An introduction to scalar Kalman filters.pdf

8/9/2019 An introduction to scalar Kalman filters.pdf

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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 Problem

Discrete time linear systems are often represented in a state variable format given by the equation:

Equation 1

where, for this discussion, the state, x j, is a scalar, a and b are constants and the input u j is a scalar; j represents 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 k to

represent time. I have chosen to use j to represent the time variable because we use the variable k for the Kalman filter gain

later in the text. The equation states that the current value of the variable ( x j) is equal to the last value ( x j-1) multiplied by a

constant (a) plus the current input (u j) mulitiplied by another constant (b). Equation 1 can be represented pictorially as shown

below, where the block with T in it represents a time delay (the input is x j, the output is x j-1). Note: some books will use z-1 or

1/z in place of T .

Figure 1

Now imagine some noise is added to the process such that:

Equation 2

The noise, w, 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

troduction to scalar Kalman filters


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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 signal x is not directly measured, but instead we measure z which is x multiplied by a gain (h) with

noise added (v).

Equation 3

The 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 and is uncorrelatedwith the input or with the noise w. The two noise sources are independent of each other and independent of the input.

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 x j (which we will call x̂ 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 value x0), 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 arrangement 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 x j is now called x̂ j-; we will

refer to this as the a priori estimate.

Equation 4

We use this a priori estimate to predict an estimate for the output, ẑ j. The difference between this 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 x j; we call this new estimate the a posteriori estimate, x̂ j. If the residual is small, so is the

correction to the estimate. As the residual grows, so does the correction. The pertinent equation is (from the block diagram):

Equation 6

The only task now is to find the quantity k that 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 optimal estimator, 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 it turn out, is that the estimator, as shown above, is the optimal linear estimator that can be developed.

The details are here.

Finding K, the Kalman Filter Gain(you can skip the next three sections if you are not interested in the math).

To begin, let us define the errors of our estimate. There will be two errors, an a priori error, e j-, and an a posteriori error, e j.

Each one is defined as the difference between the actual value of x j and the estimate (either a priori or a posteriori).

Equation 7

Associated with each of these errors is a mean squared error, or variance:



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Equation 8

where the operator E{ } represents the expected, or average, value. These definitions will be used in the calculation of the

quantity k .

A Kalman filter minimizes the a posteriori variance, p j, by suitably choosing the value of k . We start by substituting equation

7 into equation 8, and then substituting in equation 6.

Equation 9

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 z j.

The measurement noise, v, is uncorrelated to either the input or the a priori estimate of x, so:



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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:

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 x j. Therefore our next task will be to find an estimate for the a priori covariance.

Before we move on, let's look at this equation in detail. First note that the "constant", k , changes at every iteration. For this

reason it should really be written with a subscript (i.e., k j). 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 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.



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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 themeasurement 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

so

Equation 18

We are still not finished, however, because we need an expression for p j, the a posteriori estimate.

Finding the a posteriori covariance

As with the a priori covariance, we find the a posteriori covariance by starting with its definition.



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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

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 Results

Any Kalman filter operation begins with a system description consisting of gains a, b and h. The state is x, the input to the

system is u, and the output is z. The time index is given by j.

The process has two steps, a predictor step (which calculates the next estimate of the state based only on past measurements



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of the output), and a corrector step (which uses the current value of the estimate to refine the result given by the predictor

step).

Predictor Step

We 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, k j .

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 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



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large, then we have low confidence in the measurement and our estimate will depend more

upon the previous estimates.

Alternate, More Common, Notation

The notation used in this document was taken from [1]. More common notation is given below.

VariableNotation in this

Document

More Common Notation

time variable j k

state x j x(k)

system gains a, b, h a, b, h (note: b is often 0)

input u j u(k) (note: often there is no input)

output z j z(k)

gain k j K k

a priori estimate

a posterioriestimate

a priori covariance p j- p(k|k-1) or p(k+1|k)

a posteriori

covariance p j p(k|k) or p(k+1|k+1)

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. The notation

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

and current values of the output, or the a posteriori estimate

Examples

Example of estimating a constant (along with Matlab code).1.

Example of estimating a first order process (along with Matlab code).2.

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].

References

[1] An Introduction to Kalman Filters, G Welch and G Bishop, http://www.cs.unc.edu/~welch/kalman/kalman_filter

/kalman.html. See also their other introductory information on Kalman Filters.



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[2] Handbook of Digital Signal Processing , D Elliot ed, Academic Press, 1986.

[3] Digital and Kalman filtering : an introduction to discrete-time filtering and optimum linear estimation, SM Bozic,

Halsted Press, 1994.

[4] An Engineering Approach to Optimal Control and Estimation Theory, GM Siouris, John Wiley & Sons, 1996.

[5] Statistical and Adaptive Signal Processing , DG Manolakis, VK Ingle, SM Kogon, McGraw Hill, 2000.

[6] Smoothing, Filtering and Prediction - Estimating The Past , GA Einicke, a free on-line text:http://www.intechopen.com

/books/smoothing-filtering-and-prediction-estimating-the-past-present-and-future

[7] Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation, R Faragher, IEEE SIG PROC

MAG, [128] Sept. 2012 http://www.cl.cam.ac.uk/~rmf25/papers

/Understanding%20the%20Basis%20of%20the%20Kalman%20Filter.pdf This article has a good intuitive description based

on noise distributions.

Comments or Questions?

Send me email

Erik Cheever

Engineering Department

Swarthmore College


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An introduction to scalar Kalman filters.pdf