Model Predictive Control (MPC)

Deep Dive

Model Predictive Control (MPC) is a control strategy that, at every step, uses a model of the system to predict the next several steps into the future, finds the sequence of actions that minimizes a cost while obeying constraints, applies only the first action, then measures the new state and re-plans from scratch. This matters because it is the most natural way to control a system that has hard limits — motor torque ceilings, joint angle stops, or “stay outside this obstacle” — which simpler controllers cannot enforce directly.

Prerequisites

State-Space Representation MPC predicts the future using a state-space model of the system.

LQR MPC's cost (state error + control effort) is the LQR cost — but minimized over a finite horizon with constraints.

The Receding Horizon

At each control step MPC predicts N steps ahead, optimizes the whole action sequence, but commits only to the first action. One step later it slides the window forward and repeats from the freshly measured state.

 step k:   measure x_k
           ├─ predict ──►  □ □ □ □ □   (N steps ahead)
           ├─ optimize ─►  u_k u_{k+1} ... u_{k+N-1}  (whole sequence)
           └─ APPLY only u_k  ────────────────►  (discard the rest)

 step k+1: measure x_{k+1}   (window slid forward by one)
              ├─ predict ──►   □ □ □ □ □
              ├─ optimize ─►   u_{k+1} ...
              └─ APPLY only u_{k+1}
                 ... repeat every cycle ...

Why re-plan every step? The first prediction is never perfect — the real world drifts from the model. Re-planning from the actual measured state closes the feedback loop and corrects that drift.

The Killer Feature: Hard Constraints

LQR and PID compute a control law and then clip whatever comes out — they cannot reason about limits ahead of time. Because MPC solves a constrained optimization each step, it can natively enforce:

Actuator limits — |u| ≤ u_max (torque, thrust, steering angle)
State limits — joint stops, speed caps, stay-above-ground
Safety bounds — keep distance from an obstacle or another robot

This is why MPC dominates legged locomotion, autonomous driving, and drone flight, where violating a limit means a fall, a crash, or a flip.

Example: MPC for a Double Integrator

A double integrator (position, velocity, controlled by acceleration) is the “hello world” of MPC. Here we drive it to the origin while respecting an acceleration limit, using a convex QP solved over a horizon.

# Minimal MPC. Requires: pip install cvxpy numpy
import numpy as np
import cvxpy as cp

# Discrete double integrator: state = [position, velocity], input = acceleration
dt = 0.1
A = np.array([[1.0, dt],
              [0.0, 1.0]])
B = np.array([[0.5 * dt**2],
              [dt]])

N = 20                  # prediction horizon (steps)
Q = np.diag([10.0, 1.0])  # penalize position error, then velocity
R = np.array([[0.1]])     # penalize control effort
u_max = 1.0               # HARD actuator limit

def mpc_step(x0):
    """Solve the horizon QP from current state x0, return only the first action."""
    x = cp.Variable((2, N + 1))   # predicted states
    u = cp.Variable((1, N))       # planned actions
    cost, constraints = 0, [x[:, 0] == x0]
    for k in range(N):
        cost += cp.quad_form(x[:, k], Q) + cp.quad_form(u[:, k], R)
        constraints += [x[:, k + 1] == A @ x[:, k] + B @ u[:, k]]  # model
        constraints += [cp.abs(u[:, k]) <= u_max]                  # constraint
    cost += cp.quad_form(x[:, N], Q)  # terminal cost
    cp.Problem(cp.Minimize(cost), constraints).solve()
    return float(u.value[0, 0])       # APPLY ONLY THE FIRST ACTION

# Closed loop: measure -> solve -> apply first -> step -> re-plan
x = np.array([5.0, 0.0])   # start 5 m away, at rest
for t in range(60):
    a = mpc_step(x)        # re-plan from the measured state
    x = A @ x + B.flatten() * a
print("final state (pos, vel):", np.round(x, 3))  # -> near [0, 0]

Each call to mpc_step re-solves the full horizon but the loop applies only u[0] — that is the receding horizon in code.

LQR The unconstrained, infinite-horizon limit MPC reduces to when no constraints bind.

State-Space Representation The model MPC rolls forward to predict future states.

System Identification How you obtain the accurate model MPC depends on.

Robot Dynamics The physical equations of motion that nonlinear MPC predicts over.

Sources

Rawlings, Mayne & Diehl, Model Predictive Control: Theory, Computation, and Design, 2nd ed., Nob Hill Publishing, 2017.
Borrelli, Bemporad & Morari, Predictive Control for Linear and Hybrid Systems, Cambridge University Press, 2017.
Di Carlo et al., “Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control,” IEEE/RSJ IROS, 2018.