Lead Lag Compensation

Lead Lag Compensation (gnc-autonomy/control/lead-lag-compensation)

Use when the task is classical compensator design for a flight control or GNC loop: synthesizing a phase lead network to raise the phase margin toward a specification, or a phase lag network to improve the steady state error constant without disturbing the crossover much.

Domain quick reference

• Transfer function representation: the plant is G(s) = num(s) / den(s) with coefficient lists in descending powers of s, so G(s) = 1/(s(s + 1)) is num = [1], den = [1, 1, 0]. The compensator D(s) sits in series ahead of the plant, giving the open loop L(s) = D(s) * G(s).
• Gain crossover frequency: omega_wc is the frequency where |G(j*omega_wc)| = 1 (0 dB); it is found by bisection over the monotone magnitude response of a proper plant.
• Phase margin: PM = 180 + phase(G(j*omega_wc)) in degrees. A positive phase margin means the closed loop is stable; transport design practice targets 45 degrees minimum, 60 degrees for comfort.
• Lead compensator form: D(s) = (1 + sT)/(1 + alpha * sT) with 0 < alpha < 1. The zero 1/T lies below the pole 1/(alpha*T), so the network adds positive phase and lifts the magnitude between them.
• Maximum phase boost: phi_m = asin((1 - alpha)/(1 + alpha)); solving for the ratio gives alpha = (1 - sin(phi_m))/(1 + sin(phi_m)). The peak phase occurs at omega_m = 1/(T * sqrt(alpha)).
• Gain rise at the peak: |D(jomega_m)| = 1/sqrt(alpha), which is -10log10(alpha) dB above the low frequency gain. The design places omega_m at the new crossover, where the plant magnitude is short by exactly that amount (|G(j*omega_m)| = sqrt(alpha)).
• Lead design recipe: boost = PM_desired - PM_plant + margin (5 to 12 degrees of margin for the crossover shift), alpha from the boost, omega_m where |G| = sqrt(alpha), then T = 1/(omega_msqrt(alpha)), zero = 1/T and pole = 1/(alphaT).
• Lag compensator form: D(s) = Kc * (1 + s/zero)/(1 + s/pole) with pole = zero/beta and beta > 1. The zero sits one decade below crossover so the phase lag at crossover stays near -5 degrees.
• Steady state error constants: position constant Kp = G(0) for a type-0 loop, velocity constant Kv = lim(s*G(s)) as s goes to 0 for a type-1 loop. Step error is 1/(1 + Kp), ramp error is 1/Kv.
• Lag effect on the error constant: with dc gain Kc = beta the velocity error constant is multiplied by beta, cutting the ramp error by the same factor while the crossover moves little.
• Certification context: control law design and the margins that verify it are developed and validated under ARP4754A development assurance; the compensator math here is the classical control methodology behind those checks.

Workflow

1. Write the plant as coefficient lists num and den; confirm it is proper so the magnitude falls off at high frequency.
2. Measure the plant phase margin with phase_margin_degrees; the crossover comes from gain_crossover_frequency.
3. For a phase margin specification, compute the required boost with design_lead_compensator: alpha from lead_alpha_from_phase_boost, the new crossover where the plant sits at sqrt(alpha), and the zero/pole from lead_zero_pole. Verify the boost is feasible (lead_max_phase_deg and lead_gain_boost_db bound it).
4. Assemble the lead network with lead_transfer_function and check the compensated loop with compensated_phase_margin; iterate the boost margin until the spec is met.
5. For steady state error, read the error constants with velocity_error_constant and position_error_constant, then size a lag network with design_lag_compensator (beta from the required Kv improvement). Confirm the loop stays stable with phase_margin_degrees of the series product.
6. Record the compensator transfer function, its zero and pole, the new crossover frequency, and the compensated phase margin; sanity check the phase at crossover with lead_phase_deg or lag_phase_deg.

Pitfalls

• Forgetting the boost margin: the lead peak is placed at the NEW crossover, but raising the crossover changes the plant phase; a 5 to 12 degree margin prevents the compensated loop from missing the specification.
• Asking for more boost than alpha can give: phi_m approaches 90 degrees only as alpha goes to 0; a boost near 60 degrees needs a very small alpha and a large gain rise, which stresses the loop.
• Placing the lead peak below the plant crossover: the compensator then boosts magnitude where the plant is already above 0 dB and the crossover lands well above omega_m, wrecking the phase margin.
• Putting the lag zero above crossover: the lag pole zero pair then adds several degrees of phase lag at crossover and erodes the margin instead of leaving it alone; keep the zero one decade below.
• Treating the lag as free gain: with dc gain beta the lag lifts Kv by beta, but the pair also moves the crossover slightly and adds phase lag; verify the compensated phase margin stays positive.
• Confusing lead and lag ratios: lead uses alpha in (0, 1) and a zero below the pole; lag uses beta > 1 and a pole below the zero. Swapping the forms produces a network with the wrong sign of phase.
• Using a single error constant for every loop type: the step error formula needs Kp from a type-0 loop; a type-1 loop has infinite Kp and zero step error, and the ramp error 1/Kv is the meaningful check.
• Checking the margins of G instead of the loop: the phase margin that matters is the one of the compensated open loop L = D * G, not the raw plant.

Behavior contract (gate 3)

The compensator design math is exercised by the gate 3 contract test: scripts/test_lead_lag_compensation.py against scripts/lead_lag_compensation_logic.py (stdlib unittest, offline). Run: python3 scripts/test_lead_lag_compensation.py

Compliance

• Standards referenced, not reproduced: ARP4754A frames the development assurance and verification of flight control law design that this compensator math supports; the lead lag formulas above are common classical control methodology, summary-only per standards-map.yaml. ARP4754A is proprietary (SAE), name and paraphrase only.
• compliance: STANDARDS-REF, gated: false.