control//adaptive control//gain scheduling

Gain scheduling is a control strategy that switches or interpolates controller gains according to a measured variable that indexes the plant's operating point, and it is the most widely used way of controlling plants whose dynamics change: aircraft across speed and altitude, wind turbines across blade pitch, pipelines across flow. Linearize the plant at several operating points (linearization), design a PID or an LQR at each, and interpolate according to the **scheduling variable** \(\sigma\):

Gain scheduling is a control strategy that switches or interpolates controller gains according to a measured variable that indexes the plant's operating point, and it is the most widely used way of controlling plants whose dynamics change: aircraft across speed and altitude, wind turbines across blade pitch, pipelines across flow. Linearize the plant at several operating points (linearization), design a PID or an LQR at each, and interpolate according to the scheduling variable σ\sigmaσ:

K(σ)=∑iwi(σ) Ki,∑iwi(σ)=1,K(\sigma)=\sum_i w_i(\sigma)\,K_i,\qquad \sum_i w_i(\sigma)=1,K(σ)=i∑​wi​(σ)Ki​,i∑​wi​(σ)=1,

where KiK_iKi​ is the gain designed at point σi\sigma_iσi​ and wiw_iwi​ are interpolation weights, usually linear between the two neighbouring points. It is a lookup table with ambitions.

The examples are everywhere because the physics hands you the variable. Flight control has scheduled on dynamic pressure and Mach for decades, since control-surface effectiveness changes enormously with speed. Wind turbines schedule their blade-pitch PI on the pitch angle itself, because the rotor's aerodynamic sensitivity changes with it. In a process plant the transport delay of a pipe is its volume divided by the flow, so at half flow the delay doubles and gains are scheduled on flow. In drones, Betaflight attenuates PID gains as throttle rises (its TPA), and PX4 can scale the thrust command with measured battery voltage.

It is the flyswatter of adaptive control.

Deterministic, cheap and certifiable point by point, since each table entry is validated as a fixed controller. If the variable that matters can be measured, schedule before you think of adapting.

The scheduling variable must be measured and vary slowly compared with the loop. If it changes fast, the time-varying system can go unstable even though every frozen design is stable (the frozen-time stability fallacy); points between table entries also need validation.

It learns nothing. A change that does not show up in σ\sigmaσ, such as a nicked propeller, is invisible to the table; that residue is the job of adaptive control or of fault diagnosis.

It is the linearize-solve-repeat pattern done offline: one linear design per operating point, stitched together by interpolation, where an nonlinear MPC would relinearize online.

Maturity: industry for decades in aircraft, turbines, process plants and drones; it is the second rung of the ladder after a well-tuned PID (controller design).