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

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

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

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

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

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

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

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

Y ( s ) = C ( s ) G ( s ) 1 + C ( s ) G ( s ) R ( s ) + 1 1 + C ( s ) G ( s ) N ( s ) . {\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 | S | {\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 | S ( j ω ) | {\displaystyle |S(j\omega )|} is less than one are reduced by an amount equal to the distance to the critical point − 1 {\displaystyle -1} and disturbances with frequencies such that | S ( j ω ) | {\displaystyle |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 {\displaystyle M_{s}} is defined as

M s = max 0 ≤ ω < ∞ | S ( j ω ) | = max 0 ≤ ω < ∞ | 1 1 + G ( j ω ) C ( j ω ) | {\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, M s {\displaystyle M_{s}}, be in a range of 1.3 to 2.

Sensitivity circle

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

See also