FPGA-Based Computed Force Control System

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1238 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 4, APRIL 2009

FPGA-Based Computed Force Control System UsingElman Neural Network for Linear Ultrasonic Motor

Faa-Jeng Lin, Senior Member, IEEE, Ying-Chih Hung, and Syuan-Yi Chen

AbstractA field-programmable gate array (FPGA)-basedcomputed force control system using an Elman neural network(ENN) is proposed to control the mover position of a linearultrasonic motor (LUSM) in this paper. First, the structure andoperating principle of the LUSM are introduced. Then, the dy-namics of the LUSM mechanism with the introduction of a lumpeduncertainty, which include the friction force, are derived. Sincethe dynamic characteristics and motor parameters of the LUSMare nonlinear and time varying, a computed force control systemusing ENN is designed to improve the control performance forthe tracking of various reference trajectories. The ENN with bothonline learning and excellent approximation capabilities is em-

ployed to estimate a nonlinear function including the lumped un-certainty of the moving table mechanism. Moreover, the Lyapunovstability theorem and the projection algorithm are adopted toensure the stability of the control system and the convergence ofthe ENN. Furthermore, an FPGA chip is adopted to implementthe developed control algorithm for possible low-cost and high-performance industrial applications. The experimental resultsshow that excellent positioning and tracking performance areachieved, and the robustness to parameter variations and frictionforce can be obtained as well using the proposed control system.

Index TermsComputed force control, Elman neural network(ENN), field-programmable gate array (FPGA), linear ultrasonicmotor (LUSM).

I. INTRODUCTION

THE TECHNOLOGIES concerning biotechnology,

semiconductor-, nanotechnology, etc., have recently

experienced innovation in high-precision accuracy. Therefore,

the precise positioning technologies are required in many fields

particularly for the advancement of the semiconductor manu-

facturing equipment. These types of manufacturing equipment

require high-precision positioning, multidimensional drive,

miniaturization, and lightweight. Piezoelectric ceramic linear

ultrasonic motors (LUSMs) are one of the new kinds of ultra-

sonic motors which are driven by the ultrasonic vibration force

of piezoelectric ceramic elements and the mechanical friction

effect [1][3]. Different constructions and driving principles ofLUSMs have been reported [3], [4]. They permit high precision,

fast control dynamics, and large driving force in small

dimensions. Thus, some precision motion control stages using

LUSMs, which are small, light, and high-precise positioning,

have been developed for industrial applications [5], [6]. How-

Manuscript received April 23, 2008; revised September 11, 2008. Firstpublished October 31, 2008; current version published April 1, 2009. This workwas supported by the National Science Council, Taiwan, under Grant NSC95-2221-E-008-177-MY3.

The authors are with the Department of Electrical Engineering, NationalCentral University, Chungli 320, Taiwan (e-mail: [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TIE.2008.2007040

ever, the control accuracy of the LUSMs is much influenced by

the existence of uncertainties, which usually comprises parame-

ter variations, external disturbances, high-order dynamics, etc.

Moreover, the frictional force dynamics between two moving

bodies should also be considered in the linear actuator systems.

Furthermore, the mathematical models of piezoelectric ceramic

LUSMs are complex, and the motor parameters are time vary-

ing due to increasing in temperature and changing in motor

drive operating conditions. In short, the control characteristics

of the LUSMs are complicated and highly nonlinear. Therefore,

the intelligent control approaches are good candidates tocontrol the single-axis motion control stage using LUSMs

for high-performance applications due to their ability of to

approximate nonlinear and time-varying system without using

the mathematical models of the controlled system.

There have been many researchers using intelligent control

approaches to represent complex plants and construct advanced

controllers [7]. Although the Elman neural network (ENN) was

first proposed for speech processing [8] it has been widely

applied in dynamical systems identification and control due to

its distinguished dynamical characteristics [9][11]. Generally,

ENN can be considered as a special kind of feedforward

neural network with additional memory neurons [8]. Due to

the context neurons, it has certain dynamical advantages overstatic neural network, such as multilayer perceptrons and radial-

basis function networks (RBFNs). Furthermore, the ENN can

approximate a high-order system with high precision and fast

convergence speed.

The recurrent neural networks such as recurrent RBFN

and recurrent neural network (RNN), which are always self-

feedback in hidden-layer neurons or in output neurons, have

received increasing attention due to its structural advantage

in nonlinear system modeling and dynamic system control

[5], [12], [13]. The most important characteristic of recurrent

neural networks is its self-connection to memorize feedback

information of the history influence in the same neuron. In thispaper, the adopted ENN can be considered as a special type of

recurrent neural network with feedback connections from the

hidden layer to the context layer in which the context layer is

an addition layer and used as an extra memory to memorize

previous activations of the hidden neurons and feed to all the

hidden neurons after one-step time delay. Therefore, compared

to the general recurrent neural networks, ENN has a special

explicit memory to store the temporal information. Moreover,

in the recurrent RBFN and RNN, the specific self-connection

feedback of the hidden neuron or output neuron is responsible

to memorize the specific previous activation of the hidden

neuron or output neuron and feed to itself only. Therefore,

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LIN et al.: FPGA-BASED COMPUTED FORCE CONTROL SYSTEM USING ENN FOR LINEAR ULTRASONIC MOTOR 1239

the outputs of the other neurons have no ability to affect the

specific neuron. However, in the complicated and nonlinear

dynamic system, the inherent cross-coupled interference and

effect of each state are always existent and serious. Hence,

if each neuron in the recurrent neural networks is considered

as a state in nonlinear dynamic systems, the self-connection

feedback type is unable to approximate the dynamic systemsefficiently. On the other hand, the feedbacks in ENN not only

are self-connection but also store in the context neurons and

feed to all the hidden neurons. Thus, the structure of ENN is

more powerful than general recurrent neural networks to deal

with nonlinear dynamic systems due to the cross-coupled inter-

ference and effect of each state can be approximated efficiently

with the additional context layer [13], [14]. Therefore, since

the nonlinear function including the lumped uncertainty of the

LUSM dynamics is very complicated, the ENN is more suitable

than the other recurrent neural networks to be adopted as a

nonlinear function estimator.

Field-programmable gate array (FPGA) incorporates the ar-

chitecture of gate arrays and the programmability of a program-

mable logic device. It consists of thousands of logic gates, some

of which are combined together to form a configurable logic

block thereby simplifying high-level circuit design. Intercon-

nections between logic gates using software are externally de-

fined 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 fa-

vorite choice for prototyping an application-specified integrated

circuit (ASIC) [15], [16]. All the internal logic elements and all

the control procedures of the FPGA are executed continuously

and simultaneously. On the other hand, the control algorithmis executed sequentially using software in a digital signal

processor (DSP) or personal computer (PC), and the minimum

execution time is limited. Therefore, the execution time of the

FPGA is faster than DSP and PC. Furthermore, the circuits

and algorithms can be developed in the Very High Speed

Integrated Circuit Hardware Description Language (VHDL)

[15], [16]. This method is as flexible as any software solution.

Another important advantage of VHDL is that it is technol-

ogy independent. The same algorithm can be synthesized into

any FPGA and even has a direct path to an ASIC to open

interesting possibilities in industrial applications in terms of

performance and cost. However, the major disadvantage of anFPGA-based system for hardware implementation is the limited

capacity of available cells. Therefore, in the past decade, only

research on FPGA-based sliding mode, proportional-integral

differential, and fuzzy controllers can be found in control

application literatures [17][21]. Recently, some FPGA-based

motor applications using neural networks have been found [22],

[23]. Nevertheless, the FPGA-based neural network control

systems proposed by the aforementioned literatures do not

possess the online learning capability. In this paper, the FPGA

chip is adopted to implement the proposed neural network

control system with online learning capability in order to allow

possible low-cost and high-performance industrial applications.

In addition, the developed VHDL code can easily be modifiedand implemented to control any type of ac motors as well.

In this paper, an FPGA-based computed force control system

using ENN is developed to control the mover position of

an LUSM to track periodical step command, sinusoidal, and

trapezoidal reference trajectories with accuracy tracking per-

formance. First, in order to obtain the simplified control design,

the single-axis dynamics of the LUSM with the introduction

of a lumped uncertainty, which includes parameter variations,external disturbances, and friction force, are derived. Then, a

computed force control system using ENN is proposed where

the ENN with accurate approximation capability is employed to

estimate a nonlinear function including the lumped uncertainty

of the moving table mechanism. Moreover, a robust compen-

sator is proposed to confront the minimum approximation error.

Furthermore, an adaptive learning algorithm for the online

training of the ENN is derived using the Lyapunov theorem to

guarantee closed-loop tracking stability. In addition, to ensure

the convergence of the ENN, the adaptive learning algorithm is

further modified using the projection algorithm. Additionally,

the circuits design of the proposed intelligent control system

via an FPGA chip is discussed. Finally, experimentation is

carried out to investigate the effective of the proposed control

scheme.

II. LUSM MOTION CONTROL STAGE

A. Motion Control Stage

The motion control stage can be divided into two parts: the

LUSM servodrive and the motion control stage. The AB1A-

SP-E1 drive manufactured by Nanomotion Corp. is adopted for

the single-axis servodrive. Moreover, the motion control stage

comprises one SP series LUSM, the SP1-02 LUSM, which isalso manufactured by the Nanomotion Corporation. The travel

of the moving table of the motion control stage is linear. The

moving table is directly driven using the spacer of the LUSM.

The structure of the SP series LUSMs is a large face of

a relatively thin rectangular piezoelectric ceramic device, as

shown in Fig. 1(a) [3], [4]. Four electrodes (A, A, B, and B)are bounded to the front face to form a checkerboard pattern

of rectangles, and each substantially covers one quarter of this

face. The rear face is substantially fully covered with a single

electrode. Diagonally located electrodes (A, A, B, and B)are electrically connected by wires. The single electrode on

the rear face is grounded via the tuning inductor that canchange the resonant frequency. The movement of LUSM is

constrained by four support springs with large stiffness. These

support springs along a pair of long edges of the LUSM contact

the piezoelectric ceramic at points of zero movement in the

x-direction. A relatively hard ceramic spacer is attached withcement to a short edge of piezoelectric ceramic at the center

of the edge. A preload spring is preferably pressed against the

middle of a second short edge of piezoelectric ceramic opposite

to the short edge with the spacer. The preload spring supplies

pressure between the spacer and the moving table that causes

the motion of the spacer to be transmitted to the moving table.

A friction force takes place at the surface between the spacer

and the table. The face of the spacer pressed against the table isdesigned to move away from the table during part of the cycle



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Fig. 1. Operating principles of LUSM. (a) Structure of piezoelectric ceramicLUSM. (b) Vibration modes of LUSM.

when the spacer is moving opposite to the direction of motion

applied to the table.

The operation of the SP series LUSMs are based on the

double-mode vibrations, i.e., bending and longitudinal modes,

as shown in Fig. 1(b) [3], [4]. Using the bending mode in the

x-direction and the longitudinal mode in the y-direction leads

to the desired elliptical motion of the surface points in thexy-plane. The left-hand picture in Fig. 1(b) shows the defor-mation of the LUSM for the bending mode. A cross-sectional

area in the xz-plane is displaced in the x-direction and revolvedabout the z-axis. The displacement and the rotation of the cross-sectional area are denoted as x and , respectively. The vi-brating nodes of this mode are in the middle, up, and down parts

of the LUSM. The right-hand picture in Fig. 1(b) shows the

deformation of the LUSM for the longitudinal mode. A cross-

sectional area in the xz-plane is displaced in the y-direction,and the displacement of the cross-sectional area is denoted as

y. The longitudinal mode has a vibrating node in the middleof the LUSM. In order to obtain maximum displacements, the

modes have to be excited in resonance. An elliptical motion canonly be obtained if the resonant frequencies of the two modes

are equal or at least very close to each other. Thus, one has to

choose the length and the width of the piezoelectric ceramic to

determine the resonant frequency.

The output of the controller is the command voltage (con-

trol effort) of the LUSM servodrive. Moreover, the command

voltage with 10 V is applied to the drive AB1A-SP-E1. The

drive generates a fixed frequency (39.6 kHz) sine wave voltagewhose amplitude (0260 Vrms) is a function of the applied

command voltage (010 V). Furthermore, the generated sine

wave voltage drives the LUSM, and then generates thrust force

and velocity to rotate the moving table to reach the desired

position. When the command voltage is positive, the electrodes

A and A, shown in Fig. 1(a), are electrified, and B and B,shown in Fig. 1(a), are left floating (or grounded), and the

moving table moves to the right; when the command voltage

is negative, the electrodes B and B are electrified, and A andA are left floating (or grounded), and the moving table movesto the left.

B. Modeling of Motion Control Stage

For a friction drive system of single-axis LUSM mechanism,

according to the Newtonian law, the dynamic motion equation

can be assumed to take the following form:

FD = Md(t) + Dd(t) + (f(v) + FL) = KfU (1)

where M denotes the equivalent mass of the single-axis mech-anism; D denotes the viscous friction; FL is the externaldisturbance term; d denotes the position of the moving stage;

v denotes the velocity of the moving stage; d denotes thesecond-order derivative of d with respect to time; Kf denotesthe nominal voltage-to-force coefficient of an LUSM drive; Udenotes the control effort, i.e., the thrust voltage command;

and f is the lumped friction force includes the static friction,Coulomb friction, and viscous friction between the moving

table and the sliders. The friction force can be formulated as

follows [24], [25]:

f(v) = FCsgn(v) + (FS FC)e(v/vs)

2

sgn(v) + Kvv (2)

where FC is the Coulomb friction; FS is the static friction; vS is

the Stribeck velocity parameter; Kv is the coefficient of viscousfriction; and sgn() is a sign function. All the parameters in (2)are time varying.

Although the thrust force of LUSM can be simplified, as

shown in (1), however, considering the variation of system

parameters and external disturbances including friction force,

the motion control stage is a nonlinear time-varying system in

practical applications.

III. PROPOSED COMPUTED FORCE CONTROL SYSTEM

In order to control the motion control stage, a computed

force control system with ENN estimator is developed, asshown in Fig. 2. Assuming that the system parameter variations



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Fig. 2. Block diagram of computed force control system with ENN estimator.

and external disturbance are absent, then LUSM drive can be

formulated by rewriting (1) as follows:

d(t) = D

Md(t) +

Kf

MU And(t) + BnU (3)

where An = D/M; Bn = Kf/M > 0. The overbar symbolrepresents the system parameter in the nominal condition. Now,

considering the existence of parameter variations and external

force disturbances including friction force for the LUSM drive

system, then

d(t) = (An+A)d(t)+ (Bn+B)U+(Cn+C) [FL+f(v)]

= And(t)+ BnU+L(t) (4)

where Cn = 1/M; A, B, and C denote the uncertain-ties introduced by system parameters M and D; and L(t) iscalled the lumped uncertainty defined as

L(t) = Ad(t) + BU + (Cn + C) [FL + f(v)] . (5)

Here, the bound of the lumped uncertainty is assumed to be

known, i.e.,

|L(t)| (6)

where is a given positive constant. To confront these unpre-dictable uncertainties existing in the LUSM drive system, a

computed force control system is proposed.

The control problem is to find a control law so that the state

d(t) can track the desired command dm(t) accurately underthe occurrence of the uncertainties. To achieve this control

objective, define the tracking error and its derivative as follows:

e(t) = dm(t) d(t) (7)

e(t) =

dm(t)

d(t) (8)e(t) = dm(t) d(t). (9)

Substitute (9) into (4), then

e(t) = dm(t) And(t) BnU L(t). (10)

According to (10), the perfect equivalent control law U of Ucan be designed as follows:

U = B1n

dm(t) And(t) L(t)

+ z(t) (11)

where z(t) is designed as z(t) = z1e + z2e, in which z1 and z2are nonzero positive constants. Substituting (11) into (10), the

following error dynamic equation can be obtained:

e + Bnz1e + Bnz2e = 0 (12)

which implies limt0 e(t) = 0. However, the products of Bnand zi are difficult to obtain since Bn is an unknown systemparameter. Therefore, an auxiliary proportional-derivative (PD)

controller u(t) is adopted to replace z(t) to obtain the samecontrol performance. The PD controller u(t) is designed asu(t) = k1e + k2e, in which k1 and k2 are nonzero positiveconstants. Moreover, rewrite the perfect equivalent control

law (11) as

U = W + u(t) (13)

where the nonlinear function W is defined as

W = B1n

dm(t) And(t) L(t)

. (14)

Then, the proposed ideal computed force control law is

designed as

Ueq = U

= W + u(t). (15)

However, the ideal computed force control law cannot be

obtained in practical applications since the nonlinear function

W is unknown owing to the existence of the uncertainties.Therefore, the stability of the control system cannot be guar-

anteed. For this reason, the computed force control law is

redesigned to approximate the perfect equivalent control law

via the estimation ofW using ENN.

A. Design of ENN

The nonlinear function W contains the effects of uncer-

tainties including mechanical parametric variations, externaldisturbances, and friction force. Since the parameter variations

of the system are difficult to measure, and the exact values of

the external disturbances and friction force are also difficult

to know in advance for practical applications, the control law

shown in (15) cannot be implemented practically. Therefore,

a practical computed force control law Ueq is proposed toapproximate Ueq as follows:

Ueq = W + u(t) (16)

where the intelligent control law W is adopted to learn thenonlinear function W and defined as

W = WENN + Ur (17)



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Fig. 3. Network structure of ENN.

where WENN is an ENN estimator which is used to learn the

nonlinear equation W, and Ur is a robust compensator whichis designed to compensate the minimum approximation errorbetween W and WENN. Therefore, the practical computed forcecontrol law in (16) can be rewritten as

Ueq = WENN + Ur + u(t). (18)

Therefore, the practical control effort for the LUSM can be

obtained using the equation (18) based on the designed practical

computed force control law as follows:

U = Ueq. (19)

Moreover, using (18), the error dynamic equation (10) can be

rewritten as

e(t) = Bn(W Ueq)

= Bn

W WENN Ur u(t)

. (20)

Thus, an error equation is obtained as follows:

E = E+B(W WENN Ur) (21)

whereE= [e e]T; = [0 1

Bnk2 Bnk1 ] is a stable matrix;

andB = [0 Bn]T.

The structure of the ENN is shown in Fig. 3, including the

input layer (i layer), the hidden layer (j layer), the contextlayer (r layer), and the output layer (o layer). In the network,the hidden layer neurons are fed by the outputs of the context

neurons and the input neurons. Context neurons are known

as memory units as they store the previous output of hidden

neurons. Signal propagation and the basic function of each layer

are introduced in the following:

Layer 1 (Input layer): In the input layer, the node input and

the node output are represented as

ei(k) = fi(neti) = neti, i = 1, 2 (22)

where neti represents the ith input to the input layer. In thispaper, the inputs of the ENN are the tracking error and its

derivative, i.e., [e1(k) e2(k)]T = [e e]T.

Layer 2 (Hidden layer): In the hidden layer, the receptive

field function is a sigmoid function S(x), and the node inputand the node output are represented as

xj(k) = S(netj) =1

1 + enetj, j = 1, 2, . . . , 9. (23)

Moreover

netj =r

xcr(k) +i

W1ij neti, r = 1, 2, . . . , 9

(24)

where netj and xj(k) are the input and the output of the hiddenlayer, respectively; xcr(k) is output of the context layer; W

1ij is

the connective weight of input neuron to hidden neuron.

Layer 3 (Context layer): In the context layer, the node inputand the node output are represented as

xcr(k) = xj(k 1). (25)

Layer 4 (Output layer): In the output layer, the node input

and the node output are represented as

yo(k) = f(neto(k)) = neto(k) = WENN (26)

neto(k) =j

W2jo xj(k) (27)

where yo(k) is the output of the ENN; W2jo is the con-

nective weight of hidden neuron to output neuron; =[W211 W

2j1] R

1j is the adjustable weight vector of W2jo;

and = [x1(k) xj(k)]T Rj1 is the vector ofxj(k).

B. Robust Compensator

According to the universal approximation property, there

exists an optimal ENN estimator WENN to estimate thenonlinear function W such that

W = WENN

+ h = + h (28)

where h is the minimum reconstructed error and the absolutevalue ofh is assumed to be less than a small positive constant ,i.e., |h| < ; and is the optimal parameters of . Re-writing (17), one can obtain

W = WENN + Ur = + Ur (29)

where is the real value of as provided by tuning algorithm

to be introduced. Subtracting (29) from (28), an approximation

error W is defined as

W = W W = + h Ur = + h Ur(30)



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where = . According (21) and (30), the errorequation can be represented as

E=E+B(W WENN Ur)

=E+ BW

=E+B(+ h Ur). (31)

Theorem 1: Consider the LUSM drive system represented

by (4). If the computed force controller is designed as (16),

the intelligent control law is designed as (17), in which the

adaptation law of the ENN is designed as (32), and the robust

compensator is designed as (33) with an estimation algorithm

shown in (34), then the stability of the controlled system can be

guaranteed

= 1E

TPBT (32)

Ur = h(t) (33)

h(t) = 2E

TPB (34)

where 1 and 2 are positive constants; = [W211

W2j1] R1j ; and h(t) is the estimated value ofh.

Proof: Choose a Lyapunov function candidate as

VE, h(t),

=

1

2ETPE+

1

21(

T) +

1

22h2(t)

(35)

where h(t) = h h(t) denote the estimated error and

P = [P1 00 P2

] is a symmetric positive definite matrix which

satisfies the following Lyapunov equation:

TP+ P = Q (36)

and Q > 0. Differentiating (35) with respect to time andusing (31), one can obtain

VE, h(t),

=

1

2ETPE+

1

2ET

PE1

1

T

1

2h(t)

h(t)

=1

2ETP

E+B(+ h Ur)

+1

2

E+B(+ h Ur)

TPE

1

1

T

1

2h(t)

h(t)

=1

2ETPE+

1

2ETTPE

+ETPB(+ h Ur)

11

T 1

2h(t) h(t)

= 1

2ETQE

+

ETPB

1

1

T

+ETPB(hUr)1

2

h(t)h(t). (37)

If the adaptation law of the ENN is chosen as (32), and the

robust compensator is designed as (33) with the estimation

algorithm (34), then (37) can be rewritten as

VE, h(t),

=

1

2ETQE+ETPB

h h(t)

1

2h(t)

h(t)

= 1

2ETQE+ETPBh(t) h(t)ETPB

=

1

2ET

QE 0. (38)

Since V(E, h(t), ) 0, V(E, h(t), ) is negative semi-definite, i.e., V(E, h(t), ) V(E, h(0), ), which impliesthat E, h(t), and are bounded. Define function (t) =1/2ETQE V, and integrate the function (t) with respectto time

t0

()d VE, h(0),

V

E, h(t),

. (39)

Because V(E, h(0), ) is bounded, and V(E, h(t), ) is non-

increasing and bounded, one can obtain

limt

t0

()d < . (40)

Differentiate (t) with respect to time

(t) = ETQE. (41)

Since all the variables in the right-hand side of (31) are

bounded, it implies E is also bounded. Then, (t) is uniformlycontinuous [26]. By using Barbalets lemma [26], it can be

shown that limt (t) = 0. That is, E 0 as t . Asa result, the proposed control system is asymptotically stable.Moreover, the tracking error of the control system will converge

to zero according to E 0.

C. Online Parameters Learning of ENN

In order to train the ENN effectively, an online parameters

learning algorithm, which is derived using Lyapunov stability

theorem and the gradient descent method, is proposed. First,

the connecting weights between the input and the hidden layer

of ENN are adjusted online, and the adaptation law shown in

(32) can be rewritten as follows [27]:

= 1E

TPBT 1eP2BnT. (42)



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Although Bn is unknown for the LUSM drive system, anappropriate positive constant can be selected to replace P2Bn,which is a positive constant, as part of the learning rate in both

(34) and (42). According to the gradient descent method, the

update law of the connecting weights between the hidden and

output layer can be represented as follows:

= 1e

T = 1V

WENN

WENNyo(k)

yo(k)

W2jo

= 1V

WENNxj(k). (43)

Thus, the Jacobian of the controlled system V/WENN =e. The approximated error term needs to be calculated andpropagated by the following equation:

jo = V

WENN

WENNyo(k)

yo(k)

xj(k)

xj(k)

netj

= e W2joxj(k) (1 xj(k)) . (44)

The update law of W1ij can be obtained also by the gradientdescent method, i.e.,

= 1

V

W1ij

= 1 VWENN

WENNyo(k)yo(k)xj(k)

xj(k)netj

netj

W1ij

= 1joei(k) (45)

where = [W111 . . . W 11j , W

121 . . . W

12j ] R

2j1; =

[W111 . . . W11j, W

121 . . . W

12j ] R

2j1 and 1 is thelearning rate.

D. Stability Analysis Using Projection Algorithm

According to the assumption of the bound of the minimum

reconstructed error, the stability proof of the control systemcan be guaranteed using Theorem 1. However, the convergence

condition of the ENN must be satisfied. If the parameters of

the ENN are bounded, the convergence property of the ENN

can be guaranteed. Moreover, according to (26), the output of

the ENN is bounded if the weights between the output layer

and the hidden layer are bounded. Define the constraint set

b for as

b = Mb

(46)

where is a two-norm of vector and Mb is the upperbound of . Therefore, to guarantee b, the adapta-

tion law (32) is modified using the projection algorithm [28]

as follows:

=

1ETPBT

if< Mb or= Mb and E

TPBTT

0

1E

T

PBT

1ET

PB

TT

2

if= Mb and E

TPBTT

> 0

.

(47)

The upper bound is selected to achieve the best transient

control performance in the experimentation considering the

possible variation of operating conditions. According to the

projection algorithm, the convergence property of the ENN

can be guaranteed. Thus, it is reasonable to assume that the

minimum reconstructed error to be bounded. Furthermore, the

asymptotical stability of the proposed control system using

the projection algorithm can be shown by the following

theorem.Theorem 2: Consider the LUSM drive system represented

by (4). If the computed force controller is designed as (16),

the intelligent control law is designed as (17), in which the

adaptation law of the ENN is designed as (47), and the robust

compensator is designed as (33) with an estimation algorithm

shown in (34), then the stability of the controlled system can be

guaranteed.

Proof: When the conditions < Mb or ( = Mb

and ETPBTT 0) hold, the stability analysis is the

same as Theorem 1. On the other hand, when = Mb and

ETPBTT

> 0, the derivative of the Lyapunov function

shown in (37) can be rewritten as (48), which is shown at thebottom of the next page.

Use the property ( )T

= (1/2)(2 2

2) < 0, which is according to = Mb > ,

then

VE, h(t),

=

1

2ETQE+ETPB

( )T

2

T

1

2ETQE+

1

2ETPB

2 2 2

2T

1

2ETQE 0. (49)

Then, by using of the Barbalats lemma, as shown in Theorem 1,

E 0 as t can be obtained as well. Thus, the stabilityproperty also can be guaranteed.

IV. CIRCUITS DESIGN ON FPGA CHI P

The FPGA-based control system for a single-axis motion

control stage is shown in Fig. 4. The proposed control systemis realized on a 24-MHz FPGA (XC2V1000) with 1 million



8/16


Fig. 4. FPGA-based control system for motion control stage using LUSM drive.

gate counts and 10 240 flip-flops from Xilinx, Inc. using VHDL

with the Q-format arithmetic representation. Integrated Soft-

ware Environment version 8.1i from Xilinx, Inc. is adopted to

develop the FPGA chip. A linear scale with resolution 1 mis adopted to detect the position of the moving table. To

implement the computed force control system with the ENN

VE, h(t),

=

1

2ETQE+ETPB(h Ur)

1

2h(t)

h(t) +

ETPB

1

1

T

= 1

2ETQE+ETPB

h h(t)

h(t)ETPB

+

( )

ETPB

1

1

1E

TPBT 1ETPB

T

T

2

= 1

2ETQE+ETPBh(t) h(t)ETPB

+ETPB (

)T

2T

(48)



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Fig. 5. Circuits design on FPGA. (a) Timing control module. (b) Encoder interface module. (c) Auxiliary control, robust compensator, and computed forcecontroller of CFENN module. (d) Input, hidden, and context layer of CFENN module. (e) Output layer of CFENN module. (f) Online parameter learning andadaptation law of CFENN module.

estimator, the timing control module, encoder interface module,

and the computed force control and ENN (CFENN) module are

realized on the FPGA chip, respectively. Moreover, to show

the effectiveness of the control system with small number of

neurons, the ENN that has two, nine, nine, and one neurons at

the input, hidden, context, and output layers, respectively. Therequired processing time of the control algorithm is 0.341 ms

(2929 Hz). Two D/A converters are utilized to display the

reference trajectory dm, the mover position d, the controleffort U, the tracking error e, the estimated value using ENNWENN, and connected weights alternately on the oscilloscopeby changing the definition of I/O pins in VHDL code. The

entire I/O port for this chip includes two pins for the inputports and 36 pins for the output port. Furthermore, 4804 out of



10/16


Fig. 6. Experimental results of computed force control system with ENN estimator for periodical step command reference trajectory at nominal condition.(a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value ofW using ENN. (e) Connected weight W2

51between hidden layer and output

layer. (f) Connected weight W115

between input layer and hidden layer.

10 240 flip-flops (46%) have been used in the FPGA chip. In

addition, the used slices of the encoder interface module, the

data and D/A controllers, and the CFENN module are 102, 37,

and 3417 out of 5120, respectively; the used gate counts of the

encoder interface module, the data and D/A controllers, and theCFENN module are 1964, 711, and 62 162, respectively. Addi-

tionally, the circuits and control algorithm in the FPGA are de-

veloped using VHDL by a PC as the development system. In the

development of the VHDL codes for a 12-bits FPGA proces-

sor, by using fractional numbers between 1 and 1 with theQ-format representation, the results of multiplication can easily

be handled since the product of two numbers is always in the

same range. Besides, the sine function and multiplier are imple-

mented using available intellectual property (IP).

A. Timing Control Module

The block diagram of the timing control module, which iscomposed of one 13-bit counter and one events generator, is

shown in Fig. 5(a). In the timing control module, the counter

is used to generate various counts for the generation of various

control signals to enable (EN) the respective circuit. Moreover,

the sampling frequency 2929 Hz, which is used in the encoder

interface circuit, is also generated via the counter. Furthermore,various control signals are generated using the events generator

to EN the respective hardware circuit in different modules to

meet the specific timing control sequence.

B. Encoder Interface Module

The block diagram of encoder interface module is shown in

Fig. 5(b), which consists of timing control, two digital filters, a

decoder, an updown counter, a register, a command generator,

and one adder. The function of the encoder interface is to obtain

the position 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 throughtwo digital filters. The position signal d can be obtained using



11/16


Fig. 7. Experimental results of computed force control system with ENN estimator for periodical sinusoidal reference trajectory at nominal condition.(a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value ofW using ENN. (e) Connected weight W2




the PLS and DIR signals through updown counter. Moreover,

the command generator includes periodical sinusoidal IP and

periodical trapezoidal IP in order to generate the reference

trajectory dm. Furthermore, v is the velocity signal, and resultsfrom the difference between the position signal d and the time

delay of d, ddelay. In addition, the resolution of the encoder is1 mm = 1000 digital value at the sampling frequency 2929 Hz.

C. CFENN Module

To implement the control law effectively, the CFENN mod-

ule is divided into four parts which are shown in Figs. 5(c)(f):

1) Fig. 5(c) shows the block diagram of auxiliary control,

robust compensator, and computed force controller, using dm,d, WENN, and h(k) to obtain the control effort U. First, usingdm and d to obtain e and e. Moreover, the auxiliary controlinput u(t) based on the PD controller is obtained using an adder

and two multipliers. Then, the robust compensator Ur with theestimation algorithm

h(k) shown in (33) and (34) which is

equivalent to h(k) = 2e is obtained using an adder, tworegisters, and two multipliers. Finally, the control effort U basedon (18) is obtained using an adder; 2) in order to reduce the

computational load and facilitate the hardware implementation

using FPGA, a piecewise continuous function is selected as the

receptive field function for the ENN to approximate the sigmoid

function in the following:

xj(k) =

1, if netj > 0.5/a0, if netj < 0.5/aa netj + 0.5, otherwise

j = 1, 2, . . . , 9 (50)

where a is the slope of linear function. Fig. 5(d) shows theblock diagram of the input, hidden, and context layers utilized

to obtain the output of the hidden layer x1(k) x9(k) usinge, e, and xcr(k) based on (22), (24), (25), and (50). First, two

multipliers are utilized to multiply the outputs of the input layer,e and e, with the weights between the input layer and the hidden



12/16


Fig. 8. Experimental results of computed force control system with ENN estimator for periodical trapezoidal reference trajectory at nominal condition.(a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value ofW using ENN. (e) Connected weight W2




layer W111 W119 and W

121 W

129 individually. Then, add the

summation of the output of the context layer to result in net1 net9 based on (24). Moreover, the outputs of each neuron in the

hidden layerx1(k) x9(k)

are generated using the piecewise

function shown in (50) where all the slopes are 1.0; 3) Fig. 5(e)

shows the block diagram of the output layer utilized to obtain

the output of the ENN, WENN, using x1(k) x9(k) based on(26) and (27). First, two multiplexers with a multiplier are uti-

lized to multiply the outputs of the hidden layer x1(k) x9(k)with the weights between the hidden layer and the output layer

W211 W291 individually. Then, the output WENN is obtained

through a register and an adder; and 4) Fig. 5(f) shows the

block diagram of online parameters learning algorithm utilized

to achieve the adaptation law of the weights between the hidden

layer and the output layer based on (42) which is equivalent

to W2j1 = 1e xj(k), and the weights between the input

layer and the hidden layer based on (45). Since the sigmoidfunction has been approximated as (50), (44) is modified to

jo = e W2joa in the experimentation. In Fig. 5(f), first, the

variations of the weights W211 W291 are obtained by a

multiplexer, three multipliers, and a register; the variations of

the weights W1

11 W1

19 and W1

21 W1

29 are obtainedby two multiplexers, four multipliers, and two registers. Then,

the new weights W211(k + 1) W291(k + 1), W

111(k + 1)

W119(k + 1) and W121(k + 1) W

129(k + 1) are obtained by

adding the weights of current iteration W211(k) W291(k) with

W211 W291, W

111(k) W

119(k) with W

111 W

119 and

W121(k) W129(k) with W

121 W

129 individually.

V. EXPERIMENTAL RESULTS

The control objective is to control the mover of the LUSM

to move periodically according to the reference trajectories

including step command, sinusoidal, and trapezoidal com-

mands using the computed force control system with ENNestimator. Moreover, two test conditions are provided in the



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Fig. 9. Experimental results of computed force control system with ENN estimator for periodical step command reference trajectory at parameter variationcondition. (a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value of W using ENN. (e) Connected weight W2

51between hidden layer

and output layer. (f) Connected weight W115


experimentation, which are the nominal condition and the pa-

rameter variation condition. The parameter variation condition

is the addition of one iron disk with the mass 3.66 kg on the

moving table.

Some experimental results are provided to demonstrate the

effectiveness of the proposed FPGA-based control systems.

The control objective is to control the mover to move 1 mm

periodically for step command reference trajectory, 1 mmperiodically for sinusoidal reference trajectory, and 1 mm pe-

riodically for trapezoidal reference trajectory. The step com-

mand reference trajectory is obtained using a first-order transfer

function with periodical step function as input. A first-order

transfer function 2/(s + 2) with rise time 1 s is adopted tosmooth the step function. Figs. 611 show the experimental

results of the command tracking due to the periodical step com-

mand, sinusoidal, and trapezoidal reference trajectories of the

proposed computed force control system with ENN estimator.The tracking responses of the mover position due to periodical

step command reference trajectory at nominal condition and

parameter variation condition are shown in Figs. 6(a) and 9(a);

the associated control efforts are shown in Figs. 6(b) and

9(b); the tracking errors are shown in Figs. 6(c) and 9(c); the

estimated values ofW using ENN are shown in Figs. 6(d) and9(d); the responses of one of the connected weights between the

hidden layer and the output layer of the ENN, W251, are shownin Figs. 6(e) and 9(e); the responses of one of the connected

weights between the input layer and the hidden layer of the

ENN, W115, are shown in Figs. 6(f) and 9(f), respectively. Goodpositioning performance is achieved from the experimental

results, as shown in Figs. 6(a) and 9(a). Moreover, the tracking

responses of the mover position due to the periodical sinusoidal

reference trajectory at nominal condition and parameter varia-

tion condition are shown in Figs. 7(a) and 10(a); the associated

control efforts are shown in Figs. 7(b) and 10(b); the tracking

errors are shown in Figs. 7(c) and 10(c); the estimated values ofW using ENN are shown in Figs. 7(d) and 10(d); the responses



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Fig. 10. Experimental results of computed force control system with ENN estimator for periodicalsinusoidal reference trajectoryat parameter variationcondition.(a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value ofW using ENN. (e) Connected weight W2




of one of the connected weights between the hidden layer and

the output layer of the ENN, W251, are shown in Figs. 7(e) and10(e); the responses of one of the connected weights between

the input layer and the hidden layer of the ENN, W115, are shownin Figs. 7(f) and 10(f), respectively. Furthermore, the tracking

responses of the mover position due to the periodical trape-

zoidal reference trajectory at nominal condition and param-

eter variation condition are shown in Figs. 8(a) and 11(a); the

associated control efforts are shown in Figs. 8(b) and 11(b); the

tracking errors are shown in Figs. 8(c) and 11(c); the estimated

values of W using ENN are shown in Figs. 8(d) and 11(d);the responses of one of the connected weights between the

hidden layer and the output layer of the ENN, W251, are shownin Figs. 8(e) and 11(e); the responses of one of the connected

weights between the input layer and the hidden layer of the

ENN, W115, are shown in Figs. 8(f) and 11(f), respectively.From the experimental results, good tracking responses of the

mover can be obtained at both the nominal and the parametervariation conditions for the proposed computed force control

system with ENN estimator, as shown in Figs. 6(a), 7(a), 8(a),

9(a), 10(a), and 11(a). It also shows that the proposed FPGA-

based computed force control system using ENN has effective

hardware online learning ability of the network parameters and

is robust with regard to the parameter variations and friction

force.

VI. CONCLUSION

Since the dynamic characteristics of the LUSM are non-

linear and time varying, and the precise dynamic model is

difficult to obtain, an FPGA-based computed force control

system using ENN has been proposed to control the position

of the moving table of the LUSM to achieve high-accuracy

position control in this paper. First, the LUSM drive system

was introduced. Next, the dynamics of LUSM mechanism with

the introduction of a lumped uncertainty, which includes the

friction force, was derived. Since the dynamic characteristicsand motor parameters of the LUSM are nonlinear and time



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Fig. 11. Experimental results of computed force control system with ENN estimator for periodical trapezoidal reference trajectory at parameter variationcondition. (a) Tracking response. (b) Control effort. (c) Tracking error. (d) Estimated value of W using ENN. (e) Connected weight W2

51between hidden

layer and output layer. (f) Connected weight W115


varying, a computed force control system with ENN estimator

was designed to improve the control performance in reference

trajectories tracking. Then, the Lyapunov stability theorem and

the projection algorithm were adopted to ensure the stability

of the control system and the convergence of the ENN, and to

derive the adaptation laws of the control system. Moreover, the

online parameters learning algorithm of the ENN was derived

using gradient decent method for the purpose of real-time

estimation. Furthermore, to verify the effectiveness of the pro-

posed control scheme, the computed force control system with

ENN estimator was implemented in an FPGA-based control

system. Finally, the effectiveness of the proposed low-cost high-

performance FPGA-based LUSM drive has been confirmed by

some experimental results. The experimental results show that

excellent positioning and trajectories tracking performance are

achieved. In addition, the robustness to parameter variations

and friction force can be obtained as well using the proposedcontrol system.

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Faa-Jeng Lin (M93SM99) received the Ph.D.degree in electrical engineering from National TsingHua University, Hsinchu, Taiwan, in 1993.

From 1993 to 2001, he was an Associate Pro-fessor and then a Professor with the Department ofElectrical Engineering, Chung Yuan Christian Uni-versity, Chungli, Taiwan. From 2001 to 2003, he wasChairperson and a Professor with the Department

of Electrical Engineering, National Dong Hwa Uni-versity, Hualien, Taiwan. He was Dean of Researchand Development from 2003 to 2005 and Dean of

Academic Affairs from 2006 to 2007 at the same university. Currently, he isa Professor with the Department of Electrical Engineering, National CentralUniversity, Chungli. He is also the Chair, Power Engineering Division, NationalScience Council, Taiwan, from 2007 to 2009. His research interests include acand ultrasonic motor drives, DSP-based computer control systems, fuzzy andneural network control theories, nonlinear control theories, power electronics,and micromechatronics.

Dr. Lin received the Outstanding Research Professor Award from ChungYuan Christian University in 2000; the Crompton Premium Best Paper Awardfrom the Institution of Electrical Engineers (IEE), U.K., in 2002; the Outstand-ing Research Award from the National Science Council, Taiwan, in 2004; theOutstanding Research Professor Award from National Dong Hwa Universityin 2004; and the Outstanding Professor of Electrical Engineering Award in2005 from the Chinese Electrical Engineering Association, Taiwan. He was

also the recipient of the Distinguished Professor Award from National CentralUniversity in 2007. He is a Fellow of the Institution of Engineering andTechnology (formerly IEE).

Ying-Chih Hung was born in Taipei, Taiwan, in1984. He received the B.S. degree in electricalengineering from Chung Yuan Christian University,Chungli, Taiwan, in 2006, and the M.S. degree inelectrical engineering from the National Dong HwaUniversity, Hualien, Taiwan, in 2008. He is currentlyworking toward the Ph.D. degree in electrical engi-neering at National Central University, Chungli.

His research interests include field-programmablegate arrays, linear ultrasonic motors, motion control,and intelligent control theories.

Syuan-Yi Chen was born in Changhua, Taiwan,in 1982. He received the B.S. degree in electricalengineering from National Kaohsiung University ofApplied Science, Kaohsiung, Taiwan, in 2004, andthe M.S. degree in industrial educationfrom NationalTaiwan Normal University, Taipei, Taiwan, in 2006.He is currently working toward the Ph.D. degree inelectrical engineering at National Central University,Chungli, Taiwan.

His research interests include magnetic levitationsystems, active magnetic bearing systems, nonlinear

control theories, and intelligent control theories.

FPGA-Based Computed Force Control System

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