Nonlinear Model Predictive Control
Nonlinear MPC solves a constrained optimal control problem over a nonlinear model at each step, applying the first move and re-optimizing.
Receding horizon on nonlinear models
Nonlinear MPC predicts the plant's behavior over a finite horizon using a nonlinear dynamic model, minimizes a cost subject to state and input constraints, applies only the first control move, then shifts the horizon forward and repeats. The nonlinear model lets it handle systems where linear predictions would be inaccurate over the horizon, at the price of solving a nonconvex optimization each step.
Solving the optimization
The finite-horizon problem is discretized, by direct multiple shooting or collocation, into a large nonlinear program and solved with sequential quadratic programming or interior-point methods. Warm-starting from the previous solution and exploiting the sparse, banded structure of the problem make real-time solution feasible. The real-time iteration scheme performs a single Newton-type step per sample to meet tight timing.
Stability guarantees
Nominal stability is secured by a terminal cost and terminal constraint set: the terminal cost is a local control-Lyapunov function and the terminal set is control-invariant. Together they make the optimal cost a Lyapunov function for the closed loop, guaranteeing recursive feasibility and convergence. Without them, a sufficiently long horizon often suffices in practice but without a formal guarantee.
- Nonlinear prediction over a receding horizon
- Nonconvex NLP solved each step by SQP or interior point
- Terminal cost and set give stability guarantees
- Real-time iteration meets tight sampling deadlines
The nonconvexity means solvers find local optima, so a good initial guess matters; warm-starting usually keeps the solution near the previous one. Sensitivity to model accuracy motivates the robust and stochastic extensions.
For a design-stage machine with strongly nonlinear dynamics across its operating range, nonlinear MPC would enforce hard constraints while optimizing performance, all in simulation. It generalizes to economic, robust, and stochastic objectives.