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Colorado Center for Astrodynamics Research The University of Colorado 1
STATISTICALORBIT DETERMINATION
The Minimum Variance Estimate
ASEN 5070
LECTURE 17
10/05/09
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Review problemII. Given the system
23 4 0
with the state vector defined by and the deviation vector defined by
X
Χ
. Where indicates a small deviation from a reference value.
a. Write the linearized equations in state space form, b. How would you determine the state transition matrix for this system? What Additional information is needed to generate the state transition matrix?
A Χ Χ
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Review problemIII. Circle the correct answers or answers.
a. Given the observation state equation
2i i 0 i 1y(t ) = t ax + t cx i =1,2...10
0 1 ia, x , x and c are constants and t is given.Where Which of the following state vectors are observable:
100
0 0
axa xa
1. x 2. 3. 4. 5.x xc c
c
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Review problem
b. The differential equation
2 4 0xx ax t
is 1. 1st order and 1st degree2. 2nd order and 1st degree3. linear
4. nonlinear5. 2nd order and 2nd degree
c. Given two uncorrelated observations and the second is twice as accurate as the first, we would use the following weighting matrix
1 0 2 0 1 0 1 11. , 2. , 3. , 4. ,
0 1 0 1 0 2 1 2W W W W
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Review problem
d. If the differential equation for the state is given by
2 0x bx
It would not be necessary to use a state deviation vector. T or F
e. The state transition matrix will contain terms such as . The units of this partial derivative are: 2 0
x(t)
J ( )t
1. L/T, 2. L2/T, 3. 1/T, 4. It is dimensionless
f. If the state transition matrix is symplectic it can be inverted by inspection. T or F
Colorado Center for Astrodynamics Research The University of Colorado
Inclusion of Apriori Information in the Batch Processor Computational Algorithm
If apriori information and with attendant covariance is given, this Information should be maintained when iterating the batch algorithm i.e.,
0x 0P
and should be held constant to begin each iteration. Hence,for the first iteration
*X x 0P
* *0 0 1 1X x X x
but* *1 0 0ˆX X x
Hence,* *0 0 0 0 1ˆX x X x x
Solving for yields,
1 0 0ˆx x x
*0X
1x
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Inclusion of Apriori Information in the Batch Processor Computational Algorithm
Thus for the nth iteration, the apriori value of is given by nx
1 1ˆn n nx x x (4.6.4)
and
* *1 1ˆn n nX X x
1 11 0 0
1 1
ˆm m
T Tn i i i i n
i i
x H wH P H wy P x
Finally,
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4.4 THE MINIMUM VARIANCE ESTIMATE
• The least squares and weighted least squares methods do not include any information on the statistical characteristics of the measurement errors or the a priori errors in the values of the parameters to be estimated. The minimum variance approach is one method for removing this limitation.
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THE MINIMUM VARIANCE ESTIMATE
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THE MINIMUM VARIANCE ESTIMATE
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THE MINIMUM VARIANCE ESTIMATE
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THE MINIMUM VARIANCE ESTIMATE
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THE MINIMUM VARIANCE ESTIMATE
Note that both matrices on the left side of eq (4.4.11) are nxnAnd symmetric.
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THE MINIMUM VARIANCE ESTIMATE
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THE MINIMUM VARIANCE ESTIMATE
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Minimum Variance filter
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Minimum Variance filter
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Minimum Variance filter
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Minimum Variance filter
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Minimum Variance filter
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