Robust performance design of PID controllers with inverse multiplicative uncertainty

In this paper a graphical method is introduced for finding all proportional integral derivative (PID) controllers that satisfy a robust performance constraint for a given single input-single-output (SISO) linear time invariant (LTI) transfer function of any order with time-delay. This problem 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. Inverse multiplicative modeling is used to describe the uncertainty of unstable perturbed system. A key advantage of this procedure is that it only depends on the frequency response of the system and does not require the plant transfer function coefficients. If the plant transfer function is given, the procedure is still appropriate. Inverse multiplicative modeling often allows for designs with reduced conservativeness in the unstable pole uncertainty and it increases the size of the set of all PID controllers that robustly meet the performance requirements.