Tài liệu Tracking and Kalman filtering made easy P15 ppt

15
LINEAR TIME-VARIANT SYSTEM
15.1 INTRODUCTION
In this chapter we extend the results of Chapters 4 and 8 to systems having
time-variant dynamic models and observation schemes [5, pp. 99–104]. For a
time-varying observation system, the observation matrix M of (4.1-1) and
(4.1-5) could be different at different times, that is, for different n. Thus the
observation equation becomes
Y
n
¼ M
n
X
n
þ N
n
ð15:1-1Þ
For a time-varying dynamics model the transition matrix È would be different
at different times. In this case È of (8.1-7) is replaced by Èðt
n
; t
nÀ1
Þ to indicate
a dependence of È on time. Thus the transition from time n to n þ 1isnow
given by
X
nþ1
¼ Èðt
nþ1
; t
n

ÞX
n
ð15:1-2Þ
The results of Section 4.1 now apply with M, È, and T replaced by M
n
,
Èðt
n
; t
nÀi
Þ, and T
n
, respectively; see (4.1-5) through (4.1-31). Accordingly, the
least-squares and minimum-variance weight estimates given by (4.1-32) and
(4.5-4) apply for the time-variant model when the same appropriate changes are
made [5]. It should be noted that with È replaced by Èðt
n
; t
nÀi
Þ, the results
apply to the case of nonequal spacing between observations. We will now
present the dynamic model differential equation and show how it can be
numerically integrated to obtain Èðt
n
; t
nÀi
Þ.
354
Tracking and Kalman Filtering Made Easy. Eli Brookner
Copyright # 1998 John Wiley & Sons, Inc.

ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic)
15.2 DYNAMIC MODEL
For the linear, time-variant dynamic model, the differential equation (8.1-10)
becomes the following linear, time-variant vector equation [5, p. 99]:
d
dt
XðtÞ¼AðtÞXðtÞð15:2-1Þ
where the constant A matrix is replaced by the time-varying matrix AðtÞ,a
matrix of parameters that change with time. For a process described by (15.2-1)
there exists a transition matrix Èðt
n
þ ; t
n
Þ that transforms the state vector at
time t
n
to t
n
þ , that is,
Xðt
n
þ Þ¼Èðt
n
þ ; t
n
ÞXðt
n
Þð15:2-2Þ
This replaces (8.1-21) for the time-invariant case. It should be apparent that it is
necessary that

Èðt
n
; t
n
Þ¼I ð15:2-3Þ
15.3 TRANSITION MATRIX DIFFERENTIAL EQUATION
We now show that the transition matrix for the time-variant case satisfies the
time-varying model differential equation given by (15.2-1), thus paralleling the
situation for the time-invariant case; see (8.1-25) and (8.1-28). Specifically, we
shall show that [5, p. 102]
d
d
Èðt
n
þ ; t
n
Þ¼Aðt
n
þ ÞÈðt
n
þ ; t
n
Þð15:3-1Þ
The above equation can be numerically integrated to obtain È as shall be
discussed shortly.
To prove (15.3-1), differentiate (15.2-2) with respect to  to obtain [5, p. 101]
d
d
½Èðt
n

þ ; t
n
ÞXðt
n
Þ ¼
d
d
Xðt
n
þ Þð15:3-2Þ
Applying (15.2-1) (15.2-2) yields
d
d
½Èðt
n
þ ; t
n
ÞXðt
n
Þ ¼ Aðt
n
þ ÞXðt
n
þ Þ
¼ Aðt
n
þ ÞÈðt
n
þ ; t
n

ÞXðt
n
Þ
ð15:3-3Þ
Because Xðt
n
Þ can have any value, (15.3-1) follows, which is what we wanted
to show.
TRANSITION MATRIX DIFFERENTIAL EQUATION
355
One simple way to numerically integrate (15.3-1) to obtain Èðt
n
þ ; t
n
Þ is to
use the Taylor expansion. Let  ¼ mh, where m is an integer to be specified
shortly. Starting with k ¼ 1 and ending with k ¼ m, we use the Taylor
expansion to obtain [5, p. 102]
Èðt
n
þ kh; t
n
޼Ƚt
n
þðk À 1Þh; t
n
þh
d
d
Ƚt

n
þðk À 1Þh; t
n
ð15:3-4Þ
which becomes [5, p. 102]
Èðt
n
þ kh; t
n
Þ¼fI þ hA½t
n
þðk À 1ÞhgȽt
n
þðk À 1Þh; t
n

k ¼ 1; 2; 3; ...; m
ð15:3-5Þ
At k ¼ m we obtain the desired Èðt
n
þ ; t
n
Þ. In (15.3-4) m is chosen large
enough to make h small enough so that the second-order terms of the Taylor
expansion can be neglected. The value of m can be determined by evaluating
(15.3-5) with successively higher values of m until the change in the calculated
value of Èðt
n
þ ; t
n

Þ with increasing m is inconsequential.
Equation (15.2-2) is used to transition backward in time when rewritten as
Xðt
n
Þ¼Èðt
n
; t
n
þ ÞXðt
n
þ Þð15:3-6Þ
The above is obtained by letting  be negative in (15.2-2). It thus follows that
the inverse of Èðt
n
þ ; t
n
Þ is
Èðt
n
; t
n
þ Þ¼½Èðt
n
þ ; t
n
Þ
À1
ð15:3-7Þ
Thus interchanging the arguments of È gives us its inverse. In the literature the
inverse of È is written as and given by

ðt
n
þ ; t
n
Þ¼½Èðt
n
þ ; t
n
Þ
À1
ð15:3-8Þ
It is a straightforward matter to show that satisfies the time-varying associated
differential equation [5, p. 103]
d
d
ðt
n
þ ; t
n
Þ¼À ðt
n
þ ; t
n
ÞAðt
n
þ Þð15:3-9Þ
thus paralleling the situation for the time-invariant case; see (8.1-30).
356
LINEAR TIME-VARIANT SYSTEM