Gain Scheduling
Gain scheduling controls a nonlinear plant by interpolating a family of linear controllers designed at fixed operating points.
The classic recipe
A nonlinear plant is linearized at several operating points spanning its envelope. A linear controller is designed for each. During operation a scheduling variable, measured in real time, selects or interpolates among the controllers. The idea is old and pervasive: flight control, engine control, and process control all rely on it, because it reuses the mature toolkit of linear design.
Interpolation choices
Controllers can be interpolated in several ways: blending their outputs, blending their gains, or blending their state-space matrices. Interpolating output signals with smooth weights avoids abrupt switching. A key requirement is that the scheduling variable vary slowly relative to the closed-loop dynamics, because the guarantees come from freezing the parameter at each point.
Pitfalls and remedies
The central weakness is that pointwise stability at every frozen operating point does not guarantee stability during fast transitions. A schedule of individually stable controllers can go unstable when the parameter moves quickly. Hidden coupling terms, arising when the scheduling variable is itself a state, further complicate matters. The velocity-based or quasi-LPV reformulations, and rigorous LPV synthesis, address these gaps with envelope-wide certificates.
- Design linear controllers at several operating points
- Interpolate on a measured scheduling variable
- Requires slow parameter variation for validity
- LPV methods give the guarantees scheduling lacks
Good practice includes scheduling on exogenous rather than state variables where possible, ensuring smooth interpolation to avoid transients, and validating the schedule against fast-transition simulations, not just frozen points.
Gain scheduling remains the most widely used method for nonlinear control in industry because of its practicality, and its modern, provably correct form is LPV control. For a design-stage machine with strongly operating-point-dependent dynamics, scheduling is the natural first architecture, refined into LPV where guarantees are required.