For most systems, the integration must be performed numerically. A system simulator based on numerical integration can be constructed by breaking into smaller intervals and iterating classical methods for computing numerical solutions to differential equations. The Euler method is the simplest of these methods. Let denote a small time interval over which the approximation will be made. This can be considered as an internal parameter of the system simulator. In practice, this is usually much smaller than the used in the discrete-time model of Section 14.2.2. Suppose that and are given and the task is to estimate .
By performing integration over time, the state transition equation can be used to determine the state after some fixed amount of time has passed. For example, if is given and is known over the interval , then the state at time can be determined as
Using the fact that
Steven M LaValle 2012-04-20