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A Discrete Kalman-Bucy Filtering Example

Introduction
Discrete Kalman-Bucy filtering and smoothing are important data analysis tools. A lot of time is spent designing these filters for all sorts of applications. The KBF program is a graphical user interface to a discrete Kalman-Bucy filtering and smoothing tool. This section contains a review an example filter constructed using KBF. This as an introduction both to discrete Kalman-Bucy filtering and the KBF program.

Loading the Example
If you installed KBF in C:\KBF and you run the O-Matrix program C:\KBF\KBF.OMS, the KBF command window will be displayed. If you then load EXAMPLE.KBF file, the O-Matrix main window appear as follows:



The following table contains a brief description of the subwindows above. A more detailed description is contained in the other sections of this users manual.
Name Description
KBF COMMAND overall control of KBF
NOTES brief description of the problem
MEAS defines the measurement model
TRAN defines the transition  model
DATA some KBF input values
SIM used to create simulated data
Z1 Table table of first data column versus time index
Z1 PLot plot of first data column versus time index
State Table table of second state components versus time index
State PLot plot of second state component versus time index

Fitting and Plotting
For this example, the State Table window contains the variable x2. The value for this variable is originally -1e+300 because that is the bad data flag. In addition, the State Plot window is empty.

Step 1: select the Fit button in the KBF COMMAND window and the following dialog will appear:



Step 2: select the Iterated smoother option and leave the Number of iterations field at 2. Then select the Ok button.

Step 3: A dialog will be left on the screen reporting the results of the fit. Select the Close button in this dialog.

If you now inspect the State Plot window, you will see the following plot:



If you inspect the Z1 Plot window, you will see the following plot:



This is the measurement values corresponding to the first component of the measurement vector. You can add a curve corresponding to the model for the expected value of the measurements as follows:

Step 4: Select the Plot button in the KBF COMMAND window. The following dialog will appear:



Step 5: In this dialog make the following changes:,
Old Value New Value Description
z1 z1, h1 add variable corresponding to expected value for z1
na na, bl plot this variable using the color black
++ ++, so use a solid line to plot this variable


The Z1 Plot window will now contain the following plot:

Problem Statement
In the example, we measure the range from the ship to two shore stations at know locations. Our problem is to determine the location of the ship as a function of time. The following table defines some mathematical notation that we use to describe our filter:
Name Description Dimension
A position of first shore station 2 x 1
B position of second shore station 2 x 1
S position of the ship 2 x 1
V velocity of the ship 2 x 1
r1 range from S to A 1 x 1
r2 range from S to B 1 x 1
We include both the ships position and velocity in the state vector. This enable us to model the expected value of the data from the state vector and to model the expected change in the position as linear with respect to time. The corresponding state vector at time index k is 
	    / S(1) \
	x = | S(2) |
	    | V(1) |
	    \ V(2) /
 The KBF Notes window is used to store general comments about the particular Kalman-Bucy filter. The Notes window for our example is:

Input Values
The following is a list of the KBF input values described in this section:
Name Description
bad bad data flag
xi initial state estimate
Pi covariance of initial state estimate
Z matrix of measurement values
The following is a list of the rest of the KBF input values which are described in subsequent sections:
Name Description
hk expected measurement model
Rk covariance of measurement noise
hk model for expected measurement
Rk covariance of measurement noise
gk model for expected transition
Qk covariance of transition noise
The bad data flag for our example is 
	-1d+300
 Before the Fit or Sim commands are executed, some of the variables are not yet defined. These variables appear in the Table windows with the value -1e+300 and they do not appear in the Plot windows. The estimate for the value of the state vector at time index k = 1 is 
	     / S(1) \   / 50 \
	xi = | S(2) | = |  0 | 
	     | V(1) |   |  1 |
	     \ V(2) /   \  0 /
 The accuracy of this initial estimate is specified by the covariance matrix 
	     / 16  0  0  0 \
	Pi = |  0 16  0  0 |
	     |  0  0  1  0 |
	     \  0  0  0  1 /
 Note that because the matrix Pi is diagonal, the components of xi are independent. Using delta to denote the corresponding standard deviations, 
	delta S(1) = 4
	delta S(2) = 4
	delta V(1) = 1
	delta V(2) = 1
 Neither of the measurement columns are modulo a base value so the corresponding field in the Data window is 
	0  0
 The file
      KBF\DATA\TMP\DATA.VAL
contains the values 
	119.372  107.759
	122.161  104.008
	120.752  104.096
	121.608  108.792
	114.187  114.778
	109.812  120.034
	106.191  124.459
	108.104  121.789
	110.782  121.566
	115.898  115.941
 The first column corresponding to the measurements for r1 and the second column corresponds to the measurements for r2. The kth row corresponds to the measurements for the kth time point.

The Data window for our example is:

Measurement Model
Each Kalman-Bucy filter has a model that relates the state vector to the measurements. The general form of this model is 
	z  = h (x ) + v
	 k    k  k     k
 where the last term is the mean zero random measurement noise. For our example, the covariance of the measurement noise is given by 
	      / 4  0 \
	R  =  |      |
	 k    \ 0  4 /
 Note that because this matrix is diagonal, the components of the measurement noise are independent. Using delta to denote the corresponding standard deviations,
     delta v (1) = 2
            k
     delta v (2) = 2
            k
The expected value of the measurements are modelled by 
	r1(x) = |S - A|
	          __________________________________
	      = \/ [x(1) - A(1)]^2 + [x(2) - A(2)]^2 

	r2(x) = |S - B|
	          __________________________________
	      = \/ [x(1) - B(1)]^2 + [x(2) - B(2)]^2 


	       / r1(x) \ 
	h(x) = |       | 
	       \ r2(x) /
 (Note that we are able to drop the dependence of h on k for our example.) We use the notation D_j to denote the partial derivative with respect to the jth component of x. Using this notation and the fact that 
	d    __      1
	-- \/ s = ---------- 
	ds            ___
	          2 \/ s
 we obtain 
	D_1 r1(x) = [S(1) - A(1)] / r1(x)
	D_2 r1(x) = [S(2) - A(2)] / r1(x)
	D_3 r1(x) = 0
	D_4 r1(x) = 0
 A similar formula holds for the partial derivatives of r2(x). It follows that 
	d         / D_1 r1(x)  D_2 r1(x)  D_3 r1(x)  D_4 r1(x) \
	-- h(x) = |                                             |
	dx        \ D_1 r2(x)  D_2 r2(x)  D_3 r2(x)  D_4 r2(x) /

	          / [S(1)-A(1)]/r1(x)  [S(2)-A(2)]/r1(x)  0  0 \
	        = |                                            |
	          \ [S(1)-B(1)]/r2(x)  [S(2)-B(2)]/r2(x)  0  0 /
This measurement model is specified by a function called meas in the Measurement window of the KBF program. The function meas has the following input and output:
Name Description Dimension Type
k time index 1 x 1 Input
xk current state value 4 x 1 Input
hk h(xk) 2 x 1 Output
dhk derivative of h(xk) 2 x 4 Output
Rk covariance of measurement noise 2 x 2 Output
The Measurement window for our example is:

Transition Model
A Kalman-Bucy filter also has a model that relates the state vector at one time to its value at the next time. The general form of this model is 
	x   = g (x ) + w
	 k+1   k  k     k
 where the last term is the mean zero random transition noise. For our example, the covariance of the transition noise is given by 
	    / 1  0  0  0 \
	Q = | 0  1  0  0 |
	 k  | 0  0 .4  0 |
	    \ 0  0  0 .4 /
 Note that because this matrix is diagonal, the components of the transition noise are independent. Using delta to denote the corresponding standard deviations,
     delta w (1) = 1
            k
     delta w (2) = 1
            k        __
     delta w (3) = \/.4
            k        __
     delta w (4) = \/.4
            k
Let Dt spacing between time point in our example. The expected value of the next state vector as a function of the current is modelled as 
	       / x(1) + x(3) Dt \
	g(x) = | x(2) + x(4) Dt | 
	       |      x(3)      |
	       \      x(4)      /
 (Note that we are able to drop the dependence of g on k for our example.) It follows that 
	d         / 1  0  Dt  0  \
	-- g(x) = | 0  1  0   Dt |
	dx        | 0  0  1   0  |
	          \ 0  0  0   1  /
 This transition model is specified by a function called tran in the Transition window of the KBF program. The function tran has the following input and output:
Name Description Dimension Type
k time index 1 x 1 Input
xk current state value 4 x 1 Input
gk g(xk) 4 x 1 Output
dgk derivative of g(xk) 4 x 4 Output
Qk covariance of transition noise 4 x 4 Output
The Transition window for our example is: