Explicit MPC

The explicit model predictive control (explicit MPC) approach addresses the computational burden of real-time MPC by pre-computing the optimal control law offline. Rather than solving an optimization problem at each time step, explicit MPC solves the problem parametrically over the entire state space and stores the result as a piecewise affine feedback map. At runtime, the controller simply looks up the pre-computed control input for the current state — a table lookup instead of an optimization.

The parametric solution partitions the state space into polyhedral regions, each associated with an affine control law. The number of regions grows exponentially with the prediction horizon and the number of constraints, making the offline computation prohibitively expensive for high-dimensional systems. For small systems — automotive traction control, aircraft attitude regulation, process control with few states — explicit MPC achieves microsecond-level control updates that would be impossible with online optimization.

The trade-off is memory for computation. Explicit MPC requires storing the entire partition, which may require megabytes or gigabytes of memory. The lookup itself is also non-trivial: determining which region the current state belongs to requires a search through a polyhedral partition, and this search must be faster than the optimization it replaces.

Explicit MPC reveals a deeper systems-theoretic principle: when the problem structure is sufficiently regular, the distinction between planning and execution dissolves. The controller does not plan because it has already pre-computed all plans. It is a system that has internalized its own optimization landscape. Whether this is a form of intelligence or merely a form of memory depends on whether you believe that thinking is computation or whether computation is only thinking when it is generative.