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Control engineering
Sensitivity at one frequency
Magnitude of the sensitivity function at one frequency, from the magnitude and phase of the open-loop transfer function.
- Ref
- CTL-19
- Rev
- v1
- Field
- Control engineering
- Result
- |S|
S = 1/(1 + L) is the sensitivity function. It maps the reference to the tracking error and the plant disturbance to the output, so |S| below 1 means feedback reduces errors at that frequency, and above 1 means it amplifies them. |1 + L| is the distance from the point −1 to the open-loop frequency response on a Nyquist plot, so the closer the curve passes to −1, the larger |S|. S and the complementary sensitivity T = L/(1 + L) always add to 1.
Magnitude of the sensitivity function at one frequency, from the magnitude and phase of the open-loop transfer function. A value above 1 means feedback amplifies errors at that frequency rather than reducing them.
Calculator
- Distance to the critical point, |1 + L|1.474
- Complementary sensitivity, |T| = |L||S|1.357
- Sensitivity in decibels−3.368 dB
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At the defaults the sensitivity is 0.6786, so errors at this frequency are reduced to 67.9 % of what they would be without feedback. Moving the phase to −180° at |L| = 0.9, close to the critical point, gives |S| = 10.
- Peak sensitivity is a robustness measure: the smallest distance of the Nyquist curve to −1 is the reciprocal of the largest |S|. Designers commonly aim to keep the peak of |S| below about 1.5 to 2 (3.5 to 6 dB).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| |L| Open-loop gain magnitude | 0.1 to 5 | 0.2298 | 1.411 | 1.181 | +10% | −10.8 % |
| ∠L Open-loop phase | −180 to −90 ° | 0.4472 | 1 | 0.5528 | +1.8 ° | −2.02 % |
Sweep of Open-loop gain magnitude
| Open-loop gain magnitude | Sensitivity magnitude |
|---|---|
| 0.1 | 1.073 |
| 0.7125 | 1.414 |
| 1.325 | 1.065 |
| 1.938 | 0.7047 |
| 2.55 | 0.5066 |
| 3.163 | 0.3914 |
| 3.775 | 0.3176 |
| 4.388 | 0.2668 |
| 5 | 0.2298 |
Sweep one variable
Variables and how each one contributes
Sensitivity magnitude
What the equation gives you. Everything else feeds it.
Open-loop gain magnitude
The magnitude of the open-loop transfer function at the frequency of interest, as a plain ratio. At a phase well away from −180°, a larger |L| shrinks |S|. It is most dangerous when the phase is near −180° and |L| is close to 1. Raising it by 10 % (to 2.2) moves the result by −10.8 %, from 0.6786 to 0.6054.
Open-loop phase
The phase of the open-loop transfer function in degrees. The loop is close to trouble as it nears −180°. A phase closer to −180° moves L towards −1 and increases |S|. Lowering it by 20° (to −155°) moves the result by +25.7 %, from 0.6786 to 0.8529.
Worked example with the values above
- L = 2∠−135° = −1.414 + j−1.414, so 1 + L = −0.4142 + j−1.414.
- Distance from −1: |1 + L| = √(0.1716 + 2) = 1.474.
- |S| = 1 / 1.474 = 0.6786.
Assumptions and limits
- A stable closed loop with negative unity feedback, evaluated at a single frequency.
- L(jω) is known to adequate accuracy at that frequency. Modelling error matters most near |L| = 1.
Common mistakes
- Looking only at gain and phase margins at their crossover frequencies. The curve may come closer to −1 at another frequency.
- Treating |S| < 1 as universally good: the waterbed effect means that suppressing sensitivity at some frequencies raises it at others.
Statistics and uncertainty
Near L = −1 small errors become large ones. For |L| = 0.9 at −180°, a ±10 % error in the gain gives |S| from 5.26 to 100. That is why a plant model that is accurate to 10 % is not enough to trust a loop that passes this close to −1.
Where it is used
- Finding the frequencies at which a loop amplifies disturbances and measurement errors.
- Relating the closeness of the Nyquist curve to −1 with the robustness of a design.
Maths you need
- Complex numbers in polar form: magnitude and angle
- Nyquist plot: the open-loop frequency response in the complex plane