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:
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. Equation 1 can be represented pictorially as shown below, where the block with T in it represents a time delay. 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
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 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.
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 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 value x_{0}), 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.
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 x^{^}_{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, z^{^}_{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, x^{^}_{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):
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.
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:
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:
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 come up with 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.
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 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.
As 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
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
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 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 large, then we have low confidence in the measurement and our estimate will depend more upon the previous estimates.
The 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 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 posteriori estimate 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
Example of estimating a constant (along with Matlab code).
Example of estimating a first order process (along with Matlab code).
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].
[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.
[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.
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