A unified approach for robust stability design of PID controllers

In this paper a graphical technique is introduced for finding all continuous-time or discrete-time proportional integral derivative (PID) controllers that satisfy a robust stability constraint for an arbitrary order transfer function with time delay. These problems can be solved by finding all achievable PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. The key advantage of this procedure is that this method depends only on the frequency response of the system. The ability to include the time delay in the nominal model of the system will often allow for designs with reduced conservativeness in plant uncertainty and an increase in size of the set of all PID controllers that robustly stabilize the system. The delta operator is used to describe the controllers in a discrete-time model, because it not only possesses numerical properties superior to the discrete-time shift operator, but also converges to the continuous-time controller as the sampling period approaches zero. A unified approach allows us to use the same procedure for discrete-time and continuous-time robust stability design of PID controllers.