Tài liệu Tracking and Kalman filtering made easy P8 ppt

We saw that this target also can be alternatively expressed in terms of the transition equation as given by 2.4-1 with the state vector by 5.4-1 for m¼ 2 and the transition matrix by 5.4

GENERAL FORM FOR LINEAR

TIME-INVARIANT SYSTEM

FUNCTION OF TIME

8.1.1 Introduction

In Section 1.1 we defined the target dynamics model for target having a constant velocity; see (1.1-1) A constant-velocity target is one whose trajectory can be expressed by a polynomial of degree 1 in time, that is, d¼ 1, in (5.9-1) (In turn, the tracking filter need only be of degree 1, i.e., m¼ 1.) Alternately, it

is a target for which the first derivative of its position versus time is a constant

In Section 2.4 we rewrote the target dynamics model in matrix form using the transition matrix
; see (2.4-1), (2.4-1a), and (2.4-1b) In Section 1.3 we gave the target dynamics model for a constant accelerating target, that is, a target whose trajectory follows a polynomial of degree 2 so that d¼ 2; see (1.3-1)

We saw that this target also can be alternatively expressed in terms of the transition equation as given by (2.4-1) with the state vector by (5.4-1) for m¼ 2 and the transition matrix by (5.4-7); see also (2.9-9) In general, a target whose dynamics are described exactly by a dth-degree polynomial given by (5.9-1) can also have its target dynamics expressed by (2.4-1), which we repeat here for convenience:

Xnþ1¼
Xn

where the state vector Xnis now defined by (5.4-1) with m replaced by d and the transition matrix is a generalized form of (5.4-7) Note that in this text d represents the true degree of the target dynamics while m is the degree used by

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Copyright # 1998 John Wiley & Sons, Inc ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic)

the tracking filter to approximate the target dynamics For the nonlinear dynamics model case, discussed briefly in Section 5.11 when considering the tracking of a satellite, d is the degree of the polynomial that approximates the elliptical motion of the satellite to negligible error

We shall now give three ways to derive the transition matrix of a target whose dynamics are described by an arbitrary degree polynomial In the process

we give three different methods for describing the target dynamics for a target whose motion is given by a polynomial

8.1.2 Linear Constant-Coefficient Differential Equation

Assume that the target dynamics is described exactly by the dth-degree polynomial given by (5.9-1) Then its dth derivative equals a constant, that is,

while itsðd þ 1Þth derivative equals zero, that is,

As a result the class of all targets described by polynomials of degree d are also described by the simple linear constant-coefficient differential equation given

by (8.1-2) Given (8.1-1) or (8.1-2) it is a straightforward manner to obtain the target dynamics model form given by (1.1-1) or (2.4-1) to (2.4-1b) for the case where d¼ 1 Specifically, from (8.1-1) it follows that for this d ¼ 1 case

Thus

Integrating this last equation yields

Equations (8.1-4) and (8.1-5) are the target dynamics equations for the constant-velocity target given by (1.1-1) Putting the above two equations in matrix form yields (2.4-1) with the transition matrix
given by (2.4-1b), the desired result In a similar manner, starting with (8.1-1), one can derive the form of the target dynamics for d¼ 2 given by (1.3-1) with, in turn,
given

by (5.4-7) Thus for a target whose dynamics are given by a polynomial of degree d, it is possible to obtain from the differential equation form for the target dynamics given by (8.1-1) or (8.1-2), the transition matrix
by integration

8.1.3 Constant-Coefficient Linear Differential Vector Equation for State Vector X(t)

A second method for obtaining the transition matrix
will now be developed

As indicated above, in general, a target for which

can be expressed by

Assume a target described exactly by a polynomial of degree 2, that is, d¼ 2 Its continuous state vector can be written as

XðtÞ ¼

xðtÞ _xðtÞ

 xðtÞ

2 4

3

5 ¼ DxðtÞxðtÞ

D2xðtÞ

2 4

3

It is easily seen that this state vector satisfies the following constant-coefficient linear differential vector equation:

DxðtÞ

D2xðtÞ

D3xðtÞ

2 4

3

5 ¼ 00 10 01

2 4

3

5 DxðtÞxðtÞ

D2xðtÞ

2 4

3

or

d

where

2 4

3

The constant-coefficient linear differential vector equation given by (8.1-9), or more generally by (8.1-10), is a very useful form that is often used in the literature to describe the target dynamics of a time-invariant linear system As shown in the next section, it applies to a more general class of target dynamics models than given by the polynomial trajectory Let us proceed, however, for the time being assuming that the target trajectory is described exactly by a polynomial We shall now show that the transition matrix
can be obtained from the matrix A of (8.1-10)

First express Xðt þ &Þ in a vector Taylor expansion as

Xðt þ &Þ ¼ XðtÞ þ &DXðtÞ þ&

2 2!D

2XðtÞ   

¼X1

¼0

&

 !D

n

From (8.1-10)

Therefore (8.1-11) becomes

Xðt þ &Þ ¼ X1

¼0

ð&AÞ

 !

We know from simple algebra that

ex¼X1

¼0

x

Comparing (8.1-14) with (8.1-13), one would expect that

X1

¼0

ð&AÞ

Although A is now a matrix, (8.1-15) indeed does hold with exp¼ e being to a matrix power being defined by (8.1-15) Moreover, the exponent function GðAÞ has the properties one expects for an exponential These are [5, p 95]

Gð&1AÞGð&2AÞ ¼ G½ð&1þ &2ÞA ð8:1-16Þ

½Gð&1AÞk¼ Gðk&1AÞ ð8:1-17Þ d

We can thus rewrite (8.1-13) as

Comparing (8.1-19) with (8.1-7), we see immediately that the transition matrix is

for the target whose dynamics are described by the constant-coefficient linear vector differential equation given by (8.1-10) Substituting (8.1-20) into (8.1-19) yields

Xðtnþ &Þ ¼
ð&ÞXðtnÞ ð8:1-21Þ Also from (8.1-15), and (8.1-20) it follows

ð&Þ ¼ I þ &A þ&

2 2!A

2

þ&

3 3!A

3

From (8.1-17) it follows that

Therefore

By way of example, assume a target having a polynomial trajectory of degree

d¼ 2 From (8.1-10a) we have A Substituting this value for A into (8.1-22) and letting &¼ T yields (5.4-7), the transition matrix for the constant-accelerating target as desired

8.1.4 Constant-Coefficient Linear Differential Vector Equation for Transition Matrix

A third useful alternate way for obtaining
is now developed [5 pp 96–97] First, from (8.1-21) we have

Differentiating with respect to & yields

d d&
ð&Þ

Xð0Þ ¼ d

The differentiation of a matrix by & consists of differentiating each element

of the matrix with respect to & Applying (8.1-10) and (8.1-25) to (8.1-26) yields

d d&
ð&Þ

Xð0Þ ¼ AXð&Þ

d

On comparing (8.1-28) with (8.1-10) we see that the state vector XðtÞ and the transition matrix
ð&Þ both satisfy the same linear, time-invariant differential vector equation Moreover, given this differential equation, it is possible to obtain
ð&Þ by numerically integrating it This provides a third method for obtaining
ð&Þ

Define the matrix inverse of
by , that is,

The inverse  satisfies the associated differential equation [5, p 97]

d

Thus ð&Þ can be obtained by numerically integrating the above equation

To show that (8.1-30) is true, we first verify that the solution to (8.1-30) is

This we do by differentiating the above to obtain

d d&ð&Þ ¼ ð0Þ½expð &AÞA

Thus (8.1-31) satisfies (8.1-30), as we wished to show For ð0Þ let us choose

This yields for ð&Þ the following:

It now only remains to show that the above is the inverse of
To do this, we use (8.1-16), which yields

expð&AÞexpð &AÞ ¼ expð0Þ

This completes our proof that
1¼  and  satisfies (8.1-30)

For a target whose trajectory is given by a polynomial, it does not make sense to use the three ways given in this section to obtain
The
can easily be obtained by using the straightforward method illustrated in Section 2.4; see (2.4-1), (2.4-1a), and (2.4-1b) and (1.3-1) in Section 1.3 However, as shall be seen later, for more complicated target models, use of the method involving the integration of the differential equation given by (8.1-28) represents the preferred method In the next section we show that (8.1-10) applies to a more general class of targets than given by a polynomial trajectory

THE PRODUCT OF POLYNOMIALS AND EXPONENTIALS

In the preceeding section we showed that the whole class of target dynamics consisting of polynomials of degree d are generated by the differential equation given by (8.1-2) In this section we consider the target whose trajectory is described by the sum of the product of polynomials and exponentials as given by

xðtÞ ¼Xk j¼0

where pjðtÞ is a polynomial whose degree shall be specified shortly The above xðtÞ is the solution of the more general [than (8.1-2)] linear, constant-coefficient differential vector equation given by [5, pp 92–94]

ðDdþ1þ dDdþ    þ 1Dþ 0ÞxðtÞ ¼ 0 ð8:2-2Þ

We see that (8.1-2) is the special case of (8.2-2) for which 0 ¼ 1 ¼    ¼

d ¼ 0 The j of (8.2-1) are the k distinct roots of the characteristic equation

dþ1þ ddþ    þ 1þ 0¼ 0 ð8:2-3Þ The degree of pjðtÞ is 1 less than the multiplicity of the root j of the characteristic equation

By way of example let d¼ 2 Then

ðD3þ 2D2þ 1Dþ 0ÞxðtÞ ¼ 0 ð8:2-4Þ Let the state vector XðtÞ for this process defined by (8.1-8) Then it follows directly from (8.2-4) that

d

dtXðtÞ ¼

_x

 x _x

0

@

1

0 1 2

0

@

1

A x_x

 x

0

@ 1

d

where

A

0 1 2

0

@

1

This gives us a more general form for A than obtained for targets following exactly a polynomial trajectory as given in Section 8.1; see (8.1-10a)

The matrix A above can be made even more general To do this, let

^

where G is an arbitrary constant 3 3 nonsingular matrix Applying (8.2-7) to (8.2-6) yields

d

dtG

Because G is a constant, the above becomes

G 1d

or

d

or finally

d

where

Because G is arbitrary, B is arbitrary, but constant Thus, (8.2-6) applies where

A can be an arbitrary matrix and not just (8.2-6a)