Kalman Filtering

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Kalman Filtering

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Title: Kalman Filtering

1
Kalman Filtering

It is an effective and versatile procedure for
combining noisy sensor outputs to estimate the
state of a system with uncertain dynamics.
Kalman filtering is a relatively recent (1960)
development in filtering, although it has its
roots as far back as Gauss (1795).
Kalman filtering has been applied in areas as
diverse as aerospace, marine navigation, nuclear
power plant instrumentation, demographic
modeling, manufacturing, and many others.
For Kalman filter the problem is formulated is
state space and is time varying.

2
Introduction

Consider the problem of estimating the variables
of a system. In dynamic systems (that is, systems
which vary with time) the system variables are
often denoted by the term "state variables".
Since its introduction in 1960, the Kalman filter
has become an integral component in thousands of
military and civilian navigation systems.
This deceptively simple, recursive digital
algorithm has been an early-on favorite for
conveniently integrating (or fusing) navigation
sensor data to achieve optimal overall system
performance.

The Kalman filter is a multiple-input,
multiple-output digital filter that can optimally
estimate, in real time, the states of a system
based on its noisy outputs.
The purpose of a Kalman filter is to estimate the
state of a system from measurements which contain
random errors. An example is estimating the
position and velocity of a satellite from radar
data. There are 3 components of position and 3 of
velocity so there are at least 6 variables to
estimate. These variables are called state
variables. With 6 state variables the resulting
Kalman filter is called a 6-dimensional Kalman
filter.
To provide current estimates of the system
variables - such as position coordinates - the
filter uses statistical models to properly weight
each new measurement relative to past
information.

4
How Kalman Filter Works?

The Kalman filter maintains two types of
variables
Estimate State Vector The components of the
estimated state vector include the following
The variables of interest (what we want or need
to know, such as position and velocity).
Nuisance variables that may be necessary to the
estimation process.
The Kalman filter state variables for a specific
application must include all those system dynamic
variables that are measurable by the sensors used
in the application.
A Covariance Matrix a measure of estimation
uncertainty. The equations used to propagate the
covariance matrix (collectively called the
Riccati equation) model and manage uncertainty,
taking into account how sensor noise and dynamic
uncertainty contribute to uncertainty about the
estimated system state.

By maintaining an estimate of its own estimation
uncertainty and the relative uncertainty in the
various sensor outputs, the Kalman filter is able
to combine all sensor information optimally in
the sense that the resulting estimate minimizes
any quadratic loss function of estimation,
including the mean-squared value of any linear
combination of static estimation errors.
The Kalman gain is the optimal weighting matrix
for combining new sensor data with a prior
estimate to obtain a new estimate.

6
Lecture 1 The Start

For the Kalman filter, the problem is formulated
in state space.
Consider a linear system u(t) and y(t) could be
scalars or vectors.
Each element of u(t) is a white noise.
We want to model y(t) as the response of a linear
system, where the system input is the unity power
spectrum white noise u(t). This implies E x(t)
0.

Linear System
u(t)
y(t)
7
Suppose we have a fourth order systemAssume n
4, then m n -1 3Based on the information
given in Chapter 3 of the State Space we can
write the matrix equation
Forward path
y(t)
u(t)
Feedback path
8
b3
b1
b2
1/s
1/s
1/s
1/s
bo
1
x1
x3
U(s)
Y(s)
x2
x4
-a1
-a0
-a3
-a2
9
Discrete-Time State Space ModelIn the Kalman
filtering it is customary to write w(k) and
y(k)We will repeat the process as before
Forward path
y(t)
u(t)
Feedback path
10
b3
b1
b2
1/s
1/s
1/s
1/s
bo
1
x1
x3
U(s)
Y(s)
x2
x4
-a1
-a0
-a3
-a2
11
Writing the former equations together using
matrix notation we obtain the controllable state
variable
12
Development of the Discrete Kalman Filter

There should be a discrete linear system.
The input is white noise.
The observations are the system output plus a
white noise called the measurement noise.
The system input noise and the measurement noise
are uncorrelated to each other.
You should know
The state space model for the system.
The second order statistics of the input noise.
The second order statistic of the measurement
noise.
The problem Given the noisy observations of the
output, find estimates of the system state vector.

13
What Makes Kalman Filter Different?

It is kind of like a mathematical proof by
induction.
Assume that we have obtained a prediction for the
state vector at time k and that this estimate is
based on the first k-1 observations.
In other words, assume that we have an estimate
of Xk given Zk-3, Zk-2, Zk-1. This is called a
priori estimate or prior of Xk the true state
vector at time k.
In books or other resources you may see it as

14
The Predicted State Vector Error Covariance Matrix
15
Lecture 2 State and Covariance Correction

The Kalman filter is a two-step process
prediction and correction.
The filter can start with either step but will
begin by describing the correction step first.
The correction step makes corrections to an
estimate, based on new information obtained from
sensor measurements.
The Kalman gain matrix K is the crown jewel of
Kalman filter. All the efforts of solving the
matrix is for the sole purpose of computing the
optimal value of the gain materix K used for
correction of an estimate x .

16
Filter Operation
Measurement Update (Correct)
Time Update
17
Gaussian Probability Density Function

PDFs are nonnegative integrable functions whose
integral equals unity. The density function of
Gaussian probability distributions have the form
given. Where n is the dimension of P (n?n
matrix), ? is the mean of the distribution. The
parameter P is the covariance matrix of the
distribution.

18
Likelihood Functions

Likelihood functions are similar to probability
density functions, except that their integrals
are not constrained to equal unity, or even
required to be infinite. They are useful for
comparing relative likelihoods and for finding
the value of the unknown independent variable x
at which the likelihood function achieves its
maximum.
Y is called the information matrix of the
likelihood function. It replaces P-1 in the
Gaussian probability density function. If the
information matrix Y is nonsingular, then its
inverse Y-1 P.

19
The purpose of a Kalman filter is to optimally
estimate the values of variables describing the
state of a system from a multidimensional signal
contaminated by noise
System Unknown multiple state variables
Multiple noise inputs
Sampled multiple output

Multiple noises
Multiply noisy outputs
Multidimensional signal plus noise
Multiple state Variable estimates
Kalman filter
20

The following figure illustrates the Kalman
filter algorithm itself. Because the state (or
signal) is typically a vector of scalar random
variables (rather than a single variable), the
state uncertainty estimate is a
variance-covariance matrix-or simply, covariance
matrix. Each diagonal term of the matrix is the
variance of a scalar random variable-a
description of its uncertainty. The term is the
variable's mean squared deviation from its mean,
and its square root is its standard deviation.
The matrix's off-diagonal terms are the
covariances that describe any correlation between
pairs of variables.
The multiple measurements (at each time point)
are also vectors that a recursive algorithm
processes sequentially in time. This means that
the algorithm iteratively repeats itself for each
new measurement vector, using only values stored
from the previous cycle. This procedure
distinguishes itself from batch-processing
algorithms, which must save all past
measurements.

Starting with an initial predicted state estimate
(as shown in the figure) and its associated
covariance obtained from past information, the
filter calculates the weights to be used when
combining this estimate with the first
measurement vector to obtain an updated "best"
estimate. If the measurement noise covariance is
much smaller than that of the predicted state
estimate, the measurement's weight will be high
and the predicted state estimate's will be low.
Also, the relative weighting between the scalar
states will be a function of how "observable"
they are in the measurement. Readily visible
states in the measurement will receive the higher
weights. Because the filter calculates an updated
state estimate using the new measurement, the
state estimate covariance must also be changed to
reflect the information just added, resulting in
a reduced uncertainty. The updated state
estimates and their associated covariances form
the Kalman filter outputs.

Finally, to prepare for the next measurement
vector, the filter must project the updated state
estimate and its associated covariance to the
next measurement time.
The actual system state vector is assumed to
change with time according to a deterministic
linear transformation plus an independent random
noise.
Therefore, the predicted state estimate follows
only the deterministic transformation, because
the actual noise value is unknown. The covariance
prediction ac-counts for both, because the random
noise's uncertainty is known.
Therefore, the prediction uncertainty will
increase, as the state estimate prediction cannot
account for the added random noise. This last
step completes the Kalman filter's cycle.

23
Compute weights from predicted states Covariance
and measurement noise covariance
Predicted initial state Estimate and covariance
New measurements each cycle
Predict state estimates and Covariance to next
time step
Update state estimates as Weighted linear blend
of predicted state estimates new measurement
Updated state estimates
Compute new covariance of updated state estimates
24
Mathematical Definitions

The variance and the closely-related standard
deviation are measures of how spread out a
distribution is. It is a measure of estimation
quality.
The covariance is a statistical measure of
correlation of the fluctuations of two different
quantities. Intuitively, covariance is the
measure of how much two variables vary together.
Least squares is a mathematical optimization
technique which, when given a series of measured
data, attempts to find a function which closely
approximates the data (a "best fit"). It attempts
to minimize the sum of the squares of the
ordinate differences (called residuals) between
points generated by the function and
corresponding points in the data. It is sometimes
called least mean squares.

25
Simple Example of Process Model

A simple hypothetical example may help clarify
the Kalman concepts. Consider the problem of
determining the actual resistance of a nominal
100-ohm resistor by making repeated ohmmeter
measurements and processing them in a Kalman
filter.
First, one must determine the appropriate
statistical models of the state and measurement
processes so that the filter can compute the
proper Kalman weights (or gains). Here, only one
state variable, the resistance, x is unknown but
assumed to be constant. So the state process
dynamics evolves with time as
Xk1 Xk. 1

Note that no random noise corrupts the state
process as it evolves with time. The color code
on a resistor indicates its precision, or
tolerance, from which one can deduce assuming
that the population of resistors has a Gaussian
or normal histogram that the uncertainty
(variance) of the 100-ohm value is, say, (2
ohm)2. So our best estimate of x, with no
measurements, is x0/ 100 with an uncertainty
of P0/ 4. Repeated ohmmeter measurement,
zk xk vk
2
directly yield the resistance value with some
measurement noise, vk (measurement errors from
turn-on to turn-on are assumed uncorrelated). The
ohmmeter manufacturer indicates the measurement
noise uncertainty to be Rk (1 ohm)2 with an
average value of zero about the true resistance.

27
How it Works?

Estimated State Vector including the variables of
interest, nuisance variables, and the Kalman
filter state variables for a specific
application.
A Covariance Matrix A measure of estimation
uncertainty. The equations used to propagate the
covariance matrix (called Riccatti equation)
model and manage uncertainty, taking into account
how sensor noise and dynamic uncertainty
contribute to uncertainty about the estimated
system state.
Kalman filter is able to combine all sensor
information optimally in the sense that the
resulting estimate minimizes any quadratic loss
function of estimation error.
The Kalman gain is the optimal weighting matrix
for combining new sensor data with a prior
estimate to obtain a new estimate.

28
For our purpose in ELG4152

The noisy sensors may include speed sensors
(wheel speeds of land vehicles, water speed
sensors for ships, air speed sensors for
aircraft, GPS receivers and inertia sensors, and
time sensors.
The system state may include the position,
velocity, acceleration, attitude, and attitude
rate of a vehicle on land, at sea, in the air, or
in the space.
Uncertain dynamics may include unpredictable
disturbances of the host vehicle, whether caused
by a human operator or by the medium (winds,
surface currents, turns in the road, or terrain
changes). It might include also unpredictable
changes in the sensor parameters.

29
The one dimensional Kalman Filter

Suppose we have a random variable x(t) whose
value we want to estimate at certain times t0
,t1, t2, t3, etc. Also, suppose we know that
x(tk) satisfies a linear dynamic equation.
x(tk1) Fx(tk) u(k) (the dynamic equation)
In the above equation F is state transition
matrix (in this example a known number) that
relates state at time step tk to time step tk1.
In order to work through a numerical example let
us assume F 0.9.
Kalman assumed that u(k) is a random number.
Suppose the numbers are such that the mean of
u(k) 0 and the variance of u(k) is Q. For our
numerical example, we will take Q to be 100.
u(k) is called white noise, which means it is not
correlated with any other random variables and
most especially not correlated with past values
of u.

In later lessons we will extend the Kalman filter
to cases where the dynamic equation is not linear
and where u is not white noise. But for this
lesson, the dynamic equation is linear and w is
white noise with zero mean.
Now suppose that at time t0 someone came along
and told you he thought x(t0) 1000 but that he
might be in error and he thinks the variance of
his error is equal to P. Suppose that you had a
great deal of confidence in this person and were,
therefore, convinced that this was the best
possible estimate of x(t0). This is the initial
estimate of x. It is sometimes called the a
priori estimate.
A Kalman filter needs an initial estimate to get
started. It is like an automobile engine that
needs a starter motor to get going. Once it gets
going it does not need the starter motor anymore.
Same with the Kalman filter. It needs an initial
estimate to get going. Then it will not need any
more estimates from outside.

We have an estimate of x(t0),which we will call
xe. For our example xe 1000. The variance of
the error in this estimate is defined by P E
(x(t0) -xe)2.
where E is the expected value operator. x(t0) is
the actual value of x at time t0 and xe is our
best estimate of x. Thus the term in the
parentheses is the error in our estimate. For the
numerical example, we will take P 40,000.
Now we would like to estimate x(t1). Remember
that the first equation we wrote (the dynamic
equation) was
x(tk1) Fx(tk) u(k).
Therefore, for k 0 we have x(t1) Fx(t0)
u(0).
Dr. Kalman says our new best estimate of x(t1) is
given by
New xe Fxe (Eq. 1) or in our numerical example
900.

We have no way of estimating u(0) except to use
its mean value of zero. How about Fx(t0). If our
initial estimate of x(t0) 1000 was correct then
Fx(t0) would be 900. If our initial estimate was
high, then our new estimate will be high but we
have no way of knowing whether our initial
estimate was high or low (if we had some way of
knowing that it was high than we would have
reduced it). So 900 is the best estimate we can
make. What is the variance of the error of this
estimate?
New P E (x(t1) new xe)2
Substitute the above equations in for x(t1) and
new xe and you get
New P E (Fx(t0) u - Fxe)2
E F2(x(t0) - xe)2 E u2 2F E (x(t0)-
xe)u
The last term is zero because u is assumed to be
uncorrelated with x(t0) and xe.

So, we are left with
New P PF2 Q (Eq. 2)
For our example, we have
New P 40,000 X .81 100 32,500
Now, let us assume we make a noisy measurement of
x. Call the measurement y and assume y is related
to x by a linear equation. (Kalman assumed that
all the equations of the system are linear. This
is called linear system theory.
y(1) Mx(t1) w(1)
where w is white noise. We will call the variance
of w, "R".
M is some number whose value we know. We will use
for our numerical example M 1 , R 10,000 and
y(1) 1200
Notice that if we wanted to estimate y(1) before
we look at the measured value we would use

ye Mnew xe
for our numerical example we would have ye 900
Dr. Kalman says the new best estimate of x(t1) is
given by
Newer xe new xe K(y(1) - Mnew xe)
new xe K(y(1) - ye) (Eq. 3)
where K is a number called the Kalman gain.
Notice that y(1) - ye is just our error in
estimating y(1). For our example, this error is
equal to plus 300. Part of this is due to the
noise, w and part to our error in estimating x.
If all the error were due to our error in
estimating x, then we would be convinced that new
xe was low by 300. Setting K 1 would correct
our estimate by the full 300. But since some of
this error is due to w, we will make a correction
of less than 300 to come up with newer xe. We
will set K to some number less than one.

What value of K should we use? Before we decide,
let us compute the variance of the resulting
error
E (x(t1) newer xe)2 E x new xe - K(y - M
new xe)2
E (x new xe - K(Mx w - M new xe)2
E (1 - KM) (x new xe)2 Kw2
new P(1 - KM)2 RK2
The cross product terms dropped out because w is
assumed uncorrelated with x and new xe. The newer
value of the variance is
Newer P new P (1 - KM)2 R(K2) (Eq. 5)
If we want to minimize the estimation error we
should minimize newer P. We do that by
differentiating newer P with respect to K and
setting the derivative equal to zero and then
solving for K. A little algebra shows that the
optimal K is given by
K M new P/ new P(M2) R (Eq. 4)
For our example, K .7647 Newer xe 1129
newer P 7647
Notice that the variance of our estimation error
is decreasing.

These are the five equations of the Kalman
filter. At time t2, we start again using newer xe
to be the value of xe to insert in equation 1 and
newer P as the value of P in equation 2.
Then we calculate K from equation 4 and use that
along with the new measurement, y(2), in equation
3 to get another estimate of x and we use
equation 5 to get the corresponding P. And this
goes on computer cycle after computer cycle.
In the multi-dimensional Kalman filter, x is a
column matrix with many components. For example
if we were determining the orbit of a satellite,
x would have 3 components corresponding to the
position of the satellite and 3 more
corresponding to the velocity plus other
components corresponding to other random
variables.
Equations 1 through 5 would become matrix
equations and the simplicity and intuitive logic
of the Kalman filter becomes obscured.