Pitch-Rate Autopilot Oscillation Caused by Unmodelled Lag and Light Damping
An attitude-hold autopilot for a 900 kg UAV was designed from a rigid-body model. The pitch-rate response to elevator, in (deg/s) per deg, is q/δe = 3.0/[(s/3.5)² + 0.9(s/3.5) + 1], that is, a short-period mode at 3.5 rad/s with damping ratio 0.45. A proportional rate loop of gain Kp = 1.5 deg/deg is closed around it, and the design note claims 30° phase margin and 8 dB gain margin, computed from the airframe model alone. The flight hardware adds an elevator actuator with a first-order lag of 0.080 s, a 20 ms transport delay (telemetry plus a 50 Hz sample-and-hold), and an elevator rate limit of ±40°/s. On the first flight the aircraft settles into a sustained oscillation at about 2 Hz and ±2.5 deg/s which appears at every airspeed, neither growing nor decaying. Gains, sensor and airframe match the model. Work on paper; nonlinear behaviour need only be identified, not simulated.
1. Using the stated loop gain and including the actuator lag, the transport delay and the sample-and-hold, find the actual crossover frequency and phase margin. Is the closed loop stable, and does this explain the flight observation?
Read the worked answer
Open loop: L(s) = Kp · 3.0 / D(s) · 1/(1 + 0.08s) · e^(-0.02s) · ZOH, with Kp = 1.5, so the gain is 4.5.
Magnitude: |D(jω)| = √[(1 − r²)² + (0.9r)²] with r = ω/3.5. Setting 4.5/|D| = 1 gives r = 2.24, so ωc ≈ 7.84 rad/s.
Phase budget at 7.84 rad/s:
| Element | Phase lag |
|---|---|
| Short-period (r = 2.24, ζ = 0.45) | 180° − atan(0.9r/(r²−1)) = 180° − 26.6° = 153.4° |
| Actuator, atan(0.08 × 7.84) | 32.1° |
| Transport delay, ωT = 0.157 rad | 9.0° |
| Sample-and-hold, ωT/2 = 0.078 rad | 4.5° |
| Total | 199.0° |
Phase margin = 180° − 199° = −19°. The loop is unstable, with negative gain margin as well, so the design note's 30° margin is an artefact of having dropped every element faster than the airframe mode.
The flight oscillation is at about 2 Hz = 12.6 rad/s, which is above the linear crossover but close to the actuator corner (1/0.08 = 12.5 rad/s) and where the elevator rate limit is active. A stable limit cycle at a fixed frequency and fixed amplitude, independent of airspeed, is the signature of a rate-limited nonlinearity closing an otherwise unstable linear loop.
2. Find the proportional gain that would restore a 45° phase margin, and assess whether that gain is operationally acceptable.
Read the worked answer
Solve for the frequency where the total loop phase is −135°, holding the same actuator, delay and hold:
Total lag(ω) = [180° − atan(0.9r/(r²−1))] + atan(0.08ω) + 57.3° × (0.03ω)
- ω = 4.0 rad/s: 106.5 + 17.7 + 6.9 = 131.1° → PM 48.9°
- ω = 4.2 rad/s: 112.2 + 18.6 + 7.2 = 138.0° → PM 42.0°
- ω = 4.1 rad/s: ≈ 109.5 + 18.2 + 7.1 = 134.8° → PM ≈ 45°
So the crossover must fall to about 4.1 rad/s. At r = ω/3.5 = 1.17, |D| = √[(1 − 1.371)² + (1.054)²] = 1.118, so the permissible loop gain is 1.118 and
Kp = 1.118/3.0 ≈ 0.37 deg/deg
That is a factor of four reduction from 1.5, and crossover drops from 7.8 to 4.1 rad/s.
Operational assessment: the closed-loop bandwidth would be about 4 rad/s, only 1.2 × the short-period natural frequency. The response to a pitch-rate command becomes slow and oscillatory, gust rejection and glidepath tracking degrade, and the aircraft is still flying with the airframe mode essentially at crossover - the phase margin is bought almost entirely from a lightly damped resonance sitting in the control band. It is stable on paper but not a good design: the margin depends critically on an airframe frequency that shifts with speed, altitude, mass and configuration, which is exactly why the margin is fragile. Gain reduction alone should be treated as a temporary restriction, not the fix.
3. Propose and quantify the redesign that gives 45° phase margin with crossover at or above 5 rad/s, given that the airframe gain is fixed. Rank the options you considered.
Read the worked answer
The phase table shows where the degrees are: at 7.8 rad/s the airframe supplies 153° of lag and the actuator, delay and hold supply 46°. Even with zero transport delay, PM would be −10°, so the delay is not the root cause - the root cause is crossing over at 2.2 × the short-period frequency with ζ = 0.45.
Option A - damp the mode (best). A pitch damper or rate-feedback inner loop raising ζ from 0.45 to 0.8 changes the second-order lag at a given r. Re-solving for 45° gives ωc ≈ 4.6 rad/s with |D| = 2.19, so Kp ≈ 0.73 - twice the gain of the gain-only fix for the same margin, with materially better disturbance rejection.
Option B - faster hardware. Actuator lag 0.04 s and delay 10 ms, with ζ = 0.8: total lag at 5.2 rad/s is 116.9 + 11.7 + 6.0 = 134.6°, giving PM ≈ 45° with |D| = 2.67 and Kp ≈ 0.89.
Option C - gain reduction only. Kp ≈ 0.37, crossover 4.1 rad/s; worst of the three.
Recommended: Option A now (software, no hardware change), with Option B as the growth path if bandwidth must exceed about 6 rad/s. Independently, keep a rate limit only where it cannot destabilise - shaping the command rather than limiting the actuator - and verify the design against the mode frequency moving ±30% over the envelope.