Model Predictive Control (MPC) Framework

• MPC is an optimization-based control framework that solves a receding-horizon optimal control problem using current state measurements to manage multivariable and constrained systems.
• It integrates diverse approaches such as tracking, economic, robust, and learning-augmented strategies to enhance performance and maintain stability.
• MPC’s practical applications span robotics, power electronics, and energy systems, leveraging advanced solvers and predictive models for safety-critical operations.

Model Predictive Control (MPC) is an optimization-based control framework that computes control inputs by solving finite-horizon open-loop optimal control problems at each sampling instant using current state information. By utilizing a forward model of the system, MPC can handle multivariable systems, explicit state and input constraints, and time-varying dynamics, making it a cornerstone in advanced process control, robotics, power electronics, energy systems, and many other high-performance and safety-critical applications.

1. Foundational Principles and Mathematical Formulation

At the heart of the MPC framework is a receding-horizon optimal control problem. At each time step $k$, given the current state $x_k$, MPC solves

$$\min_{\{u_i\}_{i=0}^{N-1}} \sum_{i=0}^{N-1} \ell(x_{i|k}, u_{i|k}) + V_f(x_{N|k})$$

$\text{subject to}$

$$x_{0|k} = x_k,\quad x_{i+1|k} = f(x_{i|k}, u_{i|k}),\quad (x_{i|k},u_{i|k}) \in \mathbb{X}\mathbb{U},\ x_{N|k}\in \mathbb{X}_f$$

where $N$ is the prediction horizon, $\ell$ the stage cost, $V_f$ a terminal cost, $f$ the (potentially nonlinear) system dynamics, and the constraint sets $\mathbb{X}\mathbb{U}$ and $\mathbb{X}_f$ encode admissible states, inputs, and terminal region. Only the first computed input $u_{0|k}^*$ is applied before the optimization is repeated at the next sampling period. This closed-loop design confers inherent robustness to disturbances and modeling errors as future states are repeatedly re-optimized for the updated measurement (Köhler et al., 2023).

2. Diversity of MPC Frameworks and Problem Classes

2.1 Tracking and Economic MPC

MPC can be tailored to regulate a system to a reference (tracking MPC) or to optimize a general economic criterion irrespective of setpoint stabilization (economic MPC). Tracking MPC designs typically use positive-definite quadratic costs, $Q,R$, and ensure robust stability through terminal sets and costs (often derived from local LQR approximations). Economic MPC employs stage costs that may not be positive-definite and focuses on maximizing yield, minimizing energy or operational cost, or achieving sustainability targets. Closed-loop average-cost optimality, dissipativity-based stability constructs, and explicit constraints are central for economic MPC (Köhler et al., 2023, Reinhardt et al., 2024).

2.2 Robust and Learning-Augmented MPC

Robust MPC explicitly addresses plant-model mismatch, bounded disturbances, and parameter uncertainty. Tube-based MPC (e.g., "tube-MPC," "homothetic tube"), as in (Saccani et al., 2023), constructs a "tube" of admissible trajectories surrounding a nominal, ensuring all real closed-loop states remain feasible via feedback correction. Learning-augmented MPC utilizes data-driven surrogates, model ensembles, or neural ODE augmentations for enhanced model fidelity, as exemplified by the integrated neural (KNODE-MPC (Chee et al., 2021), Real-time Neural-MPC (Salzmann et al., 2022), DT-MPC (Zheng et al., 12 Dec 2025)) and ensemble methods (Giuli et al., 26 Nov 2025). These learning-based techniques are crucial for handling nonlinear, partially known, or high-dimensional plants.

3. Key Components: Horizon, Constraints, Terminal Ingredients

The prediction horizon ($N$) and the formulation of constraint handling critically determine feasibility, performance, and stability:

• Horizon tuning: In classical tracking MPC with suitable terminal ingredients, stability and recursive feasibility can be maintained with short horizons. Economic and constraint-horizon MPC designs may require larger or carefully chosen horizons for performance and feasibility (Nascimento et al., 24 Mar 2025).
• Terminal set/cost: Terminal ingredients guarantee invariance and Lyapunov stability, especially in reference tracking and setpoint-regulation regimes. In economic or unconstrained settings, large horizons or dissipativity arguments may replace or supplement terminal constructions (Köhler et al., 2023).
• Two-horizon MPC: Introducing a constraint horizon $N_c < N$ (number of steps state constraints are enforced) enables an explicit trade-off between safety and performance, improving suboptimality bounds and compatibility with safety-critical constructs such as control barrier functions (CBF) and control Lyapunov functions (CLF) (Nascimento et al., 24 Mar 2025).

4. Computational and Algorithmic Realizations

4.1 Optimization Solvers and Integrators

Modern nonlinear MPC employs direct transcription (multiple shooting, direct collocation), with shooting nodes dictated by horizon length and sampling intervals. Integration schemes range from explicit Runge–Kutta (RK4) (Chee et al., 2021) to custom event-driven solvers for hybrid systems (Zheng et al., 12 Dec 2025). Solvers include interior-point algorithms (e.g., IPOPT), sequential quadratic programming (RTI-SQP for real-time) (Salzmann et al., 2022), and even gradient-free simplex search for non-smooth, non-differentiable cost structures (Zheng et al., 12 Dec 2025).

4.2 Data-driven and ML-embedded Dynamics Models

• Hybrid (physics-informed + neural) models: Neural ODE or MLP residuals $f_{\mathrm{NN}}(x,u;\theta)$ augment $f_{\mathrm{phy}}(x,u)$ to capture unmodeled dynamics. In KNODE-MPC, this results in improved prediction accuracy and generalization compared to GP-based residuals, with computational efficiency suitable for embedded deployment (Chee et al., 2021).
• Surrogate and ensemble models: Ensembles constructed from data-driven models with Mahalanobis- or context-adaptive weighting for high-fidelity complex systems, such as district heating, enhance coverage over diverse operation conditions (Giuli et al., 26 Nov 2025).
• Online learning and adaptation: Incorporating active learning, auto-tuning, or direct online fine-tuning of the predictive models or disturbance estimate modules improves sample efficiency and performance under model uncertainty (Tan et al., 4 Dec 2025, Wu et al., 8 Apr 2025).

5. Novel Algorithms, Framework Variations, and Theoretical Results

• Human-LLM-in-the-loop MPC (InstructMPC): Augments deterministic predictions with human/LLM-supplied scenario distributions, translating natural-language context into MPC-relevant disturbance forecasts, and leverages online preference optimization for regret-bound guarantees (Wu et al., 8 Apr 2025).
• Sampling-based and probabilistic-inference MPC: Path integral control (e.g., MPPI) reinterprets MPC as inference over control distributions, enabling sample-based optimization under non-smooth dynamics or cost, with learned updates for sample efficiency and high performance even under computational constraints (Honda, 11 Nov 2025, Sacks et al., 2022).
• Categorical frameworks and software abstractions: Category-theoretic constructs (Para(Conv)) model multistage MPC compositions as morphisms, providing a diagrammatic and algebraic syntax for automated code generation and modular design (Hanks et al., 2023).
• Time-consistent, risk-averse MPC: Incorporates nested, coherent dynamic risk measures, providing a convex programming formulation for risk-sensitive, stabilizing control policies under stochastic and multiplicative uncertainties, systematically interpolating from risk-neutral to worst-case control (Chow et al., 2015).
• Constraint horizon and safety-critical control: The two-horizon approach enables explicit trade-offs between safety (CBF constraints) and performance (CLF objectives), resolving myopia in one-step QP-CBF schemes, and guaranteeing closed-loop suboptimality bounds (Nascimento et al., 24 Mar 2025).

6. Representative Applications and Case Studies

MPC frameworks are deployed in a broad spectrum:

Domain	Example Systems (as studied in cited works)	Notable Methods
Aerial Robotics	Quadrotor trajectory tracking (Crazyflie), agile UAVs	Physics-informed NN, Neural ODE
Energy Systems	District heating networks, greenhouses (economic & environmental obj)	Ensemble, NEMPC, real-time data
Power Electronics	Dual-active-bridge converters, arbitrary power device topologies	DT-MPC (netlist → digital twin)
Autonomous Vehicles	Obstacle avoidance, aggressive maneuvers	Tube-MPC, MPPI, DNN imitation
Process Industry	Reference and economic tracking, batch and continuous plants	Artificial reference MPC
Renewable Energy	Wave energy converters, solar maximum power point tracking	LPT-based prediction, AR models

Empirical results confirm that combining model-based structure with learned and context-aware modules can yield superior open- and closed-loop system performance, improve generalization to unseen operational regimes, and reduce computation relative to baseline or classical MPC designs (Chee et al., 2021, Salzmann et al., 2022, Zheng et al., 12 Dec 2025, Wu et al., 8 Apr 2025, Giuli et al., 26 Nov 2025, Wadinger et al., 2024, Nascimento et al., 24 Mar 2025).

7. Ongoing Research Directions and Open Challenges

Despite significant advances, several areas of MPC remain active research frontiers:

• Scalable synthesis of invariant sets and terminal ingredients for high-dimensional or distributed systems, particularly for economic and time-varying objective scenarios (Köhler et al., 2023).
• Robust MPC with learning and adaptation, especially safe integration of deep models with hard constraints and theoretical guarantees (Kim et al., 25 Mar 2025, Giuli et al., 26 Nov 2025).
• Efficient sample complexity in sampling-based MPC and path integral approaches, leveraging learning-to-optimize techniques for real-world embedded deployment (Sacks et al., 2022).
• Risk-sensitive and uncertainty-aware control with tractable online solvers that bridge risk-neutral, mean-variance, and distributionally robust objectives (Chow et al., 2015).
• Integration of context-rich, human-in-the-loop information via LLMs or other AI modules for adaptive and context-aware operation (Wu et al., 8 Apr 2025).
• Software and categorical abstraction frameworks for modular, correct-by-construction MPC programming and code generation (Hanks et al., 2023).

The Model Predictive Control framework, through its continuous evolution and integration with data-driven, robust, optimization, and AI technologies, remains a central paradigm for high-performance, safety-critical, and complex system control (Köhler et al., 2023, Chee et al., 2021, Wu et al., 8 Apr 2025, Zheng et al., 12 Dec 2025, Giuli et al., 26 Nov 2025, Wadinger et al., 2024).