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 M s :
M s = max 0 ≤ ω < ∞ | S ( j ω ) | = max 0 ≤ ω < ∞ | 1 1 + G ( j ω ) C ( j ω ) |
where G ( s ) and C ( s ) denote the plant and controller's transfer function in a basic closed loop control System, using unity negative feedback.
The sensitivity function S , which appears in the above formula also describes the transfer function from external disturbance to process output. Hence, lower values of | S | suggest further attenuation of the external disturbance. The sensitivity function tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that | S ( j ω ) | is less than one are reduced by an amount equal to the distance to the critical point − 1 and disturbances with frequencies such that | S ( j ω ) | 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, M s , be in a range of 1.3 to 2.
The quantity M s is the inverse of the shortest distance from the Nyquist curve of the loop transfer function to the critical point − 1 . A sensitivity M s guarantees that the distance from the critical point to the Nyquist curve is always greater than 1 M s and the Nyquist curve of the loop transfer function is always outside a circle around the critical point − 1 with the radius 1 M s , known as the sensitivity circle.
