ABSTRACT
The basic equation on which we apply various approximations and thus various algorithms is
d d x
t f x t u t t= ( ( ), ( ), )
(17.1)
where: x(t) is the state of variable x at time t u(t) represents the value input vector at time t and the initial value of x is given by
x t x( )= =0 0 (17.2)
Such an equation can be expanded by a Taylor series. After expansion, the resulting expression can be approximated by various methods. The extent of approximation determines the accuracy of a particular approximation method. However, by applying the Taylor series expansion, we get the following expression:
x t h x t x t
t h x t
t h( ) ( ) ( ) ( )
! + = + + +d
d d d2
2
(17.3)
If xi represents the ith state, then the above expression will become
x t h x t x t
t h x t
t h
i i i i( ) ( ) ( ) ( )
! + = + + +d
d d d2
2
(17.4)
where: h is the step size
If we assume n + 1 terms of the Taylor series, the order of second derivative d2 2x t ti( ) d will be n − 2 since the second derivative is multiplied by h2. Similarly, the order of the third derivative will be n − 3 since it will be multiplied by h3, and so on. Thus, the approximation may be done up to the derivative multiplied by hnwhere n is known as the approximation order, and the integration method to be applied is said to be of the order of n.
