Model-Predictive Control: The Optimal-Control Formulation

MPC plans actuation by solving a constrained optimal-control problem over a finite horizon each cycle, the core planner for breeder shape and burner potential.

Control as constrained optimization

Model-predictive control (MPC) chooses actuator commands by solving, every control cycle, a finite-horizon optimal-control problem: minimize a cost that penalizes deviation from targets and control effort, subject to the plant model and hard operating constraints. It is the L3 planner that commands the breeder's PF coils and the burner's fueling/plug actuators, always inside a certified safe envelope.

Discrete-time MPC problem at time k:

  minimize over u_0..u_{N-1}, x_1..x_N
    sum_{i=0}^{N-1} [ (x_i - x_ref)' Q (x_i - x_ref)
                     + u_i' R u_i ]  +  (x_N - x_ref)' P (x_N - x_ref)
  subject to
    x_{i+1} = f(x_i, u_i)          (model / linearization)
    x_min <= x_i <= x_max          (state constraints, envelope)
    u_min <= u_i <= u_max          (actuator limits)
    du_min <= u_i - u_{i-1} <= du_max  (slew limits)

The receding horizon

MPC solves the whole horizon but applies only the first command, then re-solves next cycle with fresh state from the twin. This receding-horizon feedback rejects disturbances and model error: each solve corrects for what actually happened. The horizon length N trades foresight against solve time within the 1-100 ms L3 budget.

Why MPC over simple feedback

Unlike a PID loop, MPC handles multiple coupled inputs and outputs and, critically, respects hard constraints explicitly. The breeder's shape, vertical position, and current share coils with limited authority; the burner must hold the ambipolar potential without exceeding actuator and stress limits. MPC allocates effort across these under one constrained optimization rather than fighting loops.

# receding-horizon MPC loop (schematic)
while running:
    x0 = twin.estimate_state()        # from KRONOS-CTRL
    U  = solve_ocp(x0, x_ref, model, constraints, N)  # QP solve
    apply(U[0])                       # only first command
    # next cycle re-solves with new x0 (feedback)

The plant model f is a linearization from the twin (Green's-function shape response for the breeder, ambipolar sensitivity for the burner), refreshed as the operating point moves. MPC never commands outside the envelope, and the L1 hardware failsafe remains authoritative beneath it - MPC optimizes; it does not override protection.