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

for some given vector-valued function . This can be written in compressed form by using and to represent -dimensional vectors:

The expression above is often called the

Here is a simple, one-dimensional example of a state transition equation:

This is called a

(8.28) |

in which is a constant that depends on the given value for .

Steven M LaValle 2016-12-31