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Chapter 5 / Missile State Observer
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If (PIM) 2 is > 2 to 3 the measurements are discarded. If average value of
(PIM) 2 exceeds 1 the state and/or measurement noise can be adapted on-line.
5.10.8 Covariance Matrix Conditioning
There are a number of approaches to ensure that the covariance matrix [C]
remains symmetric and supposedly positive definite. On-line testing is
often restricted to ensure that the main diagonal elements are positive and,
C
2 ( i , j ) < C ( i , i ) ⋅ C ( j , j )
5-29
Equation 5.10-2
These simple tests provide a warning of ill-conditioning in [C] and the
possibility of filter divergence; they are not a definitive indicator of positive
definiteness. Sylvester’s Criteria is often used for the later where the
determinant of [C], and all its sub-minors, must be positive. This is
computationally intensive test and one rarely employed in practical systems
where symmetry is assured by computing only a triangular partition of [C],
( j , i ) : C ( i , j )
i ≠ j ⇒ C =
Equation 5.10-3
To prevent state lockout indicated by a zero on the main diagonal of [C],
and numerical instability if any of these values are negative, an artificially
lower bound can be applied, Griffin [G.15] . Such lower thresholds must
reflect any state scaling - see the thorough discourse on [C] matrix illconditioning
by Kerr [K.6] .
( i , i ) : C ( i , i )
C =
The off-diagonal terms are then modified,
*
( ≠ j ) ∧ ( j > i ) ⇒ C ( i , j )
Equation 5.10-4
2
C ( i , j )
( i , i ) ⋅ C ( j , j )
i : = *
*
C
Equation 5.10-5
Kerr warns that these ad-hoc approaches to protect [C] can significantly
alter the matrix beyond reason, and may not prevent filter divergence.
5.10.9 Filter Performance and Observability
Stochastic filters that meet the tenants of the KF are well tuned when the
63% of the actual state errors after a measurement update are matched to the
expected covariance [C]. A performance metric based on this principle is
the normalised distance (PIX) provided by the utility defined in §22.13.15,