Model Predictive Control (MPC)

• Model Predictive Control is an advanced control strategy that uses a receding horizon approach to optimize future trajectories based on system models.
• It systematically handles multivariable dynamics, state/input constraints, and performance objectives to ensure safe, efficient operations in diverse fields.
• Recent advances include FCS-MPC for power electronics, economic and data-driven MPC, and adaptive methods integrating learning and robust optimization techniques.

Model Predictive Control (MPC) is an advanced control strategy that determines optimal control actions by solving a sequence of constrained optimization problems in real time, leveraging a model of the system to predict and shape future trajectories. This technique is employed across process industries, energy systems, autonomous vehicles, robotics, aerospace, and beyond due to its systematic treatment of multivariable dynamics, state/input constraints, and performance objectives.

1. Fundamental Principles and Architecture

At its core, MPC operates on a "receding horizon" paradigm: at each control interval (sampling instant), it solves a finite-horizon optimal control problem defined by:

• prediction of future plant states using a model (which may be linear or nonlinear, and deterministic or stochastic);
• an objective (cost) function, often quadratic in deviations from target state and control signals over the forecast horizon;
• explicit constraints on states and inputs representing physical, safety, or operational limits.

After solving the optimization problem for the entire horizon, only the first control input is applied. This procedure is repeated at every time step with updated measurements, yielding a feedback control law that adapts to disturbances and modeling errors.

Mathematically, for discrete-time linear systems: $$\text{Minimize}_{u_{0:N-1}} \ \sum_{k=0}^{N-1} \|x_{k|t} - x_{\text{ref}}\|_Q^2 + \|u_{k|t}\|_R^2, \quad \text{subject to:} \ x_{k+1|t} = A x_{k|t} + B u_{k|t}, \quad (x,u) \in \mathcal{X} \times \mathcal{U}$$ where $x_{k|t}$ denotes the state prediction at future step $k$ based on information available at time $t$, given initial state $x_{0|t} = x(t)$.

2. Handling of Constraints and System Modeling

A principal strength of MPC lies in its explicit handling of constraints:

• State and input constraints (often polytopic) are directly incorporated into the optimization, ensuring that operational and safety requirements are anticipated rather than simply reacted to.
• For practical systems, models range from straightforward linear time-invariant (LTI) to highly complex nonlinear (even neural) models, with corresponding solvers such as quadratic programming (for linear/quadratic problems), or nonlinear programming (for NMPC).

In modern applications, models can also include actuator dynamics, transmission lags (Babu et al., 2018), or explicit treatment of uncertainties either via robust formulations or Bayesian learning-based updates (Wabersich et al., 2020).

3. Advances and Variants of MPC Formulations

3.1 Finite-Control-Set MPC (FCS-MPC)

For power electronic applications (e.g., converters in microgrids), control actions are inherently discrete. FCS-MPC evaluates all feasible switching states at each sampling instant via predictions (using a discrete-time model) and applies the action minimizing a cost function. This eliminates the need for traditional PI controllers, PWM stages, or droop mechanisms, enabling direct control over quantized actuators and leading to fast dynamics, robust regulation, and improved power sharing among distributed energy resources (Yi et al., 2018).

3.2 Economic MPC and Relation to MDPs

Economic MPC generalizes setpoint-tracking MPC by optimizing an economic performance index. Its relevance to Markov Decision Processes (MDPs) is rigorous when stochastic disturbances are present. If the MPC terminal cost and model are chosen so that

$$\mathbb{E}[V^*(s_+)|s, a] - V^*(f(s, a)) = \text{constant},$$

then MPC recovers (up to a constant) the Bellman recursion for the optimal value function, and thus yields an (approximately) optimal policy for the MDP (Reinhardt et al., 23 Jul 2024). This approach is computationally tractable relative to dynamic programming but rests on crucial modeling and approximation assumptions.

3.3 Data-Driven and Learning-Enhanced MPC

Learning approaches in MPC address two main challenges:

• Learning model parameters to maximize closed-loop performance rather than just prediction accuracy (identification for control), often via data-driven experiments and Bayesian optimization (Piga et al., 2019).
• Replacing online optimization with learned policies—such as explicit MPC via neural networks that embed feasibility via interior point parameterizations (Tabas et al., 2022), or by constructing low-dimensional intrinsic representations using manifold learning for efficient approximate policies (Lovelett et al., 2018).

3.4 Adaptive and Hierarchical MPC

Adaptive MPC formulations dynamically adjust the prediction horizon (Bøhn et al., 2021), sample density (Mostafa et al., 2022), or model fidelity (Norby et al., 2022) according to the local operating context, computational conditions, or task horizon to maintain feasibility and tractability. Hierarchical MPC architectures combine fast inner-loop controllers (e.g., PID) with an MPC "reference governor" that handles higher-level objectives and constraints (Piga et al., 2019).

3.5 Multi-Forecast, Edge-Assisted, and Foundation Model Augmentation

Advanced formulations consider heterogeneity and uncertainty at a systemic level:

• Multi-forecast/incremental proximal MPC plans over multiple forecast scenarios, coupling first action consistency across them and solving the resulting optimization iteratively to balance robustness and computational load (Shen et al., 2021).
• Edge-assisted MPC leverages networked edge compute resources for parallelized trajectory evaluation, integrating localized sensing/histories to expand coverage and sampling density (Lou et al., 1 Oct 2024).
• Hybrid approaches integrate high-level vision-language reasoning (via VLMs) with lower-level real-time MPC, using the VLM to generate contextually appropriate driving objectives and constraints (Long et al., 9 Aug 2024).

3.6 Diffusion Model-Based MPC

Recent developments employ diffusion models to learn coherent multi-step action and dynamics proposals from offline data. This enables high-fidelity long-horizon predictions, decoupled planning and reward adaptation, and robust transfer to new tasks by factorizing and independently updating learned models (Zhou et al., 7 Oct 2024).

4. Guaranteeing Feasibility, Stability, and Robustness

Feasibility and stability are fundamental for real-time MPC deployment:

• Recursive feasibility ensures that, if a feasible solution is found at time $t$, feasible solutions exist at all subsequent times when the control law is followed. Terminal constraints and terminal costs—solving Lyapunov or invariance equations—are standard to guarantee asymptotic stability and constraint satisfaction (Augustine, 2023), even under reference tracking with changing or non-periodic setpoints (Han et al., 26 Mar 2025).
• Robust MPC incorporates model uncertainty and external disturbances through robust control invariant sets, min-max optimization, or learning-based posterior sampling, sometimes supported by explicit regret bounds (Wabersich et al., 2020).
• Combined MPC-adaptive control architectures (e.g., $\mathscr{L}_1$ adaptive augmentation) allow performance recovery under fast, unmodeled or time-varying parameter changes (Chai et al., 2021).

5. Practical Implementation and Applications

MPC is widely deployed in industrial process control, microgrids, energy management, automotive, aerospace, and robotics. MATLAB-based implementations provide templates for both linear and nonlinear MPC, clarifying the translation of cost, model, and constraints into efficient solver calls (Augustine, 2023). Simulation studies consistently demonstrate:

• Accurate tracking (including under abrupt reference changes) and constraint satisfaction (Han et al., 26 Mar 2025).
• Rapid adaptation to disturbances, actuator dynamics, and time-varying system parameters.
• Enhanced safety and comfort in autonomous driving and aerial robotics (Babu et al., 2018, Long et al., 9 Aug 2024).
• Scalability through offline learning, explicit representation, or distributed computation.

6. Recent Theoretical Developments and Research Directions

• Category-theoretic frameworks structure MPC as compositions of convex subproblems, enabling modular, diagrammatic construction, and systematic analysis of multistage systems—facilitating correctness and extension (Hanks et al., 2023).
• Theoretical analyses connect sample complexity (e.g., in edge-assisted MPC) and regret bounds (in BMPC) with quantifiable performance gains and tractable trade-offs in multi-agent or distributed settings (Lou et al., 1 Oct 2024, Wabersich et al., 2020).
• Continued research targets deeper integration of foundation-style models, real-time adaptive architectures, and formalization of learning-based MPC guarantees on constraint satisfaction, recursive feasibility, and stability under approximation or uncertainty.

7. Limitations and Open Challenges

Despite broad maturity, MPC faces challenges including:

• High online computational demands, particularly for long horizons or nonlinear/high-dimensional models (addressed via learning-based acceleration, parallelization, or horizon/sample adaptation).
• Accurate modeling in the presence of unmodeled dynamics, nonlinearity, and uncertainty—requiring adaptive, robust, or learning-based updates.
• Ensuring closed-loop optimality when employing economic or approximate MPC to solve MDPs, particularly outside locally quadratic or linear domains (Reinhardt et al., 23 Jul 2024).

---

Model Predictive Control continues to evolve, integrating advances from robust optimization, statistical learning, distributed and edge computation, and foundation models. Developments such as FCS-MPC for power electronics, manifold learning-driven explicit MPC, diffusion model-based sequence prediction, edge-assistance, and vision-language reasoning all illustrate the diverse, rigorous pathways through which MPC adapts to contemporary cyber-physical and autonomous systems.