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

A basic closed loop control system, using unity negative feedback. C(s) and G(s) denote compensator and plant transfer functions, respectively.

Let ${\displaystyle G(s)}$ and ${\displaystyle C(s)}$ 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

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

Differentiating ${\displaystyle T}$ with respect to ${\displaystyle G}$ yields

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

where ${\displaystyle S}$ is defined as the function

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

and is known as the sensitivity function. Lower values of ${\displaystyle |S|}$ 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

Block diagram of a control system with disturbance

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

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

Hence, lower values of ${\displaystyle |S|}$ suggest further attenuation of the external disturbance. The sensitivity function tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that ${\displaystyle |S(j\omega )|}$ is less than one are reduced by an amount equal to the distance to the critical point ${\displaystyle -1}$ and disturbances with frequencies such that ${\displaystyle |S(j\omega )|}$ is larger than one are amplified by the feedback.^[1]

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 ${\displaystyle M_{s}}$ is defined as^[2]

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

and it is common to require that the maximum value of the sensitivity function, ${\displaystyle M_{s}}$, be in a range of 1.3 to 2.

Sensitivity circle

The quantity ${\displaystyle M_{s}}$ is the inverse of the shortest distance from the Nyquist curve of the loop transfer function to the critical point ${\displaystyle -1}$. A sensitivity ${\displaystyle M_{s}}$ guarantees that the distance from the critical point to the Nyquist curve is always greater than ${\displaystyle {\frac {1}{M_{s}}}}$ and the Nyquist curve of the loop transfer function is always outside a circle around the critical point ${\displaystyle -1+0j}$ with the radius ${\displaystyle {\frac {1}{M_{s}}}}$, known as the sensitivity circle. ${\displaystyle M_{s}}$ defines the maximum value of the sensitivity function and the inverse of ${\displaystyle M_{s}}$ gives you the shortest distance from the open-loop transfer function ${\displaystyle L(j\omega )}$ to the critical point ${\displaystyle -1+0j}$.^[3][4]