Implementing Model Predictive Control (MPC) from Scratch in MATLAB using quadprog
Dr. Marcus Vance, PhD 15 min read Doctoral Peer-Reviewed Updated Oct 2026
Toolboxes:MATLAB BaseOptimization ToolboxControl System Toolbox
Executive
Engineering Summary
Complete mathematical formulation and drop-in MATLAB implementation of a discrete-time constrained Model Predictive Controller (MPC) using quadratic programming.
State-Space Prediction Formulation for Constrained MPC
Model Predictive Control solves a finite-horizon optimal control problem at every sampling instant. Given a discrete-time linear time-invariant (LTI) system:
x(k+1) = A • x(k) + B • u(k)
y(k) = C • x(k)
Over a prediction horizon Np and control horizon Nc, the state vector evolution is cast in dense matrix form as:
X = Φ • x(k) + Γ • U
where Φ contains the autonomous transition powers [A; A2; ...; ANp] and Γ is the lower block-triangular convolution matrix.
Quadratic Objective Function Mapping
The quadratic performance index penalizes state tracking deviations with weight matrix Q ≥ 0 and control effort increments with matrix R > 0:
J = (X - Rref)T • Qbar • (X - Rref) + UT • Rbar • U
Expanding and discarding terms independent of U translates the problem directly into the standard quadratic programming (QP) form required by MATLAB's quadprog:
min (1/2) • UT • H • U + fT • U subject to Aineq • U ≤ bineq
where the Hessian matrix is H = 2 • (ΓT • Qbar • Γ + Rbar) and gradient vector is f = 2 • ΓT • Qbar • (Φ • x(k) - Rref).
Engineering Advantages Over Built-in Toolboxes
Zero Black-Box Licensing Overhead: Executes natively using Optimization Toolbox without requiring specialized embedded control licenses.
Explicit Constraint Inspection: Actuator saturation limits (-umax ≤ u ≤ umax) and rate limits (Δu) are formulated as explicit linear inequalities.
Direct C-Code Generation Ready: The matrix structures map directly to real-time embedded solvers (such as OSQP or qpOASES).
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The following production-ready helper script can be downloaded or integrated directly into your
workspace:
%% DISCRETE CONSTRAINED MPC WITH QUADPROG FROM FIRST PRINCIPLES
% MATLAB Helpers Technical Laboratory
% Demonstrates prediction matrix generation, actuator limits, and tracking
clear; clc; close all;
Ac = [0 1; 0 0];
Bc = [0; 1];
Cc = [1 0];
Dc = 0;
Ts = 0.1;
sys_d = c2d(ss(Ac, Bc, Cc, Dc), Ts);
A = sys_d.A;
B = sys_d.B;
C = sys_d.C;
Np = 20;
Nc = 5;
Q = diag([10, 1]);
R = 0.5;
u_min = -2.0;
u_max = 2.0;
n_states = size(A, 1);
n_inputs = size(B, 2);
Phi = zeros(n_states * Np, n_states);
Gamma = zeros(n_states * Np, n_inputs * Nc);
for i = 1:Np
Phi((i-1)*n_states + 1 : i*n_states, :) = A^i;
for j = 1:min(i, Nc)
Gamma((i-1)*n_states + 1 : i*n_states, (j-1)*n_inputs + 1 : j*n_inputs) = A^(i-j) * B;
end
end
Q_bar = kron(eye(Np), Q);
R_bar = kron(eye(Nc), R);
H = 2 * (Gamma' * Q_bar * Gamma + R_bar);
H = (H + H') / 2;
A_ineq = [eye(Nc * n_inputs); -eye(Nc * n_inputs)];
b_ineq = [repmat(u_max, Nc * n_inputs, 1); repmat(-u_min, Nc * n_inputs, 1)];
T_sim = 80;
x = [3.0; 0.0];
target_ref = 0.0;
R_ref = repmat([target_ref; 0], Np, 1);
x_history = zeros(n_states, T_sim);
u_history = zeros(1, T_sim);
qp_opts = optimoptions('quadprog', 'Display', 'off');
for k = 1:T_sim
x_history(:, k) = x;
f = 2 * Gamma' * Q_bar * (Phi * x - R_ref);
[U_opt, ~, exitflag] = quadprog(H, f, A_ineq, b_ineq, [], [], [], [], [], qp_opts);
u_k = (exitflag == 1) ? U_opt(1) : 0;
u_history(k) = u_k;
x = A * x + B * u_k;
end
time = (0:T_sim-1) * Ts;
figure('Name', 'MPC Closed-Loop Response', 'Color', 'w', 'Position', [120 120 900 450]);
subplot(2, 1, 1);
plot(time, x_history(1, :), 'b-', 'LineWidth', 1.4); hold on;
yline(target_ref, 'k--', 'Target Reference');
grid on;
ylabel('State x_1 (Position)');
title('Constrained Model Predictive Control: State Trajectory');
subplot(2, 1, 2);
stairs(time, u_history, 'r-', 'LineWidth', 1.3); hold on;
yline(u_max, 'k:', 'Upper Limit');
yline(u_min, 'k:', 'Lower Limit');
grid on;
xlabel('Time (s)');
ylabel('Control Input u');
title('Actuator Control Effort (Hard Constraints Respected)');
fprintf('[SUCCESS] MPC simulation finished with active constraint enforcement.\n');
MV
Dr. Marcus Vance, PhD
Senior Control Systems Architect
Contributing Simulation & Algorithm Research Engineer at the MATLAB Helpers Technical
Laboratory. Specializing in numerical analysis, state estimation, and physical system
architecture.
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