Nonlinear Model Predictive Control

• NMPC is a receding-horizon control paradigm that uses nonlinear prediction models to optimize multivariate dynamical systems under state and input constraints.
• It incorporates advanced computational strategies like SQP with LPV linearization, particle filtering, and approximate dynamic programming to achieve real-time feasibility.
• Recent innovations integrate learning-based methods and quantum-accelerated solvers to reduce computational complexity while ensuring robust performance under uncertainty.

Nonlinear Model Predictive Control (NMPC) is a receding-horizon optimal control paradigm for multivariate nonlinear dynamical systems subject to state and input constraints. Characterized by embedding a nonlinear prediction model into an online finite-horizon optimal control problem, NMPC formulations have driven advances in real-time constrained process control, robotics, and autonomous systems. Compared to linear MPC, NMPC directly exploits plant nonlinearities but at the expense of substantially increased computational complexity, which has motivated a wide range of algorithmic, robustness, and learning-based innovations.

1. Problem Formulation and Theoretical Foundations

Formally, a generic NMPC problem for a discrete-time nonlinear system

$$x_{k+1} = f(x_k, u_k), \quad x_k \in \mathbb{R}^{n_x},\ u_k \in \mathbb{R}^{n_u}$$

with state and input constraints $x_k \in \mathcal{X}$, $u_k \in \mathcal{U}$, can be posed as

$$\begin{aligned} \min_{u_{k|k},...,u_{k|k+N-1}} \quad & J(x_k, u_{k:k+N-1}) = \sum_{i=0}^{N-1} \ell(x_{k+i|k}, u_{k+i|k}) + V_f(x_{k+N|k}) \ \text{s.t.} \qquad & x_{k+i+1|k} = f(x_{k+i|k}, u_{k+i|k}), \quad i=0,...,N-1 \ & x_{k+i|k} \in \mathcal{X},\ u_{k+i|k} \in \mathcal{U},\ x_{k+N|k} \in \mathcal{X}_f \end{aligned}$$

where $\ell$ is the stage cost and $V_f$ a terminal cost. Only $u_{k|k}^*$ is applied before the process repeats at $k+1$.

Standard NMPC stability and recursive feasibility are ensured by appropriate selection of terminal sets, terminal costs, and the use of local stabilizing controllers on the terminal set. Asymptotic stability of the feasible closed-loop origin is established via Lyapunov arguments, provided the terminal ingredients and stage cost are chosen to satisfy decrease and invariance conditions (see, e.g., (Chee et al., 2022, Polver et al., 4 Feb 2025)). Invariant or positively invariant terminal sets are typically necessary for formal guarantees in the presence of constraints.

2. Computational Solution Strategies

2.1 Nonlinear Programming (NLP) and Sequential Quadratic Programming (SQP)

Approaching NMPC as a nonlinear programming (NLP) problem, the finite-horizon optimal control can be addressed by general-purpose solvers (e.g., IPOPT, fmincon). However, for real-time feasibility, methods exploiting model structure are preferred.

Sequential Quadratic Programming (SQP) with LPV Embedding: The nonlinear dynamics are embedded into a Linear Parameter-Varying (LPV) structure by local linearization: $$x_{k+1} \approx A(\rho_k)x_k + B(\rho_k)u_k + d(\rho_k), \quad \rho_k = \sigma(x_k,u_k)$$ with the scheduling parameter $\rho_k$ updated iteratively within the SQP loop. Each SQP iteration solves a Quadratic Program (QP) with Hessian and Jacobian information reused and updated through the LPV model, enabling fast convergence with lower computational burden compared to full NLP relinearization. This approach, as rigorously analyzed in (Karachalios et al., 2024), achieves near-NLP closed-loop cost and tracking behavior at up to half the computation time on challenging benchmarks.

2.2 Particle-Based and Sampling Methods

Bayesian perspectives recast NMPC as a smoothing problem: augmenting the state with input trajectories and employing particle filtering/smoothing with constraint-aware likelihoods (soft barrier functions over constraint violations). The "Constraint-Aware Particle" (CAP-NMPC) method explicitly integrates constraints by reweighting particles according to both cost and feasibility, sidestepping local minima and offering trivial parallelization. Empirically, constraint-aware sampling achieves both improved performance and strict satisfaction of constraints compared to unconstrained or vanilla particle-based MPC (Askari et al., 2022).

2.3 Approximate Dynamic Programming

Approximate dynamic programming (ADP)-MPC approaches exploit quadratic cost-to-go function approximation, often parameterized via Riccati recursion over locally linearized or switched-affine approximations of the nonlinear system. Computational advantages are pronounced: offline computation of a tree of quadratic forms enables the online control law to be realized as a sequence of low-complexity minimizations, with suboptimality in cost and tracking guaranteed to converge in the limit of dense quantization (Chacko et al., 2023).

2.4 Quantum-Accelerated and Emerging Solvers

Recent work leverages quantum annealers to directly solve NMPC-encoded QUBO problems. Polynomial approximation of the nonlinear map and cost function, along with binary encoding of control variables, allows the formation of quadratic unconstrained binary optimization (QUBO) problems suitable for quantum sampling. While empirical speedups and solution-quality guarantees remain hardware-dependent, this direction holds promise for scaling NMPC to extremely high-dimensional or fast applications (Novara et al., 2024).

3. Robustness, Uncertainty, and Distributional Guarantees

Robustness to model mismatch and uncertainty is central in NMPC research.

3.1 Robust and Distributionally Robust NMPC

Robust Tube-Based NMPC: Envelops the nominal trajectory within tubes that account for worst-case bounded disturbances, with constraints on the tightened state/input sets guaranteeing safety and recursive feasibility. Contractive Lyapunov or contraction metrics replace classical terminal constraints, enabling stability without explicit terminal sets (Polver et al., 4 Feb 2025).

Distributionally Robust NMPC: Recent advances address distributional ambiguity in process disturbances using ambiguity sets defined via Wasserstein balls centered at empirical disturbance distributions. The resulting controller minimizes the worst-case expected cost (over the ambiguity set), thus providing probabilistic constraint satisfaction and improved closed-loop robustness against deviation from assumed noise models ((Zhong et al., 2022), abstract only).

Convex Restriction Approaches: For general nonlinear dynamics with unknown but bounded uncertainty, tube self-mapping via convex-enveloped residuals yields tractable robust MPC formulations. Sequential convex restriction iterations can efficiently compute safe control sequences satisfying worst-case constraints over the uncertainty set, with operational complexity comparable to standard QP-based MPC and empirical safety verified on vehicle navigation benchmarks (Lee et al., 2020).

3.2 Stochastic and Chance-Constrained NMPC

Uncertainty propagation through nonlinear predictions generally prohibits closed-form solutions for chance constraints. Gaussian process (GP)- and polynomial chaos expansion (PCE)-based surrogates can deliver tractable approximations of the mean and variance of arbitrary nonlinear functions of random variables, enabling the use of Chebyshev-type or conservative risk constraints in NMPC. The GPPCE approach, for instance, accurately estimates required probability distributions and achieves zero constraint violations in demanding chemical process case studies, with performance and computational load far surpassing both nominal and state-of-the-art scenario-based alternatives (Bradford et al., 2021).

4. Learning-Enhanced and Data-Driven NMPC

Increasing model complexity and real-time constraints have driven substantial interest in learning-based NMPC.

4.1 Neural and Hybrid Model Inclusion

Deep Neural Networks (NNs) in Dynamics: When the plant nonlinearity is encapsulated by a high-capacity ReLU NN (motivated by nonlinear system ID), solving the resulting NMPC is itself NP-hard. Exact formulations using mixed-integer programming (MIP) encode the ReLU layers explicitly, while convex relaxations—particularly enhanced linear relaxations with penalty terms—can deliver near-exact solution performance with millisecond-level computation, as rigorously detailed for inverted pendulum control by (Lan, 2024).

Hybrid Physics-Learning Models: Knowledge-based neural ODEs (KNODEs), blending first-principles models with NN-parameterized residuals, enhance prediction accuracy for systems with incomplete modeling. Deep ensemble approaches further reduce variance and improve closed-loop performance. Sufficient conditions for asymptotic stability are provided via standard local terminal cost/set constructions and contractive terminal laws in (Chee et al., 2022).

4.2 Learning the Policy – Constrained DNN MPC

Policy surrogate approaches—training deep neural networks to approximate the NMPC policy itself—enforce constraints through output projection, barrier penalties, or direct constrained training. Empirical studies confirm that DNN surrogates with explicit projection exhibit only a marginal performance gap to the exact NMPC controller, with orders-of-magnitude lower online computation and dramatic reduction in constraint violations compared to unconstrained learning (Asadi, 2021).

5. Adaptation, Output-Feedback, and Extension Frameworks

5.1 Output-Feedback NMPC

Output-feedback NMPC, as instantiated in (Kamaldar et al., 2023), extends the pseudo-linear receding-horizon framework to allow iterative QP-based optimization over horizon segments with state- and control-dependent coefficients. Employing alternative state representations (block-observable canonical form), the scheme achieves output feedback performance on non-minimum phase and nonholonomic examples.

5.2 Online Parameter and Weight Adaptation

Online weight-adaptive NMPC enables dynamic tuning of quadratic cost weights to balance tracking, control smoothness, and robustness, casting weight selection itself as a min–max optimization within the NMPC loop. Real-time iteration (RTI) schemes ensure closed-loop stability and computational tractability (Kostadinov et al., 2020).

5.3 Path-Following and Contraction-Based NMPC

Nonlinear MPC for output path following (NMPC-PF) augments the system with additional virtual states to describe path parameters, enabling stabilized progression along geometric curves without pre-specified timing. Structure theorems guarantee recursive feasibility and convergence using geometric (transverse) normal forms for terminal set construction (Faulwasser et al., 2015).

Contraction-based NMPC provides reference-independent, robustness-oriented stabilization by enforcing state contraction along a Riemannian metric at each prediction step, eliminating the requirement for a Lyapunov-based terminal region and permitting the use of arbitrary (including economic) costs (McCloy et al., 2022).

6. Applications and Empirical Performance

NMPC has demonstrated efficacy in a wide spectrum of benchmark and real-world systems, including:

• Automated vehicle trajectory and lane-keeping (CAP-NMPC, (Askari et al., 2022); Set-membership methods, (Boggio et al., 2022), abstract only)
• Constrained industrial process control, including multi-tank systems and chemical reactors (ADP-MPC, (Chacko et al., 2023); GPPCE-SNMPC, (Bradford et al., 2021))
• Nonholonomic and underactuated robotics, such as mobile robots and the Kapitza pendulum (Kamaldar et al., 2023, Bobiti et al., 2017)
• Power electronics and renewable energy systems (Polver et al., 4 Feb 2025)
• Vehicle traction control with real-time preview and adaptation (Tavolo et al., 2024)
• Neural systems manipulation with data-driven surrogate models (Fehrman et al., 2023)

Table: Representative NMPC Solution Complexities and Empirical Benchmarks

Category	Online Time (per step)	Tracking/Closed-Loop Error	Reference
SQP-LPV NMPC	6–14 ms (N=20)	≈8% lower cost v. NLP	(Karachalios et al., 2024)
CAP-NMPC (PF-smooth)	<20 ms	≈1–2% lower RMSE	(Askari et al., 2022)
ADP-MPC	0.01–0.07 ms	5–25% higher ISE	(Chacko et al., 2023)
DNN-NMPC (policy surrogate)	<1 ms	≤5% higher error	(Asadi, 2021)
Robust Convex Restriction	0.7 s (N=50)	≤8.4% higher worst-case	(Lee et al., 2020)
MIP for ReLU-NN MPC	0.01–>1 s	exact (small N, small NN)	(Lan, 2024)

Note: Results refer to specific testbeds and computational platforms; performance will vary according to system scale, horizon, and complexity.

7. Open Problems and Future Directions

Despite significant advances, research challenges remain:

• Scalability and Real-Time Feasibility: Efficient NMPC for high-dimensional, fast-sampling, or distributed systems.
• Robust Learning: Integration of learning-based models with certified robust control, addressing distribution shift, and uncertainty quantification.
• Optimality vs. Computational Complexity: Closing the performance gap between approximate solutions (ADP, sampling, learning surrogates) and full NLPs, particularly for fast, safety-critical applications.
• Advanced Constraint Handling: Extending guarantees to nonconvex, mixed-integer, or high-dimensional state and input constraints.
• Quantum and Emerging Hardware: Deployment of quantum-accelerated NMPC beyond proof-of-concept on larger-scale systems.
• Unified Robustness Frameworks: Development of systematic contraction-based, tube-based, and distributionally-robust formulations with tractable implementations for nonlinear plant models.

NMPC continues to serve as a cross-disciplinary nexus, uniting optimization, control theory, learning, and computational mathematics in pursuit of safe, effective high-performance control for nonlinear dynamical systems.