Robust stability design of PID controllers for arbitrary-order transfer functions with uncertain time delay

In this paper a graphical technique is introduced for finding all proportional integral derivative (PID) controllers that satisfy the robust stability constraint of a given single-input-single-output (SISO) linear time-invariant (LTI) system of any order system with time delay. This problem can be solved by finding all PID controllers that simultaneously stabilize the closed-loop characteristic polynomial and satisfy constraints defined by a set of related complex polynomials. A key advantage of this procedure is that it is only depends on the frequency response of the system and does not require the plant transfer function coefficients. 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.