and the asymptotic estimation error is unbiased,
lim
t
E
~
xt 0: 7:3
Example 7.1: Some typical behavior patterns of suboptimal ®lter convergence are
depicted by the plots of Pt in Figure 7.2a, and characteristics of systems with these
symptoms are given here as examples.
Case A: Let a scalar continuous system equation be given by
_
xtFxt; F > 0; 7:4
in which the system is unstable, or
_
xtFxtwt7:5
in which the system has driving noise and is unstable.
Case B: The system has constant steady-state uncertainty:
lim
t
_
Pt0: 7:6
Case C: The system is stable and has no driving noise:
_
xtFxt; F > 0: 7:7
Example 7.2: Behaviors of Discrete-Time Systems Plots of P
k
are shown in
Figure 7.2b for the following system characteristics:
Case A: Effects of system driving noise and measurement noise are large relative
to P
0
t (initial uncertainty).
Case B: P
0
P
(Wiener ®lter).
Case C: Effects of system driving noise and measurement noise are small relative
to P
0
t:
Fig. 7.2 Asymptotic behaviors of estimation uncertainties.
274 PRACTICAL CONSIDERATIONS
Example 7.3: Continuous System with Discrete Measurements A scalar exam-
ple of a behavior pattern of the covariance propagation equation P
k
;
_
Pt and
covariance update equation P
k
,
_
xtFxtwt; F < 0;
ztxtvt;
is shown in Figure 7.2c.
The following features may be observed in the behavior of Pt:
1. Processing the measurement tends to reduce P:
2. Process noise covariance Q tends to increase P:
3. Damping in a stable system tends to reduce P:
4. Unstable system dynamics F > 0 tend to increase P:
5. With white Gaussian measurement noise, the time between samples T can be
reduced to decrease P:
The behavior of P represents a composite of all these effects (1±5) as shown in
Figure 7.2c.
Causes of Predicted Nonconvergence. Nonconvergence of P predicted by
the Riccati equation can be caused by
1. ``natural behavior'' of the dynamic equations or
2. nonobservability with the given measurements.
The following examples illustrate these behavioral patterns.
Example 7.4: The ``natural behavior'' for P in some cases is for
lim
t
PtP
(a constant): 7:8
For example,
_
x w; covwQ
z x v; covvR
I
e
F 0in
_
x Fx Gw;
G H 1 z Hx v:
7:9
Applying the continuous Kalman ®lter equations from Chapter 4, then
_
P FP PF
T
GQG
T
KRK
T
7.2 DETECTING AND CORRECTING ANOMALOUS BEHAVIOR
275
and
K PH
T
R
1
become
_
P Q
K
2
R
and
K
P
R
or
_
P Q
P
2
R
:
The solution is
Pta
P
0
cosh bta sinh bt
P
0
sinh bta cosh bt
; 7:10
where
a
RQ
p
; b
Q=R
p
: 7:11
Note that the solution of the Riccati equation converges to a ®nite limit:
1. lim
t
Pta > 0; a ®nite, but nonzero, limit. (See Figure 7.3a.)
2. This is no cause for alarm, and there is no need to remedy the situation if the
asymptotic mean-squared uncertainty is tolerable. If it is not tolerable, then the
remedy must be found in the hardware (e.g., by attention to the physical
sources of R or QÐor both) and not in software.
11
11
Fig. 7.3 Behavior patterns of P.
276 PRACTICAL CONSIDERATIONS
Example 7.5: Divergence Due to ``Structural'' Unobservability The ®lter is
said to diverge at in®nity if its limit is unbounded:
lim
t
Pt: 7:12
As an example in which this occurs, consider the system
_
x
1
w;
_
x
2
0;
z x
2
v;
cov wQ;
cov vR;
7:13
with initial conditions
P
0
s
2
1
0
0 s
2
2
45
P
11
0 0
0 P
22
0
45
: 7:14
The continuous Kalman ®lter equations
_
P FP PF
T
GQG
T
KRK
T
;
K PH
T
R
1
can be combined to give
_
P FP PF
T
GQG
T
PH
T
R
1
HP; 7:15
or
_
p
11
Q
p
2
12
R
;
_
p
12
p
12
p
22
R
;
_
p
22
p
2
22
R
; 7:16
the solution to which is
p
11
tp
11
0Qt; p
12
t0; p
22
t
p
22
0
1 p
22
0=R
ÂÃ
t
; 7:17
as plotted in Figure 7.3b. The only remedy in this example is to alter or add
measurements (sensors) to achieve observability.
Example 7.6: Nonconvergence Due to ``Structural'' Unobservability Parameter
estimation problems have no state dynamics and no process noise. One might
reasonably expect the estimation uncertainty to approach zero asymptotically as
more and more measurements are made. However, it can still happen that the ®lter
will not converge to absolute certainty. That is, the asymptotic limit of the estimation
7.2 DETECTING AND CORRECTING ANOMALOUS BEHAVIOR 277
uncertainty
0 < lim
k
P
k
< 7:18
is actually bounded away from zero uncertainty.
Parameter estimation model for continuous time. Consider the two-dimensional
parameter estimation problem
_
x
1
0;
_
x
2
0; P
0
s
2
1
0 0
0 s
2
2
0
!
; H 11;
z H
x
1
x
2
!
v; covvR;
7:19
in which only the sum of the two state variables is measurable. The difference of the
two state variables will then be unobservable.
Problem in discrete time. This example also illustrates a dif®culty with a
standard shorthand notation for discrete-time dynamic systems: the practice of
using subscripts to indicate discrete time. Subscripts are more commonly used to
indicate components of vectors. The solution here is to move the component indices
``upstairs'' and make them superscripts. (This approach only works here because the
problem is linear. Therefore, one does not need superscripts to indicate powers of the
components.) For these purposes, let x
i
k
denote the ith component of the state vector
at time t
k
. The continuous form of the parameter estimation problem can then be
``discretized'' to a model for a discrete Kalman ®lter (for which the state transition
matrix is the identity matrix; see Section 4.2):
x
1
k
x
1
k1
x
1
is constant; 7:20
x
2
k
x
2
k1
x
2
is constant; 7:21
z
k
11
x
1
k
x
2
k
45
v
k:
7:22
Let
^
x
0
0:
The estimator then has two sources of information from which to form an optimal
estimate of x
k
:
1. the a priori information in
^
x
0
and P
0
and
2. the measurement sequence z
k
x
1
k
x
2
k
v
k
for k 1; 2; 3; :
In this case, the best the optimal ®lter can do with the measurements is to ``average
out'' the effects of the noise sequence v
1
; ; v
k
: One might expect that an in®nite
number of measurements z
k
would be equivalent to one noise-free measurement,
that is,
z
1
x
1
1
x
2
1
; where v
1
0 and R covv
1
0: 7:23
278 PRACTICAL CONSIDERATIONS
Estimation uncertainty from a single noise-free measurement. By using the
discrete ®lter equations with one stage of estimation on the measurement z
1
, one
can obtain the gain in the form
K
1
s
2
1
0
s
2
1
0s
2
2
0R
s
2
2
0
s
2
1
0s
2
2
0R
P
T
T
T
R
Q
U
U
U
S
: 7:24
The estimation uncertainty covariance matrix can then be shown to be
P
1
s
2
1
0s
2
2
0Rs
2
1
0
s
2
1
0s
2
2
0R
s
2
1
0s
2
2
0
s
2
1
0s
2
2
0R
s
2
1
0s
2
2
0
s
2
1
0s
2
2
0R
s
2
1
0s
2
2
0Rs
2
2
0
s
2
1
0s
2
2
0R
P
T
T
T
T
R
Q
U
U
U
U
S
p
11
p
12
p
12
p
22
45
; 7:25
where the correlation coef®cient (de®ned in Equation 3.138) is
r
12
p
12
p
11
p
22
s
2
1
0s
2
2
0
s
2
1
0s
2
2
0Rs
2
1
0s
2
1
0s
2
2
0Rs
2
2
0
p
; 7:26
and the state estimate is
^
x
1
^
x
1
0K
1
z
1
H
^
x
1
0 I K
1
H
^
x
1
0K
1
z
1
: 7:27
However, for the noise-free case,
v
1
0; R covv
1
0;
the correlation coef®cient is
r
12
1; 7:28
and the estimates for
^
x
1
00,
^
x
1
1
s
2
1
0
s
2
1
0s
2
2
0
x
1
1
x
2
1
ÀÁ
;
^
x
2
1
s
2
2
0
s
2
1
0s
2
2
0
x
1
1
x
2
1
ÀÁ
;
are totally insensitive to the difference x
1
1
x
2
1
. As a consequence, the ®lter will
almost never get the right answer! This is a fundamental characteristic of the
7.2 DETECTING AND CORRECTING ANOMALOUS BEHAVIOR 279
problem, however, and not attributable to the design of the ®lter. There are two
unknowns (x
1
1
and x
2
1
) and one constraint:
z
1
x
1
1
x
2
1
ÀÁ
: 7:29
Conditions for serendipitous design. The conditions under which the ®lter will
still get the right answer can easily be derived. Because x
1
1
and x
2
1
are constants, their
ratio constant
C
def
x
2
1
x
1
1
7:30
will also be a constant, such that the sum
x
1
1
x
2
1
1 Cx
1
1
1 C
C
x
2
1
:
Then
^
x
1
1
s
2
1
0
s
2
1
0s
2
2
0
1 Cx
1
1
ÂÃ
x
1
1
only if
s
2
1
01 C
s
2
1
0s
2
2
0
1;
^
x
2
1
s
2
2
0
s
2
1
0s
2
2
0
1 C
C
x
2
1
x
2
1
only if
s
2
2
01 C
s
2
1
0s
2
2
0C
1:
Both these conditions are satis®ed only if
s
2
2
0
s
2
1
0
C
x
2
1
x
1
1
0; 7:31
because s
2
1
0 and s
2
2
0 are nonnegative numbers.
Likelihood of serendipitous design. For the ®lter to obtain the right answer, it
would be necessary that
1. x
1
1
and x
2
1
have the same sign and
2. it is known that their ratio C x
2
1
=x
1
1
.
Since both of these conditions are rarely satis®ed, the ®lter estimates would rarely be
correct.
What can be done about it. The following methods can be used to detect
nonconvergence due to this type of structural unobservability:
Test the system for observability using the ``observability theorems'' of Section
2.5.
Look for perfect correlation coef®cients (r 1) and be very suspicious of
high correlation coef®cients (e.g., r > 0:9).
280 PRACTICAL CONSIDERATIONS
Perform eigenvalue±eigenvector decomposition of P to test for negative
characteristic values or a large condition number. (This is a better test than
correlation coef®cients to detect unobservability.)
Test the ®lter on a system simulator with noise-free outputs (measurements).
Remedies for this problem include
attaining observability by adding another type of measurement or
de®ning
x x
1
x
2
as the only state variable to be estimated.
Example 7.7: Unobservability Caused by Poor Choice of Sampling Rate The
problem in Example 7.6 might be solved by using an additional measurementÐor
by using a measurement with a time-varying sensitivity matrix. Next, consider what
can go wrong even with time-varying measurement sensitivities if the sampling rate
is chosen badly. For that purpose, consider the problem of estimating unknown
parameters (constant states) with both constant and sinusoidal measurement sensi-
tivity matrix components:
Ht 1 cosot;
as plotted in Figure 7.4. The equivalent model for use in the discrete Kalman ®lter is
x
1
k
x
1
k1
; x
2
k
x
2
k1
; H
k
HkT; z
k
H
k
x
1
k
;
x
2
k
45
v
k
;
where T is the intersample interval.
What happens when Murphy's law takes effect. With the choice of intersampling
interval as T 2p=o and t
k
kT; the components of the measurement sensitivity
matrix become equal and constant:
H
k
1 cosokT
1 cos2pk
11;
as shown in Figure 7.4 (This is the way many engineers discover ``aliasing.'') The
states x
1
and x
2
are unobservable with this choice of sampling interval (see Figure
x
1
(t)
x
2
cos(wt)
102
12
k
k
Fig. 7.4 Aliased measurement components.
7.2 DETECTING AND CORRECTING ANOMALOUS BEHAVIOR 281
7.4). With this choice of sampling times, the system and ®lter behave as in the
previous example.
Methods for detecting and correcting unobservability include those given in
Example 7.6 plus the more obvious remedy of changing the sampling interval T to
obtain observability, for example,
T
p
o
7:32
is a better choice.
Causes of unpredicted nonconvergence. Unpredictable nonconvergence may be
caused by
1. bad data,
2. numerical problems, or
3. mismodeling.
Example 7.8: Unpredicted Nonconvergence Due to Bad Data ``Bad data'' are
caused by something going wrong, which is almost sure to happen in real-world
applications of Kalman ®ltering. These veri®cations of Murphy's law occur princi-
pally in two forms:
The initial estimate is badly chosen, for example,
^
x0x
2
~
x
2
trace P
0
: 7:33
The measurement has an exogenous component (a mistake, not an error) that is
excessively large, for example,
v
2
trace R: 7:34
Asymptotic recovery from bad data. In either case, if the system is truly linear, the
Kalman ®lter will (in theory) recover in ®nite time as it continues to use measure-
ments z
k
to estimate the state x: (The best way is to prevent bad data from getting
into the ®lter in the ®rst place!) See Figure 7.5.
Practical limitations of recovery. Often, in practice, the recovery is not adequate
in ®nite time. The interval 0; T of measurement availability is ®xed and may be too
K
^
()t
Fig. 7.5 Asymptotic recovery from bad data.
282 PRACTICAL CONSIDERATIONS
short to allow suf®cient recovery (see Figure 7.6). The normal behavior of the gain
matrix
K may be too rapidly converging toward its steady-state value of K 0. (See
Figure 7.7.)
Remedies for ``heading off'' bad data:
Inspection of Pt and Kt is useless, because they are not affected by data.
Inspection of the state estimates
^
xt for sudden jumps (after a bad measure-
ment has already been used by the ®lter) is sometimes used, but it still leaves
the problem of undoing the damage to the estimate after it has been done.
Inspection of the ``innovations'' vector z H
^
x for sudden jumps or large
entries (before bad measurement is processed by the ®lter) is much more
useful, because the discrepancies can be interpreted probabilistically, and the
data can be discarded before it has spoiled the estimate (see Section 7.3).
The best remedy for this problem is to implement a ``bad data detector'' to reject the
bad data before it contaminates the estimate. If this is to be done in real time, it is
sometimes useful to save the bad data for off-line examination by an ``exception
handler'' (often a human, but sometimes a second-level data analysis program) to
locate and remedy the causes of the bad data that are occurring.
Arti®cially increasing process noise covariance to improve bad data recovery.If
bad data are detected after they have been used, one can keep the ®lter ``alive'' (to
pay more attention to subsequent data) by increasing the process noise covariance Q
in the system model assumed by the ®lter. Ideally, the new process noise covariance
should re¯ect the actual measurement error covariance, including the bad data as
well as other random noise.
^
Fig. 7.6 Failure to recover in short period.
^
()t
Fig. 7.7 Failure to recover due to gain decay.
7.2 DETECTING AND CORRECTING ANOMALOUS BEHAVIOR 283
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