Sensitivity (control systems)
# Sensitivity (control systems)

In control engineering, the sensitivity (or more precisely, the sensitivity function) of a control system measures how variations in the plant parameters affects the closed-loop transfer function. Since the controller parameters are typically matched to the process characteristics and 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. Moreover, the sensitivity function is also important to analyse how disturbances affects the system.

## Sensitivity function

Let $G(s)$ and $C(s)$ denote the plant and controller's transfer function in a basic closed loop control system written in the Laplace domain using unity negative feedback.

### Sensitivity function as a measure of robustness to parameter variation

The closed-loop transfer function is given by

$T(s)={\frac {G(s)C(s)}{1+G(s)C(s)}}.$

Differentiating $T$ with respect to $G$ yields

${\frac {dT}{dG}}={\frac {d}{dG}}\left[{\frac {GC}{1+GC}}\right]={\frac {C}{(1+CG)^{2}}}=S{\frac {T}{G}},$

where $S$ is defined as the function

$S(s)={\frac {1}{1+G(s)C(s)}}$

and is known as the sensitivity function. Lower values of $|S|$ implies that relative errors in the plant parameters has less effects in the relative error of the closed-loop transfer function.

### Sensitivity function as a measure of disturbance attenuation

The sensitivity function 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

$Y(s)={\frac {C(s)G(s)}{1+C(s)G(s)}}R(s)+{\frac {1}{1+C(s)G(s)}}N(s).$

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 $|S(j\omega )|$ is less than one are reduced by an amount equal to the distance to the critical point $-1$ and disturbances with frequencies such that $|S(j\omega )|$ is larger than one are amplified by the feedback.

## Sensitivity peak and sensitivity circle

### Sensitivity peak

It is important that the largest value of the sensitivity function be limited for a control system. The nominal sensitivity peak $M_{s}$ is defined as

$M_{s}=\max _{0\leq \omega <\infty }\left|S(j\omega )\right|=\max _{0\leq \omega <\infty }\left|{\frac {1}{1+G(j\omega )C(j\omega )}}\right|$

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.

### Sensitivity circle

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 ${\frac {1}{M_{s}}}$ and the Nyquist curve of the loop transfer function is always outside a circle around the critical point $-1+0j$ with the radius ${\frac {1}{M_{s}}}$, known as the sensitivity circle. $M_{s}$ defines the maximum value of the sensitivity function and the inverse of $M_{s}$ gives you the shortest distance from the open-loop transfer function $L(j\omega )$ to the critical point $-1+0j$.

