One MPC Optimization Step
Solve a single model-predictive-control problem: minimize a quadratic cost over a short horizon, apply only the first control, then repeat.
Problem
Model predictive control (MPC) optimizes a sequence of future control inputs over a finite horizon using a model, applies only the first input, then re-solves at the next step with fresh measurements. This receding horizon gives feedback and lets you enforce constraints directly.
Setup
Take a scalar system x_{k+1} = x_k + u_k with current x0=2 and target 0. Over a horizon N=3 we minimize sum of x_k^2 + 0.1 u_k^2. We solve the small quadratic program for the optimal control sequence and keep only u0.
import numpy as np
N=3; x0=2.0; r=0.1
# build prediction x = x0 + cumulative sum of u
# minimize ||x||^2 + r||u||^2 over u in R^N
L=np.tril(np.ones((N,N))) # x_k depends on u_0..u_{k-1}
H=2*(L.T@L + r*np.eye(N))
f=2*(L.T@(x0*np.ones(N)))
u=np.linalg.solve(H,-f)
print('optimal sequence',np.round(u,3))
print('applied u0',round(u[0],3))Result
The optimizer front-loads control effort: the first move is the largest because acting early drives the state toward zero for the rest of the horizon. Only u0 is applied; at the next step the state is remeasured and the whole plan is recomputed, which is what makes MPC robust to model error and disturbances.
- Longer horizons give smoother, more anticipatory control but cost more computation each step.
- Adding inequality constraints (actuator limits) turns the linear solve into a constrained QP, still solvable in real time for modest sizes.
- Kronos coil-current control studies use MPC to respect voltage and ramp-rate limits while tracking a target plasma shape.