Mathematical Problems in Engineering

Volume 2015, Article ID 106707, 17 pages

http://dx.doi.org/10.1155/2015/106707

## Hybrid Recurrent Laguerre-Orthogonal-Polynomial NN Control System Applied in V-Belt Continuously Variable Transmission System Using Particle Swarm Optimization

Department of Electrical Engineering, National United University, No. 1, Lienda, Kung-Jing Li, Miaoli City, Miaoli County 36003, Taiwan

Received 18 May 2014; Accepted 1 September 2014

Academic Editor: Kang Li

Copyright © 2015 Chih-Hong Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Because the V-belt continuously variable transmission (CVT) system driven by permanent magnet synchronous motor (PMSM) has much unknown nonlinear and time-varying characteristics, the better control performance design for the linear control design is a time consuming procedure. In order to overcome difficulties for design of the linear controllers, the hybrid recurrent Laguerre-orthogonal-polynomial neural network (NN) control system which has online learning ability to respond to the system’s nonlinear and time-varying behaviors is proposed to control PMSM servo-driven V-belt CVT system under the occurrence of the lumped nonlinear load disturbances. The hybrid recurrent Laguerre-orthogonal-polynomial NN control system consists of an inspector control, a recurrent Laguerre-orthogonal-polynomial NN control with adaptive law, and a recouped control with estimated law. Moreover, the adaptive law of online parameters in the recurrent Laguerre-orthogonal-polynomial NN is derived using the Lyapunov stability theorem. Furthermore, the optimal learning rate of the parameters by means of modified particle swarm optimization (PSO) is proposed to achieve fast convergence. Finally, to show the effectiveness of the proposed control scheme, comparative studies are demonstrated by experimental results.

#### 1. Introduction

A V-belt continuously variable transmission (CVT) [1–8] is typically composed of two hydraulically, or spring, actuated variable radii pulleys and a chain, or metal pushing, belt. To launch a vehicle from rest, the input pulley radius will be smaller than the output pulley radius, resulting in a speed reduction and torque multiplication transmitted to the drive shaft. For increased output shaft speed, the pulley radii are inversely manipulated simultaneously (i.e., input pulley radius increases as the output pulley radius decreases) to maintain constant belt length. A CVT may operate at a specific speed while changing the pulleys’ radii to achieve torque multiplication, acceleration, and speed as per the vehicle’s velocity, load requirements, engine power, and gear ratios. This operating profile provides the research motivation for CVT dynamics and nonlinear control algorithms. CVT-based vehicles have been traditionally regulated using a standard proportional integral derivative- (PID-) based controller with measurements of the gear ratio [3]. It has also been demonstrated that this control strategy provides satisfactory performance using gain-scheduling with a large set of points. In addition, numerous fuzzy logic controllers [4] have also been proposed. However, V-belt continuously variable transmission (CVT) system driven by alternating current (AC) motor is yet not shown in any commercial reports so that it provides the research motivation in this study.

The AC motor has several types such as permanent magnet synchronous motor (PMSM), switched reluctance motor (SRM), and induction motor (IM). In order to select the appropriate AC motor for driven V-belt CVT system, high efficiency is one of the most important factories to be selected. The PMSM provides higher efficiency, higher power density, and lower power loss for their size compared to SRM and IM. In addition, field-oriented control is one the most popular control techniques for the PMSM servo-driven system. As a result, torque ripple of the PMSM is lower than the SRM and IM. On the other hand, the PMSM controlled by field-oriented control, which can be achieved fast by four-quadrant operation, is much less sensitive to the parameters variation of the motor [9–11]. Therefore, the PMSM has been widely used in many industrial applications such as robotics, electric power steering, and other mechatronics [9–11].

Artificial neural networks (ANNs) have emerged as a powerful learning technique to perform complex tasks such as highly nonlinear approximations and the control of dynamical systems [12–16]. Some of the prime advantages of using NN are their ability to learn based on optimization of an appropriate error function and their excellent performance for approximation of nonlinear functions. There are different paradigms of NNs proposed by different researchers for the tasks of system identifications and controls [13–16]. One of the major drawbacks of the NN is that it is computationally intensive and needs large number of iterations for its training. In order to reduce the computational complexity, a functional-link NN, which has shown that it is capable of producing similar performance as that of NN but with much less computational cost, was reported in [17–19]. These functional-link NNs [17–19] with faster convergence and lesser computational complexity were executed in the identification and controls of nonlinear dynamic system with satisfactory results. Recently, the Laguerre-functional-expansions combined with NN, which was applied in highly nonlinear approximations and the control of dynamical systems, have been proposed [20–24]. Aadaleesan et al. [20] proposed the Laguerre filter combined with the wavelet network in order to approximate the memoryless nonlinearity. Approximation the linear and nonlinear parts of a Wiener structure by means of the Laguerre filter and the general feed-forward NN was reported in [21]. The Laguerre-functional-expansions feed-forward NN, which employed Laguerre-orthogonal-polynomials in the activation functions of the hidden neurons in order to identify models of the chaotic time series, was proposed by Zou and Xiao [22]. Patra et al. [23] proposed a computationally efficient Laguerre NN, which is based on Laguerre-functional-expansions to autocompensate for the associated nonlinearity and environmental dependence for intelligent sensors, and provide linearized sensor readout even when the motes are operated in harsh environments. Patra et al. [24] present an intelligent technique by means of novel computationally efficient Laguerre NN to compensate for the inherent sensor nonlinearity and the environmental influences. Since the Laguerre NN is a single-layer NN, its computational complexity is found to be much lower than a multilayer perception (MLP). However, these Laguerre-functional-expansions feed-forward NNs without a feedback loops can be used for static function approximation, but they cannot adequately approximate dynamic behaviors found in PMSM servo-driven V-belt CVT system with nonlinear and time-varying characteristics.

The recurrent NN has received increasing attention due to its structural advantage in the modelling of the nonlinear system and dynamic control of the system [25–29]. These networks are capable of effective identification and control of complex process dynamics, but with the expense of large computational complexity. Hence, if each neuron in the recurrent NN is considered as a state in the nonlinear dynamic systems, the self-connection feedback type is able to approximate the dynamic systems efficiently [25–29]. In order to improve the ability of identifying high order systems and reduce computational complexity, the recurrent Laguerre-orthogonal-polynomial NN, which has more advantages than the Laguerre-orthogonal-polynomial NN including better performance, higher accuracy, and dynamic robustness, has been proposed to control the PMSM servo-driven V-belt CVT system with nonlinear and time-varying characteristics in this paper.

Particular swarm optimization (PSO) is a population-based, self-adaptive search optimization technique first introduced by Kennedy and Eberhart [30]. Similar to genetic algorithms [31], an evolutionary algorithm approach, the PSO is an evolutionary optimization tool of swarm intelligence field based on a swarm (population), where each member is seen as a particle, and each particle is a potential solution to the problem under analysis. The motivation for the development of this method was based on the simulation of simplified animal social behaviors such as fish schooling, bird flocking, and so forth. However, unlike in other evolutionary optimization methods, in PSO there is no direct recombination of genetic material between individuals during the search. The PSO algorithm works on the social behavior of particles in the swarm. Therefore, it finds the global best solution by simply adjusting the trajectory of each individual toward its own best location and toward the best particle of the entire swarm at each time step (generation) [30, 32]. Clerc and Kennedy [32] introduced the concept of inertia weight to the original version of PSO, in order to balance the local and global search during the optimization process. Thus, PSO has been widely applied in mathematical modeling, dynamic programming, and system control [33–36] due to simple structure, simple parameter setting, and fast convergence speed. How to improve the convergence speed and how to guarantee the convergence of PSO are the main problems of PSO improvement [37] and are gradually turning into a hot topic in this field. In order to weigh the relationship between local search and global search, Clerc and Kennedy [32] and Eberhart and Shi [38, 39] proposed improved particle swarm optimization with inertial weight to control the exploitation and exploration. Meanwhile, some researchers [40–42] have proposed the topical improved particle swarm algorithm with inertia factor, which is called topical particle swarm optimization. However, the PSO existed in premature convergence problem and the modified PSO is proposed to prevent premature convergence and to acquire optimal learning rate with better convergence in this paper.

In this study the hybrid recurrent Laguerre-orthogonal-polynomial NN control system is developed to control the V-belt CVT system with many nonlinear dynamics [1–8, 43–46], which is driven by PMSM. The hybrid recurrent Laguerre-orthogonal-polynomial NN control system has fast learning property and good generalization capability. The control method, which is not dependent upon the predetermined characteristics of the motor, can adapt to any change in the motor characteristics. The hybrid recurrent Laguerre-orthogonal-polynomial NN control system, which is composed of the inspector control, the recurrent Laguerre-orthogonal-polynomial NN control with adaptive law, and the recouped control, is applied to the V-belt CVT system driven by PMSM. The adaptive law of the online parameter in the recurrent Laguerre-orthogonal-polynomial NN can be derived according to the Lyapunov stability theorem and the gradient descent method. The recurrent Laguerre-orthogonal-polynomial NN has the online learning ability to respond to the system’s nonlinear and time-varying behaviors under the occurrence of the lumped nonlinear external disturbances with parameters variation. Furthermore, two optimal learning rates of the parameters by means of modified PSO are proposed to achieve fast convergence. Finally, the control performances of the proposed hybrid recurrent Laguerre-orthogonal-polynomial NN control system are verified by experimental results.

The paper is structured as follows: Section 2 provides the configuration of the V-belt CVT system driven by PMSM. Section 3 develops the proposed novel hybrid recurrent Laguerre-orthogonal-polynomial NN control system for controlling the V-belt CVT system driven by PMSM. Section 4 presents the experimental results for comparisons between the proposed control method and PI control method at three cases. Section 5 provides the conclusions.

#### 2. Configuration of System

Since the electric scooter system has much unknown nonlinear uncertainties and parameter variations, such as load torque, rolling resistance, wind resistance, and braking force, the V-belt CVT and clutch in the scooter model 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 transition between modes is 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 the electric scooter.

##### 2.1. Structure of the V-Belt CVT System Driven by PMSM

The development of the V-belt CVT began with rubber V-belts [5]. Despite the fact that rubber V-belt CVTs are not well suited for automotive applications because of their limited torque capacity, there are some interesting concepts on the market. The V-belt CVT consists of a segmented rubber V-belt and two shafts with conical pulleys. The V-belt is clamped between two pairs of conical sheaves. In the V-belt CVT, the transmission ratio is determined by simultaneous adjustment of the running radii of the belt on the pulleys. On each shaft, there is one fixed and one axially moveable sheave. Axial movement of the moveable sheave adjusts the gap between the sheaves and thereby the belt running radius. The input shaft of the V-belt CVT is called the primary shaft which mounted the PMSM, and the output shaft is the secondary shaft which mounted the wheel. The structure of the V-belt CVT is shown in Figure 1. The appearance of the primary pulley side and the secondary pulley side in the V-belt CVT is shown in Figure 1(a), and the cross-section view of the secondary shaft in the V-belt CVT is shown in Figure 1(b). The wheel of electric scooter is connected to the output shaft of the final reduction using a torsion spring, which models the combined stiffness of both the drive shafts. The electric scooter inertia is connected to the wheel using linear damper, which models the tire force. The rolling resistance on the electric scooter is modeled as a load torque. In order to reduce system complexity, the torque dynamic equations in the primary drive shaft and the secondary drive shaft of the V-belt CVT shown in Figure 2 can be simplified as [1–8] in which [1–8] is the lumped nonlinear external disturbances of the secondary drive side on the wheel; is the drive torque of the primary pulley shaft; is the drive torque of the secondary pulley shaft; is the conversion ratio with respect to secondary pulley shaft transferred to primary pulley shaft of V-belt arc length; is the rolling resistance; is the wind resistance; is a braking force; is the total wind velocity; represents the total frictional coefficient of ground surface; and represent the viscous frictional coefficients of the PMSM and the wheel, respectively; and are the inertias of the PMSM and the wheel, respectively; and are the speeds of the PMSM and the wheel, respectively. Then using speed ratio and sliding ratio [1–8], the torque equation can be transformed from the secondary pulley side to the primary pulley side. Therefore, the resultant dynamic equation of the PMSM driven V-belt CVT system from (1) can be simplified as [1–8, 41–45]in which [1–8] is the resultant lumped nonlinear external disturbances with parameter variations; is the fixed load torque; is the resultant parameter variation; is the resultant rolling resistance; is the resultant wind resistance; is the resultant braking force; is the resultant unknown nonlinear load torque; is the resultant viscous frictional coefficient; is the resultant moment of inertia.