PID vs. Model-Based Control for the Double Integrator Plus Dead-Time Model: Noise Attenuation and Robustness Aspects

One of the most important contributions of modern control theory from the 1960s was the separation of the dynamics of state-space controller design from the dynamics of state reconstruction. However, because modern control theory predates the mass spread of digital controllers and was predominantly focused on analog solutions that avoided modeling dead-time elements, it cannot effectively cover all aspects that emerged with the development of programmable devices and embedded systems. The same historical limitations also characterized the development of proportional-integral-derivative (PID) controllers, which began several decades earlier. Although they were used to control time-delayed systems, these solutions, which are most commonly used in practice today, can also be referred to as simplified disturbance observers that allow the avoidance of the the direct use of dead-time models. Using the example of controlling systems with a double integrator plus dead-time model, this article shows a novel controller design that significantly improves control performance compared to conventional PID controllers. The new control structure is a combination of a generalized state-space controller, interpreted as a higher-order derivative controller, and a predictive disturbance observer that uses the inversion of double integrator dynamics and dead-time models. It enables the elimination of the windup effect that is typical for PID control and extends the separation of the dynamics of setpoint tracking from the dynamics of state and disturbance reconstruction to time-delayed processes as well. The novelty of the presented solution offers several orders of magnitude lower amplification of measurement noise compared to traditional PID control. On the other hand, it offers high robustness and a stable transient response despite the unstable internal feedback of processes like the magnetic levitation system. The improvements achieved are so high that they call into question the classical solutions with PID controllers, at least for DIPDT models. In addition to the comparison with PID control, the relationship with traditional state space controllers, which today form the basis of active disturbance rejection control (ADRC), is also discussed and examined for processes including dead time.