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 ${\dot x}= f(x,u)$ is given in which the dimension of $X$ is $n$. The model can be enhanced as follows:

1. Select an action variable $u_i$.
2. Rename the action variable as a new state variable, $x_{n+1} = u_i$.
3. Define a new action variable $u'_i$ that takes the place of $u_i$.
4. Extend the state transition equation by one dimension by introducing ${\dot x}_{n+1} = u'_i$.

This enhancement will be referred to as placing an integrator in front of $u_i$. 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.