Model Predictive Control

Receding-horizon control and guarantees

  • Constrained MPC - Mayne et al. study repeated finite-horizon optimization: apply the first control, observe the new state and replan. State/input constraints, recursive feasibility and closed-loop stability are central design questions. Mayne et al. 2000.
  • Terminal ingredients - Terminal costs, terminal sets and local controllers can establish feasibility and stability under stated assumptions. Merely solving a finite-horizon problem does not supply these guarantees. Rawlings et al. 2017.
  • State estimation - Output-feedback MPC acts on estimated states. Moving-horizon estimation fits recent measurements under dynamics and constraints; estimation error must enter the closed-loop analysis. Rawlings et al. 2017.
  • Economic MPC - Optimizes an operating objective directly instead of only tracking a setpoint. Closed-loop economic performance and stability require their own analysis; a cheaper predicted trajectory is not a universal stability certificate. Rawlings et al. 2017.

Uncertainty and explicit control

  • Explicit MPC - Bemporad et al. compute constrained linear-quadratic feedback offline as a continuous piecewise-affine law. Online optimization becomes region lookup and evaluation; the partition must be stored. Bemporad et al. 2002.
  • Real-time iteration - Diehl et al. interleave Newton-type optimization with feedback, refining approximate solutions as the plant evolves. Contractivity and suboptimality bounds justify this local scheme; control updates need not solve the nonlinear problem to convergence. Diehl et al. 2005.
  • Tube-based robust MPC - Mayne et al. control constrained linear systems with bounded disturbances using a nominal trajectory and invariant error set. Robust stability is established for the disturbance-invariant set under their assumptions. Mayne et al. 2005.
  • Scenario-based constraints - Samples uncertain convex constraints to form a finite problem. Calafiore and Campi bound violation probability under independent sampling; this risk/confidence guarantee does not establish safety over an entire control trajectory. Calafiore and Campi 2006.
  • Stochastic MPC - Uses disturbance distributions to optimize expected performance and impose probabilistic constraints. Chance constraints permit specified violation probabilities; they differ from worst-case bounds for every admissible disturbance. Mesbah et al. 2017.
  • MPPI / information-theoretic MPC - Williams et al. derive sampling-based control updates from an information-theoretic objective and test autonomous driving. Sampling evaluates candidate trajectories without requiring a differentiable optimizer. Williams et al. 2018.

Learning dynamics, costs and feasible behavior

  • Control-barrier-function filters - Enforces inequalities in the control input to keep a specified state set forward invariant. Ames et al. combine barrier and Lyapunov conditions in a quadratic program; feasibility and model assumptions are part of the safety claim. Ames et al. 2017.
  • Learning MPC for iterative tasks - Rosolia and Borrelli build terminal sets and costs from previous successful trajectories. Their construction gives recursive feasibility and non-increasing iteration cost under the stated iterative-task assumptions. Rosolia and Borrelli 2018.
  • Koopman MPC - Lifts observations into a feature space and fits approximately linear controlled dynamics by least squares. Linear MPC can then optimize nonlinear-system predictions; the finite lifting determines approximation quality. Korda and Mezić 2018.
  • PETS - Combines probabilistic dynamics ensembles with trajectory sampling to plan under model uncertainty. Chua et al. report strong sample efficiency on studied control tasks; uncertainty-aware planning alone does not enforce state constraints. Chua et al. 2018.
  • Differentiable MPC - Amos et al. differentiate through the controller’s convex approximation at a fixed point to learn costs and dynamics. The gradient construction depends on that solution structure, not arbitrary differentiation through any solver. Amos et al. 2018.
  • Learning an MPC approximation - Hertneck et al. imitate a robust MPC law, then use statistical validation of approximation error to support closed-loop guarantees. The robust design tolerates bounded action errors; low average imitation loss alone is insufficient. Hertneck et al. 2018.
  • DeePC - Predicts trajectories directly from recorded input/output data. Coulson et al. establish equivalence to MPC for deterministic linear time-invariant systems with suitable data; noisy nonlinear extensions use regularization. Coulson et al. 2019.
    • Distributionally robust DeePC - Coulson et al. relate regularization to worst-case expected cost over distributional uncertainty for stochastic linear systems. The equivalence holds for specified costs; it supplies a rationale for regularization under noisy trajectory data. Coulson et al. 2019.
  • Learning-based MPC - Hewing et al. distinguish learning prediction models, learning controller components and using MPC to constrain learning. Model accuracy, uncertainty and closed-loop guarantees are separate parts of the design. Hewing et al. 2020.
  • Cautious GP-MPC - Adds a Gaussian-process residual to nominal dynamics and propagates approximate state uncertainty into chance constraints. Hewing et al. test racing hardware; the uncertainty approximation is part of the controller, not just a prediction interval. Hewing et al. 2020.
  • Bayesian MPC - Wabersich and Zeilinger use posterior sampling to trade exploitation against model exploration and derive finite-time regret bounds under assumptions. Learning performance and constraint satisfaction remain distinct criteria. Wabersich and Zeilinger 2020.
  • Stable data-driven MPC - Berberich et al. add terminal equalities and regularized slack for noisy LTI data. Their multi-step scheme has practical exponential stability with persistently exciting data and small bounded output noise. Berberich et al. 2021.
  • Predictive safety filters - Check a proposed action against a model-based backup plan and modify it when needed. Wabersich and Zeilinger’s safety argument depends on uncertainty bounds and feasible recovery, not on the upstream policy alone. Wabersich and Zeilinger 2021.

Latent planning and comparative evidence

  • TD-MPC - Jointly learns rewards, values and action-conditioned latent dynamics. Matching future encodings regularizes the model without observation reconstruction. Short-horizon planning uses a terminal value estimate for longer-term return. Hansen et al. 2022.
    • TD-MPC2 - Extends latent MPC and reports strong results across 104 tasks with one hyperparameter set. Its multi-task scaling experiments assess control return; these benchmarks do not certify physical constraint satisfaction. Hansen et al. 2024.
  • V-JEPA 2-AC - Learns action-conditioned dynamics on frozen video-JEPA features and plans toward image goals. Demonstrates robot grasping and pick-and-place in two labs without task-reward training. Pick-and-place uses supplied subgoal images. Assran et al. 2025; code.
  • DINO-WM - Learns action-conditioned dynamics over frozen DINOv2 patch features without pixel reconstruction and plans toward image goals. On simulated PushT with goals reachable within 25 steps, success is 90% versus 44% with a global DINO embedding. Zhou et al. 2025.
    • ID-WM - Searches diffusion noise for nearby subgoals, then plans actions with a local dynamics model. On simulated PushT with random goals and state observations, success is 77% versus 28% for a flat planner given matched inference time. Zhou et al. 2026.
  • MPC-RL combinations - Reiter et al. distinguish MPC as an expert for training, as part of the deployed policy, or as part of the critic. This separates learning from planned demonstrations, learning a controller’s parameters and learning value estimates. Reiter et al. 2026.

Software

  • acados - Modular C solvers for embedded optimal control, with Python/Matlab interfaces and CasADi-compatible models. Solver choice, tolerances and target hardware determine measured latency. Verschueren et al. 2021.
  • do-mpc - Python tools for nonlinear/economic MPC, robust multi-stage MPC and moving-horizon estimation. Includes simulation and differential-algebraic models; the toolbox supplies implementations, not automatic safety certification. docs.