. 8(a) has better tracking performance.

Fig. 9.

Experimental results for the SynRM servo-drive CVT system obtained using the three-layer feedforward NN control system at 314 rad/s (3000 rpm) case under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

Moreover, the tracking response of speed shown in Fig. 9(a) leads to good tracking at bigger nonlinear disturbances, i.e., rolling resistance, wind resistance, and parameter variation at high speed operation. Meanwhile, the tracking responses of the speed error e shown in Figs 8(b) and 9(b) appear small tracking errors. Furthermore, the electromagnetic torque T _e emerges small torque ripple shown in Figs 8(c) and 9(c). However, because of online adaptive mechanism of the three-layer feedforward NN control system, accurate tracking responses of the speed can be obtained.

Thirdly, the experimental results of the proposed novel AIRHPNN control system for the SynRM servo-drive CVT system at 157 rad/s (1500 rpm) case under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}=∼{\rm\Delta}T+T_{u}$ and 314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ are shown in Fig. 10 and Fig. 11, respectively.

Fig. 10.

Experimental results for the SynRM servo-drive CVT system obtained using the novel AIRPHPNN system at 157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}:T_{1}^{l}={\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

The tracking responses of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r are presented in Figs 10(a) and 11(a). The tracking responses of the speed error e are shown in Figs 10(b) and 11(b). The responses of the command electromagnetic torque T _e are presented in Figs 10(c) and 11(c). The low-speed operation was the same as the nominal case which is $T_{e}\cong T_{1}^{l}$ operation because of smaller disturbances; hence, the speed tracking response in Fig. 10(a) shows higher tracking performance. The higher speed tracking response is shown in Fig. 11(a) under larger lumped nonlinear external disturbances and parameter variations. Meanwhile, the tracking responses of the speed error e shown in Figs 10(b) and 11(b) appear smaller tracking errors. Furthermore, the electromagnetic torque T _e emerges smaller torque ripple shown in Figs 10(c) and 11(c).

Fig. 11.

Experimental results for the SynRM servo-drive CVT system obtained using the novel AIRHPNN control system at 314 rad/s (3000 rpm) caseunder lumped nonlinear external disturbancesand with twice the parameter variations $T_{1}^{l}=∼2{\rm\Delta}T+T_{u}$ : (a) tracking response of the command rotor speed ω^∗, desired command rotor speed ω, and measured rotor speed ω_r; and (b) response of the tracking error e; (c) response of electromagnetic torque T _e.

The dynamic response of command electromagnetic torque T _e brings about lower torque ripple by online adjustment of the IRHPNN to cope with the high-frequency unmodelled dynamic of the electric scooter with CVT system’s nonlinear disturbances, such as V-belt shaking friction, action friction between the front pulley and the rear pulley.

The experimental results show that the accurate tracking performance was achieved for the SynRM servo-drive CVT system when the AIRHPNN control system was used, because of the online adaptive mechanism of the IRHPNN control and the operation of the reimbursed controller. Therefore these experimental results show that the novel AIRHPNN control system has better control performance than the well-known PI controller and the three-layer feedforward NN control system under high speed perturbation for the SynRM servo-drive CVT system.

Fig. 12.

Experimental results of speed-adjusted response of the command rotor speed ω^∗ and the measured rotor speed ω_r under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load at 314 rad/s (3000 rpm): (a) by use of the well-known PI controller; (b) by use of the three-layer feedforward NN control system; (c) by use of the novel AIRHPNN control system.

Finally, the condition under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load at measured rotor speed responses is tested by using the well-known PI controller, the three-layer feedforward NN control system and the novel AIRHPNN control system. The experimental result of load adjustment when the well-known PI controller, the three-layer feedforward NN control system, and the novel AIRPHNN were used under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load at 314 rad/s (3000 rpm) are shown in Figs 12 and 13.

Fig. 13.

Experimental results of response of the measured current i _a in phase a under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u with adding load at 314 rad/s (3000 rpm): (a) by use of the well-known PI controller; (b) by use of the three-layer feedforward NN control system; (c) by use of the novel AIRHPNN control system.

The experimental results of the measured rotor speed response when the well-known PI controller, the three-layer feedforward NN control system, and the novel AIRHPNN control system was used under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load at 314 rad/s (3000 rpm) are shown in Figs 12(a), 12(b), and 12(c), respectively. The experimental results of the measured current in phase a when the well-known PI controller, the three-layer feedforward NN control system, and the novel AIRHPNN was used under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load at 314 rad/s (3000 rpm) are shown in Figs 13(a), 13(b), and 13(c), respectively. The experimental results show that the degenerated responses under load torque disturbance and parameter variations $T_{1}^{l}=∼2$ Nm (T _a) + T _u with adding load are considerably improved when the novel AIRHPNN control system is used. Moreover, the transient response of the novel AIRHPNN control system shows faster convergence and more favorable load regulation than that of the well-known PI controller and the three-layer feedforward NN control system.

In addition, a control performance comparison of the well-known PI controller, three-layer feedforwardNN control system, and the novel AIRHPNN control system is presented in Table 1 for experimental results of three test cases. Clearly, the use of the novel AIRHPNN control system control system results in smaller tracking errors. The tabulated measurements show that the proposed novel AIRHPNN control system indeed shows the superior control performance compared with the well-known PI controller and the three-layer feedforward NN control system.

Furthermore, a characteristic performance comparison between the well-known PI controller, the three-layer feedforward NN control system and the novel AIRHPNN control system is presented in Table 2 on the basis of experimental results. In Table 2, the various performances of the novel AIRHPNN control system is superior to those of the well-known PI controller and the three-layer feedforward NN control system.

Table 2

Characteristic performance comparison of control systems

Characteristic performance	Control system
	Well-known PI controller	Three-layer feedforward NN control system with two fixed learning rates	Novel AIRHPNN control system
Dynamic response	Slow	Fast	Faster
Load regulation capability	Poor	Good	Best
Convergence speed	Low	Middle	High
Two learning rates	Not applicable	Fixed	Variable
Torque ripple (V-belt shaking friction, action friction between the primary pulley and the secondary pulley)	High	Middle	Low

5. Conclusion

A novel AIRHPNN control system with robust control characteristics has been successfully applied in controlling the SynRM servo-drive CVT system. The major contributions of this study are as follows. (1) The simplified dynamic and kinematic models of the CVT system driven by the SynRM with unknown nonlinear and time-varying characteristics were successfully derived. (2) The adaptive law of online parameters tuning in the IRHPNN and the appraised law of the reimbursed controller were successfully derived using the Lyapunov stability theorem. (3) Two optimal learning rates of connective weights and recurrent weights in the IRHPNN according to the increment Lyapunov function were successfully derived for achieving faster convergence. Finally, the proposed novel AIRHPNN control system has better control performances than the well-known PI controller and the three-layer feedforward NN control system, such as torque ripple, dynamic response, load regulation and convergence speed.

Footnotes

Acknowledgements

The author would like to acknowledge the financial support of the Ministry of Science and Technology of Taiwan under grant MOST 107-2221-E-239-021.

References

Kim

Y.H.

and Lee

J.H.

, Optimum design criteria of an ALA-SynRM for the maximum torque density and power factor improvement, Intl. J. Applied Electromagnetics and Mechanics 53(S2) (2017), S279–S288.

Chai

Zhao

and Kwon

, Optimal design of wound field synchronous reluctance machines to improve torque by increasing the saliency ratio, IEEE Trans. Magnetics 99(99) (2017), 00–99, doi:10.1109/TMAG.2017.2707459.

Betz

R.E.

Lagerquist

Jovanovic

Miller

T.J.E.

and Middleton

R.H.

, Control of synchronous reluctance machines, IEEE Trans. Industry Applications 29(6) (1993), 1110–1122.

Uezato

Senjyu

and Tomori

, Modeling and vector control of synchronous reluctance motors including Stator Iron Loss, IEEE Trans. Industry Applications 30(4) (1994), 971–976.

Matsuo

Antably

A.E.

and Lipo

T.A.

, A new control strategy for optimum-efficiency operation of a synchronous reluctance motor, IEEE Trans. Industry Applications 33(5) (1997), 1146–1153.

Rashad

E.M.

Radwan

T.S.

and Rahman

M.A.

, A maximum torque per ampere vector control strategy for synchronous reluctance motors considering saturation and iron losses, IEEE Industry Applications Conf. (2004), 2411–2417.

Tseng

C.Y.

Lue

Y.F.

Lin

Y.T.

Siao

J.C.

Tsai

C.H.

and Fu

L.M.

, Dynamic simulation model for hybrid electric scooters, IEEE Int. Symp. Industrial Electronics (2009), 1464–1469.

Guzzella

and Schmid

A.M.

, Feedback linearization of spark-ignitionengines with continuously variable transmissions, IEEE Trans. Control Systems Technology 3(1) (1995), 54–58.

Kim

and Vachtsevanos

, Fuzzy logic ratio control for a CVThydraulic module, Proc. of the IEEE Symp. Intelligent Control (2000), 151–156.

10.

Carbone

Mangialardi

Bonsen

Tursi

and Veenhuizen

P.A.

, CVT dynamics: theory and experiments, Mechanism and Machine Theory 42(4) (2007), 409–428.

11.

Toshihiko

Noriyuki

and Shi

, Vibration suppression of a structure with an MR damper using a neural network controller, Intl. J. Applied Electromagnetics and Mechanics 21(3,4) (2005), 209–219.

12.

and Zheng

, Fuzzy neural network multi-modal vibration control of thin cylindrical shells laminated with photostrictive actuators, Intl. J. Applied Electromagnetics and Mechanics 46(4) (2014), 951–963.

13.

Nouicer

and Feliachi

, Neural network-DFT based model for magnetostrictive hysteresis, Intl. J. Applied Electromagnetics and Mechanics 42(3) (2013), 343–348.

14.

Fusaomi

Shohei

Tomoya

Akimasa

and Keigo

, Application of fuzzy reasoning and neural network to feed rate control of a machining robot, Intl. J. Applied Electromagnetics and Mechanics 52(3,4) (2016), 897–905.

15.

Lin

C.H.

, Novel adaptive recurrent Legendre neural network control for PMSM servo-drive electric scooter, J. Dynamic Systems, Measurement, and Control-Trans. ASME 137(2) (2015), 011010-1, 12.

16.

Lin

C.H.

, Application of hybrid recurrent Laguerre-orthogonal-polynomial NN control in V-belt continuously variable transmission system using modified particle swarm optimization, J. Mechanical Science and Technology 29(9) (2015), 3933–3952.

17.

and Khorasani

, Constructive feedforward neural networks using Hermite polynomial activation functions, IEEE Trans. Neural Networks 16(4) (2005), 821–833.

18.

and Khorasani

, Adaptive constructive neural networks using Hermite polynomials for image compression, 2nd Intl. Symp. Neural Networks (2005), 713–722.

19.

Rigatos

G.G.

and Tzafestas

S.G.

, Feed-forward neural networks using Hermite polynomial activation functions, 4th Helenic Conf. Advances in Artificial Intelligence (2006), 323–333.

20.

Siniscalchi

S.M.

and Lee

C.H.

, Hermitian polynomial for speaker adaptation of connectionist speech recognition systems, IEEE Trans. Audio, Speech, and Language Processing 21(10) (2013), 2152–2161.

21.

Astrom

K.J.

and Hagglund

, PID Controller: Theory, Design, and Tuning, Instrument Society of America, Research Triangle Park, North Carolina, USA, 1995.

22.

Hagglund

and Astrom

K.J.

, Revisiting the Ziegler–Nichols tuning rules for PI control, Asian J. Control 4(4) (2002), 364–380.

23.

Hagglund

and Astrom

K.J.

, Revisiting the Ziegler–Nichols tuning rules for PI control-part II: the frequency response method, Asian J. Control 6(4) (2004), 469–482.

24.

Slotine

J.J.E.

and Li

, Applied Nonlinear Control, EnglewoodCliffs, Prentice-Hall, New Jersey, 1991.

25.

Astrom

K.J.

and Wittenmark

, Adaptive Control, Addison-Wesley, New York, 1995.

26.

Lewis

F.L.

Campos

and Selmic

, Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities, SIAM Frontiers in Applied Mathematics 2002.

Performance	Control system and three test cases
	Well-known PI controller
	157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u
Maximum error of e	9.5 rad/s	22.8 rad/s	42.7 rad/s
	(91 rpm)	(218 rpm)	(408 rpm)
RMS error of e	5.0 rad/s	6.5 rad/s	5.5 rad/s
	(48 rpm)	(62 rpm)	(53 rpm)
Performance	Control system and three test cases
	Three-layer feedforward NN control system
	157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u
Maximum error of e	7.5 rad/s	9.5 rad/s	21.3 rad/s
	(72 rpm)	(91 rpm)	(204 rpm)
RMS error of e	3.5 rad/s	3.5 rad/s	3.0 rad/s
	(33 rpm)	(33 rpm)	(29 rpm)
Performance	Control system and three test cases
	Novel AIRHPNN control system
	157 rad/s (1500 rpm) under lumped nonlinear external disturbances and with parameter variations $T_{1}^{l}={\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) under lumped nonlinear external disturbances and with twice the parameter variations $T_{1}^{l}=2{\rm\Delta}T+T_{u}$	314 rad/s (3000 rpm) with adding load under load torque disturbance and parameter variations $T_{1}^{l}=2$ Nm (T _a) + T _u
Maximum error of e	4.5 rad/s	5.2 rad/s	7.6 rad/s
	(43 rpm)	(50 rpm)	(73 rpm)
RMS error of e	2.0 rad/s	2.5 rad/s	2.0 rad/s
	(19 rpm)	(24 rpm)	(19 rpm)