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.

Ms = ||S(jw)||
              inf

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 by courtesy of NICONET e.V. to calculate the infinity norm of the sensitivity function.

References
[1] Aström, K. and Hägglund, T. (1995) PID Controllers: Theory, Design and Tuning, Second Edition. Instrument Society of America.

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