##

13.2.4 Extending Models by Adding Integrators

The differential models from Section 13.1 may seem
unrealistic in many applications because actions are required to
undergo instantaneous changes. For example, in the simple car, the
steering angle and speed may be instantaneously changed to any value.
This implies that the car is capable of instantaneous acceleration
changes. This may be a reasonable approximation if the car is moving
slowly (for example, to analyze parallel-parking maneuvers). The
model is ridiculous, however, at high speeds.

Suppose a state transition equation of the form
is
given in which the dimension of is . The model can be enhanced
as follows:

- Select an action variable .
- Rename the action variable as a new state variable,
.
- Define a new action variable that takes the place of
.
- Extend the state transition equation by one dimension by
introducing
.

This enhancement will be referred to as *placing an integrator in
front of *. This procedure can be applied incrementally as many
times as desired, to create a chain of integrators from any action
variable. It can also be applied to different action variables.

**Subsections**
Steven M LaValle
2020-08-14