Modern Control System Theory and Design, 2nd Edition

5.3.  SENSITIVITY

Sensitivity is a measure of the dependence of a system’s characteristics on those of a particular element. The differential sensitivity of a system’s closed-loop transfer function H(s) with respect to the characteristics of a given element K(s) is defined as

where

H(s) = C(s)/R(s).

A more meaningful definition can be obtained by rewriting Eq. (5.2) as

Equation (5.2) states that the differential sensitivity of H(s) with respect to K(s) is the percentage change in H(s) divided by that percentage change in K(s) that has caused the change in H(s) to occur. This definition is valid only for small changes. It is important to note that sensitivity is a function of frequency and an ideal system has zero sensitivity with respect to any parameter.

In order to illustrate the concept of sensitivity, consider the general control system shown in Figure 5.1a. Here K₁ represents the transfer function of the input transducer, K₂ represents the transfer function of the feedback transducer, and G(s) represents the combined transfer function of an amplifier, stabilizing network, motor, and gear train in the forward part of the feedback loop.

The overall system transfer function H(s) is given by

Let us now determine the sensitivity of the overall system transfer function with respect ...