Using the definition of the Kalman gain, the equation for the updated covariance

matrix can be reduced to

P

k

(+)   =   (I      K

k

  H

k

)  P

k

( )

Although much simpler than the Joseph form, caution must be exercised if this

equation is used since it is susceptible to numerical problems.  Since P is a

covariance matrix, theoretically it is symmetric and nonnegative definite (all

eigenvalues   0).  One approach to insure that P is always nonnegative definite is

to factor the initial P as a product of upper (or lower) unit triangular matrices and a

diagonal matrix as

P   =   U D U

T

Here U is unit triangular and D is diagonal.  If the initial P is nonnegative definite,

then all elements of D will be   0.  Algorithms exist to propagate and update the

factors U and D instead of P so that P need never explicitly be formed.  These

algorithms operate on U and D in a manner that guarantees that the elements of D

are always   0, implying that P is always nonnegative definite.  Other algorithms can

also be used to ensure the positive definiteness of P.  The matrix P can be factored

into a product of lower triangular and diagonal matrices, exactly equivalent to the

UD factorization, or P may be factored into its square root as P  = W W

T

 (square

root formulation).

Although the equation for the Kalman gain seems complex, a simple example will

help develop an intuitive feel for this gain calculation.  Note first that the updated

state estimate can be rewritten as

x (+)   =   (I     

)  x ( )  + 

  z

k

K

k

H

k

k

K

k

k

Assume that the state and measurement are scalars and that the measurement

matrix H is 1. Then the Kalman gain is

K   =   P  /  (P  +  R)

For large uncertainty in the state model (P   R), as P    , then K   1.  As K   1, then

x

k

(+)   z

k

.  In other words, given the large uncertainty in the state, the new

measurement is assumed to be a much better estimate of the state than is the

propagated estimate.  On the contrary, for large uncertainty in the measurement

compared with the estimated state, R   P, then as R    , K   0.  As K   0, then 

x

k

(+)

x

k

( ).  Thus the new information is essentially ignored since the apriori estimate is

deemed much better than the

9 5