Robust RBFN Control for Linear Induction

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2170 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008

Robust RBFN Control for Linear InductionMotor Drive Using FPGA

Faa-Jeng Lin , Senior Member, IEEE , Li-Tao Teng, Cheng-Yan Chen, and Chih-Kai Chang

Abstract—A field-programmable gate array (FPGA) basedrobust radial basis function network (RBFN) control system isproposed to control the mover position of a linear induction motor(LIM). First, the indirect field-oriented mechanism is adoptedfor the control of LIM. Next, an equivalent control law bases onsliding-mode control is designed, in which the uncertainties arelumped by a conservative constant. However, the lumped uncer-tainty is unknown and difficult to obtain in advance in practicalapplications. Therefore, a RBFN is derived to approximate theequivalent control law in real-time, and a robust RBFN controlsystem with online training ability is resulted. Then, a FPGA chipis adopted to implement the indirect field-oriented mechanismand the developed control algorithms for possible low-cost and

high-performance industrial applications. The effectiveness of the proposed control scheme is verified by some simulated andexperimental results. With the robust RBFN control system,the mover position of the FPGA-based LIM drive possesses theadvantages of good transient control performance and robustnessto uncertainties in the tracking of periodic reference trajectories.

Index Terms—Field programmable gate array (FPGA), indirectfield-oriented control, linear induction motor (LIM), radial basisfunction network (RBFN).

I. INTRODUCTION

FIELD programmable gate array (FPGA) incorporates thearchitecture of gate arrays and the programmability of a

programmable logic device (PLD). It consists of thousands of

logic gates, some of which are combined together to form a

configurable logic block (CLB) thereby simplifying high-level

circuit design. Interconnections between logic gates using

software are externally defined through SRAM and ROM,

which will provide flexibility in modifying the designed circuit

without altering the hardware. Moreover, concurrent operation,

simplicity, programmability, a comparatively low cost and rapid

prototyping make it the favorite choice for prototyping an appli-

cation specific integrated circuit (ASIC) [1], [2]. Furthermore,

all the internal logic elements and all the control procedures

of the FPGA are executed continuously and simultaneously.

The circuits and algorithms can be developed in the VHSIC

hardware description language (VHDL) [1], [2]. This method

Manuscript received February 28, 2007; revised April 24, 2007. PublishedJune 13, 2008. This work was supported by the National Science Council of Taiwan, R.O.C., through Grant NSC 95-2221-E-259-042-MY3. Recommendedfor publication by Associate Editor P. Luk.

F.-J. Lin is with the Department of Electrical Engineering, National CentralUniversity, Chungli 320, Taiwan, R.O.C. (e-mail: [email protected]).

L.-T. Teng, C.-Y. Chen, and C.-K. Chang are with the Department of Elec-tricalEngineering,National DongHwa University, Hualien 974,Taiwan, R.O.C.(e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TPEL.2008.924604

is as flexible as any software solution. Another important

advantage of VHDL is that it is technology independent. The

same algorithm can be synthesized into any FPGA and even

has a direct path to an ASIC, opening interesting possibilities

in industrial applications in terms of performance and cost.

However, the major disadvantage of a FPGA-based system for

hardware implementation is the limited capacity of available

cells. Therefore, only research on FPGA-based sliding-mode

or fuzzy controllers can be found in high-performance control

application literature [3]–[5]. In addition, some FPGA-based

applications of various motor drives can be found in [6]–[9].

However, the FPGA-based control systems proposed by above

literatures do not have online learning ability.

Intelligent control approaches such as neural network and

fuzzy system do not require mathematical models and have the

ability to approximate nonlinear systems. Therefore, there were

many researchers using intelligent control approaches to rep-

resent complex plants and construct advanced controllers [10].

Moreover, the locally tuned and overlapped receptive field is a

well-known structure that has been studied in regions of cerebral

cortex, visual cortex, and so on [10]. Based on the biological re-

ceptive fields, the radial basis function network (RBFN) that em-

ploys local receptive fields to perform function mappings was

proposed in [11]. Furthermore, the RBFN has a faster conver-gence property than a multilayer perceptron (MLP) since only

the connected weights between the hidden layer and the output

layer of the network are adjusted during training to reduce the

computational requirements. In addition, the RBFN has a sim-

ilar feature to the fuzzy system: first, the output value is cal-

culated using the weighted sum method; second, the number of

nodes in the hiddenlayer of the RBFN is the same asthe number

of if-then rules in the fuzzy system; finally, the receptive field

functions of the RBFN are similar to the membership functions

of the premise part in the fuzzy system. Therefore, the RBFN is

very adaptable to the design of a FPGA chip and very useful to

be applied to control the dynamic systems [12], [13].A linear induction motor (LIM) has many desirable perfor-

mance features including high-starting thrust force, no need for

a gear between motor and the motion devices, the reduction of

mechanical losses and the size of motion devices, a high-speed

operation, silence, and so on [14], [15]. Due to these advan-

tages LIMs have been used widely in industrial processes and

transportation applications [16]–[18]. The driving principles of

a LIM are similar to those of a traditional rotary induction motor

(RIM). However, the motor parameters are time varying due

to changes in operating conditions, such as the speed of the

mover, temperature and rail configuration. Moreover, there are

significant parameter variations in reaction rail resistivity, the

0885-8993/$25.00 © 2008 IEEE



LIN et al.: ROBUST RBFN CONTROL 2171

dynamics of the air gap, slip frequency, phase unbalance, satu-

ration of the magnetizing inductance, and end-effects [14], [17].

Therefore, its mathematical model is difficult to derive com-

pletely. Furthermore, since the operation of a LIM involves two

contacting bodies, a friction force is inevitably among the forces

of motion. In addition, this friction characteristic may be easily

varied due to change in normal forces in contact, and also thetemperature and humidity. Additionally, since friction is a nat-

ural phenomenon that is quite difficult to model, it is impossible

to obtain a precise friction model for practical applications [19],

[20]. On the other hand, the dynamic model of a LIM can be

modified from the dynamic model of the RIM at certain low

speed since a LIM can be visualized as an unrolled RIM. Thus,

field-orientated control [21], [22] can be adopted to decouple

the dynamics of the thrust force and the flux amplitude of the

LIM.

The motivation of this study is to propose a hardware online

learning robust RBFN control system due to its robustness to

confront the uncertainties that exist in a field-oriented control

LIM drive including the friction force. Moreover, the FPGA

chip is adopted to implement the proposed controller in order

to allow possible low-cost and high-performance industrial ap-

plications. The proposed control algorithms are realized on a

24-MHz FPGA (XC2V1000) with 1 million gate counts and

10240 flip-flops from Xilinx, Inc using VHDL. The design and

implementation of the FPGA-based control IC will be described

in detail. Compared with a DSP or a PC-based robust RBFN

control system, the merits of the FPGA-based robust RBFN con-

trol system are parallel processing and small size in addition to

low-cost. Furthermore, the developed VHDL code can be easily

modified and implemented to control any type of ac motors as

well.

II. INDIRECT FIELD-ORIENTED LINEAR

INDUCTION MOTOR DRIVE

The primary (mover) of the adopted three-phase LIM is

simply a cut open and rolled flat rotary-motor primary. The

secondary usually consists of a sheet conductor using aluminum

with an iron back for the return path of magnetic flux. The

primary and secondary form a single-sided LIM. Moreover, a

simple linear encoder is adopted for the feedback of the mover

position. The dynamic model of the LIM is modified from

the traditional model of a three-phase, Y-connected inductionmotor in synchronous rotating reference frame [14]. The block

diagram of an indirect field-oriented LIM system is shown in

Fig. 1, where is the position of mover; is the reference

trajectory; is the velocity of mover; is the derivative of

reference trajectory; the -axis primary command current

is the command of flux current; the -axis primary command

current is the control effort. The indirect field-oriented LIM

system consists of a LIM, a ramp comparison current-controlled

pulsewidth modulation (PWM) voltage-source-inverter (VSI),

an indirect field-oriented mechanism, a coordinate translator,

and generator, where is the position of the sec-

ondary flux, a speed control loop, and a position control loop.Three-phase current commands, , and are generated

from the coordinate translator for the ramp comparison cur-

rent controller. By using of the indirect field-oriented control

technique, the electromagnetic force can be simplified by the

following equations [7], [21]:

(1)

(2)

where is the magnetizing inductance per phase; is

the secondary inductance per phase; , are the -axis

and -axis primary current; is the pole pitch. The curve

fitting technique based on step response is applied to find the

drive model off line at the nominal case ( 0). Moreover,

the scaling is necessary for the digital implementation of the

proposed FPGA-based LIM drive. The position resolution

of the encoder is 0.1(m)/1000 (digital) 0.1(mm) and the

velocity resolution is 1.6104(m/s)/22(digital) 0.073(m/s) at

the sampling frequency 732 Hz, and the conversion range of

the adopted D/A converter is 2047 to 2048 digital 5 V

to 5 V, i.e., 1 V 409 digital. Therefore, by 1.6104/22

/409, the scaling is 29.9388 (m/s)/V. Then the results

are:

(3)

where is the total mass of the moving element; is the vis-

cous friction and iron-loss coefficient. The “-” symbol repre-

sents the system parameters in the nominal condition. Though

the electromagnetic force can be simplified as (1) via the field-

oriented control, considering the variations of system param-

eters and external nonlinear and time-varying disturbance in-

cluding friction force, the LIM drive system is a nonlinear time-

varying system in practical applications. Furthermore, since the

gains of the current controllers are large enough, the three-phase

current tracking responses are very close to three-phase current

commands. Therefore, the ideal current control is assumed in

the following proof to simplify the analysis. In practical appli-

cations, the current control is not ideal. However, the non-ideal

current control can be considered as external disturbance and

included in the lumped uncertainty, which will be described in

the next section, and confronted by the proposed RBFN control.

III. PROPOSED CONTROL SYSTEM

Consider a drive system with parameter variations, external

force disturbance and friction force for the actual LIM drive

system, then

(4)

(5)

where , 0, 1 ;

and denote the uncertainties introduced by system param-

eters and ; is the control input to the motor drivesystem; is the external disturbance; isthe friction force.




Fig. 1. System configuration of LIM drive.

Considering Coulomb friction, viscous friction and Stribeck ef-

fect, the friction force can be formulated as follows [19], [20]:

(6)

where is the Coulomb friction; is the static friction; is

the Stribeck velocity parameter; is the coefficient of viscous

friction; is a sign function. Reformulate (5), then

(7)

where is the lumped uncertainty and is assumed to be

bounded. Moreover, the is defined by

(8)

A. Sliding-Mode Control

First, the position tracking error is defined as follows:

(9)

and its derivative is

(10)

Now, an auxiliary signal is designed as follows:

(11)

where is a positive constant. Then a switching function is de-

fined as

(12)

If the sliding mode occurs, i.e., , then

(13)

Substituting (11) into the derivative of (13), it can be obtained

(14)

which implies that .

In the design of the sliding-mode control system, the equiva-

lent control law, which will determine the dynamic of the system

on sliding surface, is found. The equivalent control law is de-

rived by

(15)

Taking the derivative of (12) and using (7), then

(16)

Solving (15) with (16), one can obtain

(17)

However, the lumped uncertainty is unknown in practical

applications. Therefore, a RBFN is adopted to approximatethe equivalent control law to ensure the control system is

asymptotically stable even if parametric uncertainty, external

force disturbance and friction force exist.

B. Description of RBFN

The architecture of a RBFN with receptive field units is

shown in Fig. 2(a). It comprises input layer, hidden layer and

output layer. The signal propagation and the basic function in

each layer of the RBFN are introduced as follows.

Layer 1: Input Layer

In the input layer, the input vector of the RBFN is thesliding surface and its derivative, .

Layer 2: Hidden Layer

In the hidden layer, the receptive field function is usually a

Gaussian function or a logistic function [11]. In order to re-

duce the computational requirements in FPGA, a triangular

function shown in Fig. 2(b) is selected as the receptive field

function in the following:

if

if

if

(18)




Fig. 2. (a) Network structure of RBFN. (b) Triangular function in hidden layerof RBFN.

where is the number of hidden nodes; is the center

of the triangle; is the center’s width of the triangular;

is the output of the hidden layer.

Layer 3: Output Layer In the output layer, the weighted sum method is used to

calculate the output of the RBFN, then the output becomes

(19)

where is the connected weight between the hidden layer

and the output layer.

For ease of discussion, the output of the RBFN is rewritten as

follows:

(20)

Fig. 3. Robust RBFN control system.

where ; is the

weighting vector as provided by tuning algorithm to be intro-

duced later. By universal approximation theorem, there exists

an optimal RBFN approximator in the form of (19) such that

(21)

where is the optimal weighting vector that achieve the per-fect approximation; is the minimum reconstructed error. The

absolute value of is assumed to be less than a small positive

constant , i.e., . The control block diagram of robust

RBFN control system is shown in Fig. 3, which consists of a

RBFN approximator and a robust controller . The

proposed robust RBFN control system is designed in the fol-

lowing theorem to ensure the high-performance applications.

Theorem 1: Considering the system dynamic equation rep-

resented by (7), if the robust RBFN control system is designed

as (22) and (23) with the adaptation law shown in (24), then the

stability of the control system can be guaranteed

(22)

(23)

(24)

where is a positive constant.

Proof: Define a Lyapunov function candidate as

(25)

where . Taking the time derivative of the Lya-

punov function and using (16), one can obtain

(26)

The approximative error is define as

(27)




Substituting (27) into (26), it can be concluded that

(28)

To satisfy , the robust controller (23) and the adaptation

law (24) are substituted into (28), then

(29)

Since 0, is negative semi-definite [i.e., ],

which implies and are bounded. Define function

, and integrate function with

respect to time

(30)

Since is bounded, and is non-increasing and

bounded, the following result can be concluded:

(31)

Differentiate with respect to time

(32)

Since the lumped uncertainty is assumed to be bounded, all

the variables on the RHS of (16) are bounded, which means that

is also bounded. Moreover, since all the variables on the

RHS of (32) are bounded, is bounded. Then, is uni-

formly continuous [23]. By using the Barbalat’s Lemma [23],

[24], it can be shown that 0. That is, 0 as

. As a result, the tracking performance and stability of

the proposed robust RBFN control system for the LIM can be

guaranteed.

IV. CIRCUITS DESIGN ON FPGA CHIP

The block diagram of the FPGA-based control system for a

LIM drive using current-controlled technique is shown in Fig. 4.

The current-controlled PWM voltage source inverter is imple-

mented by an intelligent power module (IPM) switching com-

ponent (MUBW30-06A7) manufactured by IXYS Co. with a

switching frequency of 15 KHz. The timing control module,

encoder interface module, the field-oriented control module andthe robust RBFN control module are realized on the FPGA chip.

Fig. 4. Block diagram of FPGA-based control system.

Fig. 5. Photograph of experimental setup.

The sampling interval of the control algorithms is 1.366 ms

(732 Hz). Three-phase current commands, , and are gen-erated from the coordinate translator and sent to three D/A con-

verters for the ramp comparison current control. Moreover, two

D/A converters are utilized to display the reference trajectory

, the mover position , the control effort and the weight

alternately on the oscilloscope. The adopted D/A converters

are 12 b in size with an output voltage of 5 V. The entire I/O

port for this chip includes 2 pins for the input ports and 36 pins

for the output port. Furthermore, 2101 of 10240 flip-flops (20%)

have been used in the FPGA chip. In addition, the used gate

counts of the encoder interface module, the field-oriented con-

trol module, the data and D/A controller and the robust RBFN

control module are 67553, 84030, 677, and 39963, respectively.Additionally, the circuits and control algorithms in the FPGA




(a) (b)

(c) (d)

Fig. 6. Circuits design on FPGA. (a) Encoder interface module. (b) Field-oriented control module. (c) Sliding surface generator in robust RBFN control module.(d) Input and hidden layer in robust RBFN control module.

are developed using VHDL by a personal computer (PC) as the

development system. In the development of the VHDL codes, all

the sine, cosine, multiplier and divider are implemented using

available intellectual property (IP). A photograph of the experi-

mental setup including the FPGA chip, the development system,

the motor drive and the LIM is shown in Fig. 5.

A. Encoder Interface Module

The block diagram of encoder interface module is shown in

Fig. 6(a), which consists of timing control, two digital filters,

a decoder, an up-down counter, two clock (CLK) generators, a

register, a command generator and one adder. The function of

the encoder interface is to obtain theposition and speed values of

the mover. The pulse count signal PLS and the rotating direction

signal DIR are obtained using the A, B pulse input signals from

the decoder through two digital filters. The position signal can

be obtained using the PLS and DIR signals through up-down

counter. Moreover, the command generator includes periodic

sinusoidal IP and periodic trapezoidal IP in order to generate thereference trajectory . Furthermore, is th velocity signal, and

results from the difference between the position signal and the

time delay of , .

B. Field-Oriented Control Module

The field-oriented control module shown in Fig. 6(b) is com-

posed of a generator, a coordinate translator, a and

generator and timing control. The is obtained using

the estimated slip velocity signal , the control effort signal, the velocity signal and an integrator. Then, and

signals are obtained through the and gen-

erator. Moreover, three-phase current commands, , and

are generated from the coordinate translator, which consists of

six multipliers and five adders, and sent to three D/A converters

for the ramp comparison current controller. Each D/A converter

needs 12 pins in the output port.

C. Robust RBFN Control Module

To implement the control law effectively, the robust RBFN

control module is divided into five parts which are shown in

Fig. 6(c)–(g). 1) Fig. 6(c) shows the block diagram of slidingsurface generator designed to obtain the sliding surface and




(e) (f)

(g)

Fig. 6. (Continued ) Circuits design on FPGA. (e) Output layer in robust RBFN control module. (f) On-line learning algorithm in robust RBFN control module.(g) Robust controller in robust RBFN control module.

its derivative using , and . First, the auxiliary signal

is obtained. Moreover, is obtained by a differentiator, which

is composed of a register and an adder, and is obtained by

an adder and a multiplier. Then, is obtained based on (12)

by using an adder. Finally, is obtained by a differentiator. 2)

Fig. 6(d) shows the block diagram of input and hidden layers

utilized to obtain the output of the hidden layer using

and based on (18). First, the ouputs of each neuron are gen-

erated using triangular functions where all the means are dis-

tributed among [ 1, 1] (digital value [ 409, 409]) and all thewidths of the triangulars are 1.0 (digital value 409). Then, two

multiplexers with a multiplier are utilized to multiply the output

of each neuron based on (18). 3) Fig. 6(e) shows the block di-

agram of output layer utilized to obtain the ultimate output of

the RBFN using based on (19). First, two mul-

tiplexers with a multiplier are utilized to multiply the outputs

of the hidden layer and the weights indi-

vidually. Then, the output is obtained through a register

and an adder. 4) Fig. 6(f) shows the block diagram of on-line

learning algorithm utilized to achieve the adaptation laws of the

weights between the hidden layer and the output layer based on

(24). First, the variations of the weights are ob-tained by a multiplexer, three multipliers and a register. Then,

the new weights are obtained by

adding the weights of current iteration with

. 5) Fig. 6(g) shows the block diagram of robust

controller utilized to achieve the robust control law using

based on (23). First, the sliding surface is operated using a sign

function . Then, isobtained bymultiplya constantgain

to the output of the sign funtion.

V. SIMULATED AND EXPERIMENTAL RESULTS

The control objective is to control the mover of the LIM tomove periodically according to reference trajectories including

sinusoidal and trapezoidal commands. To investigate the effec-

tiveness of the proposed robust RBFN control system, two cases

including the nominal condition (Case 1) and the parameter vari-

ation condition (Case 2) are considered here. In Case 2, one

iron disk with 5.71kg weight is added to the mass of the mover.

Moreover, to show the effectiveness of the control system with

small number of neurons, the RBFN has two, nine and one neu-

rons at the input, hidden and output layers, respectively.

A. Simulated Results

The simulation is carried out using “Matlab” package. Todemonstrate the control performance of the control system with




Fig.7. Simulatedresultsof robust RBFNcontrolsystem for periodic sinusoidalreferencetrajectory (0.2 Hz): (a) Trackingresponseat Case 1. (b)Trackingerrorat Case 1. (c) Control effort at Case 1. (d) Connected weights between hiddenlayer and output layer at Case 1. (e) Tracking response at Case 2. (f) Trackingerror at Case 2. (g) Control effort at Case 2. (h) Connected weights between

hidden layer and output layer at Case 2.

different reference trajectories, the simulated results due to pe-

riodic sinusoidal and trapezoidal commands are given. The con-

trol objective is to control the mover to move 0.1 m periodi-

cally for sinusoidal reference trajectory and 0.1 m periodically

for trapezoidal reference trajectory. The gain, threshold value

and learning rate of the robust RBFN control system are given

as follows:

(33)

These values are chosen to achieve the best transient control per-formance in both the simulation and experimentation consid-

ering the requirement of asymptotical stability. Moreover, the

coefficients of the friction model used in this study are selected

as follows:

(34)

The simulated results of the robust RBFN control system

due to the periodic sinusoidal and trapezoidal reference tra-

jectories with frequency 0.2 Hz are shown in Figs. 7 and 8.

The tracking responses of the mover position due to the peri-

odic sinusoidal reference trajectory at Case 1 and Case 2 are

shown in Fig. 7(a) and (e); the tracking errors are shown inFig. 7(b) and (f); the associated control efforts are shown in

Fig. 8. Simulated results of robust RBFN control system for periodic trape-zoidal reference trajectory (0.2 Hz): (a) Tracking response at Case 1. (b)Tracking error at Case 1. (c) Control effort at Case 1. (d) Connected weightsbetween hidden layer and output layer at Case 1. (e) Tracking response at Case2. (f) Tracking error at Case 2. (g) Control effort at Case 2. (h) Connected

weights between hidden layer and output layer at Case 2.

Fig. 7(c) and (g); and two of the connected weights between

the hidden layer and the output layer of the RBFN, and ,

are shown in Fig. 7(d) and (h). Moreover, the tracking responses

of the mover position due to the periodic trapezoidal reference

trajectory at Case 1 and Case 2 are shown in Fig. 8(a) and (e);

the tracking errors are shown in Fig. 8(b) and (f); the associated

control efforts are shown in Fig. 8(c) and (g); and two of the con-

nected weights between the hidden layer and the output layer of

the RBFN, and , are shown in Fig. 8(d) and (h). From the

simulated results, satisfactory tracking performance and robustcontrol characteristics under the occurrence of the uncertainties

can be obtained for the proposed robust RBFN control system.

Furthermore, the connected weights of the RBFN are also well

trained to achieve excellent command tracking performance.

B. Experimental Results

Some experimental results are provided to demonstrate the ef-

fectiveness of the proposed FPGA-based control system. Figs. 9

and 10 depict the experimental results of the command tracking

due to the periodic sinusoidal and trapezoidal reference trajecto-

ries with frequency 0.2 Hz of the proposed robust RBFN control

system. The tracking responses of the mover position due to theperiodic sinusoidal reference trajectory at Case 1 and Case 2




Fig. 9. Experimental results of robust RBFN control system for periodic si-nusoidal reference trajectory (0.2 Hz). (a) Tracking response at Case 1. (b)Tracking error at Case 1. (c) Control effort at Case 1. (d) Connected weights be-tween hidden layer and output layer at Case 1. (e) Tracking response at Case 2.(f) Tracking error at Case 2. (g) Control effort at Case 2. (h) Connected weights

between hidden layer and output layer at Case 2.

are shown in Fig. 9(a) and (e); the tracking errors are shown

in Fig. 9(b) and (f); the associated control efforts are shown in

Fig. 9(c) and (g); and two of the connected weights between the

hidden layer and the output layer of the RBFN, and , are

shown in Fig. 9(d) and (h). Moreover, the tracking responses of

the mover position due to the periodic trapezoidal reference tra-

jectory at Case 1 and Case 2 are shown in Fig. 10(a) and (e);

the tracking errors are shown in Fig. 10(b) and (f); the associ-

ated control efforts are shown in Fig. 10(c) and (g); and two of

the connected weights between the hidden layer and the outputlayer of the RBFN, and , are shown in Fig. 10(d) and (h).

Since the parameters of the RBFN are initially set

to zero, accurate tracking control performance of the LIM drive

can be obtained after half cycle of on-line training of the RBFN

for periodic reference trajectories. In Fig. 10(a), the command

velocity is about 0.1 m/s. Though the command velocity is close

to the velocity resolution, however, it does not affect the tracking

performance since only position feedback signal with resolution

0.1 mm is used in the proposed robust RBFN control system.

Furthermore, from the experimental results, good tracking re-

sponses of the mover shown in Figs. 9(a), (e) and 10(a), (e)

can be obtained at both the nominal and the parameter variationconditions. It also depicts that the proposed FPGA-based robust

Fig. 10. Experimental results of robust RBFN control system for periodictrapezoidal reference trajectory (0.2 Hz). (a) Tracking response at Case 1. (b)Tracking error at Case 1. (c) Control effort at Case 1. (d) Connected weightsbetween hidden layer and output layer at Case 1. (e) Tracking response at Case2. (f) Tracking error at Case 2. (g) Control effort at Case 2. (h) Connected


RBFN control system has hardware online learning ability. In

addition, the tracking errors are about 1 and 2 mm after half

cycle of on-line training of the RBFN for the sinusoidal and

trapezoidal reference trajectories, respectively. Comparing to

the simulated results, the tracking errors are larger than the re-

spective simulated results due to the uncertainties including fric-

tion force, parameter variations and external disturbance in the

experimentation. On the other hand, since the precisely friction

model is difficult to obtain and the uncertainties are also difficult

to formulate in the simulation, therefore, the connected weightsshown in Figs. 7(d), (h) and 8(d), (h) are not the same as the

experimental results shown in Figs. 9(d), (h) and 10(d), (h).

To further demonstrate the effectiveness of the proposed con-

trol system, the experimental results of the command tracking

due to the periodic sinusoidal and trapezoidal reference tra-

jectories with frequency 0.3 Hz are shown in Figs. 11 and 12.

In Fig. 12(a), the command velocity is about 0.15(m/s). From

the experimental results, the proposed robust RBFN control

system still has good tracking control performance. Moreover,

since the connected weights in the RBFN are initially set to

zero, the position overshoots are unavoidable in the experi-

mentation even with the nominal system parameters as shownin Figs. 9(a), 10(a), 11(a), and 12(a). The position overshoots




Fig. 11. Experimental results of robust RBFN control system for periodic si-nusoidal reference trajectory (0.3 Hz): (a) Tracking response at Case 1. (b)Tracking error at Case 1. (c) Control effort at Case 1. (d) Connected weights be-tween hidden layer and output layer at Case 1. (e) Tracking response at Case 2.(f) Tracking error at Case 2. (g) Control effort at Case 2. (h) Connected weightsbetween hidden layer and output layer at Case 2.

can be avoided by the pre-training of the RBFN and using

the pre-training weights as the initial values of the connected

weights in the RBFN. However, to show the learning ability

of the FPGA-based RBFN control system, the pre-training

process is not adopted in this study.

VI. CONCLUSION

This study has successfully demonstrated the design and im-

plementation of a FPGA-based robust RBFN control system for

the position control of a LIM drive system. First, the indirect

field-oriented mechanism was introduced. Then, an equivalent

control law based on sliding-mode control methodology was de-

signed. However, since the lumped uncertainty is unknown and

difficult to obtain, a robust RBFN control system with online

training ability was proposed. Moreover, the robustness of the

proposed FPGA-based robust RBFN control system is achieved

under the occurrence of parameter variations and friction force

at various reference trajectories. Finally, the effectiveness of the

proposed low-cost high-performance FPGA-based LIM drivehas been confirmed by some simulated and experimental results.

Fig. 12. Experimental results of robust RBFN control system for periodictrapezoidal reference trajectory (0.3 Hz): (a) Tracking response at Case 1. (b)Tracking error at Case 1. (c) Control effort at Case 1. (d) Connected weightsbetween hidden layer and output layer at Case 1. (e) Tracking response at Case2. (f) Tracking error at Case 2. (g) Control effort at Case 2. (h) Connected


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dision-Wesley, 1995.

Faa-Jeng Lin (M’93–SM’99) received the B.S. and

M.S. degrees in electrical engineering from NationalCheng Kung University, Tainan, Taiwan, R.O.C., in1983 and 1985, respectively, and the Ph.D. degree inelectrical engineering from National Tsing Hua Uni-versity, Hsinchu, Taiwan, R.O.C., in 1993.

From 1993 to 2001, he was an Associate Professorand then a Professor in the Department of ElectricalEngineering, Chung Yuan Christian University,Chung Li, Taiwan. From 2001 to 2003, he wasChairperson and a Professor in the Department of

Electrical Engineering, National Dong Hwa University, Hualien, Taiwan. From2003 to 2005, he was Dean of Research and Development, National Dong HwaUniversity, Hualien, Taiwan. From 2006 to 2007, he was Dean of AcademicAffairs at the same University. Currently, he is Professor in the Departmentof Electrical Engineering, National Central University, Chung Li, Taiwan. Heis also the Chair, Power System Division, National Science Council, Taiwan,2007 to 2009. His research interests include ac and ultrasonic motor drives,

DSP-based computer control systems, fuzzy and neural network control theo-ries, nonlinear control theories, power electronics, and micro mechatronics.

Dr. Lin received the Outstanding Research Professor Award from the ChungYuan Christian University, Taiwan, in 2000; the Excellent Young Electrical En-gineer Award from the Chinese Electrical Engineering Association, Taiwan, in2000; the Crompton Premium Best Paper Award from the Institution of Elec-trical Engineers (IEE), United Kingdom, in 2002; the Outstanding ResearchAward from the National Science Council, Taiwan, in 2004; the OutstandingResearch Professor Award from the National Dong Hwa University, Taiwan, in2004, and the Outstanding Professor of Electrical Engineering Award in 2005from the Chinese Electrical Engineering Association, Taiwan. He is Fellow of the Institution of Engineering and Technology (formerly IEE), U.K..

Li-Tao Teng was born in Taipei, Taiwan, R.O.C., in1982. He received the B.S. degree in electrical en-gineering from the National Dong Hwa University,Hualien, Taiwan, R.O.C., in 2004 where he is cur-rently pursuing the Ph.D. degree.

His research interests include nonlinear controltheories, artificial intelligence control theories, mag-netic levitation systems, and wind driven induction

generator systems.

Cheng-Yan Chen was born in Kaohsiung, Taiwan,R.O.C., in 1983. He received the B.S. degree in elec-trical engineering from National Formosa University,Yunlin, Taiwan, R.O.C., in 2005 and the M.S. degreein electrical engineering from National Dong HwaUniversity, Hualien, Taiwan, R.O.C., in 2006.

His research interests include ac motor drives andFPGAs.

Chih-Kai Chang was born in Kaohsiung, Taiwan,R.O.C., in 1981. He received the B.S. and M.S. de-grees in electrical engineering from National DongHwa University, Hualien, Taiwan, R.O.C., in 2004and 2006, respectively.

His research interests include nonlinear controltheories, artificial intelligence control theories, andFPGAs.

Documents

Robust RBFN Control for Linear Induction