Pid Control Design

PID Control Design (gnc-autonomy/control/pid-control-design)

Use when the task is PID controller design for an aerospace flight or GNC loop: computing the output from the proportional, integral, and derivative error terms, deriving gains from the plant model (first or second order) by Ziegler-Nichols tuning or pole placement, protecting the integrator with anti-windup clamping, or checking the loop's gain and phase margins.

Domain quick reference

• The PID output is u = kpe + kiint(e) + kd*de/dt where e is the error, int(e) its accumulated integral, and de/dt its derivative. kp acts on the error magnitude, ki removes steady-state error by integrating it, kd anticipates the error trend and adds damping.
• Ziegler-Nichols continuous-cycling tuning uses the ultimate gain ku and ultimate period tu measured at the stability boundary. Classic rules: P gives kp = 0.5ku; PI gives kp = 0.45ku with Ti = tu/1.2; PID gives kp = 0.6*ku with Ti = tu/2 and Td = tu/8.
• Pole placement for the first-order plant G(s) = b/(s + a) with a PI controller: matching the closed loop to s^2 + 2zetawns + wn^2 gives kp = (2zeta*wn - a)/b and ki = wn^2/b.
• Pole placement for the second-order plant G(s) = b/(s^2 + a1s + a0) with a PID controller: matching to (s^2 + 2zetawns + wn^2)(s + p3) gives kd = (2zetawn + p3 - a1)/b, kp = (wn^2 + 2zetawnp3 - a0)/b, and ki = wn^2p3/b.
• Anti-windup: when the actuator saturates, the integrator keeps accumulating and drives the loop into a long overshoot. Conditional integration clamps the accumulated integral to +/-limit each step, so the integrator cannot wind up beyond the actuator's authority.
• Margins for the type-1 open loop K/(s(s + a)): crossover where K^2 = wc^2(wc^2 + a^2), phase margin 90 - atan(wc/a) in degrees. The phase reaches -180 deg only at infinite frequency, so the gain margin is infinite for this loop.
• Discrete implementation: on a sampled flight computer the derivative term uses the backward difference (e_k - e_{k-1})/dt at sample time dt, and the integral accumulates kiedt; dt must be positive and the sample rate consistent with the loop bandwidth.

Workflow

1. Write the error equation: e = command - measured state in plant units (angle, rate, position).
2. Model the plant: first order b/(s + a) or second order b/(s^2 + a1*s + a0) from the flight dynamics (see the flight-mechanics stability and control leaves).
3. Choose a tuning route: Ziegler-Nichols from a measured ku/tu (scripts/pid_control_design_logic.py: ziegler_nichols) or pole placement from a target wn/zeta (pole_placement_first_order, pole_placement_second_order).
4. Add anti-windup with integrator_clamp on the accumulated integral.
5. Verify the margins of the loop (stability_margins_type1) and sanity-check the gains (kp > 0, ki >= 0, kd >= 0).
6. Implement in discrete time with discrete_derivative and the sample time dt; recompute the gains if dt changes materially.

Pitfalls

• Tuning with Ziegler-Nichols from an unvalidated ku/tu: the ultimate values come from the stability boundary, not from any operating point.
• Placing poles without checking that the resulting gains are physically sane; negative integral or derivative gains are a red flag.
• Omitting anti-windup on an actuator-limited loop: the integrator winds up during saturation and the loop overshoots on release.
• Reading the gain margin as finite for a type-1 loop; its phase only reaches -180 deg at infinite frequency.
• Using a continuous derivative on a sampled computer; the backward difference needs dt and produces a gain that changes with sample rate.
• Mixing units across error, integral, and derivative terms; the gains must be dimensioned per term.

Behavior contract (gate 3)

The PID output, Ziegler-Nichols, pole placement, anti-windup, margin, and discrete-derivative logic is exercised by the gate 3 contract test: scripts/test_pid_control_design.py against scripts/pid_control_design_logic.py (stdlib unittest, offline). Run: python3 scripts/test_pid_control_design.py

Compliance

• ARP4754A is proprietary (SAE); name + paraphrase only per standards-map.yaml and brief 06 (revision note: ARP4754B supersedes; this skill keys to A, the certification-baseline revision).
• compliance: STANDARDS-REF, gated: false.