An Introduction to the Kalman Filter

TL;DR: The discrete Kalman filter as mentioned in this paper is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error.

Abstract: In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

Summary

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.
• For continuous rotation at the highest torque and e ciency, the stator ¯ux is rotated in the desired direction of motion, keeping 90° ahead of the rotor ¯ux.
• It is also a challenging problem because the measured winding currents are strongly a ected by electrical noise in the motor drive.

2.1. Current estimation

• Consider the problem of estimating a discrete-time signal fxg corrupted by noise.
• (The determination of the rate estimate is discussed in Section 2.2.).
• The fuzzy rule base for the mapping g was chosen as shown in Table 1.
• Triangular input and output membership functions are used.
• An appropriate initial knowledge base is critical, because without an initial knowledge the authors cannot proceed further with any optimization schemes.

2.2. Current rate estimation

• One of the inputs to the fuzzy estimator discussed above is the current rate estimate v̂ (see Eq. (4)).
• The method of undetermined coe cients is a simple but elegant approach to deriving formulas for numerical di erentiation.
• The ®rst part of Eq. (17) re¯ects the e ect of using current estimates that are separated too much in time to estimate the current rate.
• (The current is measured by an analog-to-digital converter [ADC] on the motor drive, so the acquired voltage is directly proportional to the motor winding current.).
• Again based on their knowledge of the current waveforms, the authors will assume that the standard deviation of the fourth derivative of the current is about 80 V/(ms)4.

3. Optimization

• If the fuzzy membership functions are triangular as shown in Fig. 1, gradient descent can be used to optimize the centroids and the widths of the input and output membership functions.
• The work in this section builds on and extends similar e orts in [5,8].
• The authors can optimize E by using the partial derivatives of E with respect to the centroids and half-widths of the input and output fuzzy membership functions.

3.4. Output half-widths

• Note from the above equations that if the authors start with symmetric output membership functions (i.e., bÿj b j ), then ox̂=obÿj ox̂=ob j .
• Therefore, if the authors start their optimization with symmetric output membership functions, they will always have symmetric output membership functions because the derivatives of the error function with respect to the lower and upper half-widths will always be equal.

4. Rule base reduction

• Wang et al. [9] have recently used an SVD method to reduce the dimension of the input space of a fuzzy system, assuming that the membership functions are B-splines.
• Consider a fuzzy rule base with two inputs a and b and a single fuzzy consequent r.
• Actually, UU and UV can be chosen as any invertible matrices whose row sums are equal to the column sums of Ur and Vr, respectively [10].
• The authors choose the above forms for ease of computation.
• The centroids of the consequents for the reduced rule base are de®ned in the matrix R.

5. Experimental results

• The fuzzy estimator and optimizer discussed in this paper was implemented in Visual Basic and was used to ®lter the winding currents of a permanent magnet synchronous motor.
• The fuzzy ®lter was causal and was implemented as described earlier in this paper.
• Fig. 3 shows the training data that was used for the gradient descent optimization.
• Fig. 4 shows the seven initial membership functions for the two inputs and the output.
• A comparison with Fig. 4 shows that the membership functions did not change dramatically during the optimization process.

6. Conclusion

• A fuzzy ®lter has been applied to the estimation of motor winding currents.
• The gradient descent optimization discussed in this paper is attractive because of its conceptual straightforwardness, but one of its primary disadvantages is its convergence to a local minimum.
• Further work on the topic of this paper is focusing on optimization methods that do better at ®nding the global minimum (e.g., genetic algorithms), integration of the ®ltering scheme with motor control, and real time implementation issues.
• This reduction could be important for real time implementation where cycle time is at a premium.
• A MATLAB m-®le for rule base reduction (based on the algorithms presented in [10] and summarized here) of a general two-input, oneoutput fuzzy logic system can be downloaded from http://csaxp.csuohio.edu/ simon/reduce/.

Figures

Fig. 8. Reduced membership functions for (a) input 1, (b) input 2, (c) output. For input 1 and input 2, dashed line negative, dotted line zero, and solid line positive.

Table 2 Reduced rule base for fuzzy ®ltera

Table 3 Error function values for test data

Fig. 6. Optimized membership functions for (a) input 1, (b) input 2, (c) output.

Fig. 2. Standard deviation of current rate estimation error.

Table 1 Rule base for fuzzy ®ltera

Fig. 4. Default membership functions for input 1, input 2, and output.

Fig. 5. Gradient descent training of membership functions.

Fig. 7. Test data: (a) un®ltered; (b) ®ltered with fuzzy estimator.

Fig. 1. Triangular membership function.

Fig. 3. Training data: (a) un®ltered; (b) 51-point moving average.

Full text

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 eective
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 ecient 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 eciency, 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 eective 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 aected 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
k1
 f x
k
v
k
; 1
z
k
 hx
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
gz
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
 gz
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
ÿinput 1
kÿ1
: 7
So the correction mapping depends on the dierence between the measurement
and the a priori estimate, and the amount by which that dierence 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 dicult 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 ÿx 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].
gz
k
;
^
x
ÿ
k

P
n
j1
my
j
y
j
J
j
P
n
j1
my
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 my is computed as
myfuzzy 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
yw
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

Frequently asked questions

Q1. What have the authors contributed in "Design and rule base reduction of a fuzzy ®lter for the estimation of motor currents" ?

This paper discusses and demonstrates the application of fuzzy ®ltering to motor winding current estimation in permanent magnet synchronous motors. The methods discussed in this paper are demonstrated on real motor winding currents that were collected with a digital oscilloscope. 

Q2. What have the authors stated for future works in "Design and rule base reduction of a fuzzy ®lter for the estimation of motor currents" ?

The fuzzy estimator o ers the possibility of training if a nominal current history is known a priori. Further work on the topic of this paper is focusing on optimization methods that do better at ®nding the global minimum ( e. g., genetic algorithms ), integration of the ®ltering scheme with motor control, and real time implementation issues. It is not di cult to program a general purpose rule base reduction algorithm if the authors can make the following assumptions: ( 1 ) There are an odd number of membership functions for the two inputs and the output ; ( 2 ) the membership functions are symmetric triangles ; and ( 3 ) they desire to keep the two largest singular values in the R matrix of Eq. ( 56 ). A MATLAB m-®le for rule base reduction ( based on the algorithms presented in [ 10 ] and summarized here ) of a general two-input, oneoutput fuzzy logic system can be downloaded from http: //csaxp. csuohio. edu/ simon/reduce/.