Disturbance Rejection

Disturbance rejection is a control loop's ability to hold the output near its target despite unwanted inputs, governed by the sensitivity function.

Holding Steady Against Perturbations

A disturbance is any unwanted input that pushes the output away from its target: a wind gust on an aircraft, a load change on a motor, an ambient temperature swing on a reactor. Disturbance rejection is the closed loop's ability to counteract these perturbations and keep the output near its setpoint.

The role of sensitivity

How well a loop rejects disturbances at each frequency is measured by the sensitivity function S = 1/(1 + L), where L is the loop gain. A disturbance entering at the output is multiplied by S. High loop gain makes S small, so the disturbance is strongly attenuated. Rejection is therefore best at low frequencies, where loop gain is high, and weakens as frequency rises.

The internal model principle

• To reject a disturbance of a given type completely at steady state, the loop must contain a model of that disturbance's dynamics.
• An integrator in the loop rejects constant (step) disturbances, driving steady-state error to zero.
• A resonant term at a known frequency rejects a persistent sinusoidal disturbance at that frequency.
• This internal model principle explains why integral action is essential for eliminating steady offset.

Feedforward for measured disturbances

When a disturbance can be measured directly, feedforward cancels it before it reaches the output, achieving rejection faster and more completely than feedback alone. Feedback then handles the unmeasured disturbances and the imperfection of the feedforward cancellation.

Fundamental limits

Disturbance rejection cannot be improved without bound. The waterbed effect, encoded in the Bode sensitivity integral, means that pushing sensitivity down at some frequencies forces it up at others. Non-minimum-phase zeros and time delays tighten this limit further. A loop cannot reject fast disturbances beyond its bandwidth, and widening the bandwidth eventually conflicts with stability and noise.

Good disturbance rejection is often the real purpose of a control loop: not just to follow commands but to keep a plant on target as the world perturbs it. In a fusion device, rejecting the disturbances that shift plasma position and shape is central to holding a stable discharge.