Implementing a deadband on a PID controller for a dc motor
It's a common fact that a drive system has a resonance frequency due to moment of inertia and the elasticity of the transmission. If you have a high closed loop gain, then the system is always "nervous" waiting for dynamic action. To overcome this small oscillations problem in a professional equipment you set the notch filters to eliminate this resonance frequency. This is done by injecting signals and draw Bode plot.  You can add a lowpass filter, but you will loose the dynamics, or you might decrease the gain.

Introducing the dead band means introducing the nonlinearity in the system. Perhaps there are better solutions rather a dead band for the drive control. 

The professional drive controllers I have used, have an adaptation of Kp and Ti with regard the input value. At low error, the Kp is low and Ti is slightly higher. This could be also a way to go.



For example, you do the abs() on input then you make a linear function with minimal Kp at low input error and nominal Kp at high input error.

If you want a deadband anyway, you have a formula already in the article (29) and (30) give the new transformed input as depicted in figure 16.
 

  

To be continued...