adaptive neural network–based synchronized control of dual-axis

Special Issue Article

Advances in Mechanical Engineering2016, Vol. 8(7) 1–17� The Author(s) 2016DOI: 10.1177/1687814016654603aime.sagepub.com

Adaptive neural network–basedsynchronized control of dual-axis linearactuators

Wei-Lung Mao, Suprapto and Chung-Wen Hung

AbstractSynchronized motion control with high accuracy becomes very essential part in industry. Due to some possible effectsuch as unknown disturbance or unmatched system model, it is difficult to obtain the precision of synchronous controlusing the conventional proportional–integral control method with parallel architecture. The adaptive compensator mustbe employed to eliminate tracking errors. The objective of this research is to propose the modified cross-coupling archi-tecture using single-neuron proportional–integral controller and a synchronous compensator for dual-axis linear actua-tor. The single-neuron proportional–integral control strategy with delta learning algorithm can adjust the weightingcoefficients of controllers to provide the robustness for each single-axis DC linear actuator system. A back-propagationneural network compensator is designed to adaptively reduce position and velocity errors between the two-axis servosystems. Both simulation and experimental results are developed to demonstrate that the synchronous position trackingperformances in terms of root mean square error and sum of absolute error can be substantially improved, and therobustness to linear actuator uncertainties can be obtained as well. The proposed coupling strategy which uses themicrochip platform and pulse–width modulation control technique is realized and implemented, and the synchronizationperformances to external disturbance load are illustrated by several experimental results.

KeywordsSynchronization control, adaptive back-propagation neural network controller, linear actuators, single-neuronproportional–integral controller, synchronous compensator

Date received: 2 October 2015; accepted: 20 May 2016

Academic Editor: Stephen D Prior

Introduction

Synchronized motion control of actuator has beenapplied in various applications such as precisionmachines, health care, and home appliances.1 Since thecontrol systems must have some aspects such as robust-ness, safety, and low-cost in maintenance, they can beobtained by conventional or modern control methodthat appropriate with the linear actuators. Linearactuators are used in machine tools, industrial machin-ery, and in many other places where linear motion isrequired. It is an important problem that the motion ofmultiple axes linear actuators must be controlled in asynchronous manner.2,3

Architectures of the synchronous motion controlmethod1 can be classified into four categories: (1)master–slave motion control, (2) cross-couplingtechnique (or synchronous master motion control), (3)

Graduate School of Engineering Science and Technology and Department

of Electrical Engineering, National Yunlin University of Science and

Technology, Yunlin, Taiwan, R.O.C.

Corresponding author:

Chung-Wen Hung, Graduate School of Engineering Science and

Technology and Department of Electrical Engineering, National Yunlin

University of Science and Technology, 123 University Road, Section 3,

Douliou, Yunlin 64002, Taiwan, R.O.C.

Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

bi-axial cross-coupled technique, and (4) relativedynamic stiffness motion control. In multi-axes syn-chronization motion control, the system performancedepends more on the coordination of multiple motionactuators than on individual motion actuator.4 A linearquadrature controller and an adaptive controller wereused to synthesize the synchronization compensator.However, each axis of multiple motion actuator controlhas a separate closed-loop control, so that the controlloop of one axis receives no information regarding theother.5 Therefore, the individual actuator system needsa synchronous compensator to coordinate and elimi-nate different position errors among multiple actuators.The synchronization motion control can use couplingas synchronous compensator, which can be tackled bymechanical or control algorithm to compensate theeffect of backlash, friction, and mechanical defects. Theparallel synchronous control scheme6,7 is proposed toconstruct the coupled model for a gantry machine tool.The proportional–integral (PI) synchronous compensa-tor is designed to verify the positioning accuracy.However, conventional control is inappropriate totackle many problems such as steady-state error, speedchanges, and load disturbances. Several advance intelli-gent control methods,7–9 such as fuzzy method, neuralnetwork (NN), and particle swarm optimization (PSO),are proposed. One of the proposed methods is tuningparameters of PI controller using NN. The objectivesusing NN are stability robustness, good tracking per-formance, and robustness against plant modelinguncertainty and environmental uncertainty.8–13 Newmethods for proportional–integral–derivative (PID)controller using single neuron11 is proposed and testedin speed control system with DC servo motor.Furthermore, the single-neuron proportional–integral(SNPI) controller has properties such as self-learning,self-adaptive, and strong robustness. Moreover, the sin-gle neuron is a simple structure and a powerful tool inPI parameter optimization.

Due to the inherent behavior of each single-axisservo actuator, it is difficult to model the complicatednonlinear property of the synchronous error betweendual-axis servo systems. For each DC feed actuator,the identification process is the first step to have anapproximated system modeling. The adaptive SNPIcontroller is developed in single servo loop to improvethe tracking performance and position accuracy. Theproposed modified cross-coupling technique, whichuses an additional synchronous controller to reflect thesystem mismatch and load variations, can eliminate therelative servo position error signal. A back-propagationneural network (BPNN) compensation controller isproposed, and the online learning rule is utilized toreduce the synchronous error between the two actua-tors. The BPNN regularizes input signals from synchro-nous position error and velocity error and generates

output signal via the NNs, and the output signal ascompensated force is fed to controller of each servo sys-tems. The online learning rule minimizes the definederror using delta algorithm to adjust weighting of NN.The simulation and experiment results show that theproposed architecture can outperform parallel architec-ture in terms of better tracking performances.

This article is organized as follows: dynamic modelof DC servo linear actuator is constructed in section‘‘Dynamic model of DC servo linear actuator.’’ Designof synchronous architecture using NN is described insection ‘‘NN-based synchronous compensator.’’ SNPIcontroller is stated in section ‘‘Adaptive neuron control-ler.’’ Section ‘‘Simulation and experimental results’’ pro-vides the simulation and experimental results. Finally, abrief conclusion is given in section ‘‘Conclusion.’’

Dynamic model of DC servo linearactuator

The electric linear actuators can provide an easy way tomount and operate into an automated process. Theyoffer advantages over mechanical and hydraulic systemsince their functions are self-contained, rugged, andmaking them ideal anywhere you want to lift, push,pull, and position a load. The linear actuator is com-posed of a DC motor, two gears, and ball screw, asshown in Figure 1. A traveler on the ball screw is forcedeither toward or away from the motor, essentially con-verting the rotating motion to a linear motion. It isdesired to have an application need for synchronousmotion for coordinated moves and precise control ofposition, velocity, and acceleration. The synchronousoperations of multiple actuators are widely used inpatient lifts or beds, handicap adapted vehicles, con-veyor belts, and door controlling.

Mathematical model of linear actuator is derived bydeveloping equation between the motor and

Figure 1. The mechanism of DC motor and linear actuatorsystem.

2 Advances in Mechanical Engineering

components of the feed drive system. Severalresearches6,14 on the dynamic model of ball screw stagewith DC motor have been conducted. The system blockdiagram for single-axis DC servo linear actuator isdepicted in Figure 2. For the input signal is the refer-ence voltage u(t), the mathematical model of ball screwdrive system is expressed as follows

KtKe_u1 = �Rmte +Ktu ð1:1Þ

J€u1 +B _u1 +Ku1 = te +KRx ð1:2Þ

M€x+D _x+Kx+FL =KRN1

N2u1 ð1:3Þ

withR= L2p, u2 =

N1

N1u1

where te is the motor output torque, Kt is the is torqueconstant, Ke is the DC motor back-electromotive force(EMF) constant, Rm is the motor armature resistance,J is the moment inertia of DC motor rotor, B is the vis-cous damping coefficient of a motor shaft, u1 is therotational angle of gear 1, u2 is the rotational angle ofgear 2, tl is the driving torque of the motor, x is the dis-placement of linear actuator, K is the equivalent stiff-ness of the ball screw, R is the conversion ratio oflinear-to-rotational motion, L is the ball screw lead, Mis the mass, D is the viscous damper coefficient, and FL

is the load disturbance. In this experiment, DC servomotor is used to drive a ball nut linear actuator. Themotor armature inductance is small and reasonable toneglect in amplifier dynamics.

In general, DC motor has a high speed and low tor-que, but a linear actuator needs high torque with lowspeed to drive load. To obtain the high torque, preci-sion velocity, and position, the contact mechanismusing gear is employed between DC servos and linearactuator. Transfer function of the feed drive systembetween the voltage command and the actuator velo-city v(t) is represented as

T sð Þ= V sð ÞU sð Þ =

n0s3 +d2s2 +d1s+d0

ð2Þ

withv (t)= _x(t),n0=KtRKN1

RmJMN2, d2=

RmBMN2+RmJDN2+KtKeMN2

RmJMN2, d1=

RmBDN2+RmDKN2+KtKeDN2+JKRmN2

RmJMN2, and d0=

RmKDN1+RmKBN2+KtKeKN2

RmJMN2

Table 1 shows the specification of DC linear actua-tor. The control method performed by the actuator isby means of pulse–width modulation (PWM) controldriven by an H-bridge module. Since the overall drivesystem has nonlinear effects such as saturation, deadzone, friction, and backlash effects, it is difficult toidentify an accurate electromechanical system model.The MATLAB System Identification Toolbox15 is usedto obtain the approximated system transfer functionfor each axis. The identification process shown inFigure 3 is applied to construct the transfer function. Itis equipped with dsPIC30 microcontroller (MCU), H-bridge drive circuit, digital input–output (I/O) interfacecard, and a personal computer. The duty cycle of thepulse generated by peripheral interface controller (PIC)MCU is programmable and can be set at 50%, 70%,and 100%. Figure 4(a) and (b) shows time-domainvelocity responses with different duty cycles for A- andB-axis actuators, respectively. The input PWM com-mand and output rotation speed data are measured andcollected. The properties of linear parametric modelsand first-order model are selected in the identificationtoolbox. From the measurement results, it is confirmedthat the system models can be obtained. For simplifica-tion, the first-order transfer functions can be approxi-mated and used to describe the two-axis system

Figure 2. Block diagram of linear actuator.

Table 1. Specification of the DC linear actuator.

Specification Value Unit

Weight 1.96 kgRating voltage 24 V (DC)Rating current 5 AThrust force (maximum) 8000 NPulling force (maximum) 4000 NHolding force (maximum) 6000 NSpeed (load = maximum/load = 0) 2/3.5 mm/sHall sensor resolution 0.08 mm/pulseDuty cycle 100 %

Mao et al. 3

TA sð Þ= Va sð ÞUa sð Þ =

3:2257

0:10549s+1ð3:1Þ

TB sð Þ= Vb sð ÞUb sð Þ =

3:3998

0:0894s+1ð3:2Þ

where TA(s) and TB(s) are transfer functions of linearactuators A and B, respectively.

In practical applications, the DC actuator modelvaries when the drive system operates under differentconditions. The mathematical dynamic model in equa-tion (1.3) can be expressed as

€x=F _x+Gx+CFL +UI ð4Þ

with F= � D

M;G= � K

M;C= � 1

M;UI =

KRN1

MN2u1

By considering the parameter variations and loaddisturbance, the system dynamics is considered as

€x= Fn +DFð Þ _x+ Gn +DGð Þx+ Cn +DCð ÞFL +UI =Fn _x+Gnx

+CnFL +UI +Dt

ð5Þ

with Dt =DF _x+DGx+DCFL

where Fn, Gn, and Cn are nominal system values of F,G, and C, respectively; DF, DG, and DC are the uncer-tainties of the mechanical parameters, respectively; andDt is the lumped disturbance parameter which isbounded.

NN-based synchronous compensator

The synchronized motion control method can bedivided into some categories.1,6 The framework of syn-chronous master motion control is illustrated inFigure 5. It is also called the parallel synchronous con-trol architecture. In this system, the SNPI controller isutilized in position and velocity feedback loops to per-form a precision control of position. It includes themaster command generator scheme, and each axis isdriven separately and independently. The SNPI con-troller with the characteristics of adaptive and onlineadjustment can replace conventional PI controller to

improve performance of motion control. Thus, it doesnot have synchronous compensator between two actua-tors, and the synchronization error cannot be correctedby each other.

To improve the performance of system, cross cou-pling1,3,6 has been applied between actuator A andactuator B in SNPI controller methods. In this research,the modified cross-coupling technique is designed asshown in Figure 6. An additional feedback signal gener-ated by the relative position signal between two subsys-tems is employed. This arrangement can reflect anyload variations and system uncertainty presented inboth subsystems by the applied extra signal. The syn-chronous compensator that used an adaptive neuroncontrol is developed in the modified cross-couplingstructure to eliminate the synchronization error.

A three-layer feedforward NN is implemented inour BPNN controller, as shown in Figure 7. Due to thedifferent position responses of the two axes, the track-ing error can be derived by the position error and thevelocity error. The two synchronization errors are theinput variables of NN and defined as the input vectorE, which is given as

E=xa � xbva � vb

� �=

emvm

� �ð6Þ

where em denotes the tracking position error and vmdenotes the tracking velocity error between the twoaxes. The NN controller has subscripts i, j, and k asinput layer, hidden layer, and output layer, respectively.A tangent hyperbolic function is applied as the activa-tion function of the nodes in the hidden and outputlayers. The total number of units in hidden layer equalsto Jt. The number of units in the input layer equals to2, and the number of units in the output layer equals to1. The net input node to a node j in the hidden layer isgiven as

netj =X

WjiOi

� �+zj i=1, 2 and j=1, 2, . . . , Jt

ð7Þ

where Wji is the connective weights between the inputand the hidden layers and zj is the bias input. The out-put of hidden node j is given as

Figure 3. System identification architecture.


Figure 4. The velocity measurements for (a) A-axis actuator system and (b) B-axis actuator system, for 100%, 70%, and 50% dutycycle.

Figure 5. The parallel architecture with SNPI controller.

Mao et al. 5

Oj = f netj� �

=tanh b1netj� �

ð8Þ

where f( � ) is the activation function with a tanh ( � )activation function and b1.0 is a constant. The netinput to a node k at the output layer is

netk =X

WkjOj

� �+zk j=1, 2, . . . , Jt and k=1 ð9Þ

where Wkj is the connective weights between the hiddenand the output layers and zk is the bias input. The cor-responding output node is

Ok = f netkð Þ=tanh b2netkð Þ ð10Þ

where b2.0 is a constant. The output layer parameterOk is applied as the control input of the linear actuator

dynamic system. The synchronous controller design canbe derived to define the error function, so the energyfunction can be expressed as

EN =1

2emNð Þ2 ð11Þ

where emN is the position error and vnN is the velocityerror between the master-axis and slave-axis at the Nthiteration. Within each interval from (N 2 1) to N or Nto (N + 1), the back-propagation algorithm is used toupdate the weights matrix. That is

DWN =WN+1 �WN = � h∂E

∂WN+aDWN�1 ð12Þ

Figure 6. (a) The proposed modified cross-coupling architecture with SNPI and BPNN compensators and (b) the design flowchartof the network learning rate in our proposed system.


where h is the learning rate parameter and a is themomentum parameter. Using the chain rule, it followsthat the required gradient of E between the output andhidden layers is determined by

∂E

∂Wkj=

∂E

∂netk

∂netk∂Wjk

=∂E

∂netkOj ð13Þ

and dk is defined as

dk =∂E

∂netk=

∂E

∂Ok

∂Ok

∂netk=

∂E

∂Okb2 1�O2

k

� �ð14Þ

The gradient of E between the hidden and inputlayers is determined by

∂E

∂Wij=

∂E

∂netj

∂neti∂Wji

=∂E

∂netjOi ð15Þ

and dj is defined as

dj =∂E

∂netj=Xm

∂E

∂netk

∂netk∂Oj

∂Oj

∂netj

=Xm

dkWkjf0 netj� �

=Xm

dkWkjb1 1�O2j

� � ð16Þ

It is difficult to calculate the Jacobian (∂E=∂Ok) ofthe system in an online fashion because the uncertaintyand disturbance always occur. This problem can beapproximated using a delta adaptation law.16 It can berepresented as follows

∂E

∂Ok=Q1emN+Q2vmN ð17Þ

where Q1 and Q2 are positive constants. The BPNNcontroller can be online trained effectively to compen-sate the synchronization error between the master andslave actuators.

Adaptive neuron controller

P/PI controller and stability analysis

A better control system is developed to provide goodcharacteristics such as stable, simple structures, goodtracking performance, and easy to design without theneed for heavy computational complexity. One bettercontrol structure is to include a velocity loop inside aposition loop for position control. Often, the velocityloop uses PI control, and the position loop applies Pcontrol structure. It is usually called the P/PI con-trol.17,18 Traditionally, the PI velocity loop is tuned formaximum performance, omitting high-frequency effectsdue to the feedback velocity signal formed by differen-tiating a sampled position signal. The design of thecontrollers for the position loop and velocity loop in S-domain is expressed as

Ca sð Þ=KP ð18:1Þ

Cb sð Þ=KV +KI

sð18:2Þ

where Ca(s) is the position controller for the positionloop, Cb(z) is the velocity controller for the velocityloop, KP.0 is the position controller gain, and KV.0

and KI.0 are the proportional (P) and integral gainsof the velocity loop, respectively. The aim of the motioncontrol is to provide the precise velocity and positioncontrol. The inner velocity loop is designed first. Theequivalent velocity loop transfer function TV(s) for asingle axis is described as

TV sð Þ= Cb sð ÞTA sð Þ1+Cb sð ÞTA sð Þ =

Kvs+KIn0s2 + n0Kv +d0ð Þs+KIn0

=Kvs+KIn0

s2 +2jvns+v2n

ð19Þ

withTA(s)=n0

s+d0, vn=

ffiffiffiffiffiffiffiffiffiffiffiKIn0p

; and j=2�1(KIn0)�0:5

(n0Kv+d0)where TA(s) is the equivalent first-order dynamic sys-tem model of motor. This second-order closed-loop sys-tem can be considered as a stable velocity loop system,because two poles of the transfer function TV(s) are inthe left half plane. In position loop analysis, the velocityloop above can be treated as unity gain to reduce theorder of the position loop to first-order system.7,17,18 Itis assumed that the bandwidth of the velocity loop issufficiently larger than that of the position loop one.For a P control structure, the equivalent transfer func-tion of position loop TP(s) becomes

TP sð Þ=Ca sð Þ 1

s

� �1+Ca sð Þ 1

s

� � = KP

s+KPð20Þ

The pole of the position loop is also in the left halfplane, so the feedback loop is stable. The stability ofthe proposed P/PI control method is guaranteed as theP gain and integral gain are .0.

Adaptive single-neuron controller

Figure 8(a) shows the single-axis position control loop.The feedback loop included the position controller andvelocity controller in digital form is given by

C1 zð Þ=Kpp ð21:1Þ

C2 zð Þ=Kvp+Kviz

z� 1ð21:2Þ

where C1(z) is the equivalent position controller for theposition loop, C2(z) is the equivalent velocity controllerfor the velocity loop, Kpp is the position controller gain,

Mao et al. 7

and Kvp and Kvi are the P and integral gains of velocity,respectively. Ts is the sampling time.

The common structure for high-performance electricmotor is to enclose a cascaded velocity loop within aposition loop. Using the traditional P/PI control, thecontrol gains are fixed and cannot be changed duringthe overall control process. The uncertainty, such asparameter variations, varying load, friction, and deadzone, is not taken into consideration. To deal withthese uncertainties, adaptive SNPI controller with self-adapting and self-learning abilities is employed toreplace the PI controller and improve the tracking per-formances. The SNPI controller is designed and evalu-ated to retain the desired performance automatically inmany dynamic systems.11,19,20 An adaptive SNPI con-trol system is represented by the block diagram inFigure 8(b). In our architecture, a single-neuron P con-troller is designed in the position loop, and an SNPIcontroller is developed in the velocity loop. It offersbetter learning ability and ensures that the system cantrack the trajectory under external disturbances. Thecomplete structure of the SNPI controller can beexpressed by

C1p zð Þ=wpp ð22:1Þ

C2v zð Þ=wvp+wviz

z� 1ð22:2Þ

where C1p(z) is the adaptive neuron position controller,wpp.0 is the adaptive P gain, C2v(z) is the adaptive neu-ron velocity controller, and wvp.0 and wvi.0 are theadaptive P and PI gains, respectively. In Figure 8(a),

c(n) is the reference position input, g( � ) is a linear func-tion, parameter A is the P coefficient of the neuron, andx1(n) and x2(n) are the inputs of neuron. For the posi-tion loop controller, the input signal is the positionerror ep(n), and one synaptic weight wpp is used, whichare given as

x1 nð Þ=ep nð Þ=c nð Þ � x nð Þ ð23:1Þ

wpp nð Þ=wpp n� 1ð Þ+hPep nð Þup nð Þx1 nð Þ ð23:2Þ

where hP is the learning rate of the P control in theposition loop. The SNPI output can be written as

up nð Þ=up n� 1ð Þ+Apwpp nð Þxp nð Þ ð24Þ

where up(n) is the adaptive control output for the velo-city loop and Ap is the scaling parameter. For the velo-city control loop, two error signals are used as neuroninput. They are

x1v nð Þ=De nð Þ=ev nð Þ � ev n� 1ð Þ ð25:1Þ

x2v nð Þ=ev nð Þ ð25:2Þ

where ev(n) is the velocity error signal. The weights ofneuron are obtained using the delta learning algorithmas shown in equation (23). The weighting parameterscan be updated as

wvp nð Þ=wvp n� 1ð Þ+hvPev nð Þuv nð Þx1v nð Þ ð26:1Þ

wvi nð Þ=wvi n� 1ð Þ+hvIev nð Þuv nð Þx2v nð Þ ð26:2Þ

where hvP and hvI are the learning rate of the propor-tion control and integral control, respectively. TheSNPI output can be written as

uv nð Þ=uv n� 1ð Þ+Av w0vp nð Þxvp nð Þ+w0vi nð Þxvi nð Þ

ð27Þ

with w0vp(n)=wvp(n)

wvp(n)+wvi(n)

and w0vi(n)=wvi(n)

wvp(n)+wvi(n)

where uv(n) is the adaptive control output for the velo-city loop and Av is the scaling parameter.

Simulation and experimental results

In this section, simulation results and experimentalresults are provided to verify the effectiveness of ourproposed control methods. In the simulation andexperiment, two architectures are conducted to evalu-ate the performances of the proposed control system.Two architectures, (1) the parallel architecture withSNPI method and (2) the modified cross-couplingarchitecture with SNPI controller and adaptive BPNN

Figure 7. Adaptive back-propagation neural networkcontroller structure.


synchronous compensator, are shown in Figures 5 and6, respectively.

The BPNN compensation architecture for dual-axislinear actuator is simulated to demonstrate the synchro-nization performance. To measure the control perfor-mance of the proposed control system, the root meansquare error (RMSE) and sum of absolute error (SAE)for the synchronized tracking are employed. These aredefined as follows:

1. RMSE. The metric adopted to verify the track-ing error performance is the RMSE, whichmakes an excellent general-purpose error metric.This is defined and given by

RMSE=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi=1 e

2m, i

N

sð28:1Þ

where em, i denotes the tracking position error for theith time instant and N is the total simulation point.

2. SAE. The SAE metric can be the second indica-tor to decide the control system performance. Itis defined as

SAE=XNi=1

em,ij j ð28:2Þ

The control objective is to control the position ofthe two-axis linear actuator to track the reference posi-tion profile with minimum tracking error using the

proposed adaptive BPNN compensator under differentload conditions.

Simulation results

In the simulation, MATLAB or Simulink toolbox isapplied to simulate and design of dual-axis linear actua-tors. The system models of actuator A and actuator Bare obtained from the system identification architecturein Figure 5. The transfer functions are approximatedand described in equations (3.1) and (3.2). In BPNNsynchronous compensator, the number of node in thehidden layer is selected as 5, the learning rate h is set as0.9, and the constants b1 and b2 in activation functionare set as 0.5. The more numbers of nodes in the hiddenlayer we used, the better the RMSE and SAE perfor-mances obtained in this control structure. When thenode number is .5, the improvement tends to satu-rated. By consideration of the computational complex-ity and RMSE/SAE performances, the number ofnodes is set as 5 here.

The learning rate of the NN is obtained using theoptimization of the metrics RMSE and SAE. Thesynaptic weights of single-neuron P/PI controllers areoptimized with respect to the position tracking errorfor each A-axis and B-axis models. The learning rateparameters of the single-neuron controllers are selectedas follows:

A-axis and B-axis

hP =0:9, Ap =0:12, wpp 0ð Þ= 0 position loopð ÞhvP =0:91, hvI = 0:9, Av = 0:01,

wvp 0ð Þ= 0, wvi 0ð Þ= 0 velocity loopð Þ

Figure 8. (a) Block diagram of the traditional single-axis position control loop and (b) the proposed structure of SNPI controller.

Mao et al. 9

Then, the RMSE and SAE performance metrics areapplied for the BPNN compensator optimization. Theparameters Q1 and Q2 in delta adaptation law forBPNN are set as 0.72 and 1, respectively. In simulation,the length of the stroke of linear actuators is 200 mm,and the total simulation point N is 5000, and the equiv-alent DC voltage of motor is 24 V. The designflowchart of the network learning rate is shown inFigure 6(b).

The comparisons of simulation results betweenSNPI controller in parallel architecture and SNPI con-troller using synchronous compensator under zero dis-turbance condition can be described in Figure 9(a). Thetrajectory is a step reference input, and the displace-ment moves to 200 mm as the time equal to zero. Ascan be seen, the tracking error of synchronous compen-sation scheme is lower than that of parallel architecture.Figure 9(b) shows the simulation result of synchroniza-tion error under nonzero disturbances. The equivalentload disturbance with db =5V is applied at 20 s. It isshown that the synchronous compensation architectureappears to provide a better performance than that ofthe other methods in terms of synchronization error. InFigure 10, a ramp input signal is applied at time equalto zero. The equivalent external disturbance conditionis db =5V occurring at 20 s in A-axis and B-axis,respectively. It is shown that the tracking performanceof BPNN compensation architecture is better than theparallel architecture in transient and steady states. Asthe external disturbance is applied, the synchronizederror using our method can be reduced significantly tozero, as illustrated in Figure 10(c). Table 2 shows thesimulation results of RMSE and SAE under two differ-ent load conditions. On average, the SNPI controllerwith BPNN compensator achieves the 41.35% reduc-tion in RMSE and 55.35% reduction in SAE thanthose of the conventional parallel architecture.

In addition, the effect of the system model variationis considered to present the dynamic response of ourproposed method. The 10% parameter difference ofthe nominal plant model is studied for the A-axis andB-axis actuators, respectively. Table 3 shows theperformances of the synchronization errors in termsof RMSE and SAE values. It can be seen that ourproposed architecture with SNPI and BPNN

compensators achieves the better synchronization capa-bility and outperforms the conventional method. Onaverage, the proposed architecture offers the 0.02037-mm reduction in RMSE and 80.0257-mm reduction inSAE compared with the traditional method.

Experimental results

Experiments are performed to verify the resultsobtained from the simulations. The experimental setupis composed of the dsPIC30f4011 MCU board, theMOSFET driver A3941 module, the two-axis linearactuators, and the development system using personalcomputer. The experimental setup of dual-axis linearactuators is illustrated in Figure 11. The dsPIC30Fdevice family employs a powerful 16-bit architecturethat integrates the control feature of an MCU with thecomputational capabilities of a digital signal processor(DSP). In the MCU resources, the PWM module, uni-versal asynchronous receiver–transmitter (UART)module, and quadrature encoder interface (QEI) areused in our system. This research demonstrates thetracking performance of SNPI controller and BPNNcompensator which are experimentally implemented indsPIC30F4011 MCU using C language. Figure 12shows the block diagram of the hardware experimentsetup. All the C language codes are developed usingMPLAB IDE in the personal computer (PC) Windowsenvironment and then downloaded to the Flash ROM.In the main function, all the parameters and digital I/Oinitialization, such as UART, PWM, and QEI function,are arranged first. The interrupt service routine (ISR) isdesigned using 1-ms sample period to connect withQEI for position feedback. The ISR program can com-pute the tracking position error and velocity error andgenerate the PWM commands to A3941 moduleaccording to the proposed control algorithm. The con-trol algorithm implemented in MCU includes (1)BPNN compensator, (2) single-neuron P controller forthe position loop, and (3) SNPI controller for the velo-city loop. The A3941 drive module is a full-bridge con-trol circuit which includes the external N-channelpower MOSFETs. This application module is specifi-cally designed for automotive applications with high-power inductive motors and brush DC motors.

Table 2. Simulation results of synchronization error.

Disturbance Parallel architecture with SNPI controller Modified cross-coupling architecturewith SNPI and BPNN compensators

RMSE (mm) SAE (mm) RMSE (mm) SAE (mm)

Zero load disturbance 0.04775 132.996 0.0289 60.3606Nonzero load disturbance 0.05688 181.1384 0.0323 79.4450

SNPI: single-neuron proportional–integral; BPNN: back-propagation neural network; RMSE: root mean square error; SAE: sum of absolute error.


Figure 13 shows the H-bridge circuit using A3941 mod-ule. The A3941 can be driven with a single PWM inputfrom the dsPIC MCU, and four low-voltage level digi-tal inputs can provide control for the gate drives. Pulsewidth modulation high (PWMH) and pulse width

modulation low (PWML) pins can be used to controlcurrent in the power bridge. PHASE pin can determinethe positive direction of load current. Synchronous rec-tification (SR) pin enables or disables synchronousrectification.

Figure 9. Simulation results of (a) the synchronization error under zero disturbance condition and (b) the synchronization errorunder the equivalent disturbance db = 5 V at 20 s.

Table 3. Simulation results of synchronization error under plant model parameter variations.

System parameter Parallel architecture with SNPI controller Modified cross-coupling architecturewith SNPI and BPNN compensators


Nominal value 0.04775 132.996 0.0289 60.360610% model variation in actuator A 0.05233 150.292 0.0329 67.331210% model variation in actuator B 0.05824 153.224 0.0354 68.7432


Mao et al. 11

In the experiments, the C language program isdesigned in dsPIC30 MCU to demonstrate the perfor-mance of dual-axis linear actuators. In BPNN compen-sator, the number of node in the hidden layer is selectedas 5, the learning rate h is set as 0.91, and the constantsb1 and b2 in activation function are set as 0.6. Theparameters Q1 and Q2 in delta adaptation law forBPNN are set as 0.05 and 1, respectively. The learningrate parameters of the single-neuron controllers aregiven as follows:

A-axis and B-axis

hP =0:85, Ap = 0:6 position loopð ÞhvP =0:91, hvI =0:8, Av =0:85 velocity loopð Þ

In this experiment, the PWM frequency is set as1 kHz, and the stroke of linear actuators is 200 mm.The sampling frequency is 1 kHz measured from the

Hall sensor built in the servo mechanism. The velocityfeedback signal is formed by differentiating a sampledposition signal. It is obtained and calculated by dsPICMCU with the sample frequency of 1 kHz. The sam-pling frequencies for current, velocity, and positionloop are designed as 1 kHz, 200 Hz, and 100 Hz, respec-tively. We have built and tested the speed sensor in theproposed control system. The speed information is cal-culated by differentiating a sampled position data fromHall sensor. The velocity control loop is tested bydownsampling the original sampling frequency of1 kHz. The minimum speed measurement is 200 Hz inour experiment setup. These designed parameters aredeveloped and tested in our system.

Based on the plant model obtained from systemidentification, simulation results are conducted anddemonstrated in Figure 14(a) and (b) for the parallelarchitecture and the synchronous compensator

Figure 10. Simulation results of ramp input signal: (a) position response of parallel architecture with nonzero disturbance, (b)position response of the modified cross-coupling BPNN compensation architecture with nonzero disturbance, and (c) thesynchronization error comparisons.


architecture, respectively. Figure 14(c) and (d) showsthe experimental results of position responses of actua-tor A and actuator B with these two structures. In thisstudy, the ramp reference input is used to verify thecontrol performance of our proposed method. Thedesired output position moves from 0 to 220 mm with amaximum velocity of 3.5 mm/s. Figure 15(a) and (b)shows the transient responses of synchronization errors.Clearly, the transient synchronization error of compen-sation architecture is significantly lower than that ofparallel architecture. As can be seen, the synchronouserror is always within 1.6 mm for our method. Figure15(c) shows the simulation results corresponding to theexperiment results above. The structure that utilized theapproximated plant model from system identificationcan offer a better reference design for practical system.Using the parameter adjusting mechanism, the adaptivecompensator improves the transient and steady-stateperformance, and the position response error can bereduced well.

The simulation and experimental results are pro-vided to further demonstrate the robustness of the pro-posed control system under the external disturbance.The equivalent external disturbance condition isdb =5V in A-axis occurring at 0 s. The simulationresults for the ramp tracking response under the exter-nal load disturbance are described in Figure 16(a) and(b). The conventional parallel structure yields a largersynchronous error before 15 s. Figure 16(c) and (d)shows the experimental ramp responses of parallelstructure and synchronous compensator architectures.Under the external disturbance, parallel architectureproduces a large amount of synchronous error betweenthe two axes, as shown in Figure 17(a). As can be seen,the synchronization error is always within 2.5 mm andcan decrease gradually, as shown in Figure 17(b). It is

Figure 11. The experimental setup.

Figure 12. Blok diagram of the hardware experiment setup.

Figure 13. The H-bridge circuit using A3941 module.

Mao et al. 13

Figure 14. (a) Simulation results of position response with the parallel architecture, (b) simulation results of position response withthe proposed modified cross-coupling architecture, (c) experimental results of position response with the parallel architecture, and(d) experimental results of position response with the proposed modified cross-coupling architecture.

Figure 15. (a) Experimental results of synchronization error of parallel architecture, (b) experimental results of synchronizationerror of the modified cross-coupling architecture with BPNN compensator, and (c) simulation results of synchronization error forthe parallel architecture and BPNN compensator with zero disturbance.


shown that the synchronous compensator with SNPIcontroller structure can reduce error significantly andprovide a robust control performance under distur-bance and uncertainty. The corresponding simulationresults are illustrated in Figure 17(c). The BPNN com-pensation structure provides the better tracking perfor-mances which is similar to the experimental resultsabove. Furthermore, the comparison results can be

depicted in Table 4. It is composed of the measurementdata from the MCU-based control system under no-load and equivalent disturbance mode. On average, theSNPI controller with BPNN compensator provides the0.7903 mm in RMSE and 3070.24 mm in SAE. Fromthe experimental results, it can be observed that thecompensator architecture offers a better performanceand robustness in terms of RMSE response and SAE

Figure 16. (a) Simulation results of position response of the parallel architecture with nonzero disturbance, (b) simulation resultsof position response of the proposed modified cross-coupling architecture with nonzero disturbance, (c) experimental results ofposition response of parallel architecture with nonzero disturbance, and (d) experimental results position response of the modifiedcross-coupling architecture with nonzero disturbance.

Table 4. Experimental results of synchronization error.

Disturbance Parallel architecture with SNPI controller Modified cross-coupling architecturewith SNPI and BPNN compensators


Zero load disturbance 2.9953 13178.96 0.7058 2502.64Nonzero load disturbance 33.4 199028.3 0.875 3637.84


Mao et al. 15

response than those of the conventional parallel archi-tecture. It is worth considering our controller designfrom the performance requirements, such as toleranceerror bound and load disturbance.

Conclusion

This article presents the adaptive NN-based synchroni-zation architecture for dual-axis linear actuator system.An integrated model of DC motor drive system includ-ing mechanical ball screw subsystem is derived to repre-sent the motion dynamics of control system. The novelsingle-neuron architecture with simple structure andbetter approximation property is applied for P/PI con-trollers to improve the tracking performance of thesingle-axis position and velocity control loops. TheBPNN synchronous compensator can be online trainedeffectively and adaptively to compensate the synchroni-zation error between the master and slave actuators.An accurate identification process of electrical andmechanical system has been conducted to verify the

effectiveness of the proposed architecture in simulationdesign. The new SNPI controller and BPNN compen-sator are implemented and built in a dsPIC30 MCU toevaluate the tracking performances. From the theoreti-cal and experiment results, the control performances interms of RMSE and SAE are well achieved, and syn-chronization accuracy is much improved. Also, ourproposed method with adaptive learning algorithm is afeasible solution in practical implementation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The authors would like to thank the Ministry of Science andTechnology of the Republic of China, Taiwan, for financiallysupporting this research under Contract No. MOST 105-2622-E-224-010 -CC3, MOST 104-2622-E-224-016-CC3, andMOST 104-2218-E-224-002.

Figure 17. Experimental results of (a) synchronization error of parallel architecture with nonzero disturbance, (b) synchronizationerror of the modified cross-coupling architecture with nonzero disturbance, and (c) the simulation results of synchronization errorsfor parallel architecture and modified cross-coupling architecture.


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