Page 112 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONTINUOUS STATE VARIABLES 101

Using these approximations, Kalman-like filters become within reach.
These solutions are suboptimal since there is no guarantee that the
approximations are close.
An obvious way to get the approximations is by application of a
Taylor series expansion of the functions. Ignorance of the higher order
terms of the expansion gives the desired approximation. The Taylor
series exists by virtue of the assumed smoothness of the nonlinear func-
tion; it does not work out if the nonlinearity is a discontinuity, i.e.
saturation, dead zone, hysteresis, and so on. The Taylor series expan-
sions of the system equations are as follows:

1 M 1 T m
X
fðx þ eÞ¼ fðxÞþ FðxÞe þ e m e F ðxÞe þ HOT
xx
2
m¼0
ð4:32Þ
1 N 1 T n
X
hðx þ eÞ¼ hðxÞþ HðxÞe þ e n e H ðxÞe þ HOT
xx
2
n¼0
e m is the Cartesian basis vector with appropriate dimension. The m-th
element of e m is one; the other are zeros. e m can be used to select the m-th
m
T
element of a vector: e x ¼ x m . F(x)and H(x) are Jacobian matrices. F (x)
m
xx
n
and H (x) are Hessian matrices. These matrices are defined in Appendix
xx
B.4. HOT are the higher order terms. The quadratic approximation arises
if the higher order terms are ignored. If, in addition, the quadratic term is
ignored, the approximation becomes linear, e.g. f(x þ e) ffi f(x) þ F(x)e.
The linearized Kalman filter
The simplest approximation occurs when the system equations are lin-
¼
earized around some fixed value x of x(i). This approximation is useful if
the system is time invariant and stable, and if the states swing around an
equilibrium state. Such a state is the solution of:
¼ ¼
x ¼ fðxÞ ð4:33Þ
¼
Defining e(i) ¼ x(i) x, the linear approximation of the state
¼ ¼
equation (4.31) becomes x(i þ 1) ffi f(x) þ F(x)e(i) þ w(i). After some
manipulations:

¼ ¼ ¼
xði þ 1Þffi FðxÞxðiÞþ I FðxÞ x þ wðiÞ
ð4:34Þ

¼ ¼ ¼
zðiÞffi hðxÞþ HðxÞ xðiÞ x þ vðiÞ