Tube Model Predictive Control
Tube MPC keeps the true state inside a bounded tube around a nominal trajectory by combining a nominal optimizer with an ancillary error-rejecting feedback.
Separating nominal and error
Tube MPC decomposes the control into two parts. A nominal MPC plans a disturbance-free trajectory. An ancillary feedback controller drives the difference between the real state and the nominal state, keeping that error inside an invariant set called the tube cross-section. Because the error stays bounded, the real trajectory stays within a tube surrounding the nominal plan.
The invariant tube
The tube cross-section is a robust positively invariant set for the error dynamics under the ancillary feedback: once the error is inside, the ancillary controller keeps it inside despite any disturbance in the bounded set. The nominal MPC then optimizes subject to constraints tightened by exactly this tube, so that even in the worst case the true state satisfies the original constraints.
Why it is attractive
Tube MPC recovers most of the guarantees of full robust MPC at roughly the cost of nominal MPC, because the expensive worst-case reasoning is precomputed once into the tube and the tightened constraints. The online optimization is a standard nominal problem. The control applied is the nominal input plus the ancillary correction.
- Nominal plan plus ancillary error feedback
- Error confined to a robust invariant tube
- Constraints tightened by the tube cross-section
- Near-nominal online cost with robust guarantees
Design choices are the ancillary feedback gain, which shapes the tube, and the disturbance set. A tighter tube frees more of the state space for the nominal problem but demands more aggressive ancillary feedback. Homothetic and elastic-tube variants let the tube size adapt online.
For a design-stage plant with bounded disturbances and hard constraints, tube MPC offers a practical route to guaranteed constraint satisfaction, evaluated in simulation. It is the most widely used constructive form of robust MPC.