Tài liệu Tracking and Kalman filtering made easy P15 ppt

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

LINEAR TIME-VARIANT SYSTEM

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

For a time-varying dynamics model the transition matrix
would be different

at different times In this case
of (8.1-7) is replaced by
ðtn; tn1Þ to indicate

a dependence of
on time Thus the transition from time n to nþ 1 is now given by

The results of Section 4.1 now apply with M,
, and T replaced by Mn,

ðtn; tniÞ, and Tn, 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
replaced by
ðtn; tniÞ, 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
ðtn; tniÞ

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

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
ðtnþ ; tnÞ that transforms the state vector at time tn to tnþ , that is,

Xðtnþ Þ ¼
ðtnþ ; tnÞXðtnÞ ð15:2-2Þ This replaces (8.1-21) for the time-invariant case It should be apparent that it is necessary that

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
ðtnþ ; tnÞ ¼ Aðtnþ Þ
ðtnþ ; tnÞ ð15:3-1Þ The above equation can be numerically integrated to obtain
as shall be discussed shortly

To prove (15.3-1), differentiate (15.2-2) with respect to  to obtain [5, p 101]

d d½
ðtnþ ; tnÞXðtnÞ ¼ d

Applying (15.2-1) (15.2-2) yields

d

d½
ðtnþ ; tnÞXðtnÞ ¼ Aðtnþ ÞXðtnþ Þ

¼ Aðtnþ Þ
ðtnþ ; tnÞXðtnÞ

ð15:3-3Þ

Because XðtnÞ 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
ðtnþ ; tnÞ is to use the Taylor expansion Let ¼ mh, where m is an integer to be specified shortly Starting with k¼ 1 and ending with k ¼ m, we use the Taylor expansion to obtain [5, p 102]

ðtnþ kh; tnÞ ¼
½tnþ ðk  1Þh; tn þ h d

d
½tnþ ðk  1Þh; tn ð15:3-4Þ which becomes [5, p 102]

ðtnþ kh; tnÞ ¼ fI þ h A½tnþ ðk  1Þhg
½tnþ ðk  1Þh; tn

At k¼ m we obtain the desired
ðtnþ ; tnÞ 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
ðtnþ ; tnÞ with increasing m is inconsequential

Equation (15.2-2) is used to transition backward in time when rewritten as

XðtnÞ ¼
ðtn; tnþ ÞXðtnþ Þ ð15:3-6Þ The above is obtained by letting  be negative in (15.2-2) It thus follows that the inverse of
ðtnþ ; tnÞ is

ðtn; tnþ Þ ¼ ½
ðtnþ ; tnÞ1 ð15:3-7Þ Thus interchanging the arguments of
gives us its inverse In the literature the inverse of
is written as and given by

ðtnþ ; tnÞ ¼ ½
ðtnþ ; tnÞ1 ð15:3-8Þ

It is a straightforward matter to show that satisfies the time-varying associated differential equation [5, p 103]

d d ðtnþ ; tnÞ ¼  ðtnþ ; tnÞAðtnþ Þ ð15:3-9Þ thus paralleling the situation for the time-invariant case; see (8.1-30)

356 LINEAR TIME-VARIANT SYSTEM