The compensator will calculate and control the solenoids' input force. However, the exogenous force is something the compensator has no control over. It is caused by the linear potentiometer's friction and the springs' reaction force. The compensator must be able to estimate it in order to calculate a suitable input force. Thus, the exogenous force must be the third state variable. We previously modeled the exogenous force as a constant. Now, for the dynamic plant, we will model the exogenous force as a time derivative of white noise. White noise is a random signal that has equal intensity across all frequencies, giving it a constant power spectral density. See Calculating the Kalman Gains for a more detailed description of white noise.
q is formally called the process white noise. This is the white noise that will interfere with the internal workings of the plant.
The linear potentiometer can only measure the displacement of the payload. It cannot measure the payload velocity or acceleration. The attachment of a speedometer or accelerometer to the payload will interfere with the operation of the plant. At the very least, it will add a lot more mass and friction to it. Here is the equation of the output.
r is the sensor white noise. It is a model for any inaccuracies the potentiometer itself may suffer from.
We would express the state equations as follows:
F is now the continuous-time exogenous input matrix for the dynamic plant. Note that it adds white noise to the state vector.