FPGA Implementation of a Fuzzy Controller for Neural Network Based Adaptive Control of a Flexible Joint with Hard Nonlinearities

Abstract

A control strategy based on artificial networks (ANN) has been proposed for a positioning system with a flexible transmission element, taking into account Coulomb friction for both motor and load, and using a variable learning rate for adaptation to parameter changes and to accelerate convergence. The control structure consists of an ANN that approximates the inverse of the model and of a reference model which defines the desired error dynamics. A fuzzy rule based supervisor for on-line adaptation of the reference model bandwidth parameter is used to accelerate the convergence rate of the controller and enhance the stability to the system. The fuzzy controller is implemented on a Virtex2 Pro 2VP30 field programmable gate array (FPGA) from Xilinx. A pipelined implementation is used to speed-up the process. Simulation results highlight the performance of the controller

Full text

Page 1

FPGA Implementation of a Fuzzy Controller for
Neural Network Based Adaptive Control of a
Flexible Joint with Hard Nonlinearities

Hicham Chaoui1, Mustapha C.E. Yagoub2 and Pierre Sicard3
1,2 School of Information Technology and Engineering (SITE), Ottawa, Ontario, Canada, K1N 6N5
3 Groupe de recherche en électronique industrielle, Université du Québec à Trois-Rivières, Canada, G9A 5H7
1 Hicham.Chaoui@uqtr.ca , 2 Myagoub@site.uottawa.ca , 3 Pierre.Sicard@uqtr.ca

Abstract – A control strategy based on artificial networks
(ANN) has been proposed for a positioning system with a
flexible transmission element, taking into account Coulomb
friction for both motor and load, and using a variable learning
rate for adaptation to parameter changes and to accelerate
convergence. The control structure consists of an ANN that
approximates the inverse of the model and of a reference model
which defines the desired error dynamics. A fuzzy rule based
supervisor for on-line adaptation of the reference model
bandwidth parameter is used to accelerate the convergence rate
of the controller and enhance the stability to the system. The
fuzzy controller is implemented on a Virtex2 Pro 2VP30 Field
Programmable Gate Array (FPGA) from Xilinx. A pipelined
implementation is used to speed-up the process. Simulation
results highlight the performance of the controller.

I. INTRODUCTION

onlinear friction is an important aspect for many control
systems for high quality servomechanisms and can lead
to tracking errors, limit cycles, etc. [1] Many control laws
have been proposed for flexible joints, including classical,
robust and adaptive control laws, using techniques such as
singular perturbations and energy methods (e.g. [2]) but they
are generally limited to the presence of viscous friction only.
Similarly, many publications deal with modeling and
compensation of friction in positioning systems (e.g. [3], [4])
and new results are published regularly. Several models and
compensation schemes have been proposed, adaptive control
being one of the main avenues of research. However, few
results are available for systems that exhibit both flexibility
and hard nonlinearities.
The presence of hard nonlinearities such as Coulomb
friction on a load driven through a flexible shaft significantly
changes the control problem [5], [6]: the inverse model of
the system is not realizable as discontinuity in motor
position would be needed to track exactly a load position
reference and the motor position is not uniquely defined at
standstill. This last condition also illustrates that motor state
cannot be observed continuously from the load output.
Henceforth, only an approximate inverse model can be
found and model based exact compensation schemes cannot
be applied to this case. The ability of artificial neural
networks (ANNs) to approximate nonlinear systems is
exploited in this work.
A control strategy proposed in [7] and [8] uses ANNs as
black boxes, as opposed to be model specific, to take
advantage of their approximation abilities. This scheme
consists of feedforward ANN based adaptive controllers that
can be viewed as a model reference adaptive controller (Fig.
1). The reference model is implemented in a similar way as a
sliding line in sliding mode control and its output is used as
an error signal to adapt the weights of the ANN controller. It
consists of a first order model that defines the desired
dynamics of the error between desired and actual load
positions and between motor and load velocities to assure
internal stability. The feedforward ANNFF approximates the
inverse model of the positioning system.
The contribution of this paper is to propose a VLSI
implementation of a fuzzy mode controller to adapt the
bandwidth parameter ρ that represents the desired closed
system loop to improve the convergence properties of the
adaptation process and increase the stability region of the
controller in [7] and [8] (Fig. 1). For the sake of clarity, the
description of the system under study and its model, the
basic control strategy and the reference model described in
[7] and [8] are presented in Section 2, 3 and 4. Section 5
describes the supervisory
implementation is described in Section 6, and simulation
results are presented and commented in Section 7.
Concluding remarks are given in Section 8.

II. FLEXIBLE JOINT MODEL

The flexible joint model under study consists of a dc
motor driving a load through a gearbox represented by an
equivalent linear torsion spring (Fig. 2). The electrical time
constant of the motor is neglected, but we consider the inner
hard nonlinearities relative to the motor and load caused by
Coulomb friction and stiction.

loop approach. FPGA
** ,
LLθθ
&
,
LLθθ
&
M
θ&
S
Reference
Model
Motor+load
T τ
** ,
LLθθ
&
,
LLθθ
&
ANNFF

Fig. 1. Feedforward control structure.
N

Page 2

FMMM
τ
,
ω
,
θ

(k)
(N:1)
FLLL
τωθ
,,
Motor
Load
Transmission
JL
JM

Fig. 2. Flexible joint model.

The mathematical model is described by

ω
FM
M
J
dt
θ
M
dt

ω
T FL
L
J
dt
θ
L
dt

)/(
LMT
Nk
θθτ−=

where subscripts L, M and T are relative to the load, motor and
transmission quantities; subscript F denotes friction; time
dependant variables are angular position θ, velocity ω and
torque τ; N is the gear ratio; efficiency η is assumed unity
and the stiffness coefficient k is supposed constant.
Superscript * will denote the desired trajectory.
Several studies have been published on friction modeling,
e.g. [4]. The most widely used model for memoryless
friction is described by
[
⎪⎩
(
0
uF
Fu
uuF
>
if
]/[
1
ττ
η
τ
MT
M
N
d
+−−=
−

ω
M
d
=
][
1
ττ
L
d
+−=
−

ω
L
d
=


−
()
()
]
⎪⎨
⎧
=
≠⋅⋅−++
=
0)( if)
0)( if))(()(
)(
)(
2
tv
tvtvsigneFFFtvF
t
svtv
cscv
F
τ
s
ss
s
s
s
Fu
FuF
F
F
≤≤
−<
⎪⎩
⎪⎨
⎧−
=
-if
if
)(
0


where Fc, Fv and Fs are the Coulomb, viscous and static
friction parameters respectively, ν is the velocity, νs is the
rate of decrease of the static friction term and u is the
effective torque input:
M
ητ−
for the load. No a priori knowledge of the parameters is
assumed.
N
T
τ
for the motor and
T τ

Remarks:
- In a single stage speed reduction system, the assumed
lumped flexibility model can be used if the dominant flexion
appears in the gear teeth; then, the inertia of the input gear is
added to the motor inertia JM and the inertia of the output
gear to the load inertia JL. This same approximation has been
used for multi-stage reduction systems such as planetary
gears (e.g. [7], [8] and references therein).
- Multi-mass flexibility models have also been used, for
example for harmonic drives, resulting into more accurate
but higher order models.
- At zero velocity, the effective friction value depends
explicitly on control input value. Henceforth, the value of
the control input to maintain zero velocity is not unique.

III. CONTROL STRATEGY

ANN’s parallel structure, learning and generalization
capabilities provide interesting properties for application to
advanced intelligent control. ANNs have been proven to be
universal approximators of nonlinear functions, which make
them highly suitable for development and implementation of
inverse control laws.
A feedforward signal can be obtained to compensate for
the nonlinearities and for flexion by training an ANN
(ANNFF in Fig. 1) to learn an inverse model of a positioning
mechanism. Due to the presence of Coulomb friction and of
the flexible coupling, the inverse model of the system is not
realizable in some operating conditions so that only an
approximate model can be found.
A first order reference model that is described in Section 4
defines the desired dynamics of the error between the
desired and actual load positions, with an additional term to
improve internal stability. The system cannot follow exactly
the reference model as it would entitle a reduction of the
relative order of the system. Our control objectives are
asymptotic tracking and internal stability.

IV. REFERENCE MODEL

The control strategy is based on the design of a feedback
term that must not only improve precision of the system, but
also improve internal stability. To insure output stability, we
would like for

S1 =
)()(
LLLL
θθρθ
&
θ
&
−+−
∗∗
(1)

to converge to zero, where ρ represents the desired
bandwidth, whilst for internal stability, we would like for

S2 =
)()(
MMMM
θθρθ
&
θ
&
−+−
∗∗
(2)

to converge to zero, assuming that the motor position
reference signal, which is not defined by the user, is stable.
Given that the motor position and velocity reference signals
are not readily available and that their evaluation is highly
dependent on the system model, we define the desired
behavior of the drive, using (1) and (2), as S→0 with

(
)1 (
MLL
S
θλθλθ−−−=
&&&
)
)(/
LL
N
θθρ−+
∗∗
(3)

where ρ represents the desired bandwidth of the closed loop
system and λ is a tuning parameter that trades off output
tracking performance for internal stability: decreasing λ
improves internal stability at the cost of lost of performance

Page 3

Z PS PM.PB
0 0.033 0.066 0.1 0.133
Z
0 3.33 6.66 1013.33
L
θ&
∆
B Z S M
0 100
ρ
L
θ∆
NS NM
NB NS NM
-0.133 -0.1 -0.066 -0.033
-13.33 -10 -6.66 -3.33
PS PMPB
NB

LL
θ
&
θ
∆∆
;
NB = Negative Big
NM = Negative Medium
NS = Negative Small
Z = Zero
PS = Positive Small
PM = Positive Medium
PB = Positive Big
Z = Zero
S = Small
M = Medium
B = Big
ρ

Fig. 3. Fuzzy sets definitions.

in the output response. This model is inspired from the
classical control laws for geared motors that include a motor
velocity feedback loop in addition to the load position and
velocity loops. Equation (3) defines the reference model for
the feedback controller.

V. FUZZY CONTROLLER APPROACH TO ADAPT ρ

Sliding mode control can deal well with parameter
uncertainties but is known for its sensitivity to unmodeled
dynamics. The motor dynamics can be seen as unmodeled
dynamics in (3) and cause oscillations in the response that
would not be present with a rigid joint. To achieve desired
performance, we propose to tune the parameter ρ in (3).
Bandwidth parameter ρ has an effect on control chattering
and with the flexible joint, it will have effects on motor state
oscillations. A fuzzy sliding mode control is introduced to
overcome this problem. The idea is to design a non-linear
gain ρ that varies with the dynamics of the system and
assumes large gain values only when it is really needed.
The fuzzy rules were chosen using heuristics and can be
refined by an expert (Table 1). The main idea is: (i) when S
is far away from the nominal sliding surface, then parameter
ρ takes a high value, giving a large weight to the error in
position; (ii) when S approaches the nominal sliding surface,
the gain is adjusted to a smaller value for a smoother
approach; (iii) once S is close on the nominal sliding line,
then parameter ρ takes a high value that corresponds to its
nominal value, for convergence along the sliding line.
The range and the functions for the input and output fuzzy
variables are defined in Fig. 3. Center of gravity method is
used for defuzzification.

VI. FPGA IMPLEMENTATION OF FUZZY CONTROLLER

Fuzzy logic algorithms are easily implemented on
computers. However, while custom fuzzy circuits require
long development time, they offer higher performance. As a
compromise or for validation purpose, FPGA-based systems
can reduce development time and operate at high speed.
Algorithms are described in languages like VHSIC hardware
description language (VHDL) and are verified by simulation.
Fig. 4 presents the structure of the fuzzy logic controller
for adaptation of parameter ρ as described in section 5.
The fundamental operation in fuzzy logic is fuzzification
which consists of the determination of the degree of
association of a variable to a fuzzy set. A key element in the
design of a high speed fuzzy processor is the availability of a
high speed fuzzifier to allow maximum processor speed.
For fuzzification, the inputs
produce the membership functions of the fuzzy sets. There
are 7 resulting fuzzy sets {NB, NM, NS, Z, PS, PM, PB}. So
we code the fuzzy sets into 3-bits fuzzy set code stored in the
variable state (Fig. 5), two states being active for each crisp
value of
L
θ∆
or
L
θ&
∆
. For each state, the value of the
membership function is stored in the variable power. The
fuzzification module of Fig. 5 can be divided into two parts.
The first part (Fig. 6 and 7) represents the triangular
membership function, which was chosen since it can be
easily modeled using the elementary line equation;
y=slope*x, where y represents the fuzzy value, slope
represents the gradient of the membership function, x
symbolizes the difference between the crisp input and the
intersection with the y-axis.

TABLE 1. ADAPTATION RULES OF MODEL BANDWIDTH PARAMETER ρ
L
θ∆
and
L
θ&
∆
are digitized to
L
θ∆

L
θ&
PB
∆
NB NM NS Z PS PM PB
B M M M M B B
PM
S B M S M M B
PS
Z S B S S M M
Z
M Z M B M Z M
NS
M M S S B S Z
NM
B M M S M B S
NB
B B M M M M B




Fuzzy logic
controller
ρ
L
θ∆
L
θ&
∆
Rst
Clk

Fig. 4. Fuzzy controller structure.

Page 4

Fuzzification
L
θ∆

Rst
Clk
1_ state
L
θ∆
1_ state
L
θ&
∆
2_ state
L
θ&
∆
2_ state
L
θ∆
1_ power
L
θ∆
1_ power
L
θ&
∆
2_ power
L
θ&
∆
2_ power
L
θ∆
L
θ&
∆


Fig. 5. Fuzzification structure.


IN
<
middle
OUT
MIN
<
OUT
IN
<
MAX

Fig. 6. Combo2/Combo1 structures for fuzzy set classification.


Combo
logic 2
P_x
MAX
IN
MIN
∑
∑
-
+
-
+
Combo
logic 1
X
M
U
X
slope
M
U
X
MAX
IN
MIN
middle

Fig. 7. Structure for fuzzy set membership evaluation.

The circuits in Fig. 6 are used to determine conditions
required to define the state variables. In particular, the
circuit Combo2 on the left determines if the crisp input value
is in a specific fuzzy set and the circuit Combo1 on the right
determines if this value is left or right of the center of the
specific fuzzy set.
The circuits of Fig. 6 are also used in the calculation of
membership values P_x, as illustrated in Fig. 7. The circuits
in Fig. 6 and 7 are repeated for the different fuzzy sets and x
corresponds to {NB, NM, NS, Z, PS, PM, PB}.
The second part of the fuzzification module is described in
Fig. 8 and 9. Combo3 and Combo4 in Fig. 8 are used to
define respectively variables state1 and state2. Variables
state1 and state2 are used along with the membership values
evaluated in Fig. 7 to determine the specific membership
function power1 and power2 of the active fuzzy sets in
state1 and state2. This is repeated for each input.
The fuzzy rules of Table 1 are evaluated by the circuit of
Fig. 10. For all four combinations of active input fuzzy sets
defined by the state variables, the active output fuzzy set is
defined in Fig. 11. A Min operation is performed on the

P_NS
P_PS
P_PB
Enable_NS
Enable_PS
Enable_PB
P_NM
P_Z
P_PM
Enable_MN
Enable_Z
Enable_PM

Fig. 8. Combo3/Combo4 structures.


Combo
logic 3
P_NB
power1
M
U
X
P_NS
P_PS
P_PB
M
U
X
M
U
X


Combo
logic 4
power2
P_NM
P_Z
P_PM
M
U
X
M
U
X
M
U
X

Fig. 9. Fuzzification module structure.



Rst
Clk
1_ state
L
θ∆
1_ state
L
θ&
∆
2_ state
L
θ&
∆
2_ state
L
θ∆
1_ power
L
θ∆
1_ power
L
θ&
∆
2_ power
L
θ&
∆
2_ power
L
θ∆
Fuzzy rules
Max_zero
Max_small
Max_medium
Max_large

Fig. 10. Fuzzy rules structure.

combination pairs of membership function values in Fig. 12.
The Min operation simply takes the minimum of two values.
A Max-Min composition is then carried out in Fig. 13 to
obtain output membership functions for ref = {Z, S, M, B}.
The defuzzification module of Fig. 14, detailed in Fig. 15,

Page 5

Rst
Clk
Look-up table
control
state
L_
θ∆

state
L_
θ&
∆


Fig. 11. Fuzzy table structure.




power
L_
θ∆
power
L_
θ&
∆
<
Inter_power
M
U
X

Fig. 12. Fuzzy rules module structure – Min operation.

M
U
X
>
Inter_power1
control
Inter_power2
M
U
X
ref
=
Max_ref

Fig. 13. Fuzzy rules module structure – Max-Min operation.



Rst
Clk
Defuzzification
Max_zero
Max_small
Max_medium
Max_large
ρ

Fig. 14. Defuzzification structure.

accepts four inputs each from the fuzzy rules module and
produces the defuzzified output signal. The defuzzification
method that is implemented is the Center of gravity, as
explained previously.

VII. SIMULATION RESULTS

The online adaptation scheme and the model of the system
were implemented in SimulinkTM and the fuzzy logic
controller algorithms were coded as a C program. The
feedforward neural network, the reference model, a learning
rate supervisor [8] and the fuzzy logic controller hardware
algorithms for adaptation of the reference model were
designed using VHDL.
Functional simulations were carried-out using ModelSim
and hardware synthesis was accomplished using Precision
RTL from Mentor Graphics targeting a Xilinx Virtex II Pro
FPGA device. Synthesis results are presented in Table 2.
Synthesis results without the fuzzy logic controller are
presented in Table 3. We note that the fuzzy controller only
requires an additional 13% of resources and that it does not
significantly add to the computation time.
Max_zero
zero
Max_small
small
Max_medium
Max_large
large
X
ρ
X
X
X
+
+

+
+
/
Max_zero
Max_small
Max_medium
Max_large
medium

Fig. 15. Defuzzification module details – center of gravity method.

TABLE 2: SYNTHESIS RESULTS OF THE COMPLETE CONTROLLER
Minimum period:
Maximum frequency:
Minimum input required time before clock:
Minimum output required time after clock:
FPGA resources used:

TABLE 3: SYNTHESIS RESULTS EXCLUDING THE FUZZY CONTROLLER
Minimum period:
Maximum frequency:
Minimum input required time before clock:
Minimum output required time after clock:
FPGA resources used:
25 ns
38 MHz
20 ns
18 ns
89 %
23.5 ns
42.5 MHz
17 ns
13 ns
76 %

The nominal parameters for the mechanism and
controllers are listed in Appendix. The input position and
velocity reference signals are presented in Fig. 16.
Simulation results for the nominal parameters with and
without the fuzzy logic controller are shown in Fig. 17.
Without the fuzzy logic controller, the reference model
could not be chosen with a large bandwidth without causing
internal instabilities or unacceptable control activity. The
value ρ = 40 has been chosen through repeated simulations
with variable operating conditions, in particular for different
values of inertia, as an acceptable upper value. We observe
that the maximum position error is comparable with or
without the fuzzy controller. However, the steady state error
decreased significantly with the use of the fuzzy logic
controller. We note a high activity of ρ during transients and
a stationary value in steady state.
Fig. 18 gives a representation of the response to interpret
the reference model in the context of a sliding line as
encountered in variable structure control; the velocity terms
in (3) are plotted as a function of the position term in (3).
The trajectory remains close to the line in the plane with
slope of about -40s-1, as defined by the fuzzy logic
controller. However, this is obtained without a high
frequency control signal as in sliding mode control.

VIII. CONCLUSION

An ANN based control structure VLSI implementation on