Explicit Model Predictive Control
Explicit MPC precomputes the optimal control law offline as a piecewise-affine function of the state, replacing online optimization with a lookup.
Moving the computation offline
For a linear system with a quadratic cost and linear constraints, the MPC optimization is a multiparametric quadratic program with the current state as the parameter. Its solution, as a function of the state, turns out to be piecewise affine over a partition of the state space into polyhedral regions. Explicit MPC computes this partition and the affine gain for each region offline, once.
Online lookup
At runtime the controller identifies which polyhedral region contains the current state and applies the corresponding affine law: a matrix multiply and add. This eliminates the online solver entirely, making it attractive for systems with fast sampling or limited computing hardware where running a QP each step is infeasible.
The region explosion
The catch is that the number of regions grows rapidly, often exponentially, with the horizon length, the number of constraints, and the state dimension. For small problems, a few states and short horizons, explicit MPC is ideal; for larger problems the region count and the point-location search become impractical, and online implicit MPC is preferred. Approximate explicit methods trade optimality for a manageable partition.
- Solution is piecewise affine in the state
- Regions and gains precomputed offline once
- Online cost is region lookup plus affine evaluation
- Region count explodes with problem size
Point location uses binary search trees or hash schemes to keep the online lookup fast. Memory to store the regions is the main hardware cost, trading storage for computation relative to implicit MPC.
Explicit MPC suits small, fast subsystems, such as a local actuator regulator in a design-stage plant, where microsecond response and no solver are required. It is the same optimal law as implicit MPC, merely precomputed.