Page 100 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

CONTINUOUS STATE VARIABLES 89

general, a direct implementation of the scheme is difficult. Fortunately,
there are circumstances which allow a fast implementation. For instance,
in the special case, where the models are linear and the disturbances have
a normal distribution, an implementation based on an ‘expectation and
covariance matrix’ representation of the probability densities is feasible
(Section 4.2.1). If the models are nonlinear, but the nonlinearity is
smooth, linearization techniques can be applied (Section 4.2.2). If the
models are highly nonlinear, but the dimensions N and M are not too
large, numerical methods are possible (Section 4.2.3 and 4.4).

4.2.1 Optimal online estimation in linear-Gaussian systems

Most literature in optimal estimation in dynamic systems deals with the
particular case in which both the state model and the measurement
model are linear, and the disturbances are Gaussian (the linear-Gaussian
systems). Perhaps the main reason for the popularity is the mathematical
tractability of this case.

Linear-Gaussian state space models

The state model is said to be linear if the transition from one state to the
next can be expressed by a so-called linear system equation (or: linear
state equation, linear plant equation, linear dynamic equation):

xði þ 1Þ¼ FðiÞxðiÞþ LðiÞuðiÞþ wðiÞ ð4:11Þ

F(i) is the system matrix.Itisan M M matrix where M is the dimen-
sion of the state vector. M is called the order of the system. The vector
u(i) is the control vector (input vector) of dimension L. Usually, the
vector is generated by a controller according to some control law. As
such the input vector is a deterministic signal that is fully known, at least
up to the present. L(i) is the gain matrix of dimension M L. Sometimes
the matrix is called the distribution matrix as it distributes the control
vector across the elements of the state vector.
w(i) is the process noise (system noise, plant noise). It is a sequence of
3
random vectors of dimension M. The process noise represents the

3
Sometimes the process noise is represented by G(i)w(i) where G(i) is the noise gain matrix.
With that, w(i) is not restricted to have dimension M. Of course, the dimension of G(i) must be
appropriate.