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 [math]\displaystyle{ G(s) }[/math] and [math]\displaystyle{ C(s) }[/math] 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

[math]\displaystyle{ T(s) = \frac{G(s)C(s)}{1 + G(s)C(s)}. }[/math]

Differentiating [math]\displaystyle{ T }[/math] with respect to [math]\displaystyle{ G }[/math] yields

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

where [math]\displaystyle{ S }[/math] is defined as the function

[math]\displaystyle{ S(s) = \frac{1}{1 + G(s)C(s)} }[/math]

and is known as the sensitivity function. Lower values of [math]\displaystyle{ |S| }[/math] 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

[math]\displaystyle{ Y(s) = \frac{C(s)G(s)}{1+C(s)G(s)} R(s) + \frac{1}{1+C(s)G(s)} N(s). }[/math]

Hence, lower values of [math]\displaystyle{ |S| }[/math] suggest further attenuation of the external disturbance. The sensitivity function tells us how the disturbances are influenced by feedback. Disturbances with frequencies such that [math]\displaystyle{ |S(j \omega)| }[/math] is less than one are reduced by an amount equal to the distance to the critical point [math]\displaystyle{ -1 }[/math] and disturbances with frequencies such that [math]\displaystyle{ |S(j \omega)| }[/math] 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 [math]\displaystyle{ M_s }[/math] is defined as^[2]

[math]\displaystyle{ M_s = \max_{0 \leq \omega \lt \infty} \left| S(j \omega) \right| = \max_{0 \leq \omega \lt \infty} \left| \frac{1}{1 + G(j \omega)C(j \omega)} \right| }[/math]

and it is common to require that the maximum value of the sensitivity function, [math]\displaystyle{ M_s }[/math], be in a range of 1.3 to 2.

Sensitivity circle

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

References

↑ K.J. Astrom, "Model uncertainty and robust control," in Lecture Notes on Iterative Identification and Control Design. Lund, Sweden: Lund Institute of Technology, Jan. 2000, pp. 63–100.
↑ K.J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning, 2nd ed. Research Triangle Park, NC 27709, USA: ISA - The Instrumentation, Systems, and Automation Society, 1995.
↑ A. G. Yepes, et al., "Analysis and design of resonant current controllers for voltage-source converters by means of Nyquist diagrams and sensitivity function" in IEEE Trans. on Industrial Electronics, vol. 58, No. 11, Nov. 2011, pp. 5231–5250.
↑ Karl Johan Åström and Richard M. Murray. Feedback systems : an introduction for scientists and engineers. Princeton University Press, Princeton, NJ, 2008.