Design and rule base reduction of a fuzzy ®lter
for the estimation of motor currents
Dan Simon
Department of Electrical and Computer Engineering, Cleveland State University,
Stilwell Hall Room 332, 1960 E. 24th Street, Cleveland, OH 44115-2425, USA
Received 1 September 1999; accepted 1 June 2000
Abstract
Fuzzy systems have been used extensively and successfully in control systems over
the past few decades, but have been applied much less often to ®ltering problems. This
is somewhat surprising in view of the dual relationship between control and estima-
tion. This paper discusses and demonstrates the application of fuzzy ®ltering to motor
winding current estimation in permanent magnet synchronous motors. Motor winding
current estimation is an important problem because in order to implement eective
closed-loop control, a good estimation of the current is needed. Motor winding
currents are notoriously noisy because of electrical noise in the motor drive. We use a
fuzzy system with correlation-product inference and centroid defuzzi®cation for motor
winding current estimation. With the assumption that the membership functions are
triangular (but not necessarily symmetric), we then optimize the membership functions
using gradient descent. Next we use singular value decomposition to reduce the rule
base for the fuzzy ®lter. Rule base reduction can be important for fuzzy systems in
those cases where the fuzzy system needs to be implemented in real time. This is
especially true with regard to fuzzy ®ltering in a real time motor controller. The
methods discussed in this paper are demonstrated on real motor winding currents that
were collected with a digital oscilloscope. It is demonstrated that fuzzy techniques
provide a feasible approach to motor current estimation, that gradient descent
optimization improves the performance of the ®lter, and that rule base reduction
results in a relatively small degradation of ®lter performance. Ó 2000 Elsevier Science
Inc. All rights reserved.
International Journal of Approximate Reasoning 25 (2000) 145±167
www.elsevier.com/locate/ijar
E-mail address: d.simon@ieee.org (D. Simon).
0888-613X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 8 8 8 - 613X(00)00054-2
Keywords: Fuzzy logic; Filtering; Estimation; Optimization; Gradient descent; Rule
base reduction; Singular value decomposition; Motor
1. Introduction
The electrical windings of a permanent magnet synchronous motor are
spaced on the stator (the ®xed part of the motor) at regular angles. When
excited with current, the windings produce magnetic ¯uxes that add vectorially
to produce the stator ¯ux. The controlling variables are the proportions of
currents in the motor windings, which determine ¯ux magnitude and orienta-
tion. Rotating rotor magnets produce the rotor ¯ux and interact with the stator
¯ux to produce torque. When the stator and rotor ¯uxes are aligned, the
magnetic ®elds are in equilibrium at the minimum energy position and no
torque is produced. When the stator and rotor ¯uxes are not aligned, the rotor
magnets are pulled toward the stator electromagnets. This torque is maximum
when the rotor ¯ux is 90° behind the stator ¯ux in the direction of motion. At
this point the ¯ux vectors are said to be ®eld-oriented for maximum torque at a
given current. This is also the most ecient operating region of the motor,
because in this mode the power input to the mechanical side of the motor is
maximized. For continuous rotation at the highest torque and eciency, the
stator ¯ux is rotated in the desired direction of motion, keeping 90° ahead of
the rotor ¯ux. The stator ¯ux is produced by controlling the current in the
stator windings. Krause and Wasynczuk [1] provide a good overview of per-
manent magnet synchronous motors.
In order to implement an eective closed-loop current controller we need an
accurate estimate of the current [2]. Current estimation is thus an important
problem. It is also a challenging problem because the measured winding cur-
rents are strongly aected by electrical noise in the motor drive.
The motor's winding currents are generally shaped like sinusoids. Knowing
this, we can formulate common sense fuzzy membership functions for use in a
predictor±corrector type of estimator. The fuzzy winding current estimator is
recursive and non-linear. Its inputs comprises past estimates, and present and
past measurements. The use of fuzzy logic for motor winding current estima-
tion was ®rst explored by Simon [3].
We begin the fuzzy ®lter design process by gathering noisy experimental
motor winding current data from a motor. Next we construct initial mem-
bership functions for a fuzzy current estimator on the basis of common sense
and experience. We then use human expertise to guess the true motor currents
underlying the experimental data. Finally we use these ``true'' motor currents
as the basis with which to ®ne-tune the membership functions of the fuzzy
current estimator. The membership functions are ®ne-tuned (i.e., optimized)
using an iterative gradient-descent method.
146 D. Simon / Internat. J. Approx. Reason. 25 (2000) 145±167
After the membership functions are optimized, we can use singular value
decomposition (SVD) to reduce the rule set of the fuzzy estimator. Rule base
reduction is important in view of the challenge of real time implementation in a
digital signal processor. The SVD method of rule reduction generates appro-
priate linear combinations of membership functions in order to obtain new
membership functions for a reduced rule base.
The fuzzy estimator is applied to real motor winding currents in this paper.
The results presented establish fuzzy estimation as a viable option for stator
winding current estimation.
Section 2 gives a general algorithm for estimating a signal in the presence of
noise using a fuzzy ®lter. Section 3 presents a technique for optimizing fuzzy
membership functions using gradient descent, and Section 4 summarizes an
algorithm that can be used to reduce a fuzzy rule base. Section 5 contains
experimental results, and Section 6 contains some concluding remarks and
suggestions for further research.
2. Fuzzy estimation
We begin with a standard discrete, time-invariant system given by
x
k1
f x
k
v
k
; 1
z
k
hx
k
w
k
; 2
where k is the time index, x
k
the state vector, z
k
the measurement, and v
k
and w
k
are the noise processes. The problem of ®nding an estimate
^
x
k
for the state
vector based on past and present measurements is known as the a posteriori
®ltering problem. One commonly used estimator architecture is the recursive
predictor±corrector, given by
^
x
k
^
f
^
x
kÿ1
gz
k
;
^
x
kÿ1
; 3
where
^
f is an estimate of f , and g is the correction function. The
process model f is often known, or it can be found using system identi®-
cation methods. If
^
f is available, only the correction mapping g needs to
be determined. Various analytic methods have been used for obtaining the
correction mapping [4]. As an alternative to analytic methods, the correction
mapping could be implemented as a fuzzy function [5].
2.1. Current estimation
Consider the problem of estimating a discrete-time signal fxg corrupted by
noise. The fuzzy estimator structure that we use to obtain an estimate of the
signal is given by
D. Simon / Internat. J. Approx. Reason. 25 (2000) 145±167 147
^
x
ÿ
k
^
x
kÿ1
T
^
v
kÿ1
; 4
^
x
k
^
x
ÿ
k
gz
k
;
^
x
ÿ
k
; 5
where
^
x
ÿ
k
denotes the estimate of x at time k before the measurement at time k is
processed, and
^
x
k
denotes the estimate of x at time k after the measurement at
time k is processed. T is the update period of the estimator, z
k
the noisy
measurement of the winding current, and
^
v is the estimate of current rate. (The
determination of the rate estimate is discussed in Section 2.2.) The fuzzy cor-
rection mapping g has two inputs
input 1
k
z
k
ÿ
^
x
ÿ
k
; 6
input 2
k
input 1
k
ÿinput 1
kÿ1
: 7
So the correction mapping depends on the dierence between the measurement
and the a priori estimate, and the amount by which that dierence has changed
since the last time step. The output of the correction mapping is a fuzzy
variable which is determined by correlation-product inference. The fuzzy rule
base for the mapping g was chosen as shown in Table 1. In this paper, tri-
angular input and output membership functions are used.
The initial rule base and triangular membership functions were constructed
on the imprecise basis of experience, and trial and error. An appropriate initial
knowledge base is critical, because without an initial knowledge we cannot
proceed further with any optimization schemes. In spite of its importance, the
generation of initial knowledge remains a dicult and ill-de®ned task in the
construction of fuzzy logic systems.
In general, we denote the centroid and the two half-widths of the ith fuzzy
membership function of the jth input by c
ij
, b
ÿ
ij
, and b
ij
. The membership
function attains a value of 1 when the input is c
ij
. As the input decreases from
c
ij
, the membership function reaches a value of 0 at c
ij
ÿ b
ÿ
ij
. As the input
Table 1
Rule base for fuzzy ®lter
a
Input 1
Input 2
NL NM NS Z PS PM PL
NL NL NL NM NM NS NS Z
NM NL NM NM NS NS Z PS
NS NM NM NS NS Z PS PS
Z NM NS NS Z PS PS PM
PS NS NS Z PS PS PM PM
PM NS Z PS PS PM PM PL
PLZ PSPSPMPMPLPL
a
NL negative large, NM negative medium, NS negative small, Z zero, PS positive small,
PM positive medium, PL positive large.
148 D. Simon / Internat. J. Approx. Reason. 25 (2000) 145±167
increases from c
ij
, the membership function reaches a value of 0 at c
ij
b
ij
.A
generic triangular membership function is shown in Fig. 1. The degree of
membership of a crisp input x in the ith category of the jth input is given by
f
ij
x
1 x ÿ c
ij
=b
ÿ
ij
if ÿ b
ÿ
ij
6 x ÿ c
ij
6 0;
1 ÿx ÿ c
ij
=b
ij
if 0 6 x ÿ c
ij
6 b
ij
;
0 otherwise:
8
<
:
8
The fuzzy output is mapped into a crisp numerical value using centroid de-
fuzzi®cation [6].
gz
k
;
^
x
ÿ
k
P
n
j1
my
j
y
j
J
j
P
n
j1
my
j
J
j
; 9
where y
j
and J
j
are the centroid and area of the jth output fuzzy membership
function and n is the number of fuzzy output sets. (Note that for the triangular
membership functions that we are using, J
j
is equal to one-half of the sum of
the two half-widths of the jth output fuzzy membership function.) The fuzzy
output function my is computed as
myfuzzy output function
X
i;k
m
ik
y; 10
where m
ik
y is de®ned as the consequent fuzzy output function when input 1 is
in class i and input 2 is in class k.
m
ik
yw
ik
m
oik
y; 11
w
ik
is de®ned as the activation level of the consequent when input 1 is in class i
and input 2 is in class k, and m
oik
y is the fuzzy function of the consequent that
is activated when input 1 is in class i and input 2 is in class k.
Fig. 1. Triangular membership function.
D. Simon / Internat. J. Approx. Reason. 25 (2000) 145±167 149
