Frequency Response Design

Frequency Response Design (gnc-autonomy/control/frequency-response-design)

Use when the task is Bode frequency response analysis of an open loop transfer function for flight control design: magnitude and phase at a frequency, gain crossover and phase crossover frequencies, gain margin and phase margin, and the closed loop stability verdict from the margins.

Domain quick reference

For the canonical type-1 plant G(s) = K/(s(s+1)(s+2)), represented as numerator num = [K] and denominator den = [1, 3, 2, 0] in descending powers of s, evaluated on the imaginary axis s = j*w:

• Magnitude: |G(j*w)| = K / (w * sqrt(w^2 + 1) * sqrt(w^2 + 4)).
• Unwrapped phase: arg G(j*w) = -90 deg - atan(w) - atan(w/2).
• Worked anchor, K = 2 at w = 1 rad/s: |G| = 2/sqrt(10) ~= 0.6325 (-3.98 dB) and phase = -161.57 deg.
• Phase crossover w_pc: unwrapped phase reaches -180 deg, so atan(w) + atan(w/2) = 90 deg, giving w^2/2 = 1 and w_pc = sqrt(2) ~= 1.4142 rad/s. At that frequency |G| = K/6.
• Gain margin = 1/|G(j*w_pc)| = 6/K. Worked anchors: K = 2 gives 3.0 (9.54 dB); K = 6 gives 1.0 (0 dB); K = 8 gives 0.75 (-2.50 dB).
• Gain crossover w_gc: |G(j*w_gc)| = 1, so w^2(w^2+1)(w^2+4) = K^2. Worked anchor K = 2: w_gc ~= 0.7493 rad/s.
• Phase margin = 180 deg + phase(w_gc). Worked anchors: K = 2 gives ~32.61 deg; K = 6 gives 0 deg; K = 8 gives ~-7.5 deg.
• Stability verdict: stable iff gain margin > 0 dB AND phase margin

  0 deg. K = 2 stable, K = 6 marginal (0 dB, 0 deg), K = 8 unstable (negative gain margin).

• Units: omega in rad/s, magnitude linear or dB (20*log10), phase and phase margin in degrees, gain margin linear or dB.

Workflow

1. Write the open loop transfer function as numerator and denominator coefficient lists in descending powers of s, e.g. G(s) = K/(s(s+1)(s+2)) as num = [K], den = [1, 3, 2, 0].
2. Evaluate magnitude and phase at the frequency of interest with frequency_response(num, den, omega), or magnitude_db and phase_deg directly.
3. Find the 0 dB crossing with gain_crossover_frequency(num, den).
4. Find the -180 deg crossing with phase_crossover_frequency(num, den); with no crossing the result is infinite.
5. Compute margins(num, den) to get the gain margin in dB and the phase margin in degrees plus their pass flags.
6. Read stability_verdict(num, den) and iterate on K until the verdict is stable with both margins above zero.

Pitfalls

• Confusing this leaf with root-locus-design: root locus traces closed loop pole motion as gain varies; Bode analysis works on the open loop frequency response s = j*w and judges stability from the margins, never from pole coordinates.
• Confusing this leaf with lead-lag-compensation: the compensator leaf sizes lead/lag networks to boost a deficient phase margin; this leaf only measures the margins of the given open loop transfer function and does not synthesize compensators.
• Confusing this leaf with pid-control-design: PID tuning picks gains (Ziegler-Nichols, pole placement, anti-windup); this leaf reads the margins of whatever open loop transfer function is supplied, with no tuning law of its own.
• Confusing this leaf with state-space-analysis: state space works with A/B/C matrices, controllability, and eigenvalues; this leaf works with polynomial transfer function coefficients and frequency response only.
• Reading the gain margin at the gain crossover: GM is 1/|G(j*w_pc)| evaluated at the phase crossover, while the phase margin uses the gain crossover. Swapping the two frequencies produces wrong margins.
• Forgetting that atan2 wraps phase into (-180, 180]: the phase crossover search needs the unwrapped continuous phase, otherwise the -180 deg crossing is invisible on the wrapped plot.
• Applying the margin verdict to non-minimum-phase plants: the verdict assumes a minimum-phase open loop with monotonically falling phase; otherwise the Nyquist criterion is required.
• Mixing units: omega in rad/s, magnitude in dB with the 20*log10 convention, and margins in dB and degrees; a linear magnitude used inside the dB gain margin formula gives wrong results.

Behavior contract (gate 3)

The magnitude, phase, crossover, margin, and stability logic is exercised by the gate 3 contract test: scripts/test_frequency_response_design.py against scripts/frequency_response_design_logic.py (stdlib unittest, offline). Run: python3 scripts/test_frequency_response_design.py

Compliance

• FAR-25 (14 CFR Part 25) is US government public domain and CS-25 (EASA) is a free download; both are referenced for the airworthiness context of stability margins, name and paraphrase only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.