Phase Margin

1 September 1, 2024

The phase margin measures how much phase variation at the crossover frequency to lose stability. The phase margin determines how much the phase can lag before the system becomes neutrally stable, or unstable. A marginal phase margin indicates frequencies where a controller might tend to oscillate. The phase margin is computed where the gain crosses 0. A common phase margin is 30-45 degrees. The phase margin can be modified by changing the gain to move the crossover point to a lower frequency. Measurement latencies, control system delays, and IMU dynamics may decrease the phase margin. To find the phase margin you find the frequency \(f_c\) where the magnitude of the loop gain is 1. Then you determine the phase margin at that frequency
$$\phi_m=180deg+\angle T(f_c)$$
If there are no poles in the right-hand plane and there is exactly one crossover frequency, then the loop \(T/(1+T)\) contains no right-hand plane poles if and only if the phase margin is positive.

[[Nyquist Stability Criterion]]
[[Frequency Domain System Characteristics]]
[[Performance Robustness]] – classical robustness uses phase margin.
[[MVS Hardover Fail]] – phase error remains at the value of the highest branch.
[[MVS Fail to Zero]] – phase error jumps between branches at zero.
[[Sensitivity Function]] – can be used to calculate the phase margin with the equation \(PM=2sin^{-1}(\frac{1}{2s_{min}})\)
[[Modeling Errors]] – modeling errors are most problematic when they are near the gain crossover point.
F-14 Flight Control System– found decreased phase margin for FCS at sampling rates below 25 Hz
[[Servo Control Loop Stability]] – should have phase margin of 40 deg or higher.
[[RASCAL Baseline Control System]] – aimed to reach 45 deg phase margin
[[BP3.1]] – gain/phase margin are good indicators of control robustness
[[Closed-loop Synchronous Buck Converter]] – Phase margin should be between 45 and 76 degrees
[[Controller Design]] – you should maximize low-frequency gain while maintaining the phase margin
[[X-43 Flight Control System]] – specifies phase margin in different flight regions.
ADS-33E– specifies a 45 degree phase margin
[[F-117 Air Data Probes]] – needed a small time constant to meet phase margins
[[Singular Values of a Matrix]] – Phase margin of a unity singular value is $\pm60$ degrees
[[MCLAWS-2 Roll Axis Broken-Loop Response]] – shows good prediction of phase margin
[[MCLAWS-2 Roll Axis]] – Shows EH-60L and CONDUIT phase margin
[[X-29 relaxed stability]] – phase margin of 41 degrees
[[F-117 Frame Rate Management]] – lowest phase margin for various flight conditions used
[[F-117 Autopilot Design Process]] – 3DOF simulation used to determine initial phase margin
[[F-22 FCS design Philosophy]] – phase margin was 60 degrees

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Backlinks

[[Actuator Time Constant Reduction]]
[[Bode Plot]]
[[Stability Analysis of Systems]]
[[Stability Margins]]
[[Step Response]]
[[System Robustness]]