Condition
number.
Square root filtering.
Potter’s algorithm.
M. Sami Fadali
Professor EBME Department
UNR
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 Singular
 For
How close is a matrix to being singular?
 Singular value decomposition gives the
singular values.
 MATLAB: condition number
>> cond(P)

value decomposition
∗
 Singular
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∗
∗
∗
∗
∗
∗
Values
symmetric positive semidefinite.
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
Properties
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


Error covariance matrix can become illconditioned.

Occurs if one state variable has a much
higher uncertainty than another.
For symmetric positive definite, use
the Cholesky decomposition.

MATLAB
Matrix is computationally singular.
>> Ps= chol(P,'lower')

Ps = lower triangular matrix such that
P=Ps.PsT
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
A priori error covariance matrix
=square root of


Example:

Ps better conditioned than P
A posteriori error covariance matrix
=square root of
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

= state-transition matrix
= zero mean white Gaussian
process noise with covariance matrix.

Factorization,
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QR Factorization: factorize
in to the product (here

orthogonal nonunique.
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
Find any orthogonal matrix T such that

Methods to find are QR factorization
methods: Householder transformation,
Givens transformation, Gram-Schmidt,
modified Gram-Schmidt.
matrix
)
orthogonal matrix
upper triangular matrix
We only need R from the factorization.
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Rows of measurement matrix
diagonal
 Gain
,
,
,
,
,

Rewrite
as:
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 Can
show
,
,
 Recursion
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for square root matrix
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1.
Compute the a priori square root matrix and
estimate and initialize
2.
Calculate for
,
,
,
3.
Set
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 Approximation
Computing
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,
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 Next
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gain matrix
,
,
 Using:
 Actual
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value
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



Square root filtering increases the precision
of the Kalman filter.
It is computationally more costly and harder
to program.
Was more important in the early days of
Kalman filtering with less powerful
computers.
Is still useful for practical implementation in
embedded systems.
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 D.
Simon, Optimal State Estimation: Kalman,
H∞, and Nonlinear Approaches, WileyInterscience, Hoboken, NJ, 2006.
 J. W. Demmel, Applied Numerical Linear
Algebra, SIAM, Philadelphia, 1997.
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