PID controller design using Bode´s integrals

The feedback relay test is usually used to identify one point on the Nyquist diagram of the plant model. It is shown that the derivatives of amplitude and phase of the plant model with respect to frequency at that point can be approximated by the Bode´s integrals without any model of the plant. The precision of the approximation for typical industrial plant models is studied. The derivatives are used to design a PID controller for slope adjustment of the loop Nyquist diagram and improve the closed-loop performance. Simulation examples illustrate the effectiveness and the simplicity of the proposed method to design the PID controllers.