Target Tracking - MATLAB Simulations for Radar Systems Design - page 425

steady state ratio of the output variance (auto-covariance) to the input measure-
ment variance.

In order to determine the stability of the tracker under consideration, con-
sider the Z-transform for Eq. (9.72),

– 1

x () A z = x () K y n + ()

(9.78)

Rearranging Eq. (9.78) yields the following system transfer functions:

x ()
y n

– 1

IA z 1

–

h () = ------------ = ( – ) K (9.79)

()

IAz 1

–

where is called the characteristic matrix. Note that the system trans-
fer functions can exist only when the characteristic matrix is a non-singular
matrix. Additionally, the system is stable if and only if the roots of the charac-
teristic equation are within the unit circle in the z-plane,

( – )

IA z 1

–

( – ) = 0 (9.80)

The filterÓs steady state errors can be determined with the help of Fig. 9.19.
The error transfer function is

y ()

e () = -------------------

(9.81)

1 + h ()

and by using AbelÓs theorem, the steady state error is

z –

z

1

 

-----------

e ∞ =

lim

e () = lim e ()

(9.82)

 

t ∞ z → 1

Substituting Eq. (9.82) into (9.81) yields

y ()

z –

z

1

e ∞ = lim ----------- -------------------

(9.83)

1 + h () z → 1

y ()

x ()

e ()

h ()

Σ

+

-

Figure 9.19. Steady state error computation.