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A PMSM Driven Electric Scooter System with a V-Belt Continuously Variable Transmission Using a Novel Hybrid Modified Recurrent Legendre Neural Network Control
A PMSM Driven Electric Scooter System with a V-Belt Continuously Variable Transmission Using a Novel Hybrid Modified Recurrent Legendre Neural Network Control
Journal of Power Electronics. 2014. Sep, 14(5): 1008-1027
Copyright © 2014, The Korean Institute Of Power Electronics
  • Received : March 28, 2014
  • Accepted : June 10, 2014
  • Published : September 28, 2014
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About the Authors
Chih-Hong Lin
Department of Electrical Engineering, National United University, Miaoli, Taiwan
jhlin@nuu.edu.tw

Abstract
An electric scooter with a V-belt continuously variable transmission (CVT) driven by a permanent magnet synchronous motor (PMSM) has a lot of nonlinear and time-varying characteristics, and accurate dynamic models are difficult to establish for linear controller designs. A PMSM servo-drive electric scooter controlled by a novel hybrid modified recurrent Legendre neural network (NN) control system is proposed to solve difficulties of linear controllers under the occurrence of nonlinear load disturbances and parameters variations. Firstly, the system structure of a V-belt CVT driven electric scooter using a PMSM servo drive is established. Secondly, the novel hybrid modified recurrent Legendre NN control system, which consists of an inspector control, a modified recurrent Legendre NN control with an adaptation law, and a recouped control with an estimation law, is proposed to improve its performance. Moreover, the on-line parameter tuning method of the modified recurrent Legendre NN is derived according to the Lyapunov stability theorem and the gradient descent method. Furthermore, two optimal learning rates for the modified recurrent Legendre NN are derived to speed up the parameter convergence. Finally, comparative studies are carried out to show the effectiveness of the proposed control scheme through experimental results.
Keywords
Ⅰ. INTRODUCTION
There are many countries trying to develop electric vehicles in order to reduce both their dependence on petroleum and the effects of air pollution. Scooters are popular for personal transportation. This is partly due to the fact that they are cheap to buy and operate and convenient to park and store. Because scooters are easier and cheaper to own than cars in most parts of the world, the development of control methods is one of the good research areas for electric scooters. Electric scooters are driven by AC motors, so the selection of the AC motors drive systems is important. There are several basic types of AC servo motors such as PMSMs, switched reluctance motors (SRMs) and induction motors (IMs). PMSMs provide higher efficiency, higher power density, and lower power loss for their size when compared to SRMs and IMs. Direct torque control (DTC) [1] - [4] and field-oriented control [5] - [7] are the two most popular control techniques used with PMSMs. The classical DTC [3] , [4] has some drawbacks. One of these drawbacks is the higher torque ripple and high current ripple generated in steady state operation when compared to the field-oriented control system. On the other hand, the field-oriented control is much less sensitive to parameter variations and has faster operation in the four quadrants [5] - [7] . Hence, field-oriented control is used to reduce the torque ripple influence in this paper.
Artificial neural networks (ANNs) have become a formidable learning skill for performing many complex tasks such as highly nonlinear approximations [8] , [9] and the control of dynamical systems [10] , [11] . One of the major drawbacks of NNs is that they are computationally intensive and need a large number of iterations for their training. In order to reduce the computational complexity, a functional-link NN, which has shown that it is capable of producing performance that is similar to that of NNs but with a much lower computational cost, was introduced in [12] , [13] . Moreover, a functional-link NN-based nonlinear dynamic system identification with satisfactory results was reported in [14] . It is shown that the performance of the functional-link NN is similar to that of a NN but with a faster convergence and less computational complexity. Moreover, a comprehensive survey on various applications for functional-link NNs has been presented in [15] .The functional-link NN has emerged as a powerful learning technique for both nonlinear approximations and the dynamic control of systems. A novel Legendre polynomial based linear NN for static function approximation was proposed by Yang et al . [16] . It was demonstrated that this network has a fast convergence, and provide high accuracy. Recently, Legendre polynomial NNs for channel equalization problems have been proposed in [17] , [18] . The superior performance of this Legendre NN equalizer over the NN-based and RBF-based equalizers for different nonlinear channel models has been demonstrated. Moreover, the computational complexity of the Legendre NNs is lower than that of the functional-link NN since the evaluation of Legendre polynomials involves less computation than the evaluation of trigonometric functions [19] . In addition, the predictive approach is based on a Legendre neural network and a random time strength function via a promising data mining technique used in machine learning and proposed in [20] . Furthermore, a computationally efficient Legendre NN for nonlinear active noise cancellation was shown in [21] . However, the proposed Legendre NN is only a single layer network structure without a feedback loop. It cannot adequately approximate dynamic behaviors found in PMSM servo-driven electric scooter systems with nonlinear and time-varying characteristics.
The recurrent NN has received an increasing amount of attention due to its structural advantage in the modeling of nonlinear systems and the dynamic control of these systems [22] - [26] . These networks are capable of effective identification and control of complex process dynamics but at the expense of increased computational complexity. The typical Elman NN, which is a partial recurrent network model that was first proposed by Elman [27] , has one hidden layer with a delayed feedback. The Elman NN is capable of providing standard state-space representation for dynamic systems. In order to improve the ability for identifying high order systems, some recurrent modified Elman NNs [28] - [30] have been proposed recently. They have been shown to have more advantages than the basic Elman NN, including good performance, high accuracy, dynamic robustness and fast transient performance. However, the parameters of these recurrent modified Elman NNs [28] - [30] have a slower convergence speed due to their higher computational complexity and complex network structure. Additionally, the recurrent wavelet NNs proposed by [31] - [33] , which combine the properties of the attractor dynamics of the recurrent NN and the good convergence performance of the wavelet NN, can reduce the computational complexity. However, they also have slower convergence speeds due to the adopted fixed learning rates for the parameters in these recurrent wavelet NNs. Therefore, in order to reduce the network complexity and speed-up the convergence, the novel simpler NN structure of modified recurrent Legendre NN, which has a self-feedback connection in the hidden layer and a recurrent connection between the output layer and the input layer, is more powerful than the three-layer recurrent NNs for dealing with time-varying and nonlinear dynamic systems. Two optimal learning rates for the parameters in the modified recurrent Legendre NN are proposed to enhance the convergence speed in this paper.
In electric scooters [34] - [37] with nonlinear uncertainties, the hybrid recurrent fuzzy NN controller may not provide satisfactory control performance when operated over a wide range of operating conditions. In [34] , a hybrid recurrent fuzzy NN controller using a rotor flux estimator was used to control the PMSM without a shaft encoder to drive electric scooter. In [35] a hybrid recurrent fuzzy NN control was used for the feedback control of PMSM driven electric scooters with a shaft encoder. A favorable speed tracking responses can only be achieved by using a hybrid recurrent fuzzy NN controller at 1200 rpm in the nominal case and in the parameter variation case. However, poor speed tracking responses are encountered at 2400 rpm due to uncertain perturbations. On the other hand, if the controlled plant has a highly nonlinear uncertainty, the linear controller may not provide satisfactory control performance. Thus, to ensure the control performance and robustness, a V-belt CVT driven electric scooter using a PMSM servo drive is controlled by the novel hybrid modified recurrent Legendre NN control system developed in this paper. The novel hybrid modified recurrent Legendre NN control system, which is composed of an inspector control, a modified recurrent Legendre NN control with an adaptation law and a recouped control with an estimation law, is applied to a V-belt CVT driven electric scooter by using the PMSM servo drive system. The novel hybrid modified recurrent Legendre NN control system has a fast convergence ability and good generalization capability. In addition, the adaptation law of the on-line parameters in the modified recurrent Legendre NN and the estimation law of the recouped controller can be derived according to the Lyapunov stability theorem and the gradient descent method. Furthermore, two optimal learning rates for the modified recurrent Legendre NN are proposed to enhance the convergence speed. The control method, which is not dependent upon the predetermined characteristics of the system, can adapt to any change in the system characteristics. Finally, the control performance of the proposed novel hybrid modified recurrent Legendre NN control system is verified by experimental results.
This paper is organized as follows. The system structure of a V-belt CVT driven electric scooter using a PMSM servo drive system is reviewed in Section Ⅱ. The design method of the novel hybrid modified recurrent Legendre NN control system is presented in Section Ⅲ. Experimental results are illustrated in Section Ⅳ. Some conclusions are given in Section Ⅴ.
Ⅱ. STRUCTURE OF A V-BELT CVT DRIVEN ELECTRIC SCOOTER SYSTEM USING A PMSM SERVO DRIVE
Due to electric scooter systems with unknown nonlinear uncertainties, i.e., load torque, rolling resistance, wind resistance and braking force, the V-belt CVTs and clutches in scooters can be categorized as functioning in one of two operating modes depending on the speed of the V-belt CVT output axis: disengaged or completely coupled. At the start of the PMSM drive cycles, the scooter is in an idle state. The clutch is initially disengaged, and subsequent transitions between the modes are controlled by the clutch axis rotational speed. Except for the mechanical losses, the PMSM power is transmitted through the V-belt CVT and clutch to the wheel in an electric scooter. For the sake of a convenient design, the stator voltage equations of a V-belt CVT driven electric scooter using a PMSM servo drive in the synchronously rotating reference frame can be described as follows [5] - [7] :
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where vqr and vdr are the d -axis and q -axis stator voltages, iqr and idr are the d -axis and q -axis stator currents, Lqr and Ldr are the d -axis and q -axis stator inductances, λfd is the d -axis permanent magnet flux linkage, Rr is the stator resistance, and ωf is the electrical angular speed.
The electromagnetic torque Te of a V-belt CVT driven electric scooter using a PMSM servo drive can be described as:
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where Pr is the number of poles. Then, the dynamic equation of a V-belt CVT driven electric scooter using a PMSM servo drive [34 - 37] can be represented as:
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where Tl ( Ta, Δ Tp , Fl ( Bg ), υa ( νr , Bg ),τ a ( νr ), ωr 2 ) = Ta + Δ Tp + Tun [36] , [37] represents the lumped nonlinear external disturbances and parameter variations, (e.g. the electric scooter system including the fixed load torque Ta , the variation of the parameters Δ Tp , the rolling resistance υa ( vr , Bg ), the wind resistance τ a ( vr ) and the braking force Fl ( Bg ) ),
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represents the variation of the parameters, Tun = Fl ( Bg )+ υa ( vr , Bg )+ τa ( vr ) ωr 2 represents the unknown nonlinear load torque, vr is the total wind velocity, Bg is the total frictional coefficient of the ground surface, Br is the total viscous frictional coefficient, and Jr is the total moment of inertia including the electric scooter itself. ωr = 2 ωf / Pr is the rotor speed. Due to Ldr = Lqr , for a surface-mounted PMSM, the second term of (3) is zero. Moreover, λfd is a constant for the surface-mounted PMSM. The rotor flux is produced in the d -axis only, while the current vector is generated in the q -axis for the field-oriented control. When the d -axis rotor flux is a constant and the torque angle is π / 2 [6 - 7] , the maximum torque per ampere can be reached for the field-oriented control. The electromagnetic torque Te is linearly proportional to the q -axis current iqr , which is determined by the closed-loop control. The control principle of the PMSM servo-drive system is based on the field orientation. A V-belt CVT driven electric scooter using a PMSM servo drive system with the implementation of field-oriented control can be reduced as:
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in which kr = 3 Prλfd /4 is the torque constant. A block diagram of a V-belt CVT driven electric scooter using a PMSM servo drive system is shown in Fig. 1 .
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Block diagram of a V-belt CVT driven electric scooter using a PMSM servo drive.
The whole system of a V-belt CVT driven electric scooter using a PMSM servo drive can be indicated as follows: a field-oriented institution, a PI current loop control, one set of sinusoidal pulse-width-modulation (PWM) control circuits, one set of interlock and isolated circuits, a voltage source inverter with three sets of insulated-gate bipolar transistor (IGBT) power modules, and a speed controller [33] - [35] . The PI current loop controller is a current loop tracking controller. In order to attain a good dynamic response, all of the gains for a well-known PI current loop controller are listed as follows: kpc = 9.5 and kic = kpc / Tic = 2.8 through some heuristic knowledge [38] - [40] of the tuning of the PI controller. The field-oriented institution consists of a coordinate transformation, sin θf /cos θf generation and lookup table generation. A TMS320C32 digital signal processor (DSP) control system is used to implement the field-oriented institution control and speed control. A V-belt CVT driven electric scooter using a PMSM servo drive is manipulated under the lumped nonlinear external disturbances
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Ⅲ. DESIGN OF THE CONTROL SYSTEM
The nonlinear uncertainties of a V-belt CVT driven electric scooter using a PMSM servo drive such as the nonlinear friction force of the transmission belt and clutch, rolling resistance, wind resistance and braking force, lead to degenerate tracking responses in the command current and speed. These nonlinear uncertainties cause variations in the rotor inertia and friction. For a convenient novel hybrid modified recurrent Legendre NN design, the dynamic equation of the V-belt CVT driven electric scooter using the PMSM servo drive from (4) can be rewritten as:
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where un = i*qr is the command current of the stator of the PMSM. Aa = -Br/Jr , Ba = kr/Jr and Ca = -1/Jr are three known constants. When uncertainties including the lumped nonlinear external disturbances occur, the parameters are assumed to be bounded, i.e., │ Aaωr │ ≤ D 1 ( ωr ) , │ CaTl (Ta,ΔTp ,Fl (Bg ), νar , Bg ), τar ),ωr2 ) │≤ D 2 , and D 3 Ba , where D 1 ( ωr ) is a known continuous function, and D 2 and D 3 are two known constants. Then, the tracking error can be defined as:
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where ω * represents the desired command rotor speed, and e is the tracking error between the desired rotor speed and the actual rotor speed. If all of the parameters of the V-belt CVT driven electric scooter using the PMSM servo drive system including the lumped nonlinear external disturbances and parameter variations are well known, the ideal control law can be designed as:
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in which k 1 is a positive constant. Substituting (8) into (6), the error dynamic equation can be obtained as:
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The system state can track the desired trajectory gradually if e ( t )→0 as t →∞in (9). However, a novel hybrid modified recurrent Legendre NN control system is proposed to control the V-belt CVT driven electric scooter using the PMSM servo drive under uncertain perturbations.
- A. Novel Hybrid Modified Recurrent Legendre NN Control
The configuration of the proposed novel hybrid modified recurrent Legendre NN control system, which is composed of an inspector control system, a modified recurrent Legendre NN controller and a recouped controller, is described in Fig. 2 .
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Block diagram of the novel hybrid modified recurrent Legendre NN control system.
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where uin is the proposed inspector control that is capable of stabilizing around a predetermined bound area in the states of the controlled system. The modified recurrent Legendre NN control urc is a major tracking controller, which is used to imitate an ideal control law. The recouped control ure is designed to recoup the difference between the ideal control law and the modified recurrent Legendre NN control. Since the inspector control caused an overdone and chattering effort, the modified recurrent Legendre NN control and the recouped control are proposed to reduce and smooth the control effort when the system states are inside a predetermined bound area. When the modified recurrent Legendre NN approximation properties cannot be ensured, the inspector control is able to take action. For the condition of the divergence of states, the design of the novel hybrid modified recurrent Legendre NN control system is essential to stretch the divergent states back to the predestinated bound area. The novel hybrid modified recurrent Legendre NN control system can uniformly approximate the ideal control law inside the bound area. Then the stability of the system can be warranted. Error dynamic equations from (6) to (10) can be acquired as:
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Firstly, the inspected control uin can be designed as:
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in which sgn(·) is a sign function. When the modified recurrent Legendre NN approximation properties cannot be ensured, the inspector control law is able to take action, i.e., Iin = 1. Due to inadequate bound values, e.g., D 1 ( ωr ), D 2 , D 3 and the sign function, the inspected control can produce an overdone and chattering effort. However, the modified recurrent Legendre NN control and the recouped control can be devised to eliminate this flaw. The modified recurrent Legendre NN control is raised to imitate an ideal control un * . Then the recouped control is posed to recoup the difference between the ideal control un * and the modified recurrent Legendre NN control urc .
- B. Modified Recurrent Legendre NN
The architecture of the proposed three-layer modified recurrent Legendre NN is depicted in Fig. 3 . It is composed of an input layer, a hidden layer and an output layer. The activation functions and signal actions of the nodes in each layer of the modified recurrent Legendre NN can be described as follows:
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Structure of the three-layer modified recurrent Legendre NN.
1) First layer – input layer: Each node i in this layer is indicated by using ∏, which multiplies all of the input signals by one another. The output signals are the results of the product. The inputs and outputs for each node i in this layer can be expressed as:
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x 1 1 = ω * - ωr = e is the tracking error between the desired speed ω * and the rotor speed ωr . x 2 1 = e (1-z -1 )=Δ e . is the tracking error change.
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is the recurrent weight between the output layer and the input layer. N denotes the number of iterations.
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is the output value of the output layer in the modified recurrent Legendre NN.
2) Second layer – hidden layer: The single node j th in this layer is labeled with Σ. The net input and the net output for node j th of this layer are expressed as:
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Legendre polynomials [17] - [19] are selected for the activation function of the hidden layer. The Legendre polynomials are denoted by Ln ( x ), where n is the order of expansion, -1< x <1 is the argument of the polynomial, m is the number of nodes, and γ is the self-connecting feedback gain of the hidden layer which is selected to be between 0 and 1. The zero, the first and the second order Legendre polynomials are given by L 0 ( x ) = 1, L 1 ( x ) = x and L 2 ( x ) = (3 x 2 -1)/2, respectively. The higher order polynomials are given by L 3 ( x ) = (5 x 3 -3 x )/2 and L 4 ( x ) = (35 x 4 -30 x 2 +3)/8. The higher order Legendre polynomials are generated by the recursive formula given by: L n+1 ( x ) = [(2 n +1) xLn ( x ) - nLn-1 ( x )]/( n +1).
3) Third layer – output layer: The single node k th in this layer is labeled with Σ. It computes the overall output as a summation of all of the input signals. The net input and the net output for node k th in this layer are expressed as:
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where
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is the connective weight between the hidden layer and the output layer;
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is the activation function, which is selected as a linear function; and x 3 j ( N ) = y 2 j ( N ) represents the j th input to the node of the output layer. The output value of the modified recurrent Legendre NN can be represented as y 3 k ( N ) = urc . Then the output value of the modified recurrent Legendre NN, urc , can be denoted as:
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in which
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is the adjustable weight parameter vector between the hidden layer and the output layer of the recurrent Legendre NN.
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is the input vector in the output layer of the modified recurrent Legendre NN, in which
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is determined by the selected Legendre polynomials.
- C. Recouped Control with Estimation Law
By assuming that the sampling frequency is sufficiently high and that the back-EMF is constant during a sampling period, the recouped control with estimation law of a PMSM are given as follows: In order to evolve the recouped control ure , the minimum approximation error δ is defined as:
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in which
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is an ideal weight vector to reach the minimum approximation error. It is assumed that the absolute value of δ is less than a small positive value λ , i.e., │ δ │ < λ . Then, the error dynamic equation from (11) can be rewritten as:
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Then, the Lyapunov function is selected as:
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in
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is the bound estimated error, μ a >0 is the learning rate, and η is the adaptation gain. By differentiating the Lyapunov function with respect to t and using (18), (19) can be rewritten as:
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To satisfy
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the adaptation law
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and the recouped controller ure with the estimation law
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can be designed as follows:
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By substituting (12), (21) and (22) into (20) and using (12) with I in = 0, (20) can be represented as:
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By substituting (23) into (24), (24) can be obtained as:
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From (25),
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is a negative semi-definite value, i.e. V 1 ( t )≤ V 1 (0). This implies that e and
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are bounded. For proof that the hybrid modified recurrent Legendre NN control system can gradually become stable, the function is defined as:
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Integrating (26) with respect to t results in:
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Since V 1 (0) is bounded, and V 1 ( t ) is non-increasing and bounded:
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Differentiating (26) with respect to t gives:
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All of the variables in the right side of (18) are bounded. This implies that
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is also bounded. Then, ε ( t ) is a uniformly continuous function [41 - 42] . It is denoted that
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by using Barbalat’s lemma [41 - 42] . Therefore e ( t )→0 as t →∞. From the above proof, the novel hybrid modified recurrent Legendre NN control system becomes gradually stable. Moreover, the tracking error e ( t ) of the system will converge to zero. Furthermore, in order to avoid the chattering phenomenon of recouped controller ure , the sign the function sgn( Bae ) can be replaced by the equation
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where
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in which ρ 0 and τ are positive constants.
- D. Parameter Tuning Methodology and Convergence Analysis
According to the Lyapunov stability theorem and the gradient descent method, an on-line parameters training methodology for the modified recurrent Legendre NN can be derived and trained effectively. Then the parameter of adaptation law
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shown in (21) can be computed by the gradient descent method for the proper choice of two learning rates. The parameter convergence can be guaranteed but the convergent speed is reduced due to a smaller value for the learning rate. On the other hand, the parameter convergence may have an oscillation due to a larger value for the learning rate. In order to train the modified recurrent Legendre NN efficiently, two optimal learning rates will be derived to achieve a fast convergence of the output tracking error. The adaptation law
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shown in (21) can be rewritten as:
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To train the parameters of the modified recurrent Legendre NN effectively, the method to obtain a recursively gradient vector is very important. Each component can be defined as a derivative of the cost function in the training algorithm. The gradient vector is calculated in the direction opposite the flow of the output of each node. This is done by means of the chain rule. In order to describe the on-line training algorithm of the modified recurrent Legendre NN, a cost function is defined as [43] :
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The adaptation law of the connective weight by means of the gradient descent method can be represented as:
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where μα is the learning rate. Comparing (30) with (32), yields ∂ V 2 /∂ y 3 k = - eBa . The convergence analysis in the following theorem is used to derive a specific learning rate to assure the convergence of the output tracking error. The adaptation law of the recurrent weight
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by using the gradient descent method can be calculated as:
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in which μb is the learning rate. Then, two optimal learning rates are derived to assure the convergence of the output tracking error. The convergence analysis is provided in the following two theorems.
Theorem 1: Assume that μα is the learning rate of the modified recurrent Legendre NN connective weight between the hidden layer and the output layer. Meanwhile, define P 1wmax as P 1wmax maxN P 1w ( N )∥, where P 1w ( N ) = y 3 k w 2 kj , and ∥·∥ is the Euclidean norm in ℜ n . μα is chosen as [30 , 43] :
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Then, the convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate, which achieves a fast convergence, can be obtained as:
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Proof: Since:
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The discrete-type Lyapunov function is selected as:
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The change in the Lyapunov function is obtained by:
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Next, the error difference can be represented by:
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in which Δ e ( N ) is the output error change, and Δ
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represents change in the weight. (39) can be obtained by means of (30), (31), (32) and (36):
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Thereby:
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By means of (38) to (42),the Δ V 2 ( N ) can be rewritten as:
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If μα is chosen as 0 < μα < 2 /{( P 1wmax ) 2 [ eBa / e ( N )] 2 }, the Lyapunov stability of V 2 ( N ) > 0 and Δ V 2 < 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 a fast convergence, corresponds to [30 , 43] :
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i.e.:
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which comes from the derivative of (43) with respect to μα , and is equal to zero. This produces interesting results for the optimal learning rate which can be on-line tuned at each instant.
Theorem 2: Assume that μb is the learning rate of the modified recurrent Legendre NN recurrent weight between the output layer and the input layer. Meanwhile, P 2wmax should be defined as P 2wmax maxN P 2w ( N )∥ , where P 2w ( N ) =∂ yk 3 /∂ w 1 ik , and ∥·∥ is the Euclidean norm in ℜ n . μb is chosen as [30 , 43] :
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Then, the convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate which achieves a fast convergence can be obtained as:
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Proof: Since:
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The discrete-type Lyapunov function can be selected as (36) and the change in the Lyapunov function is obtained by (37). Then, the error difference can be represented by:
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where Δ e ( N ) is the output error change, and Δ
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represents change in the weight. (49) by using (31), (33) and (48), can be represented as:
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Thereby:
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By using (38), (49) to (52), Δ V 2 ( N )can be rewritten as:
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If μb is chosen as 0 < μb < 2 /{( P 2wmax ) 2 [ Bae / e ( N )] 2 }, the Lyapunov stability of V 2 ( N ) > 0 and Δ V 2 ( N ) < 0 is guaranteed. Then, 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 fast convergence, corresponds to [30 , 43] :
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i.e.:
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which comes from the derivative of (53) with respect to μb , and is equal to zero. This produces interesting results for the optimal learning rate which can be on-line tuned at each instant. In summary, the online tuning algorithm for the modified recurrent Legendre NN controller is based on the adaptation laws (32) and (33) for the connective weight adjustment and the recurrent weights adjustment with the two optimal learning rates in (35) and (47), respectively. Moreover, the modified recurrent Legendre NN weight estimation errors are fundamentally bounded [44] . The modified recurrent Legendre NN weight estimation errors are bounded, which is required to ensure that the control signal is bounded.
Ⅳ. EXPERIMENTAL RESULTS
The whole system of the DSP-based control system for a V-belt CVT driven electric scooter using a PMSM servo drive system is shown in Fig. 1 . A photo of the experimental setup is shown in Fig. 4 . The control algorithm was executed by a TMS320C32 DSP control system including multi-channels of D/A, eight channels programmable PWM and an encoder interface circuit. The methodologies proposed for the real-time control implementation in the DSP control system are composed of the main program and the interrupt service routine (ISR), as shown in Fig. 5 . In the main program, the parameters and input/output(I/O) initializations are processed first. Then, the interrupt interval for the ISR is set. After enabling the interrupt, the main program is used to monitor the control data. The ISR with a 2 ms sampling interval is used for reading the rotor position of the PMSM servo-driven electric scooter from the encoder, and the three-phase currents from the A/D converter. This is done by calculating the rotor position and speed, executing the lookup table and the coordinate translation, executing the PI current control, executing the novel hybrid modified recurrent Legendre NN control system, and outputting the three-phase current commands to the sinusoidal PWM circuit for switching the three-sets of IGBT power module inverters via the interlock and isolated circuits. The IGBT power module voltage source inverter is executed by a current-controlled sinusoidal PWM with a switching frequency of 15 k Hz by using a triangular carrier wave. In addition, the measured bandwidth of the speed control loop is about 24Hz, and the measured bandwidth of the current control loop by no load is about 240Hz for a PMSM servo-drive system without mounted V-bet CVT and wheel systems. The specifications of the surface-mounted PMSM are a three-phase 48V, 750W,16.5A,3600 rpm .The parameters of the PMSM are given as follows:
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kr = 0.86 Nm/A , R 1 = 2.5Ω, and L d1 = L q1 = 6.53 mH by using an open circuit test, short test, rotor block test, and loading test. To show the control performance of the proposed novel hybrid modified recurrent Legendre NN control system, three cases are provided in the experimentation. These cases are the 125.6 rad/s (1200 rpm ) case, the 251.2 rad/s (2400 rpm ) case and 376.8 rad/s (3600 rpm ) case. With this application it is necessary to decrease the bandwidth of the speed due to high stiffness between the motor and the nonlinear load. This is especially true with an electric scooter mounted with V-bet CVT and wheel systems. Therefore, the adopted three bandwidths of the speed control loop with the three reference models in the 125.6 rad/s (1200 rpm ) case, the 251.2 rad/s (2400 rpm ) case, and the 376.8 rad/s (3600 rpm ) case are about 6.25Hz, 2.8Hz, and 2.0Hz, respectively.
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Photo of the experimental setup.
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Flowchart of the executing program by using the DSP control system.
Since an electric scooter is a nonlinear and time-varying system, all of the gains of the well-known PI controller are kps =13.5, kis = kps/Tis =1.8 through some heuristic knowledge [38 - 40] on the tuning of the PI controller in the 125.6 rad/s (1200 rpm ) case for speed tracking in order to achieve good transient and steady-state control performances. The experimental results of the well-known PI controller for the V-belt CVT driven electric scooter using the PMSM servo drive system in the125.6 rad/s (1200 rpm )case and in the 251.2 rad/s (2400 rpm ) case are shown in Fig. 6 and Fig. 7 , respectively.
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Experimental result of the well-known PI controller controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e amplification; (c) current command ia* and measured current ia in phase a; (d) electromagnetic torque Te.
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Experimental result of the well-known PI controller controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e amplification ; (c) current command ia* and measured current ia in phase a; (d) electromagnetic torque Te.
The responses of the rotor speed command ωr * , the desired rotor speed command ω * and the measured rotor speed ωr are shown in Fig. 6 (a) and Fig. 7 (a). The response of the tracking error e amplification is shown in Fig. 6 (b) and Fig. 7 (b). The tracking response of the current command ia * and the measured current ia in phase a is shown in Fig. 6 (c) and Fig. 7 (c). The dynamic response of the electromagnetic torque Te is shown in Fig. 6 (d) and Fig. 7 (d).
In addition, the experimental result of the well-known PI controller for the V-belt CVT driven electric scooter using the PMSM servo drive system in the 376.8 rad/s (3600 rpm )case under high speed perturbation is shown in Fig. 8 , where responses of the rotor speed command ωr * , the desired otor speed command ω * and the measured rotor speed ωr are shown in Fig. 8 (a). The response of the tracking error e is shown in Fig. 8 (b). The response of the tracking error e amplification is shown in Fig. 8 (c). The dynamic response of the electromagnetic torque Te is shown in Fig. 8 (d). Since the low speed operation is the same as the nominal case due to smaller lumped nonlinear external disturbances, the response of the speed shown in Fig. 6 (a) has better tracking performance. In addition, the degenerate tracking response of the speed shown in Fig. 7 (a) and Fig. 8 (a) is obvious due to bigger lumped nonlinear disturbances (e.g. rolling resistance and parameter variations)in the 251.2 rad/s (2400 rpm ) case and in the 376.8 rad/s (3600 rpm ) case. Moreover, the electromagnetic torque brings in a greater torque ripple, as shown in Fig. 6 (d), Fig. 7 (d) and Fig. 8 (d), due to a V-belt CVT system with highly nonlinear external disturbances, such as V-belt shaking friction, and the action friction between the primary pulley and the second pulley. Moreover, the sluggish tracking response of the current shown Fig. 7 (c) is obtained for the well-known PI controlled V-belt CVT driven electric scooter using the PMSM servo drive system. The linear controller has weak robustness under larger lumped nonlinear external disturbances in high speed operations due to the lack of appropriately gains tuning or degenerate nonlinear effect.
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Experimental results of the well-known PI controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e; (c)response of tracking error e amplification ; (d) electromagnetic torque Te.
For comparison, the control performance with the proposed novel hybrid modified recurrent Legendre NN control system, for the three-layer recurrent NN control system is adopted in this paper. The three-layer recurrent NN has the recurrent loop between the output layer and the input layer and it has the sigmoid activation function in the hidden layer. It has two, three and one neurons in the input layer, the hidden layer and the output layer, respectively. All of the control gains of the three-layer recurrent NN control system are chosen to achieve the best transient control performance while considering the requirement of stability. For simplicity, the recurrent weight between the output layer and the input layer in the three-layer recurrent NN is to set 1. Moreover, the connective weight between the input layer and the hidden layer, and the connective weight between the hidden layer and the output layer in the three-layer recurrent NN are initialized with random numbers. In addition, all of the learning rates are set as fixed constants. Furthermore, the normalized inputs and references have zero and unity, respectively. Also the network outputs should be converted back to the original units of reference. The parameter adjustment process remains continually active for the duration of the experiment. The experimental results of the three-layer recurrent NN control system for the V-belt CVT driven electric scooter using the PMSM servo drive system in the 125.6 rad/s (1200 rpm ) case and the 251.2 rad/s (2400 rpm ) case are shown in Fig. 9 and Fig. 10 . The response of the rotor speed command ωr * , the desired rotor speed ω * and the measured rotor speed ωr are shown in Figs. 9 (a) and Fig. 10 (a). The response of the tracking error e amplification is shown in Fig. 9 (b) and Fig. 10 (b). The tracking response of the current command ia * and the measured current ia in phase a are shown in Fig. 9 (c) and Fig. 10 (c). The dynamic response of the electromagnetic torque Te is shown in Fig. 9 (d) and Fig. 10 (d). In addition, the experimental results of the three-layer recurrent NN control system for the V-belt CVT driven electric scooter using the PMSM servo drive system in the 376.8 rad/s (3600 rpm ) case under a high speed perturbation is shown in Fig. 11 , where response of the rotor speed command ωr * , the desired rotor speed command ω * and the measured rotor speed ωr are shown in Fig. 11 (a). The response of the tracking error e is shown in Fig. 11 (b). The response of the tracking error e amplification is shown in Fig. 11 (c). The dynamic response of the electromagnetic torque Te is shown in Fig. 11 (d).
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Experimental results of the three-layer recurrent NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e amplification ; (c) current command iα* and measured current iα in phase α; (d) electromagnetic torque Te.
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Experimental results of the three-layer recurrent NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e amplification ; (c) current command iα* and measured current iα in phase α; (d) electromagnetic torque Te.
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Experimental results of the three-layer recurrent NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr; (b) response of tracking error e; (c)response of tracking error e amplification ; (d) electromagnetic torque Te.
However, due to the on-line adaptive mechanism of the three-layer recurrent NN and the recouped controller, accurate tracking responses of the speed and current can be obtained. These experimental results show that the three-layer recurrent NN control system has better control performance than the well-known PI controller for the V-belt CVT driven electric scooter using the PMSM servo drive system in the 251.2 rad/s (2400 rpm ) case and in the 376.8 rad/s (3600 rpm ) case under the occurrence of large lumped nonlinear disturbances. Meanwhile, the electromagnetic torque Te exhibits a lower torque ripple, as shown in Fig. 9 (d), Fig. 10 (d) and Fig. 11 (d).
The control gains of the proposed novel hybrid modified recurrent Legendre NN control system are η = 0.1, λ = 0.5 according to the estimated uncertainty bound for the recouped controller compensating perturbations in order to achieve the best transient control performance. All of the control gains of the novel hybrid modified recurrent Legendre NN control system are chosen to achieve the best transient control performance, while considering the requirement of stability. Usually, some heuristics can be used to roughly initialize the parameters of the novel hybrid modified recurrent Legendre NN control system for practical applications. The effects of the inaccurate selection of the initialized parameters can be retrieved by the online parameter training methodology. For simplicity, all of the recurrent weights between the output layer and the input layer in the modified recurrent Legendre NN are to set to 1. Moreover, the connective weights between the hidden layer and the output layer in the modified recurrent Legendre NN are initialized with a random number. Furthermore, the normalized inputs and references are zero and unity, respectively. In addition, the network outputs should be converted back to the original units of the references. The parameter adjustment process remains continually active for the duration of the experiment. The structure of the modified recurrent Legendre NN controller has 2 nodes, 3 nodes and 1 node in the input layer, the hidden layer and the output layer, respectively. The experimental results of the novel hybrid modified recurrent Legendre NN control system for the V-belt CVT driven electric scooter using the PMSM servo drive system in the125.6 rad/s (1200 rpm ) case and in the 251.2 rad/s (2400 rpm ) case are shown in Fig. 12 and Fig. 13 . The response of the rotor speed command ωr * , the desired rotor speed command ω * and the measured rotor speed ωr are shown in Fig. 12 (a) and Fig. 13 (a). The response of the tracking error e amplification is shown in Fig. 12 (b) and Fig. 13 (b). The tracking response of the current command ia * and the measured current ia in phase a are shown in Fig. 12 (c) and Fig. 13 (c). The dynamic response of the electromagnetic torque Te is shown in Fig. 12 (d) and Fig. 13 (d). In addition, the experimental results of novel hybrid modified recurrent Legendre NN control system for the V-belt CVT driven electric scooter using the PMSM servo drive system in the 376.8 rad/s (3600 rpm ) case under high speed perturbations is shown in Fig. 14 . The response of the rotor speed command ωr * , the desired rotor speed command ω * and the measured rotor speed ωr are shown in Fig. 14 (a). The response of the tracking error e is shown in Fig. 14 (b). The response of the tracking error e amplification is shown in Fig. 14 (c).The dynamic response of the electromagnetic torque Te is shown in Fig. 14 (d).
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Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr ; (b) response of tracking error e amplification ; (c) current command iα* and measured current iα in phase α; (d) electromagnetic torque Te.
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Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr ; (b) response of tracking error e amplification ; (c) current command ia* and measured current iα in phase α; (d) electromagnetic torque Te.
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Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the V-belt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ωr*, desired rotor speed command ω* and measured rotor speed ωr ; (b) response of tracking error e ; (c)response of tracking error e amplification ; (d) electromagnetic torque Te.
From the experimental results, accurate tracking performance is obtained for the novel hybrid modified recurrent Legendre NN controller for the V-belt CVT driven electric scooter using the PMSM servo drive owing to the on-line adaptive mechanism of the modified recurrent Legendre NN and action of the recouped controller. Therefore, these results show that the novel hybrid modified recurrent Legendre NN control system has better control performance than the well-known PI controller for speed perturbations for the V-belt CVT driven electric scooter using the PMSM servo drive system.
Additionally, the small chattering phenomena of the currents in phase α, shown in Fig. 9 (c), Fig. 10 (c) and Fig. 12 (c) and Fig. 13 (c), are induced by on-line adjustments of the three-layer recurrent NN and the novel hybrid modified recurrent Legendre NN, respectively, to cope with the highly nonlinear dynamics of the system.
Furthermore, the speed response, shown in Fig. 13 (a) and Fig. 14 (a) from using the novel hybrid modified recurrent Legendre NN control system due to its lower computational complexity, has a faster convergence than the speed response shown in Fig. 10 (a) and Fig. 11 (a) by using the three-layer recurrent NN control system in the 251.2 rad/s (2400 rpm ) case and in the 376.8 rad/s (3600 rpm ) case. Moreover, the dynamic response of electromagnetic torque Te , shown in Fig. 12 (d), Fig. 13 (d) and Fig. 14 (d),from using the novel hybrid modified recurrent Legendre NN control system has a lower torque ripple than that obtained using the well-known PI control and the three-layer recurrent NN control system shown in Fig. 5 (d), Fig. 6 (d), Fig. 7 (d) and in Fig. 8 (d), Fig. 9 (d), Fig. 10 (d), respectively.
Moreover, the experimental results of the learning rate variation in the modified recurrent Legendre NN are shown in Fig. 15 , where the response of the learning rate μα of the connection weights in the 125.6 rad/s (1200rpm) case is shown in Fig. 15 (a). The response of the learning rate μb of the recurrent weights in the 125.6 rad/s (1200rpm) case is shown in Fig. 15 (b). The response of the learning rate μα of the connection weights in the 251.2 rad/s (2400rpm) case is shown in Fig. 15 (c). The response of the learning rate μb of the recurrent weights in the 251.2 rad/s (2400rpm) case is shown in Fig. 15 (d). The learning rate μα variation of the connection weights and the learning rate μb variation of the recurrent weights in the 251.2 rad/s (1200rpm)case and in the 251.2 rad/s (2400rpm)case have fast convergences to be bounded from the experimental results.
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Experimental result of the learning rate variations in the modified recurrent Legendre NN: (a) response of the learning rate μα the connection weights at 125.6 rad/s (1200 rpm) case; (b) response of the learning rate μb of the recurrent weights at 125.6 rad/s (1200 rpm) case; (c) response of the learning rate μα of the connection weights at 251.2 rad/s (2400 rpm) case; (d) response of the learning rate μb of the recurrent weights at 251.2 rad/s (2400 rpm).
Finally, the measured rotor speed response under step load torque disturbances is given. The condition under a Tl = 2 Nm ( Ta ) + Tun load torque disturbance with adding a load and shedding a load is tested while using the well-known PI controller, the three-layer recurrent NN control system and the novel hybrid modified recurrent Legendre NN control system. The experimental result under a Tl = 2 Nm ( Ta ) + Tun load torque disturbance with adding a load and shedding a load at the command rotor speed 251.2 rad/s (2400 rpm ) is shown in Figs. 16 - 18 . The experimental result of the measured rotor speed response and measured current in phase a while using the well-known PI controller under a Tl = 2 Nm ( Ta ) + Tun load disturbance with adding a load and shedding a load at 251.2 rad/s (2400 rpm ) is shown in Fig. 16 (a), and the amplified speed error ec response is shown in Fig. 16 (b). The experimental results of the measured rotor speed response and the measured current in phase a while using the three-layer recurrent NN control system under a Tl = 2 Nm ( Ta ) + Tun load disturbance with adding a load and shedding a load at 251.2 rad/s (2400 rpm ) is shown in Fig. 17 (a), and the amplified speed error ec response is shown in Fig. 17 (b). The experimental result of the measured rotor speed response and the measured current in phase a while using the novel hybrid modified recurrent Legendre NN control system under a Tl = 2 Nm ( Ta ) + Tun load disturbance with adding a load and shedding a load at 251.2 rad/s (2400 rpm ) is shown in Fig. 18 (a), and the amplified speed error ec response is shown in Fig. 18 (b). From the experimental results, the degenerated responses due to a Tl = 2 Nm ( Ta ) + Tun load disturbance are greatly improved when using the novel hybrid modified recurrent Legendre NN control system. From the experimental results, it can be seen that the transient response of the novel hybrid modified recurrent Legendre NN control system is better than that of the well-known PI controller and the three-layer recurrent NN control system in terms of load regulation.
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Experimental results under Tl = 2Nm (Tα) +Tun load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the well-known PI controller: (a) command rotor angular speed ωr*, measured rotor angular speed ωr and measured current ia in phase α; (b) amplified speed error e .
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Experimental results under Tl = 2Nm (Tα) +Tun load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the three-layer recurrent NN control system: (a) command rotor angular speed ωr*, measured rotor angular speed ωr and measured current ia in phase α; (b) amplified speed error e .
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Experimental results under Tl = 2Nm (Tα) +Tun load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the novel hybrid modified recurrent Legendre NN control system: (a) command rotor angular speed ωr*, measured rotor angular speed ωr and measured current ia in phase α; (b) amplified speed error e .
Moreover, the novel hybrid modified recurrent Legendre NN control system has a faster convergence and better load regulation than the three-layer recurrent NN control system under a Tl = 2 Nm ( Tα ) + Tun load disturbance.
In addition, control performance comparisons of the PI controller, the three-layer recurrent NN control system and then novel hybrid modified recurrent Legendre NN control system are summarized in Table I . They are shown for the experimental results under the four test cases. In Table I , the novel hybrid modified recurrent Legendre NN control system results in a smaller tracking speed error with respect to the PI controller and the three-layer recurrent NN control system. According to the tabulated measurements, the proposed novel hybrid modified recurrent Legendre NN control system yields superior control performance when compared with the PI controller and the three-layer recurrent NN control system. Furthermore, characteristic performance comparisons of the PI controller, the three-layer recurrent NN control system and the novel hybrid modified recurrent Legendre NN control system are summarized in Table II for the experimental results. In Table II , it can be seen that the various performances of the novel hybrid modified recurrent Legendre NN control system are superior to those of the PI controller and the three-layer recurrent NN control system.
PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
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PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
CHARACTERISTIC PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
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CHARACTERISTIC PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
Ⅴ. CONCLUSIONS
A novel hybrid modified recurrent Legendre NN control system has been successfully developed in this paper to control V-belt CVT driven electric scooters using a PMSM servo drive system with robust control characteristics.The adopted modified recurrent Legendre NN has a self-feedback connection in the hidden layer and a recurrent connection between the output layer and the input layer. It is more powerful than the three-layer recurrent NN for dealing with time-varying and nonlinear dynamical systems. Moreover, in order to improve the ability in terms of identifying high order systems and the convergence speed of the parameters, a modified recurrent Legendre NN with two optimal learning rates is also proposed in this paper. It has been proved through experimental results that the modified recurrent Legendre NN has more advantages than the three-layer recurrent NN with the sigmoid activation function, including better performance, higher accuracy, increased dynamic robustness and faster transient performance.
The main contributions of this paper are as follows as: (1) a dynamic models of a V-belt CVT driven electric scooter with unknown nonlinear and time-varying characteristics using a PMSM servo drive system was successful derived; (2) a novel hybrid modified recurrent Legendre NN control system for V-belt CVT driven electric scooters using a PMSM servo drive system under the occurrence of the variations in the rotor inertia and load torque disturbance was successful applied to enhance robustness; (3) the on-line parameters tuning methodology of the modified recurrent Legendre NN and the estimation law of the remunerated controller using the Lyapunov stability theorem were successful derived; (4) two optimal learning rates for the connective weight and recurrent weight of the modified recurrent Legendre NN according to the discrete-type Lyapunov function were successful derived in order to speedup convergence; (5) a novel hybrid modified recurrent Legendre NN control system, which has a fast convergence ability and fast capture in terms of the system’s nonlinear and time-varying behaviors, is successful developed; (6) the proposed novel hybrid modified recurrent Legendre NN control system has a lower torque ripple than the well-known PI controller and the three-layer recurrent NN control systems. Therefore, the control performance of the proposed hybrid modified recurrent Legendre NN control system is more suitable than the well-known PI controller and the three-layer recurrent NN control systems for V-belt CVT driven electric scooters using a PMSM servo drive system.
BIO
Chih-Hong Lin was born in Taichung, Taiwan, R.O.C. He received his B.S. and M.S. degrees in Electrical Engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1989 and 1991, respectively. He received his Ph.D. degree in Electrical Engineering from Chung Yuan Christian University, Chung Li, Taiwan, R.O.C.,in 2001.He is currently an Associate Professor in the Department of Electrical Engineering, National United University, Miao Li, Taiwan, R.O.C. His current research interests include power electronics, motor servo drives and intelligent control.
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