Nonlinear Model Predictive Control (nMPC)

• nMPC is a feedback control method that solves a finite-horizon, constrained nonlinear optimal control problem at each sampling instant.
• It employs nonlinear models, cost functions, and numerical techniques such as SQP and GMRES to achieve real-time trajectory optimization.
• nMPC is applied in UAVs, robotics, and process control, offering robust performance even under complex, nonlinear system dynamics.

Nonlinear Model Predictive Control (nMPC) is a feedback control technique that solves a constrained, nonlinear finite-horizon optimal control problem (OCP) at each sampling instant, using the current measured state as the initial condition. The solution yields a predicted input sequence, from which only the first control action is applied; the horizon then shifts forward, and the process repeats. nMPC is distinguished from linear MPC by the use of nonlinear prediction models, cost functions, and constraints in the OCP, motivating algorithmic innovations for numerical tractability, stability, and real-time feasibility.

1. Core Formulation of Nonlinear Model Predictive Control

At the core of nMPC is the repeated solution of a problem of the form: $$\begin{align*} \min_{\{u_k\}_{k=0}^{N-1}} \ & \Phi(x_N) + \sum_{k=0}^{N-1} L(x_k, u_k) \ \text{s.t. } \ \ \ \ & x_{k+1} = f(x_k, u_k),\ & h(x_k, u_k) \leq 0, \ & x_0 = x_{\text{current}}, \end{align*}$$ where $f$ is a nonlinear model, $L$ and $\Phi$ are stage and terminal costs, and the constraints $h$ encode state and input admissibility. The output is a receding, feedback policy executed in real time.

Variants such as sampled-data NMPC (Gerdpratoom et al., 10 Jan 2025), multirate NMPC (Lishkova et al., 2021), stochastic NMPC (Christensen et al., 26 Jul 2024), and learning-enhanced NMPC (Chee et al., 2022) adapt this generic formulation to more specialized contexts (e.g., sampled-data lifting, multi-timescale dynamics, stochastic DAEs, or data-driven models).

2. Numerical Methods and Algorithmic Implementation

Solving the nonlinear OCP at each sampling step presents significant computational challenges, especially for fast systems or resource-limited platforms. The literature addresses these issues with several classes of numerical techniques:

• Continuation/GMRES and Preconditioned Iteration: The CNMPC framework uses iterative solvers (GMRES or MINRES) to solve a finite-difference approximation of the necessary conditions for optimality (Knyazev et al., 2015). Preconditioning, achieved via parallel LU or LDLᵀ factorization of the finite-difference Jacobian, drastically improves convergence, enabling near real-time computation even with just 1–2 iterations per step.
• Sequential Quadratic Programming (SQP) and Real-Time Iteration: Many practical solvers use SQP, often with Gauss-Newton Hessian approximations, to convert the NLP at each step into a sequence of QPs or linearized problems, allowing for rapid convergence—typically just one SQP step per iteration—when warm-started from previous solutions (Karachalios et al., 28 Mar 2024, Nguyen et al., 28 Jul 2025).
• Sampling-Based and Particle Methods: For highly nonlinear or nonconvex problems (e.g., with neural network models), sampling-based approaches recast the OCP as a state/input estimation problem. Particle filtering/smoothing and sampling-guided update methods provide robust suboptimal solutions with complexity dependent mainly on horizon length and sample count, decoupled from state space dimension (Bobiti et al., 2017, Askari et al., 2022, Askari et al., 2022).
• Successive Convexification and SCP: Reformulating the OCP into a sequence of convex subproblems via trajectory linearization and multiple shooting offers guarantees on constraint satisfaction and computational efficiency, especially in the presence of nonconvex path constraints (Uzun et al., 26 Apr 2024).
• Mixed-Integer and Linear Relaxation for NN Models: To handle system dynamics modeled by deep neural networks, ReLU nonlinearities are encoded either exactly (via MIP with binary variables) or with efficient linear relaxations (triangle relaxations), trading off runtime and accuracy (Lan, 16 May 2024).

A summary table illustrates the major algorithmic classes and their distinguishing features:

Method	Key Feature(s)	Contexts
Continuation/GMRES	Preconditioned Krylov solvers, parallel LU, MINRES	Minimum-time, ill-conditioned problems (Knyazev et al., 2015)
SQP + Gauss-Newton	Real-time iteration, warm start	Embedded, resource-limited systems (Karachalios et al., 28 Mar 2024, Nguyen et al., 28 Jul 2025)
Particle/Sampling	Monte Carlo, constraint-aware filter	High nonlinearity, NN models, uncertainty (Askari et al., 2022, Askari et al., 2022)
Successive Convexification	Mayer lifting, multiple shooting	Path constraint enforcement (Uzun et al., 26 Apr 2024)
MIP/LR for NNs	ReLU encoding via binary or polyhedral relaxations	Deep neural network models (Lan, 16 May 2024)

3. Model Structures and Extensions

Beyond "vanilla" nMPC, varied system and cost structures drive innovation in NMPC methodologies:

• Set Membership and Robustification: Uncertainties in plant modeling can be handled via set-valued approaches, tightening the feasible set based on past measurements and Lipschitz continuity to guarantee constraint satisfaction and reduce search space dimension (Boggio et al., 2022).
• Gaussian Process (GP)-Based Models: When the system is not perfectly known, GP surrogates are employed to model residual dynamics and quantify uncertainty (Bradford et al., 2019). State-dependent constraint backoff values, computed offline via Monte Carlo, provide probabilistic constraint satisfaction that is less conservative than state-independent robustification.
• Learning-Enhanced Models (KNODEs and Deep Ensembles): Knowledge-based neural ODE ensembles blend first-principles modeling with learned residuals, offering improved prediction fidelity and, when embedded in MPC, can provide rigorous stability guarantees given proper terminal set and cost construction (Chee et al., 2022). Weight optimization via validation data allows adaptation to plant variability.
• Stochastic DAEs: NMPC has been extended to systems described by stochastic DAEs of index 1, with ESDIRK integration ensuring accurate state/covariance and gradient computations critical for Kalman filtering and OCP solution (Christensen et al., 26 Jul 2024).
• Integration of Low-Level and Actuator Dynamics: For high-speed applications such as UAV trajectory tracking, modeling the internal PID/motor dynamics and incorporating associated linear actuator constraints into the OCP yield substantial improvements in tracking accuracy and robustness to saturation (Gupta et al., 2 Jun 2025).
• Quantum Optimization for NMPC: By reducing the OCP to a polynomial approximation and re-encoding it as a QUBO problem, quantum annealing platforms can, in principle, achieve quadratic computational speedups and/or improved solution quality relative to conventional algorithms, provided sufficiently large quantum hardware (Novara et al., 25 Oct 2024).

4. Real-Time Feasibility and Embedded Implementation

Achieving real-time execution is a central theme across contemporary nMPC research:

• Parallel and Embedded Computation: Parallel formation of Jacobian/preconditioner columns and linearly independent particle/sampling updates are leveraged for multicore CPUs, GPUs, or FPGAs, enabling faster preconditioner construction and larger sample sizes (Knyazev et al., 2015, Bobiti et al., 2017, Nguyen et al., 28 Jul 2025).
• Resource-Constrained Microcontroller Deployment: NMPC formulations utilizing efficient code generation (e.g., via ACADO), hard-wired problem dimensions, static memory allocation, active-set QP solvers (qpOASES), and PID-based warm-starting enable full nonlinear control at high rates (up to 1 kHz) on microcontrollers with <512 kB RAM (Nguyen et al., 28 Jul 2025).
• Fast-Sample Fast-Hold and Lifting: Techniques such as fast-sample fast-hold (FSFH) approximations, nonlinear lifting, and multi-rate control architectures allow accurate representation of intersample system behavior and improved control performance under slow sampling rates (Gerdpratoom et al., 10 Jan 2025, Lishkova et al., 2021).
• Adaptive Time Mesh and LS Solvers: By adaptively refining the discretization mesh where high accuracy is most needed and applying efficient LS formulations, rapid trajectory generation for receding horizon control (1–10 ms for several seconds of motion) is achieved (Potena et al., 2018).

5. Stability, Feasibility, and Constraint Handling

Guarantees on recursive feasibility and closed-loop stability remain central in nMPC scheme design:

• CLF-Integrated NMPC: Explicit embedding of Control Lyapunov Function (CLF) constraints, either pointwise or as contractive level sets along the horizon, enables rigorous stability guarantees without delicate cost weight tuning (Grandia et al., 2020). Unified CLF-NMPC approaches yield less aggressive inputs and improved robustness to model mismatch.
• Tube-Based/Lifting Approaches: Successive tube-based linearization and variational integrators preserve system structure and bound linearization error, ensuring robust NMPC feasibility and conservation properties even under timescale separation (slow/fast modes) (Lishkova et al., 2021).
• Constraint-Awareness in Sampling Methods: Softplus virtual measurement embedding (barrier-style) in the particle filter/smoother ensures constraints are respected in a probabilistic sense, avoiding particle starvation and optimizing both control and feasibility (Askari et al., 2022, Askari et al., 2022).
• Continuous-Time Constraint Satisfaction: Augmenting the system state with accumulated constraint violation variables and enforcing constraints in the Mayer formulation guarantee no inter-sample constraint violations, an essential property in safety-critical control (Uzun et al., 26 Apr 2024).

6. Applications and Performance Benchmarks

nMPC approaches are validated across a spectrum of challenging, high-performance applications:

• UAVs and Robotics: High-frequency NMPC running on ARM processors and microcontrollers enables aggressive UAV maneuvering with tracking errors reduced by 22%–38% compared to standard methods and speeds up to 98.57 km/h with accelerations of 3.5g (Gupta et al., 2 Jun 2025, Nguyen et al., 28 Jul 2025, Potena et al., 2018).
• Process and Industrial Control: Nonlinear geometry-aware level regulation in conical tanks yields smoother and more accurate responses, especially at low levels where linear MPC suffers from model mismatch (Klaučo et al., 2018).
• Bioprocesses and Chemical Engineering: GP-based NMPC offers chance-constrained satisfaction in semi-batch processes with fast online evaluation, while ESDIRK-based NMPC for stochastic DAEs applies to electrolyzer stacks with measured improvements in tracking under model/measurement noise (Bradford et al., 2019, Christensen et al., 26 Jul 2024).
• Autonomous Vehicles: Sampling-based and learning-based NMPC approaches enable robust motion planning and constraint satisfaction in dynamic, uncertain driving environments—particularly when the models are neural network surrogates of complex vehicle dynamics (Askari et al., 2022, Chee et al., 2022).
• Embedded and Real-Time Systems: Demonstrations on hardware platforms confirm practical feasibility, not just in simulation, including closed-loop stabilization of Segway robots and quadrotors with computation times well within sampling intervals (Grandia et al., 2020, Nguyen et al., 28 Jul 2025).

7. Limitations, Open Challenges, and Future Directions

Despite significant progress, several limitations and research directions persist:

• Computational Burden: Even with preconditioning, warm starts, and sampling, high-dimensional or long-horizon problems can challenge real-time guarantees, especially with complex, deep neural network models (Lan, 16 May 2024). Relaxation and move-blocking may be required for scalability.
• Model Uncertainty and Learning: While integration of Gaussian processes, set membership, and deep ensembles offers improved robustness, conservatism or overfitting remains a challenge. Online/learning-based adaptation of weights (Kostadinov et al., 2020) and terminal set/linearization methods are under ongoing paper.
• Constraint Conservatism: Reliance on worst-case backoffs or set-membership bounds may induce conservatism, motivating research into tighter and adaptive uncertainty quantification (Boggio et al., 2022, Bradford et al., 2019).
• Hardware for Quantum Advantage: Quantum acceleration of NMPC hinges on the availability of quantum hardware with sufficient qubits, coherence, and gate precision; for now, this remains an assumption rather than a practical reality (Novara et al., 25 Oct 2024).
• Interpolation and Input Variability: Sampling-based nMPC may introduce non-smooth input sequences, which might necessitate smoothing, input rate penalization, or interpolation (Bobiti et al., 2017).
• Integration of Low-Level and Learning Components: Systematic methods for fusing low-level controller dynamics, actuator saturation, and learned model uncertainty within the receding horizon framework are critical for next-generation high-performance, robust nMPC (Gupta et al., 2 Jun 2025, Chee et al., 2022).

Overall, the field continues to advance through rigorous integration of algorithmic innovation, model learning and adaptation, and efficient embedded implementation to realize robust, real-time optimal control for increasingly complex, nonlinear dynamical systems.