When PD Is Not Enough: Bias, Integral Action, Saturation, and Windup

Single-Axis Spacecraft Attitude Holding

Consider a rigid spacecraft rotating about one principal axis. The state contains attitude angle \( \theta \) and angular velocity \( \omega \). The controller commands torque \( \tau_c \), while the spacecraft is also subjected to a constant disturbance torque \( \tau_d \).

Here \(J\) is the spacecraft moment of inertia about the controlled axis. The disturbance may represent a simplified persistent effect such as solar-radiation-pressure torque, aerodynamic torque, residual magnetic torque, actuator bias, or model mismatch.

Why PD Can Leave a Steady-State Pointing Error

Define the pointing error as

A PD controller commands

The proportional term produces restoring torque, while the derivative term damps angular motion. Under a constant disturbance, however, a non-zero equilibrium error may be required to generate enough proportional torque to balance the disturbance.

How Integral Action Removes the Bias

A PID-like spacecraft attitude law augments the controller with accumulated error:

In discrete time,

The integral state stores past error. Under a constant disturbance, it can build the exact control torque needed to cancel the disturbance while allowing the instantaneous pointing error to return toward zero.

Integrator Windup Under Actuator Saturation

Real actuators cannot supply unlimited torque. Reaction wheels, control-moment gyros, thrusters, magnetic torquers, and other actuators all have finite authority.

The controller may request a torque \( \tau_{cmd} \) larger than the actuator can deliver. Saturation clips the applied torque:

If the integrator continues accumulating error while the actuator is saturated, the internal control demand can grow far beyond the realizable hardware output. This is integrator windup.

The result can be larger overshoot, longer settling time, sluggish recovery after leaving saturation, and unnecessarily large internal torque demand.

Anti-Windup and Real System Constraints

The simplest educational anti-windup strategy is conditional integration. Instead of always updating the integral state, integration is paused when saturation is active and the current error would drive the actuator still farther into saturation.

This is more useful than simply freezing the integrator whenever saturation occurs, because it still allows the stored integral state to unwind when the error is helping the controller return toward the allowable actuator region.

const pushingFurtherIntoPositiveSaturation =
    rawTorque > tauMax && error > 0;

const pushingFurtherIntoNegativeSaturation =
    rawTorque < -tauMax && error < 0;

const blockIntegration =
    pushingFurtherIntoPositiveSaturation ||
    pushingFurtherIntoNegativeSaturation;

if (!blockIntegration) {
    integralError += error * dt;
}

More advanced implementations may use back-calculation, tracking anti-windup, integrator clamping, or actuator-aware dynamic compensation.

Compare PD, PID, and PID with Anti-Windup

Adjust the spacecraft inertia, target attitude (entered in degrees), controller gains, disturbance torque, and actuator torque limit. The plots update automatically. Compare the selected controller with the other two architectures using exactly the same physical conditions.

Selected Controller — Performance Metrics

Compare All Three Controllers

The next plots run PD, PID without anti-windup, and PID with anti-windup under identical spacecraft inertia, disturbance, gain, target, and actuator constraints.

Controller Comparison Metrics

How to Read the Plots

What This Problem Shows

The deeper engineering lesson is: