Reaction-Wheel Speed Loop Interacting With a Flexible Solar Array

The first flexible mode is only 0.06 Hz above the 1.02 Hz crossover, so r = 1.02/1.08 = 0.944. The second-order flexible factor has magnitude 1 / sqrt((1 − r²)² + (2ζr)²) with ζ = 0.005. Here 1 − r² = 0.108 and 2ζr = 0.00944, so the magnitude is about 1/0.108 = 9.3, or 19.3 dB. The phase contribution is only about −5° at that exact ratio, but the gain peak is large and the phase falls rapidly as the frequency approaches 1.08 Hz. A small shift in the mode frequency, due to propellant burn, thermal distortion, or array configuration, can move the loop onto the steep part of the resonance and consume far more phase.

The low damping gives Q = 1/(2ζ) = 100, or 40 dB at resonance. The loop is therefore operating near a high-Q pole with very little gain margin. The 5 Hz low-pass filter adds additional phase lag at crossover, and the PI integrator contributes its own −90° in the low-frequency region. The combination leaves only 18° phase margin and 3 dB gain margin. This is not a controller tuning detail; it is a structural interaction problem.

The notch filter is the better first choice if it can be made robust. A notch at 1.08 Hz with about 20 dB depth and a width of roughly 10% of the mode frequency removes the resonant gain peak before it reaches the loop crossover. Reducing the loop gain by 6 dB then moves the crossover to a lower frequency and restores phase margin. If the mode estimate is accurate, the result could be phase margin above 40° and gain margin above 9 dB, which meets the requirement.

The risk is that the notch is sensitive to mode-frequency uncertainty. If the actual mode moves by more than the notch width, the attenuation is lost and the loop can be less stable than before. The design should use the worst-case mode-frequency range, verify the notch depth at the extremes, and consider a gain-stabilized or adaptive notch. The notch also adds phase lag near crossover, so the final loop must be checked, not just the notch magnitude.

Lowering the crossover to 0.3 Hz is simpler and robust against mode-frequency shift, but it reduces wheel-speed loop bandwidth by more than a factor of three. The 0.01 Hz solar-torque disturbance is still below crossover, so integral action can reject it, but the transient response and momentum-dumping authority become slower. It may also interact with the attitude loop and delay wheel-desaturation recovery. I would use the notch plus gain reduction as the primary fix, with the 0.3 Hz crossover as a fallback if the mode frequency cannot be bounded.

The limit-cycle amplitude equals the tachometer quantization step of 0.02 rad/s. This is a classic quantization limit cycle: the controller sees an apparent speed error of one least-significant bit, applies a torque correction, and the wheel speed crosses the quantization boundary, producing a repeated overshoot and reversal. The notch filter does not remove it because the oscillation is generated by the nonlinear quantizer, not by the flexible mode. Higher loop gain makes the limit cycle more likely; the 6 dB gain reduction may have helped but did not eliminate it.

Fixes should break the limit cycle without destabilizing the loop. Options include: increasing tachometer resolution or using a higher-resolution encoder; adding a small amount of filtering or a deadband around zero speed error, with the deadband sized to the quantization step; adding a dither signal so the quantizer averages over levels; or using a state observer or disturbance observer to reconstruct wheel speed more smoothly. A deadband is simple but creates a steady-state speed error; filtering adds phase lag and must be included in the stability check. The best fix is usually better speed measurement or an observer, because it addresses the cause rather than masking it. Verify the final design over the full quantization and mode-frequency range.