Sensitivity (control systems)
The controller parameters are typically matched to the process characteristics and since the process may change, it is important that the controller parameters are chosen in such a way that the closed loop system is not sensitive to variations in process dynamics. One way to characterize sensitivity is through the nominal sensitivity peak formula_1: formula_2 where formula_3 and formula_4 denote the plant and controller's transfer function in a basic closed loop control system written in the Laplace domain using unity negative feedback. The sensitivity function formula_5, which appears in the above formula also describes the transfer  function from external disturbance to process output. In fact, assuming an additive disturbance "n" after the output of the plant, the transfer functions of the closed loop system are given by formula_6 Hence, lower values of formula_7 suggest further attenuation of the external disturbance. The sensitivity function tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that formula_8 is less than one are reduced by an amount equal to the distance to the critical point formula_9 and disturbances with frequencies such that formula_8 is larger than one are amplified by the feedback. It is important that  the largest value of the sensitivity function be limited for a control system and it is common to require that the maximum value of the sensitivity function, formula_1, be in a range of 1.3 to 2. Sensitivity circle. The quantity formula_1 is the inverse of the shortest distance from the Nyquist curve of the loop transfer function to the critical point formula_9. A sensitivity formula_1 guarantees that the distance from the critical point to the Nyquist curve is always greater than formula_15 and the Nyquist curve of the loop transfer function is always outside a circle around the critical point  formula_16 with the radius formula_15, known as the sensitivity circle. formula_1 defines the maximum value of the sensitivity function and the inverse of formula_1 gives you the shortest distance from the open-loop transfer function formula_20 to the critical point formula_16.