Mixed Modiﬁed Recurring Rogers-Szego Polynomials Neural Network Control with Mended Grey Wolf Optimization Applied in SIM Expelling System

: Due to a good ability of learning for nonlinear uncertainties, a mixed modiﬁed recurring Rogers-Szego polynomials neural network (MMRRSPNN) control with mended grey wolf optimization (MGWO) by using two linear adjusted factors is proposed to the six-phase induction motor (SIM) expelling continuously variable transmission (CVT) organized system for acquiring better control performance. The control system can execute MRRSPNN control with a ﬁtted learning rule, and repay control with an evaluated rule. In the light of the Lyapunov stability theorem, the ﬁtted learning rule in the MRRSPNN control can be derived, and the evaluated rule of the repay control can be originated. Besides, the MGWO by using two linear adjusted factors yields two changeable learning rates for two parameters to ﬁnd two ideal values and to speed-up convergence of weights. Experimental results in comparisons with some control systems are demonstrated to conﬁrm that the proposed control system can achieve better control performance.


Introduction
Neural networks [1][2][3][4][5][6][7][8] have good approximation performance in modeling, control, estimation, and prediction of systems. Nagamani and Ramasamy [5] proposed the dissipativity and passivity analysis for discrete-time stochastic neural networks with probabilistic time-varying delays. This paper is to reduce the conservatism of the dissipativity conditions for the considered neural networks by utilizing the reciprocally convex combination approach. Nagamani and Ramasamy [6] proposed the dissipativity and passivity analysis for discrete-time T-S fuzzy stochastic neural networks with leakage time-varying delays based on the Abel lemma approach. Nagamani et al. [7] proposed the problem of robust state estimation for discrete-time stochastic Markov jump neural networks with discrete and distributed time-varying delays based on dissipativity and passivity theory. Ramasamy et al. [8] proposed the strict (Q, S, R) − γ-dissipativity and passivity analysis for discrete-time Markovian jump neural networks involving both leakage and discrete delays expressed in terms of two additive time-varying delay components. But it is very time consuming in the online training procedure. Hence, some functional neural networks [9][10][11][12] with less computational complexities have been used in the controls and identifications of nonlinear systems. However, the adjustment mechanics of weights were not discussed in these control methods combined with neural networks that caused some errors in the controls and identification of nonlinear systems. Besides, Szego [13] proposed the Rogers-Szego and CVT organized system with negligible belt flexural effects, power, and sliding losses is illustrated in Figure 1 [38][39][40][41]. The dynamic equations for all torques under simplified kinematics of the CVT organized system in the succeeding driven shaft and the preceding driving shaft by using the law of conservancy shown in Figure 1 are simplified as τ a = (J a + J 1 ) . ω a + (B a + B 1 )ω a + τ 1 (1) where σ b (ς a , ς b ), (ς a , ς b ), (α a , α b ), (τ b , τ l b ) are the conversion ratio with regards to preceding and succeeding pulley shafts in the CVT organization, the slip arcs with regards to driving torque transmission under low speed, the wrap angles in the belt-pulley contacting arcs, the input torque and load torque on the preceding driving pulley, and the succeeding driven pulley shown in Figure 1a, respectively. J 1 , B 1 , and ω a represent the equivalent inertia, the equivalent viscous frictional coefficient, and the speed in the preceding pulley shaft, respectively. J a and B a represent the moment of inertia and the viscous friction coefficient of the SIM. τ 1 and τ 2 are the driven torque on the preceding pulley shaft and the driven torque on the succeeding pulley shaft, respectively. τ l b ( f lb (v ab , τ ab , F bl , B b , ω 2 b )) is the intact nonlinear outward disturbances on the succeeding driven side including the rolling resistance v ab , the wind resistance τ ab , and the braking force F bl . J b , B b , and ω b represent the equivalent inertia, the equivalent viscous frictional coefficient, and the speed in the succeeding pulley shaft, respectively. Then the torque equation can be transformed from the succeeding pulley side to the preceding pulley side by means of speed ratio and sliding ratio [38][39][40][41]. For simplification, the modelling of the CVT organized system can be assumed as negligible power and slip losses. Thus, the entire dynamic equation [38][39][40][41] of the SIM expelling CVT organized system from Equation (1) to Equation (4) is inferred as J c . ω a + B c ω a + τ l b (∆τ 1 , ∆τ s , f lb (v ab , τ ab , F bl , B b , ω 2 b )) + τ t (τ lr , τ f r , τ cr , τ rr ) = τ a (5) where J c = J a + J 1 is the integrated moment of inertia; B c = B a + B 1 is the integrated viscous friction coefficient; τ l b (∆τ 1 , ∆τ s , f lb (v ab , τ ab , F bl , B b , ω 2 b )) = ∆τ 1 + τ u1 + ∆τ s is the large nonlinear outward disturbances with parameter variations and torsional vibrations variations; τ t (τ lr , τ f r , τ cr , τ rr ) is the total torque including the added load torque τ lr , the Stribeck effect torque τ f r , the cogging torque τ cr , and the coulomb friction torque τ rr . ∆τ 1 = ∆J c . ω a + ∆B c ω a are the aggregated parameter variations. τ u1 = f lb (v ab , τ ab , F bl , B b , ω 2 b ) are the aggregated nonlinear outward disturbances. ∆τ s = a∆ω a + b∆ω 2 a + c∆ω b + d∆ω 2 b are the aggregated torsional vibration variations [42,43].
axes electric equation in the coordinate frames transformation from the six-phase axes of the SIM can be represented by [29][30][31]  The d 1 − q 1 −d 2 − q 2 axes electric equation in the coordinate frames transformation from the six-phase x 1 − y 1 − z 1 − x 2 − y 2 − z 2 axes to the d 1 − q 1 −d 2 − q 2 axes of the SIM can be represented by [29][30][31] u q1 = r s w q1 + ω e (L ss w d1 + L M w dr ) + (L ss u d1 = r s w d1 − ω e (L ss w q1 + L M w qr ) + (L ss where u d1 , u q1 , u d2 , u q2 are the d 1 − q 1 − d 2 − q 2 axes voltages; w d1 , w q1 , w d2 , w q2 are the d 1 − q 1 − d 2 − q 2 axes currents; w dr , w qr are the d r − q r axes currents; r s and r r are the stator resistance and rotor resistance; L ss = L ls + 3L ms , L rr = L lr + 3L ms and L M = 3L ms are the self-inductance of the stator winding, the self-inductance of rotor winding, and the mutual inductance between stator winding and the rotor winding, respectively. ω e , ω r = P r1 ω a /2 and ω a are the electrical angular speeds of synchronous flux, the electrical angular speeds of rotor, and the mechanical angular speeds of rotor, respectively. The developed torque τ a of the SIM can be represented as where λ dr , λ qr are the d r − q r axes flux linkages; P r1 is the number of poles. The developed torque of the SIM by use of the indirect field-oriented control (IFOC) [29][30][31] can be reduced as τ a = k ar (λ dr w q1 − λ qr w d1 ). The speed and torque dynamic equation for the SIM are reduced by where k ar is the torque constant.

SIM Expelling CVT Organized System
The conformation of the SIM expelling CVT organized system shown in Figure 2 includes three parts as the SIM expelling system, the CVT organized system, and the digital signal processor (DSP) control system. The CVT organized system consists of the wheel and the CVT system. The SIM expelling system consists of the current sensors and A/D converter, the interlock and isolated circuits, and the voltage source inverter with six sets of insulated-gate bipolar transistor (IGBT) power modules. The DSP control system can realize a space-vector pulse-width-modulation control (SVPWMC), an indirect-field-oriented control (IFOC), a proportional-integral (PI) current control loop, and a speed control loop. The IFOC consists of the sin θ e / cos θ e generation, the lookup table generation, and the coordinate translation. Two gains of the PI current controller are k p1 = 18.6 and k i1 = k p1 /T i1 = 7.6 by use of the Kronecker method to build a stability boundary in the k p1 and k i1 plane [44][45][46]. This method is used to narrow down the region for iterative selection of values of the parameters of k p1 and k i1 on the tuning of the PI controller so as to get fine dynamic response including some tests in an influence of time needed for electromagnetic torque shaping. The SIM expelling system was controlled by the DSP control system under aggregated parameter variations and aggregated nonlinear outward disturbances.

MMRRSPNN Control with MGWO by Using Two Adjusted Factors
For simplifying the control system design, the entire dynamic equation from (5) can be  represented by   c  a  c  rr  cr  fr  lr  t  b  b  bl  ab  ab  lb  s

MMRRSPNN Control with MGWO by Using Two Adjusted Factors
For simplifying the control system design, the entire dynamic equation from (5) can be represented by where τ w = τ l b (∆τ 1 , ∆τ s , f lb (v ab , τ ab , F bl , B b , ω 2 b )) + τ t (τ lr , τ f r , τ cr , τ rr ) = ∆τ 1 + τ u1 + ∆τ s + τ t (τ lr , τ f r , τ cr , τ rr ) are the total disturbances. h a = τ a is the control intensity, i.e., the control torque of the SIM. Q a = −B c /J c , Q c = 1/J c and Q b = −1/J c are three familiar constants. |Q a ω a | ≤ Z a (ω a ), |Q b τ w | ≤ Z b and Z c ≤ Q c are assumed to be bounded. Z a (ω a ) is the bounded value of the function. Z b and Z c are assumed to be two familiar values. The speed discrepancy is defined by where e a is the speed discrepancy. When the aggregated parameter variations and the aggregated nonlinear outward disturbances are fine familiar, the intact control rule can be conceived by where k a > 0 is a positive constant. When h * a = h a in (16) substituted into (14), then . e a + k a e a = 0 It is implied that the system's state will track the desired trajectory when t → ∞ and e a (t) → 0 . In order to improve response of speed track under uncertainty action, the MMRRSPNN control with MGWO by using two adjusted factors was developed for controlling the SIM expelling CVT organized system. The control rule of the proposed control system depicted in Figure 3 is devised by where h 1 is the intendant control that can stabilize the system's states on prescribed bound scope, h 2 is the MRRSPNN control that is the chief tracking controller, h 3 is the repay control that can compensate the error between the intact control rule and the MRRSPNN control. From Equation (14) to Equation (18), a discrepancy dynamic equation can be represented by Then, the intendant control h 1 can be represented by where sgn(·) is a sign function. Besides, the sign function sgn(Q c e a ) can be represented by the continuous function Q c e a /(|Q c e a | + ς) and ς = ς 0 , 0 , |Q c e a | < ς |Q c e a | ≥ ς with ς 0 and ς are two small constants to reduce chattering in the intendant control. When the MRRSPNN approximation properties cannot be guaranteed, the intendant control rule that is effective has a switched index of one, i.e., I a = 1. The MRRSPNN control rule is put forward to impersonate the intact control rule. Then the repay control rule is put forward to compensate the error between the intact control rule and the MRRSPNN control.  Further, the MRRSPNN with three-layer constitution composed of a first layer, a second layer, and a third layer, is pictured in Figure 4. The semaphore intentions in each node for each layer are explained in the following expression.
At the first layer, input semaphore and output semaphore are explained by   Further, the MRRSPNN with three-layer constitution composed of a first layer, a second layer, and a third layer, is pictured in Figure 4. The semaphore intentions in each node for each layer are explained in the following expression.
At the first layer, input semaphore and output semaphore are explained by where x 1 1 = ω * −ω a = e a and x 1 2 = e a (1 − z −1 ) = ∆e a are the speed discrepancy and speed discrepancy alteration, respectively. R is the iteration count. v 1 ik is the recurring weight through the third layer and the first layer. y 3 k is the output of node at the third layer. The symbol Π is a multiply factor.
At the second layer, input semaphore and output semaphore are explained by where η is the recurring gain at the second layer. Rogers-Szego polynomials function [13,47] is adopted as the activation function.
are the 0-, 1-, and 2-order Meixner polynomials functions, respectively. The first kind type of Rogers-Szego polynomials function at the recurrence relation [13,47] is given by The symbol is a summation factor. At the third layer, semaphore and output semaphore are explained by where v 2 k j is the connective weight through the second layer and the third layer. f 3 k is the linear activation function. The output at the third layer can be rewritten by y 3 k (R) = h 2 . Therefore, the MRRSPNN controller can be represented by where are the weight vector at the third layer and the input vector at the third layer, respectively. At the second layer, input semaphore and output semaphore are explained by where η is the recurring gain at the second layer. Rogers-Szego polynomials function [13,47] is adopted as the activation function.
) ; ( q x RS j is the Rogers-Szego polynomials function with are the 0-, 1-, and 2-order Meixner polynomials functions, respectively. The first kind type of Rogers-Szego polynomials function at the recurrence relation [13,47]

is given by
The symbol  is a summation factor. At the third layer, semaphore and output semaphore are explained by ( h R y k = . Therefore, the MRRSPNN controller can be represented by where are the weight vector at the third layer and the input vector at the third layer, respectively. Moreover, a minimum discrepancy ρ in order to execute the repay control can be represented by where * Ω is the intact weight vector and υ is a small positive number with υ ρ < . Equation (19) can be represented by

Second layer
Third layer Moreover, a minimum discrepancy ρ in order to execute the repay control can be represented by where Ω * is the intact weight vector and υ is a small positive number with ρ < υ. Equation (19) can be represented by So as to search for the adaptive rule of minimum discrepancy observer and adaptive rule of the MRRSPNN controller, the Lyapunov function then is represented by where γ > 0 is a fitted value. υ =υ − υ is the evaluated discrepancy. ε 1 is the learning rate. Then the Lyapunov function's differential by means of Equations (25) and (26) can be represented by e a e a + υ .
Theυ will be observed by an adaptive observer and assumed to be bounded during the observation. The above assumption is valid in practical digital processing of the observer since the sampling period of the observer is short enough compared with the variation of ρ. If Equation (20) with I a = 0, then Equation (28) by using Equations (29) and (30) can be represented by Equation (32) by using Equation (31) and ρ < υ can be represented by It is a negative semi-definite for .
H 1 (t) in Equation (33), and then e a and (Ω * − Ω) are represented as bounded. Additionally, the uniformly continuous function ξ(t) can be defined by Then take integration of Equation (34) as Since H 1 (0) and H 1 (t) are bounded, then Then take differential of Equation (34) as e a (37) where . e a is bounded at all variables of Equation (26) as bounded. By using Barbalat's lemma [48,49] lim t→∞ ξ(t) = 0, then e a (t) → 0 as t → ∞ . Furthermore, the sign function sgn(Q c e a ) can be represented by the continuous function Q c e a /(|Q c e a | + ς) and ς = ς 0 , 0 , |Q c e a | < ς |Q c e a | ≥ ς with ς 0 and ς are two small constants to reduce chattering in the countervailing controller.
A training means of parameters in the MRRSPNN can be unearthed by use of Lyapunov stability and the gradient descent skill. The MGWO by using two adjusted factors is applied to look for two better learning rates in the MRRSPNN to acquire faster convergence. The connecting weight parameter presented in Equation (29) can be represented by An objective function that explains online training procedure of the MRRSPNN is defined by The fitted learning rule of the connecting weight by use of the gradient descent skill with the chain rule is represented by It is known that ∂H 2 /∂y 3 k = −e a Q c from Equation (38) and Equation (40). The fitted learning rule of recurring weight v 1 ik by use of the gradient descent technology with chain rule then is represented by where ε 2 is the learning rate. To acquire better convergence, the MGWO is applied to look for two changeable learning rates in the MRRSPNN. Besides, for improving convergence and finding two optimal learning rates, the MGWO by using two linear adjusted factors is proposed in this study.

Results
The conformation of the SIM expelling CVT organized system by using DSP control system is shown in Figure 2. The rated constitution of the CVT organized system are below as 648.2 mm for V-belt length, 73.6 mm for preceding pulley diameter, 32.3 mm for succeeding pulley diameter, 4.1 for conversion ratio. The rated format of the SIM is below as six-phase 48 V, 2 kW, 3068 r/min. The electrical and mechanical parameters of the SIM are below as r s = 2.96 Ω , L ss = 19.38 mH, L rr = 19.32 mH, k ar = 0.202 Nm/A, J a = 19.61 × 10 −3 Nms, B a = 2.32 × 10 −3 Nms/rad. Because of inherent uncertainty in the CVT organized system (e.g., the aggregated parameter variations and the aggregated nonlinear outward disturbances) and current output limitation for DC power source, the SIM expelling CVT organized system is applied at 3000 r/min to avoid burning IGBT power modules. The flowchart of realized control methodologies with real-time implementation by means of DSP control system by using the "C" language in the experimental tests incorporates the principal program (PP) and the interrupt service routine (ISR), which is illustrated in Figure 5.
All input/output (I/O) initialization and parameters are first processed in the PP and then the interrupt time in the ISR is set. The ISR with 2 msec sampling time is applied for reading six-phase currents from A/D converters and rotor position of the SIM expelling CVT organized system from encoder, calculating rotor position and speed, executing lookup table and coordinate transformation, executing PI current control, executing the proposed MMRRSPNN control with MGWO by using two linear adjusted factors, and outputting six-phase SVPWMC signals to drive the IGBT power module voltage source inverter. Two judger g1 and g1_mx shown in the flowchart are provided to realize the IFOC. The judger g2 is provided to realize the proposed control scheme by DSP control system. Two initial values g1 and g2 are set to zero. The initial value g1_mx is set to three. When the IFOC is implemented less three times, i.e., g1 < g1_mx, the IFOC must be continuously realized. Then this process will go back to the primary start. Therefore, the IFOC will execute three times, then the proposed control scheme realized by DSP control system will execute one time. The voltage source inverter with six-sets of IGBT power modules is executed by a six-phase SVPWMC. In addition, the measured bandwidth of position control loop is about 0.1 kHz and the measured bandwidth of current control loop is about 1 kHz. So as to enhance precision of the sampling signals from A/D converter, the sampling interval of the control processing in the tested results is set at 1 msec (1 kHz) so that the DSP control system has enough time to process the control algorithm. Because of inherent uncertainty in the SIM expelling CVT organized system, current output limitation and voltage output limitation for DC bus power, the DC bus power only operated under maximum current, and maximum voltage for avoiding burning down the IGBT power modules for the SIM expelling CVT organized system. Furthermore, so as to prevent over-load operation and the voltage source inverter burn, the SIM expelling CVT organized system displayed in Figure 2 has six sets of over current protection circuits, six sets of over voltage protection circuits, and six sets of under voltage protection circuits.   The one aggregated parameter variations and aggregated nonlinear outward disturbances ∆τ 1 + τ u1 + ∆τ s case at 1500 r/min is the tested Event E1 case. The double aggregated parameter variations and aggregated nonlinear outward disturbances ∆τ 1 + τ u1 + ∆τ s case at 3000 r/min is the tested Event E2 case. Besides, for comparison control performance by using the celebrated PI controller as the controller CT1, the MRRHPNN control system with mend ACO [37] as the controller CT2 and the proposed wise dynamic control system using MMRRSPNN control and MGWO with two adjusted factors as the controller CT3 are adopted in this study.
To get good transient-state and steady-state control performance, two gains of the celebrated PI controller as the controller CT1 are k p2 = 18.6, k i2 = k p2 /T i2 = 4.3 by use of the Kronecker method to build a stability boundary in the k p2 and k i2 plane [44][45][46]. This method is used to narrow down the region for iterative selection of values of the parameters of k p2 and k i2 on the tuning of the PI controller with one aggregated parameter variations and aggregated nonlinear outward disturbances ∆τ 1 + τ u1 + ∆τ s case at 1500 r/min for the speed tracking. The MRRHPNN control system with mend ACO as the controller CT2 adopted 2, 3, and 1 nodes in the first, second, and third layers for the RRHPNN, respectively. Moreover, all gains are set to achieve better transient control performance in tests considering the requirement of stability. Besides, the control gains of the MRRHPNN control system with mend ACO are given by k a = 3.6, η = 0.10, γ = 0.21 with one aggregated parameter variations and aggregated nonlinear outward disturbances ∆τ 1 + τ u1 + ∆τ s case at 1500 r/min for the speed tracking. The parameter adjustment process remains continually active for the duration of the experimentation. The parameter's initialization of the MRRSPNN in Lewis et al. [50] is adopted to initialize the parameters in this paper. The parameter adjustment process remains continually active for the duration of the experimentation.
The proposed MMRRSPNN control with MGWO by using two linear adjusted factors as the controller CT3 adopted 2, 3, and 1 nodes in the first, second, and third layers for the MRRSPNN, respectively. Besides, all gains are set to achieve better transient control performance in experimentation considering the requirement of stability. Furthermore, the control gains of the MMRRSPNN control with MGWO by using two linear adjusted factors are given by k a = 3.6, η = 0.10, γ = 0.21 with one aggregated parameter variations and aggregated nonlinear outward disturbances ∆τ 1 + τ u1 + ∆τ s case at 1500 r/min for the speed tracking. The parameters adjustment process remains continually active for the duration of the experimentation. The parameter's initialization of the MRRSPNN in Lewis et al. [50] is adopted to initialize the parameters in this paper. The parameter adjustment process remains continually active for the duration of the experimentation.
Firstly, the tested results for the SIM expelling CVT organized system by using the celebrated PI controller as the controller CT1 at the tested Event E1 case and at the tested Event E2 case are demonstrated in Figures 6-8. Figures 6a and 7a demonstrate the speed responses in the command speed ω m , reference model speed ω * , and measured speed ω a . Figures 6b and 7b demonstrate the speed discrepancy e a responses. Figure 8a,b demonstrate the developed torque τ a responses. Good speed tracking performance at the tested Event E1 case shown in Figure 6a is due to small disturbances which is the same as the rated case. Besides, the developed torque τ a response shown in Figure 8a,b guides to large torque ripple due to the CVT system's push-pull friction. The tested results show that tardy speed responses demonstrated in Figure 7a was obtained by using the controller CT1 due to unfit gains tuning. model speed * ω , and measured speed a ω . Figure 6b and Figure 7b demonstrate the speed discrepancy a e responses. Figure 8a,b demonstrate the developed torque a τ responses. Good speed tracking performance at the tested Event E1 case shown in Figure 6a is due to small disturbances which is the same as the rated case. Besides, the developed torque a τ response shown in Figure 8a,b guides to large torque ripple due to the CVT system's push-pull friction. The tested results show that tardy speed responses demonstrated in Figure 7a was obtained by using the controller CT1 due to unfit gains tuning.     Secondly, the tested results of the MRRHPNN control system with mend ACO [37] as the controller CT2 for the SIM expelling CVT organized system at the tested Event E1 case and at the tested Event E2 case are demonstrated in Figures 9-11. Figure 9a and Figure 10a demonstrate the speed responses of measured speed a ω , reference model speed * ω and command speed m ω . Figure 9b and Figure 10b demonstrate the speed discrepancy a e responses. Figure 11a,b demonstrate the developed torque a τ responses. Figure 9a demonstrates good performance of speed tracking at the tested Event E1 case due to small disturbances which is the same as the rated case. Good response of speed tracking is demonstrated in Figure 10a at the tested Event E2 case. The tested results demonstrate that good tracking performance was achieved for the SIM expelling CVT organized system by use of the controller CT2 due to the online changeable method of the MRRHPNN control system and the compensated controller action. Secondly, the tested results of the MRRHPNN control system with mend ACO [37] as the controller CT2 for the SIM expelling CVT organized system at the tested Event E1 case and at the tested Event E2 case are demonstrated in Figures 9-11. Figures 9a and 10a demonstrate the speed responses of measured speed ω a , reference model speed ω * and command speed ω m . Figures 9b and 10b demonstrate the speed discrepancy e a responses. Figure 11a,b demonstrate the developed torque τ a responses. Figure 9a demonstrates good performance of speed tracking at the tested Event E1 case due to small disturbances which is the same as the rated case. Good response of speed tracking is demonstrated in Figure 10a at the tested Event E2 case. The tested results demonstrate that good tracking performance was achieved for the SIM expelling CVT organized system by use of the controller CT2 due to the online changeable method of the MRRHPNN control system and the compensated controller action.     Thirdly, the tested results of the wise dynamic control system using MMRRSPNN control and MGWO with two adjusted factors as the controller CT3 for the SIM expelling CVT organized system at the tested Event E1 case and at the tested Event E2 case are demonstrated in Figures 12-14. Figure 12a and Figure 13a demonstrate the speed responses of measured speed a ω , reference model speed * ω , and command speed m ω . Figure 12b and Figure 13b demonstrate the speed discrepancy a e responses. Figure 14a,b demonstrate the responses of developed torque a τ . Figure 12a demonstrates better performance of speed tracking at the tested Event E1 case because it is the same as the nominal case with smallest disturbances. The higher speed tracking response is demonstrated in Figure 13a at the tested Event E2 case. The tested results show that a more accurate tracking performance was achieved for the SIM expelling CVT organized system when the controller CT3 was used because of the online fitted mechanism of the MRRSPNN and the repay controller action. The developed torque a τ response shown in Figure 14a,b demonstrates lower torque ripple by online adjustment of the MRRSPNN control system and MGWO with two adjusted factors to process the unmodeled dynamic of a CVT organized system such as push-pull frictions. Figure 11. Tested results for the SIM expelling CVT organized system by using the controller CT2: (a) response of developed torque τ a at the tested Event E1 case, (b) response of developed torque τ a at the tested Event E2 case.
Thirdly, the tested results of the wise dynamic control system using MMRRSPNN control and MGWO with two adjusted factors as the controller CT3 for the SIM expelling CVT organized system at the tested Event E1 case and at the tested Event E2 case are demonstrated in Figures 12-14. Figures 12a and 13a demonstrate the speed responses of measured speed ω a , reference model speed ω * , and command speed ω m . Figures 12b and 13b demonstrate the speed discrepancy e a responses. Figure 14a,b demonstrate the responses of developed torque τ a . Figure 12a demonstrates better performance of speed tracking at the tested Event E1 case because it is the same as the nominal case with smallest disturbances. The higher speed tracking response is demonstrated in Figure 13a at the tested Event E2 case. The tested results show that a more accurate tracking performance was achieved for the SIM expelling CVT organized system when the controller CT3 was used because of the online fitted mechanism of the MRRSPNN and the repay controller action. The developed torque τ a response shown in Figure 14a,b demonstrates lower torque ripple by online adjustment of the MRRSPNN control system and MGWO with two adjusted factors to process the unmodeled dynamic of a CVT organized system such as push-pull frictions.    Besides, Figure 15a,b demonstrate the convergent speeds of two learning rates 1 ε and 2 ε in the MRRSPNN using MGWO with two adjusted factors at the tested Event E1 case, respectively. Figure 16a,b demonstrate the convergent speeds of two learning rates 1 ε and 2 ε in the MRRSPNN using MGWO with two adjusted factors at the tested Event E2 case, respectively. Besides, Figure 15a,b demonstrate the convergent speeds of two learning rates ε 1 and ε 2 in the MRRSPNN using MGWO with two adjusted factors at the tested Event E1 case, respectively. Figure 16a,b demonstrate the convergent speeds of two learning rates ε 1 and ε 2 in the MRRSPNN using MGWO with two adjusted factors at the tested Event E2 case, respectively.
Finally, rotor speed responses under adding load torque disturbance and aggregated nonlinear outward disturbances 2Nm(τ t ) + τ u1 + ∆τ s at 3000 r/min speed as the tested Event E3 case is tested by using the controller CT1, the controller CT2 and the controller CT3. Figures 17-19 demonstrate the tested results of speed and current w a1 in phase a at the tested Event E3 case when the controller CT1, the controller CT2, and the controller CT3 were used, respectively. Figure 17a, Figure 18a, and Figure 19a demonstrate speed-adjusted response of the command speed ω m and measured speed ω a at the tested Event E3 case when the controller CT1, the controller CT2, and the controller CT3 were used, respectively. Figure 17b, Figure 18b, and Figure 19b demonstrate the measured current w a1 in phase a at the tested Event E3 case when the controller CT1, the controller CT2, and the controller CT3 were used, respectively. The tested results demonstrate that the degenerated responses at the tested Event E3 case are considerably improved when the controller CT3 was used.   Figure 16. Tested results at the tested Event E2 case by using the controller CT3: (a) the convergent response of learning rate ε 1 , (b) the convergent response of learning rate ε 2 .  Figure 17. Tested results at the tested Event E3 case by using the controller CT1: (a) speed-adjusted response; (b) current response in phase a. Figure 17. Tested results at the tested Event E3 case by using the controller CT1: (a) speed-adjusted response; (b) current response in phase a.
Mathematics 2020, 8, x FOR PEER REVIEW 28 of 33 Figure 18. Tested results at the tested Event E3 case by using the controller CT2: (a) speed-adjusted response; (b) current response in phase a. Figure 18. Tested results at the tested Event E3 case by using the controller CT2: (a) speed-adjusted response; (b) current response in phase a.  Figure 19. Tested results at the tested Event E3 case by using the controller CT3: (a) speed-adjusted response; (b) current response in phase a.

Discussion
Furthermore, a control performance comparison of the the celebrated PI controller as the controller CT1, The MRRHPNN control system with mend ACO as the controller CT2, and the proposed MMRRSPNN control with MGWO by using two adjusted factors as the controller CT3 is presented in Table 1   Besides, transient response of the controller CT3 demonstrates better convergence and fine load regulation than the controller CT1 and the controller CT2.

Discussion
Furthermore, a control performance comparison of the the celebrated PI controller as the controller CT1, The MRRHPNN control system with mend ACO as the controller CT2, and the proposed MMRRSPNN control with MGWO by using two adjusted factors as the controller CT3 is presented in Table 1 for experimental results of three test cases. The maximum errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E1 case are 88 r/min, 69 r/min and 40 r/min, respectively. The root mean square errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E1 case are 45 r/min, 30 r/min, and 20 r/min, respectively. The maximum errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E2 case are 215 r/min, 88 r/min, and 43 r/min, respectively. The root mean square errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E2 case are 60 r/min, 31 r/min, and 22 r/min, respectively. The maximum errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E3 case are 398 r/min, 198 r/min, and 69 r/min, respectively. The root mean square errors of e a by utilized the control systems CT1, CT2, and CT3 at tested Event E3 case are 51 r/min, 27 r/min, and 17 r/min, respectively. As shown in Table 1, the control systems CT3 results in smaller tracking error in comparison with the control systems CT1 and CT2. According to the tabulated measurements, the control systems CT3 indeed yields the superior control performance. Furthermore, the characteristic performance comparisons of the control systems CT1, CT2, and CT3 are gathered up in Table 2 from tested results. In Table 2, some performances with regards to the control rule with vibration, the dynamic response, the regulation ability of load torque disturbance, the convergent speed, the speed tracking error, the rejection ability of parameter disturbance, and torque ripple (V-belt shaking action and torsional vibration variations) in the control systems CT3 are superior to the control systems CT1 and CT2.

Conclusions
The MMRRSPNN control with MGWO by using two adjusted factors has been favorably used for controlling the SIM expelling CVT organized system with good robustness. The MMRRSPNN control with MGWO by using two linear adjusted factors, which can execute intendant control system, MRRSPNN control, and the repay control, was put forward to decrease and sleek the control intensity when the system's states are within the specified bound range. The primary contributions of this study are as follows: (a) the simplified dynamic models of the CVT expelled by the SCRIM with nonlinear uncertainties were smoothly originated; (b) the MMRRSPNN control system was well employed for the SIM expelling CVT organized system under intact nonlinear outward disturbances to improve robustness; (c) the fitted learning rule in the MRRSPNN control and the evaluated rule in the repay control were successfully established by use of the Lyapunov stability theorem; (d) the MGWO by using two adjusted factors was well used for changing two varied learning rates of connective and recurring weights in the MRRSPNN to achieve good convergence; and (e) the MMRRSPNN control with MGWO by using two linear adjusted factors as the controller CT3 is superior than the celebrated PI controller as the controller CT1 and the MRRHPNN control system with mend ACO [37] as the controller CT2 in torque ripple reduction.
Finally, the controller CT3 is superior to the controller CT1 and the controller CT2 from all tested results and control performances for the SIM expelling CVT organized system.
Future related works are as follows: (a) the development of the detailed modelling in the entire models of CVT organized system; (b) the use of the more superior DSP control systems to reduce realized time; (c) the combination with more advanced control system to enhance the robustness of systems; (d) the program of the control structure for other (square) reference trajectory.