Model Predictive Control (MPC)

The core algorithmic strategy behind MPC. At each control interval, the controller solves a finite-horizon open-loop optimization problem over a prediction horizon of N steps. Only the first computed control action is applied to the plant. The horizon then 'recedes' forward one step, and the entire optimization is repeated with updated state feedback. This closed-loop feedback property provides inherent robustness against model mismatch and disturbances, distinguishing it from simple open-loop optimal trajectory planning.

The numerical optimization engine at the heart of most linear MPC formulations. When the cost function is quadratic and constraints are linear, the MPC problem reduces to a Quadratic Program. Solvers like qpOASES, OSQP, and FORCES Pro compute the optimal control vector in real-time. For nonlinear MPC, more complex Nonlinear Programming (NLP) solvers such as IPOPT or ACADOS are required, often leveraging sequential quadratic programming or interior-point methods.

MPC requires full knowledge of the system's current state vector, which is rarely fully measurable. A state observer reconstructs unmeasured states from available sensor outputs. The Kalman Filter provides statistically optimal estimation for linear systems with Gaussian noise. For nonlinear systems, the Extended Kalman Filter (EKF) or Moving Horizon Estimation (MHE)—which itself solves an optimization problem over a past window of measurements—provides the state estimate fed into the MPC controller.

The predictive accuracy of MPC is entirely dependent on the fidelity of its internal dynamic model. System identification builds these models from input-output data when first-principles physics are unknown. Techniques include:

• Subspace Identification (N4SID): For multivariable state-space models
• Prediction Error Minimization (PEM): For parametric input-output models
• Nonlinear ARX: For capturing nonlinear dynamics with regressors A poorly identified model leads to prediction errors that degrade control performance.

A defining advantage of MPC is its explicit handling of hard constraints on inputs, outputs, and states. Constraints encode physical limits—valve saturation, temperature ceilings, safety boundaries. If the optimization problem becomes infeasible due to a disturbance pushing the system outside the constraint set, constraint softening techniques introduce slack variables with heavy penalties to recover a solvable problem. Terminal constraint sets are designed to guarantee recursive feasibility and closed-loop stability.

For systems with small state and parameter dimensions, the online optimization can be pre-solved offline using multi-parametric programming. The result is a piecewise affine control law stored as a lookup table of critical regions. At runtime, the controller simply identifies which polyhedral region the current state falls into and applies the corresponding affine control law. This enables microsecond-scale execution on embedded hardware where a QP solver is impractical, but scales poorly with problem complexity.