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The Kalman Filter
The Scalar Kalman Filter
This document gives a brief introduction to the derivation of a Kalman filter when the input is a
scalar quantity. It is split into several sections:
Defining the Problem Finding K, the Kalman Filter Gain
Finding the a priori covariance
Finding the a posteriori covariance
Review of Pertinent Results
Alternate, More Common, Notation
Examples
Going further
References
Defining the ProblemDiscrete time linear systems are often represented in a state variable format given by the
equation:
= + Equation 1where the state, xj, is a scalar, a and b are constants and the input ujis a scalar; jrepresents the
time variable. Note that many texts don't include the input term (it may be set to zero), and most
texts use the variable kto represent time. I have chosen to usejto represent the time variable
because we use the variable k for the Kalman filter gain later in the text. Equation 1 can be
represented pictorially as shown below, where the block with T in it represents a time delay.
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Figure 1
Now imagine some noise is added to the process such that:
= + + Equation 2The noise, wj, is white noise source with zero mean and covariance Q and is uncorrelated with the
input. The process can now be represented as shown:
Figure 2
Given a situation like the one shown above, a typical question might be: Can we filter the signal x
so that the effects of the noise w are minimized? The answer, it turns out is yes. However, with
Kalman filters we can go one step further.
Let us assume that the signalxis not directly measured, but instead we measure z.
= + Equation 3The measured value z depends on the current value of x, as determined by the gain
h. Additionally, the measurement has its own noise, v, associated with it. The noise, v, is white
noise source with zero mean and covariance R that is uncorrelated with the input or with the noise
w. The two noise sources are independent of each other and independent of the input.
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Figure 3
The task of the Kalman filter can now be stated as: Given a system such as the one shown above,
how can we filter z so as to estimate the variable x while minimizing the effects of w and v?
It seems reasonable to achieve an estimate of the state (and the output) by simply reproducing
the system architecture. This simple (and ultimately useless) way to get an estimate of xj (which
we will call x^j), is diagrammed below.
Figure 4
This approach has two glaring weakness. The first is that there is no correction. If we don't know
the quantities a, b or h exactly (or the initial valuex0), the estimate will not track the exact value of
x. Secondly, we don't compensate for the addition of the noise sources (w and v). An improved
setup which takes care of both of these problems is shown below.
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Figure 5
This figure is much like the previous one. The first difference noted is that the original estimate of
xj is now called .; we will refer to this as thea prioriestimate = + Equation 4
We use this a prioriestimate to predict an estimate for the output, . The difference betweenthis estimated output and the actual output is called the residual, or innovation.
= =
Equation 5
If the residual is small, it generally means we have a good estimate; if it is large the estimate is not
so good. We can use this information to refine our estimate of xj; we call this new estimate the a
posteriori estimate, . If the residual is small, so is the correction to the estimate. As theresidual grows, so does the correction. The pertinent equation is (from the block diagram):
= + = + Equation 6The only task now is to find the quantity kthat is used to refine our estimate, and it is this process
that is at the heart of Kalman filtering.
Note: We are trying to find an optimalestimator, and thus far we are only optimizing the value for
the gain, k. We have assumed that a copy of the original system (i.e., the gains a, b, and h
arranged as shown) should be used to form the estimator. This begs the question: "Is the
estimator as developed above optimal?" In other words, should we simply copy the original
system in order to estimate the state, or is there perhaps a better way? The answer turns out, is
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To find the value of k that minimizes the variance we differentiate this expression with respect to k
and set the derivative to zero. Be patient here, the expression gets much messier before it
becomes simple.
Equation 10
We take this last expression and use it to solve for k.
Equation 11
This expression is still quite complicated. To simplify it we will consider the numerator and the
denominator separately.
We start with the numerator, and substitute in equation 3 for zj.
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The measurement noise, v, is uncorrelated to either the input or the a priori estimate of x, so:
Equation 12
This simplifies the expression for the numerator.
Equation 13
Now, in the same way, consider the denominator.
Equation 14
Again, we can use the orthogonality condition from equation 12 to set the last term to zero, so:
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Equation 15
where we used the simplification from equation 13 for the first term in the expression, and using
the definition of the measurement noise for the second term.
Using the expression for numerator and denominator, we finally get a simple expression for k:
Equation 16
However, there is still a problem because this expression needs a value for the a priori covariance
which in turn requires a knowledge of the system variable xj. Therefore our next task will be to
come up with an estimate for the a priori covariance.
Before we move on, let's look at this equation in detail. First not that the "constant", k, changes
at every iteration. For this reason it should really be written with a subscript (i.e., kj). We'll be
more careful about this later.
Next, and more significantly, we can examine what happens as each of the three terms in
equation 16 are varied.
If the a priori error is very small, k is correspondingly very small, so our correction is also very
small. In other words we will ignore the current measurement and simply use past estimates to
form the new estimate. This is as expected -- if our first estimate (the a priori estimate) is good
(i.e., with small error) there is very little need to correct it.
If the a priori error is very large (so that the measurement noise term, R, in the denominator is
unimportant) then k=1/h. This, in effect, tells us to throw away the a priori estimate and use the
current (measured) value of the output to estimate the state. This is made clear by substitution
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into equation 6. Again, this is as expected -- if the a priori error is large then we should disregard
the a priori estimate, and instead use the current measurement of the output to form our
estimate of the state.
If the measurement noise, R, is very large, k is again very small, so we disregard the current
measurement in forming the new estimate. This is as expected -- if the measurement noise is
large, then we have low confidence in the measurement and our estimate will depend more upon
the previous estimates.
Finding the a priori covariance
Finding the a priori covariance is straightforward starting with its definition.
The middle term drops out as before because the process noise is uncorrelated with previous
values of the either the state or its a priori estimate.
Equation 17
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so
Equation 18
We are still not finished, however, because we need an expression for pj, the a posteriori estimate.
Finding the a posteriori covarianceAs with the a priori covariance, we find the a posteriori covariance by starting with its definition.
Equation 19
The middle term drops out as before because the measurement noise is uncorrelated with the
current values of the either the state or its a priori estimate.
Equation 20
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So
Equation 21
We can simplify this by using our previous definition for k (Equation 16 rearranged)
Equation 22
Substituting Equation 22 into Equation 21 yields
Equation 23
Review of Pertinent ResultsAny Kalman filter operation begins with a system description consisting of gains a, b and h. The
state isx, the input to the system is u, and the output is z. The time index is given byj.
The process has two steps, a predictor step (which calculates the next estimate of the state based
only on past measurements of the output), and a corrector step (which uses the current value of
the estimate to refine the result given by the predictor step).
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Predictor StepWe form the a priori state estimate based on the previous estimate of the state and the current
value of the input.
We can now calculate the a priori covariance
Note that these two equations use previous values of the a posteriori state estimate and
covariance. Therefore the first iteration of a Kalman filter requires estimates (which are often just
guesses) of the these two variables. The exact estimate is often not important as the values
converge towards the correct value over time; a bad initial estimate just takes more iterations to
converge.
Corrector Step
To correct the a priori estimate, we need the Kalman filter gain, k.
This gain is used to refine (correct) the a priori estimate to give us the a posteriori estimates.
We can now calculate the a posteriori covariance
Notes about the Kalman filter gain, kj.
If the a priori error is very small, k is correspondingly very small, so our correction is also very
small. In other words we will ignore the current measurement and simply use past estimates to
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form the new estimate. This is as expected -- if our first estimate (the a priori estimate) is good
(i.e., with small error) there is very little need to correct it.
If the a priori error is very large (so that the measurement noise term, R, in the denominator is
unimportant) then k=1/h. This, in effect, tells us to throw away the a priori estimate and use the
current (measured) value of the output to estimate the state. This is made clear by substitution
into equation 6. Again, this is as expected -- if the a priori error is large then we should disregard
the a priori estimate, and instead use the current measurement of the output to form our
estimate of the state.
If the measurement noise, R, is very large, k is again very small, so we disregard the current
measurement in forming the new estimate. This is as expected -- if the measurement noise is
large, then we have low confidence in the measurement and our estimate will depend more upon
the previous estimates.
Alternate, More Common, NotationThe notation used in this document was taken from [1]. More common notation is given below.
Variable Notation in this Document More Common Notation
time variable j kstate xj x(k)
system gains a, b, h a, b, h (note: b is often 0)
input uj u(k) (note: often there is no input)
output zj z(k)
gain kj Kk
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a priori estimate
The notation
can be read as "The estimate of x at time k, based on the information from time k-1"; in other
words, the estimate based only upon the past values of the output, or the a priori estimate. Thenotation
can be read as "The estimate of x at time k, based on the information from time k"; in other words
the estimate based on past andcurrent values of the output, or the a posteriori estimate
Examples
Example of estimating a constant (along with Matlab code).
Example of estimating a first order process (along with Matlab code).
Going further
A matrix based (higher order system) Kalman filter is a simple extension of the scalar case
presented here. The results are given here, a full description of the mathematics can be found in
the reference [3].
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Is the Kalman Filter Optimal?This document is split into several sections:
Defining the Problem
Finding the Constants
Reconciling with Previous Document
Defining the Problem
In the previous document we assumed that the best linear estimate for the state, xj, was given by
where
The question to be answered is: Can we prove that the second statement is true?
If we want to estimate the state we can use only the three quantities that we know, the previous
estimate, the current input and the current measured output. We use these three variables to
form a linear estimate of the state:
where aj and bj are two unknowns to be chosen to minimize the error between the value of the
stat and its estimate. In other words we want to minimize the expected value of the error ej with
respect to the variables aj and bj.
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Finding the ConstantsTo do the minimization with respect to each variable we simply differentiate and set the result to
zero. At this point we will only try to fin
which can be rewritten
These last two expressions are often referred to as the orthogonality conditions; i.e., the error is
orthogonal to the previous estimated state, the current input and the current value of the
measured output.
Let's use the first condition to find an expression for aj that minimizes the expected value of the
error. If we add and subtract ajxj-1 from the equation (why we do this will become clear shortly),
we get:
Now we can use the facts that
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to write
Note that because of the orthogonality relationships the first term on the right can be rewritten as
We also know that the previous estimate is uncorrelated with the current value of the
measurement noise:
So we can simplify the equation to the following
This is a complicated expression that we can use a bit later, but first we need to derive one more
expression. By following the same sequence of steps as is done above (but starting with the
secondequation in which we set the derivative to zero), it is easily shown that
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We can rewrite the last two equations
or, in matrix form
For a matrix equation
we know that either
So for the matrix equation above, either
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Reconciling with Previous DocumentNow recall Equation 6 from the previous document
Equation 6
From the previous document we know that the a priori estimate of the state is given by
and if we let
we can rewrite our last equation (at the end of the previous section)
which matches Equation 6.
We have shown that the Kalman filter represents the optimal linear filter. The other document
goes on to derive the optimal value for kj.