Nonlinear Model Predictive Control (NMPC)

• Nonlinear Model Predictive Control (NMPC) is a control strategy that uses a receding horizon framework to manage nonlinear dynamics and complex constraints in time-varying systems.
• It leverages advanced numerical methods like multiple shooting, SQP, and sampling techniques to efficiently solve high-dimensional, nonconvex optimization problems.
• NMPC provides theoretical guarantees such as Lyapunov stability and recursive feasibility, enabling reliable real-time control even on resource-constrained platforms.

Nonlinear Model Predictive Control (NMPC) refers to the class of model predictive control methodologies in which the plant model, constraints, and/or objective function are nonlinear, leading to a nonlinear (typically nonconvex) optimization problem at each control step. NMPC is capable of handling systems exhibiting hard nonlinearities, complex constraints, multi-variable interactions, and constraints on state and input trajectories over a moving finite-time prediction horizon. The receding-horizon policy is implemented by repeatedly solving a nonlinear optimal control problem, applying only the first part of the solution, and shifting the horizon forward at each time step.

1. Core Problem Formulation

The general discrete-time NMPC problem is posed as follows:

$$\min_{u_0,\,\dots,\,u_{N-1}} \sum_{i=0}^{N-1} \ell(x_i, u_i) + \ell_f(x_N)$$

subject to: $$x_{i+1} = f(x_i, u_i), \quad x_0 = x^{\mathrm{meas}}, \quad x_i \in \mathcal{X},\; u_i \in \mathcal{U},\; i=0,\dots,N-1,\; x_N \in \mathcal{X}_f$$ where $f$ is typically nonlinear, $\ell$ and $\ell_f$ are possibly nonconvex, and both state and input constraint sets $\mathcal{X}, \mathcal{U}$ may be nonconvex. Only $u_0^*$ is applied to the system at each sampling instant, and the problem is solved again at the next step with the horizon receding.

This basic paradigm—nonlinear plant, constraints, and optimal cost—encompasses a variety of settings: explicit Euler- or Runge-Kutta-discretized ODEs and DAEs (Christensen et al., 26 Jul 2024), extended with logic constraints (e.g., MIP encodings for NN models (Lan, 16 May 2024)), Bayesian/sampling-based approximations (Askari et al., 2022Askari et al., 2022), or learning-augmented prediction models (Chee et al., 2022).

2. Numerical Methods for NMPC

NMPC leads to high-dimensional, nonlinear, nonconvex optimization problems subject to dynamic equality and inequality constraints. Core numerical strategies include:

• Direct multiple shooting/collocation: The prediction horizon is partitioned into intervals. State and input sequences are treated as independent variables, and dynamic constraints are enforced via equality constraints at each node. This enables use of modern SQP and interior-point NLP solvers, supporting warm-start and parallelization (Nguyen et al., 28 Jul 2025Uzun et al., 26 Apr 2024Christensen et al., 26 Jul 2024).
• Sequential Quadratic Programming (SQP): Each SQP iteration approximates the problem locally with a QP using linearizations of both dynamics and constraints. NMPC applications often use Gauss-Newton Hessians for efficiency; recent work demonstrates significant speed-ups via LPV-embedding and scheduling (Karachalios et al., 28 Mar 2024).
• First-order schemes: Forward-backward splitting and quasi-Newton embedded line-search methods, notably PANOC, eliminate the need for inner QP steps and memory-intensive Hessians, accelerating convergence in real-time and embedded contexts (Stella et al., 2017).
• Sampling and particle-based approaches: Bayesian/state-trajectory filtering and smoothing via particle methods allow the nonconvex NMPC problem to be approximated by Monte Carlo optimization, providing robustness to local minima and native uncertainty quantification (Askari et al., 2022Askari et al., 2022).
• Successive convexification: The nonlinear OCP is systematically convexified around a nominal trajectory, with convex QPs solved sequentially until convergence. Augmented dynamics ensure constraint satisfaction at all points in continuous time (Uzun et al., 26 Apr 2024).

Adaptive methods adapt the discretization mesh or prediction horizon (e.g., time-mesh refinement) to concentrate computational effort where system dynamics are most sensitive (Potena et al., 2018Gerdpratoom et al., 10 Jan 2025).

3. Handling Nonlinearity, Constraints, and Model Uncertainty

A defining feature of NMPC is its ability to explicitly handle nonlinearity and time-varying or state-dependent constraints:

• State/input constraints and soft barriers: Constraints are enforced directly (as in SQP or PANOC) or via differentiable barriers in sampling-based and Bayesian formulations (Askari et al., 2022Uzun et al., 26 Apr 2024).
• Chance constraints and stochasticity: When plant models are affected by noise or uncertainty (e.g., via Gaussian processes), chance constraints are enforced in closed loop, typically by constraint tightening using offline GP sampling and back-off calculation, balancing performance and feasibility (Bradford et al., 2019).
• DAE and index-1 inclusions: Nonlinear DAEs, such as in chemical or electrochemical process models, are handled using robust DAQ-specific integrators (ESDIRK schemes), supplemented by extended Kalman filters for real-time state estimation and covariance propagation (Christensen et al., 26 Jul 2024).
• Neural network and data-driven models: When the plant is (partly) modeled by a neural network, the resulting NMPC problem is highly nonconvex. Exact MIP encodings of ReLU activations yield mixed-integer nonlinear programs (intractable for deep nets or long horizons), while convex relaxations (LP, QP) or cost-penalty enhancements yield real-time feasible yet suboptimal solutions (Lan, 16 May 2024Chee et al., 2022Askari et al., 2022).
• Learning-augmented prediction: Hybrid models combining first-principles dynamics with learned corrections (e.g., knowledge-based neural ODEs with deep ensembles) improve closed-loop accuracy while admitting Lyapunov-based stability analysis (Chee et al., 2022).

4. Real-Time Implementation and Embedded NMPC

Achieving real-time feasibility for NMPC in high-speed applications is a central focus:

• Resource-constrained platforms: Full NMPC (with nonlinear dynamic models and constraints) has been deployed on microcontrollers such as the Teensy 4.1 by careful static code generation (ACADO, qpOASES), direct single-iteration SQP, fixed-dimension problem structures, and hot-start active-set QP solvers. Solve times <1 ms at N=10–18 for complex quadrotor dynamics have been demonstrated (Nguyen et al., 28 Jul 2025).
• Numerical optimizations: Static memory allocation, Gauss-Newton Hessians, iterative and hot-started QP, and the omission of secondary iterations are crucial to preserve timing budgets.
• Sampling-based methods: Particle/sampling-based NMPC algorithms are parallelizable by design and are well-suited for embedded GPUs or multicore processors. Tradeoffs include particle count vs. accuracy, and the need to carefully manage degeneracy and proposal distributions (Askari et al., 2022).
• Successive convexification: Prox-linear trust-region QP updates can be prematurely terminated while still achieving continuous-time constraint satisfaction and real-time solve times (e.g., 4–10 ms for horizon N=20, state-space ≥4) (Uzun et al., 26 Apr 2024).

5. Theoretical Guarantees: Stability, Recursive Feasibility, Robustness

NMPC, by virtue of its receding horizon and explicit constraint handling, admits strong theoretical properties under appropriate design:

• Lyapunov stability: With quadratic cost, suitable terminal penalty, and admissible terminal set constraints, NMPC can guarantee asymptotic stability of the origin for compact constraint sets, even with learned models, provided sufficient requirements on the terminal region and cost decrease are met (Chee et al., 2022Grandia et al., 2020).
• Recursive feasibility: Standard suboptimal NMPC, robust tube-based NMPC, and certain sampling-based methods can guarantee that if the problem is feasible at time $k$, it remains feasible at $k+1$, provided terminal and local control invariance conditions hold (Bobiti et al., 2017Lishkova et al., 2021).
• Robustness: Model uncertainty, process noise, and learning errors can be accommodated via constraint tightening, multi-model or ensemble predictions, robust positive invariant terminal sets, and tube MPC constructions (Chee et al., 2022Bradford et al., 2019Lishkova et al., 2021).
• Suboptimal control and anytime property: Sampling-based suboptimal NMPC strategies (e.g., sequential elementwise updating with warm start) provide strict monotonic cost improvement, recursive feasibility, and stability (when terminal assumptions are satisfied), with guaranteed quadratic-in-horizon complexity and linear-in-sample-count growth (Bobiti et al., 2017).

6. Emerging Directions: Nonclassical Optimization and Extensions

Recent proposals push NMPC beyond conventional optimization paradigms:

• Quantum optimization for NMPC: Reformulations of NMPC as polynomial or QUBO programs with move-blocking and binary encodings enable their solution on quantum annealers, with empirical quadratic speed-ups and nearly constant wall-clock as the problem size grows, contingent on suitable quantum hardware (Novara et al., 25 Oct 2024).
• Multirate and structure-preserving NMPC: For mechanical or multi-timescale systems, variational integrator-based and multirate NMPC (variational Euler-Lagrange) yield structure-preserving, symplectic, and energy/momentum-conserving flows, thus improving long-term prediction accuracy and computational efficiency (Lishkova et al., 2021).
• Continuous-time constraint satisfaction: Augmented-dynamics and successive convexification frameworks ensure that continuous-time constraints (on state or output) are respected throughout the entire horizon, not only at collocation points (Uzun et al., 26 Apr 2024).
• Adaptive and online weight tuning: Real-time optimization of weight matrices within the NMPC objective, via alternating QP solve and analytic weight updates, permits closed-loop optimization of performance tradeoffs without manual tuning (Kostadinov et al., 2020).

7. Applications and Benchmark Case Studies

NMPC has been validated in a variety of complex, real-world systems:

• Aerial robotics and UAVs: Adaptive time-mesh NMPC tracks aggressive quadrotor maneuvers with millisecond solve times even for multi-second horizons (Potena et al., 2018Nguyen et al., 28 Jul 2025).
• Chemical and electrochemical processes: ESDIRK-based NMPC for stochastic, index-1 SDAEs enables real-time setpoint tracking in alkaline electrolyzer stacks, using accurate gradients and covariance propagation (Christensen et al., 26 Jul 2024).
• Autonomous vehicles and motion planning: Both constraint-aware sampling-based NMPC and learning-augmented prediction have enabled safe, robust tracking and collision avoidance in complex dynamic scenarios (Chee et al., 2022Askari et al., 2022Askari et al., 2022).
• Nonlinear process control: NMPC outperforms linear MPC in conical tank level regulation, especially near geometric nonlinearities, as shown by lower offset and integral error (Klaučo et al., 2018).
• Embedded NMPC with neural network dynamics: Real-time-capable convex relaxations enable NMPC for systems modeled by deep neural networks, achieving near-exact tracking if enhanced cost-penalties are included (Lan, 16 May 2024).

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In summary, NMPC provides a flexible, theoretically grounded framework for optimal control of nonlinear, constrained systems. Continual advances in numerical optimization, learning-based model integration, robust constraint handling, and embedded implementation are expanding the applicability of NMPC to increasingly complex, high-speed, and uncertain systems across robotics, process control, and beyond (Potena et al., 2018Nguyen et al., 28 Jul 2025Chee et al., 2022Karachalios et al., 28 Mar 2024Lan, 16 May 2024Askari et al., 2022Stella et al., 2017Bobiti et al., 2017Novara et al., 25 Oct 2024Grandia et al., 2020Lishkova et al., 2021Christensen et al., 26 Jul 2024Uzun et al., 26 Apr 2024Kostadinov et al., 2020Klaučo et al., 2018).