Page 102 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONTINUOUS STATE VARIABLES 91

Some special state space models
In this section, we introduce some elementary random processes. They
are presented here not only to illustrate the properties of state models
with random inputs, but also because they are often used as building
blocks for models of complicated physical processes.

Random constants Sometimes it is useful to model static parameters
as states in a dynamic system. In that case, the states do not change in
time:

xði þ 1Þ¼ xðiÞ ð4:15Þ

Such a model is useful when the sequential measurements z(i)of x are
processed online so that the estimate of x becomes increasingly accurate
as time proceeds.

First order autoregressive models A first order autoregressive (AR)
model is of the type

xði þ 1Þ¼ xðiÞþ wðiÞ ð4:16Þ

where w(i) is a white, zero mean, normally distributed sequence
2
2
with variance . In this particular example, (i) C x (i) since x(i)
w x
2 2 2i 2
is a scalar. (i) can be expressed in closed form: (1) ¼ (0)þ
x
x
x
2
2
(1 2iþ2 ) =(1 ). The equation holds if 6¼ 1. The system is
w
2i 2
stable provided that j j < 1. In that case, the term (0) exponentially
x
fades out. The second term asymptotically reaches the steady state, i.e.
the solution of the Lyapunov equation:
1
2 2
ð1Þ ¼ ð4:17Þ
x 2 w
1
If j j > 0, the system is not stable, and both terms grow exponentially.
First order AR models are used to describe slowly fluctuating phe-
nomena. Physically, such phenomena occur when broadband noise is
dampened by a first order system, e.g. mechanical shocks damped by a
mass/dampener system. Processes that involve exponential growth are
also modelled by first AR models. Figure 4.3 shows a realization of a
stable and an unstable AR process.