Categories &

Functions List

Function Reference: sensitivity

Function File: [Ms, ws] = sensitivity (L)
Function File: [Ms, ws] = sensitivity (P, C)
Function File: [Ms, ws] = sensitivity (P, C1, C2, …)

Return sensitivity margin Ms. The quantity Ms is simply the inverse of the shortest distance from the Nyquist curve to the critical point -1. Reasonable values of Ms are in the range from 1.3 to 2. $$ M_s = ||S(j\omega)||_{\infty} $$ If no output arguments are given, the critical distance 1/Ms is plotted on a Nyquist diagram. In contrast to gain and phase margin as computed by function margin, the sensitivity Ms is a more robust criterion to assess the stability of a feedback system.

Inputs

L
Open loop transfer function. L can be any type of LTI system, but it must be square.
P
Plant model. Any type of LTI system.
C
Controller model. Any type of LTI system.
C1, C2, …
If several controllers are specified, function sensitivity computes the sensitivity Ms for each of them in combination with plant P.

Outputs

Ms
Sensitivity margin Ms as defined in [1]. Scalar value. If several controllers are specified, Ms becomes a row vector with as many entries as controllers.
ws
The frequency [rad/s] corresponding to the sensitivity peak. Scalar value. If several controllers are specified, ws becomes a row vector with as many entries as controllers.

Algorithm
Uses SLICOT AB13DD, Copyright (c) 1996-2025, SLICOT, available under the BSD 3-Clause (License and Disclaimer).

References

  1. Astr"om, K. and H"agglund, T. (1995) PID Controllers: Theory, Design and Tuning, Second Edition. Instrument Society of America.

Source Code: sensitivity