# 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](<https://experts.illinois.edu/en/publications/constrained-model-predictive-control-stability-and-optimality/> "Constrained model predictive control: Stability and optimality; Automatica").
- **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](<https://sites.engineering.ucsb.edu/~jbraw/mpc/MPC-book-2nd-edition-1st-printing.pdf> "Model Predictive Control: Theory, Computation, and Design; book").
- **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](<https://sites.engineering.ucsb.edu/~jbraw/mpc/MPC-book-2nd-edition-1st-printing.pdf> "Model Predictive Control: Theory, Computation, and Design; book").
- **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](<https://sites.engineering.ucsb.edu/~jbraw/mpc/MPC-book-2nd-edition-1st-printing.pdf> "Model Predictive Control: Theory, Computation, and Design; book").

## 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](<https://cse.lab.imtlucca.it/~bemporad/publications/papers/automatica-mpqp.pdf> "The explicit linear quadratic regulator for constrained systems; Automatica").
- **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](<https://epubs.siam.org/doi/abs/10.1137/S0363012902400713> "A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control; SIAM J. Control Optim.").
- **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](<https://www.sciencedirect.com/science/article/pii/S0005109804002870> "Robust model predictive control of constrained linear systems with bounded disturbances; Automatica").
- **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](<https://marco-campi.unibs.it/pdf-pszip/IEEE-TAC-scenario.pdf> "The Scenario Approach to Robust Control Design; IEEE Trans. Automat. Contr.").
- **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](<https://www.merl.com/publications/docs/TR2017-217.pdf> "Stochastic Model Predictive Control; MERL technical report").
- **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](<https://arxiv.org/abs/1707.02342> "Information Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving; IEEE Transactions on Robotics").

## 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](<https://arxiv.org/abs/1609.06408> "Control Barrier Function Based Quadratic Programs for Safety Critical Systems; IEEE Trans. Automat. Contr.").
- **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](<https://arxiv.org/abs/1609.01387> "Learning Model Predictive Control for iterative tasks. A Data-Driven Control Framework; IEEE Transactions on Automatic Control").
- **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](<https://arxiv.org/abs/1611.03537> "Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control; Automatica").
- **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](<https://arxiv.org/abs/1805.12114> "Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models; NeurIPS").
- **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](<https://arxiv.org/abs/1810.13400> "Differentiable MPC for End-to-end Planning and Control; NeurIPS").
- **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](<https://arxiv.org/abs/1806.04167> "Learning an Approximate Model Predictive Controller with Guarantees; IEEE Control Syst. Lett.").
- **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](<https://arxiv.org/abs/1811.05890> "Data-Enabled Predictive Control: In the Shallows of the DeePC; ECC").
    - **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](<https://arxiv.org/abs/1903.06804> "Regularized and Distributionally Robust Data-Enabled Predictive Control; CDC").
- **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](<https://www.annualreviews.org/content/journals/10.1146/annurev-control-090419-075625> "Learning-Based Model Predictive Control: Toward Safe Learning in Control; Annu. Rev. Control Robot. Auton. Syst.").
- **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](<https://arxiv.org/abs/1705.10702> "Cautious Model Predictive Control using Gaussian Process Regression; IEEE Trans. Contr. Syst. Technol.").
- **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](<https://proceedings.mlr.press/v120/wabersich20a.html> "Bayesian model predictive control: Efficient model exploration and regret bounds using posterior sampling; L4DC").
- **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](<https://arxiv.org/abs/1906.04679> "Data-Driven Model Predictive Control with Stability and Robustness Guarantees; IEEE Trans. Automat. Contr.").
- **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](<https://arxiv.org/abs/1812.05506> "A predictive safety filter for learning-based control of constrained nonlinear dynamical systems; Automatica").

## 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](<https://arxiv.org/abs/2203.04955> "Temporal Difference Learning for Model Predictive Control; ICML").
    - **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](<https://arxiv.org/abs/2310.16828> "TD-MPC2: Scalable, Robust World Models for Continuous Control; ICLR").
- **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](<https://arxiv.org/abs/2506.09985v1> "V-JEPA 2: Self-Supervised Video Models Enable Understanding, Prediction and Planning; arXiv"); [code](<https://github.com/facebookresearch/vjepa2>).
- **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](<https://proceedings.mlr.press/v267/zhou25t.html> "DINO-WM: World Models on Pre-trained Visual Features enable Zero-shot Planning; ICML").
    - **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](<https://arxiv.org/abs/2610.42146v1> "Hierarchical World Models with Implicit Dynamics; arXiv").
- **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](<https://publications.syscop.de/Reiter2026.pdf> "Synthesis of model predictive control and reinforcement learning: Survey and classification; Annual Reviews in Control").

<a id="software-and-references"></a>

## 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](<https://arxiv.org/abs/1910.13753> "acados: a modular open-source framework for fast embedded optimal control; Mathematical Programming Computation").
- **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](<https://www.do-mpc.com/en/latest/>).

## Related topics

- [Feature extraction](https://mlpapers.org/feature-extraction/)
- [Reinforcement learning](https://mlpapers.org/reinforcement-learning/)
- [Optimization](https://mlpapers.org/optimization/)
- [Symbolic regression](https://mlpapers.org/symbolic-regression/)
- [Meta-learning](https://mlpapers.org/meta-learning/)
- [Bayesian inference](https://mlpapers.org/bayesian-inference/)
- [Evaluation](https://mlpapers.org/evaluation/)