In this paper a graphical technique is introduced for finding all continuous-time or discrete-time proportional integral derivative (PID) controllers that satisfy a weighted sensitivity constraint of 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 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 weighted sensitivity design of PID controllers.

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