The second problem from (8.20) is to determine an expression for
. This is where the laws of physics, such as the acceleration of rigid bodies due to applied forces and gravity. The most common case is time-invariant dynamical systems, in which
depends only on the current state and not the particular time. This means each component
is expressed as
![$\displaystyle {\dot x}_i = f_i(x_1,x_2,\ldots,x_n) ,$](img771.gif) |
(8.25) |
for some given vector-valued function
. This can be written in compressed form by using
and
to represent
-dimensional vectors:
![$\displaystyle {\dot x}= f(x) .$](img773.gif) |
(8.26) |
The expression above is often called the state transition equation because it indicates the state's rate of change.
Here is a simple, one-dimensional example of a state transition equation:
![$\displaystyle {\dot x}= 2 x - 1 .$](img774.gif) |
(8.27) |
This is called a linear differential equation. The velocity
roughly doubles with the value of
. Fortunately, linear problems can be fully solved ``on paper''. The solution to (8.27) is of the general form
![$\displaystyle x(t) = \frac{1}{2} + c e^{2 t} ,$](img775.gif) |
(8.28) |
in which
is a constant that depends on the given value for
.
Steven M LaValle
2020-11-11