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Certified Neural Approximation of Robust Nonlinear Model Predictive Control

Hendrik Alsmeier, Maik Pfefferkorn, Rolf Findeisen

arXiv:2610.29559v1eess.SYcs.LGcs.SY

TL;DR

Repeated online MPC optimization creates computational burdens for resource-constrained implementations. This paper develops an offline framework that treats neural approximation error as a bounded disturbance and designs robust MPC to account for it. In simulation, the learned controller reproduces robust MPC’s closed-loop behavior within the certified invariant region while significantly reducing online computation.

  • Problem

    MPC’s repeated online optimization can burden resource-constrained implementations, motivating neural approximations whose safety and stability guarantees can be established.

    Source: §1 · p. 1

  • Method

    The framework bounds neural approximation error using Lipschitz properties and dataset coverage, then incorporates that bound as a disturbance in robust MPC design.

    Source: §1 · p. 2, §5 · p. 11

  • Results

    In simulation on a nonlinear isothermal CSRT model, the neural controller reproduces robust MPC’s closed-loop behavior within the certified invariant region while significantly reducing online computation time.

    Source: §5 · p. 11

  • Takeaways & Limitations

    The offline procedure links dataset coverage and Lipschitz constants to uniform approximation guarantees that preserve robust MPC’s recursive feasibility and input-to-state stability under neural replacement.

    Source: §5 · p. 11

    η-coverage alone does not ensure training samples lie in the MPC feedback law’s relevant local-Lipschitz neighborhoods, which may have arbitrary geometry.

    Source: §3.2 · p. 7

Abstract

from arXiv · show

Model predictive control is a high-performance control method, but it incurs comparatively high online computational costs across many control tasks and is challenging to realize in low-level hardware settings. To address this, we present a framework for generating neural network controllers that provably inherit the safety and stability guarantees of a robust model predictive policy. In contrast to approaches based on post-verification or additional online safety filters, our method provides constructive, a priori design conditions that ensure preservation of recursive feasibility and stability before deployment. We treat the approximation error as a bounded input disturbance, which is addressed in the MPC design. This allows the neural network controller to directly inherit robustness from the underlying model predictive controller, which enables the transfer of guarantees to the approximation. The design process is an offline, constructive synthesis pipeline and can be implemented once Lipschitz bounds are specified. We further demonstrate in simulation that the learned controller reproduces the closed-loop behavior of the original controller while reducing online computation.

1 Introduction

MPC can handle constraints and deliver near-optimal closed-loop behavior, but repeated online optimization burdens resource-constrained implementations. This work develops an offline framework that certifies neural approximations of robust nonlinear MPC so they inherit safety and stability guarantees.

1 Introduction

  • Neural networks are attractive MPC approximators because their evaluation effort stays fixed as state dimension, input dimension, and dataset size grow.Explicit MPC avoids online optimization, but its number of piecewise-affine regions can grow exponentially with the number of inequality constraints.Source: §1 · p. 1
  • Prior guarantee-transfer approaches differ in how they certify approximation error, including statistical validation, deterministic Lipschitz-and-coverage conditions, and kernel-based construction.The cited kernel method assumes RKHS membership with a known norm bound.Source: §1 · pp. 1–2
  • The framework links MPC-feedback Lipschitz properties and training-data coverage to certified approximation-error bounds.Its deterministic certificate avoids statistical validation and does not require the feedback law to belong to a prescribed RKHS with a known norm bound.Source: §1 · p. 2
  • An offline constructive procedure generates a neural approximator satisfying the prescribed error bounds.The procedure enables systematic network training.Source: §1 · p. 2
  • The neural controller inherits the robust MPC’s recursive feasibility and stability under the derived conditions.The framework designs robust MPC for a prescribed disturbance bound and certifies that the neural approximation satisfies it, without post-verification or auxiliary safety mechanisms.Source: §1 · p. 2

2 Robust MPC as a Basis for Certified Controller Approximation

The section establishes robust MPC as the basis for transferring guarantees to a neural controller: its approximation error is bounded, represented as a state disturbance, and accounted for in the MPC design. Under the stated assumptions, the resulting MPC is recursively feasible and input-to-state stable.

2.1 System Description and Control Task

  • The control task is to stabilize the origin while satisfying state and input constraints, using a neural approximation to reduce MPC’s online computational demands.The system is modeled with twice continuously differentiable dynamics, and the state and input sets are compact.Source: §2.1 · pp. 2–3

2.2 From Input Approximation Error to State Disturbance

  • A prescribed bound on neural approximation error is treated as an input disturbance and reformulated as a bounded additive state disturbance.Lipschitz continuity of the dynamics gives the state-disturbance bound Lf,uϵ; Lipschitz properties of the MPC law and network, together with training-data coverage, are used to ensure the error bound.Source: §2.1 · p. 3, §2.2 · p. 3

2.3 Robust MPC Design and Closed-Loop Guarantees

  • The robust MPC repeatedly optimizes nominal state predictions and planned inputs, with a terminal-region constraint supporting recursive feasibility and stability.The terminal cost serves as a Lyapunov function candidate for robust stability guarantees.Source: §2.3 · pp. 3–4
  • The MPC tightens input and nominal-state constraints to account for approximation error and its propagated effects.Open-loop disturbance propagation can make constraint tightening grow with prediction horizon and, in extreme cases, affect feasibility.Source: §2.3 · p. 4
  • Under the stated assumptions, initially feasible robust MPC remains recursively feasible and renders the origin input-to-state stable for the disturbed system.Its feasible initial-condition set is robustly forward invariant for disturbances bounded by Lf,uϵ.Source: §2.3 · pp. 4–5

3 Certified Neural Approximation of the MPC Feedback Law

This section develops a Lipschitz-based method for certifying neural approximations of MPC feedback and transferring robust MPC guarantees under stated approximation and regularity conditions. It also presents an offline procedure for constructing a network that approximates a robust MPC law within a prescribed error bound on its invariant set, and discusses practical limitations.

3.1 Feed-Forward Neural Networks

  • The method uses offline supervised feed-forward networks with differentiable activations and fixed online evaluation structure; certification requires the approximator to be Lipschitz continuous on X.The framework is not limited to feed-forward networks, provided the chosen approximator satisfies the Lipschitz requirement.Source: §3.1 · p. 5

3.2 Approximation Error Bounds

  • The certification is conditional on MPC regularity: LICQ, strict complementarity, and second-order sufficiency yield local Lipschitz continuity, while coverage alone may not place a sample in the relevant neighborhood.The paper notes these conditions can be restrictive, and the neighborhoods may have arbitrary geometry and nonuniform radii.Source: §3.2 · pp. 6–7
  • Theorem 13 bounds approximation error by combining training-sample accuracy with MPC and network Lipschitz constants, provided a nearby training point lies in the MPC feedback law’s regularity neighborhood.For a prescribed error ϵ greater than training error ϵD, the required distance is at most (LMPC(xk) + LNN)^-1(ϵ − ϵD).Source: §3.2 · p. 6

3.3 Recursive feasibility and Stability Guarantees

  • Under Theorem 16’s conditions, the neural controller preserves MPC feasibility and input-to-state stability almost surely despite approximation error and bounded additive disturbances.The result also preserves invariance of the feasible initial-state set when Theorem 13’s approximation conditions hold pointwise along the closed-loop trajectory.Source: §3.3 · p. 7

3.4 Procedure for Certified NN Approximation

  • The section introduces a procedure to generate a neural-network controller with the guarantees established earlier and discusses practical implementation limitations.No further procedural steps or specific limitations are provided in this passage.Source: §3.4 · p. 8
  • The network is fixed at runtime, so online evaluation time depends on its architecture rather than training-set size.Sampling and training can be performed once offline after provisional error and Lipschitz bounds are selected, provided the resulting network meets those bounds.Source: §3.4 · p. 8
  • Coverage sampling uses the specified MPC and network Lipschitz bounds and an over-coverage radius no greater than the required coverage radius to form the training dataset.The dataset is obtained by evaluating the MPC feedback law at coverage points in the robustly forward-invariant set.Source: §3.4 · p. 9
  • A trained network is admissible only if its maximum training-point approximation error and Lipschitz bound satisfy their prescribed limits.Hyperparameter tuning or retraining may be needed to meet both conditions; any network that meets them needs no further refinement.Source: §3.4 · p. 9
  • The procedure constructs a robust MPC law and trains a neural network to approximate it on the MPC’s robustly forward-invariant set with error bound ϵ.The design error bound is chosen at least as large as the expected training-point error bound; an optional state-disturbance bound can also be incorporated in the MPC design.Source: §3.4 · pp. 8–9
  • Coverage-based certification faces the curse of dimensionality: for a bounded domain, sample requirements scale as O((R/δ)^nx), and offline effort also depends on δ.The passage relates δ to the acceptable approximation-error margin and Lipschitz bounds; online evaluation remains determined by the network architecture.Source: §3.4 · p. 9

4 Example

The reactor study trains a neural-network approximation of a robust MPC controller and evaluates it in closed loop against the MPC under matched disturbances. The approximation tracks the reference without observed constraint violations while reducing computation.

4 Example

  • The example is an isothermal CSTR with reactions 2A + B → C, C → 2D, and selective D removal, discretized with Ts = 0.5 min.Source: §4 · p. 9

4.1 Controller Design

  • The robust MPC uses bounded concentration and feed sets, a quadratic cost, a 10-step horizon, and a prescribed approximation-error bound of 0.05.The reference is xref = [0.25, 0.25, 0.25, 0.25]⊤ mol L−1 with uref = 1.25 mol L−1.Source: §4.1 · p. 9

4.2 Neural Network Architecture

  • The controller uses a 4-dimensional concentration state as input and outputs the scalar feed concentration cA,f.The fully connected network has three hidden layers of 32 neurons each and Gaussian error linear unit activations.Source: §4.2 · pp. 9–10

4.3 Data Generation and Training

  • Training used 6,417,883 centers; the resulting NN achieved R² = 0.9999 on validation and stayed below the prescribed dataset-error and Lipschitz bounds.The maximum training-set error was 1.3759 × 10−3, and the trained network’s Lipschitz bound was 3.4999.Source: §4.3 · p. 10

4.4 Results

  • Across 16,000 sequential single-state evaluations on the same hardware, using one CPU thread and no NN batching, mean computation time fell from 20.73 ms for robust MPC to 0.59 ms for NN, approximately 35 times faster.The 100 closed-loop trajectories generated by robust MPC and its NN approximation remain close over the complete simulation horizon and approach the reference, with no state-constraint or input-constraint violations.Source: §4.4 · p. 10, Figure 1 · p. 11
  • 8.62 × 10−4 was the maximum observed pointwise approximation error, with no bound violations across 16,000 evaluations.The minimum observed error was 5.73 × 10−9.Source: Table 2 · p. 10, §4.4 · p. 10

5 Conclusion

The paper presents a constructive framework for certifying neural approximations of robust nonlinear MPC controllers. Its simulation study reports behavior close to the robust MPC with reduced online computation, while identifying conservatism and data demands as areas for future work.

  • Approximation error is treated as a bounded input disturbance, with Lipschitz-based bounds embedded in robust MPC to preserve recursive feasibility and input-to-state stability under neural replacement.The offline procedure links dataset coverage and Lipschitz constants to explicit uniform approximation guarantees.Source: §5 · p. 11
  • A nonlinear isothermal CSRT simulation reports that the NN reproduces robust-MPC closed-loop behavior within the certified invariant region while significantly reducing online computation.Source: §5 · p. 11
  • Future work targets conservatism from global Lipschitz bounds, uniform-coverage data requirements, scalability to higher-dimensional systems, and broader regularity assumptions.Source: §5 · p. 11

Declaration of Generative AI and AI-assisted technologies in the writing process

The authors report using Chat-GPT 5.5 and 5.6-Sol to improve language flow and wording, then reviewing and editing the content and taking full responsibility. The supplement describes an additional drone example and outlines its model and disturbance assumptions.

S1.1 Controller Design

The drone controller design specifies state and input constraints, quadratic costs, a reference, and an approximation-error bound. The terminal set is defined by a level set of the terminal cost.

  • The state constraints bound altitude to −20–40 m and vertical velocity to −25–25 m s−1; the input thrust is constrained to −30–30 N.Source: S1.1 Controller Design · p. 12
  • The controller uses quadratic stage and terminal costs, a terminal-cost level set with αv = 100, and reference altitude 25.0 m with zero vertical velocity.The reference thrust is uref = g · m = 9.81 N.Source: S1.1 Controller Design · p. 12
  • The designable approximation-error bound is ϵ = 3, described as small relative to the admissible input range.Source: S1.1 Controller Design · p. 12

S1.2 Horizon length and conservatism

The open-loop disturbance sets depend on the dynamics’ state Lipschitz constant: they remain bounded below one, grow linearly at one, and may grow geometrically above one. For the drone example, the resulting horizon-dependent tightening is substantial at N = 30, motivating a trade-off between look-ahead and conservatism.

S1.2 Horizon length and conservatism

  • Disturbance sets remain bounded for Lf,x < 1, grow linearly for Lf,x = 1, and may grow geometrically for Lf,x > 1.Source: S1.2 Horizon length and conservatism · p. 12
  • For the drone model, Lf,x = 1.05 > 1, so disturbance sets grow along the prediction horizon.With Lf,u = 0.05, ϵ = 3, and N = 30, the radius of W29 is approximately the value reported in the cited passage.Source: S1.2 Horizon length and conservatism · p. 12
  • At N = 30, tightening by W29 reduces the altitude and velocity widths by approximately 31% and 37%, respectively.The authors selected this horizon as a practical trade-off between improved look-ahead and increased conservatism.Source: S1.2 Horizon length and conservatism · p. 12
  • A tube-based MPC with ancillary feedback could stabilize error dynamics and limit set growth, but boundedness or contraction requires additional assumptions and analysis.Source: S1.2 Horizon length and conservatism · p. 12

S1.3 Neural Network Architecture

The MPC feedback law is approximated by a fully connected feed-forward neural network that maps states to control inputs. The specified architecture uses three hidden layers with 16 neurons each and hyperbolic tangent activations.

S1.3 Neural Network Architecture

S1.4 Data Generation and Training

The training pipeline estimates an invariant region and an initial approximation error, uses Lipschitz bounds to set a coverage radius, and evaluates the MPC law on coverage points to form the dataset. A newly initialized network trained on that dataset satisfies the specified initial bounds.

S1.4 Data Generation and Training

  • An initial training run used 20,814 evenly spaced seed points and 200 epochs, yielding an approximation error of 1.9876, rounded to 2 in the 2-norm.The seed points were also used to estimate a robustly forward-invariant subset by backwards pruning.Source: S1.4 Data Generation and Training · p. 13
  • A power-iteration method estimated LMPC = 19.887 and an initial-network Lipschitz constant of 21.436, leading to chosen bounds LMPC = 20 and LNN = 25.These bounds gave δ = 0.0222, and the chosen over-coverage radius was η = δ.Source: S1.4 Data Generation and Training · p. 13
  • The coverage procedure produced 1,231,052 points for the final training dataset.The η-coverage of XN is also a δ-coverage, and the neighboring coverage-point separation is at most 2η when their radius-η balls overlap.Source: S1.4 Data Generation and Training · p. 13
  • The final network achieved an R2-score of 0.9999 on 10,000 validation points distinct from the training data.Its approximation error was 1.9222 and its final Lipschitz estimate was 23.22, both satisfying the initial bounds.Source: S1.4 Data Generation and Training · p. 13
  • Training, sampling, and simulations ran in Python on a ThinkPad with an AMD Ryzen 7 PRO 5850U CPU and 32 GB RAM.MPC used IPOPT via CasADi, and the neural network used PyTorch.Source: S1.4 Data Generation and Training · p. 13

S1.5 Results

In 100 closed-loop simulations, the NN controller tracked the robust MPC closely, with trajectories converging near the reference and no constraint violations. The reported approximation errors stayed below the allowed bound, and computation time was significantly reduced with the NN controller.

S1.5 Results

  • All 100 trajectories remained inside X̂N without constraint violations, and both controllers’ trajectories stayed close and converged to a neighborhood of the reference.Source: S1.5 Results · p. 13
  • The results indicate that the MPC’s safety and stability guarantees transfer to the neural network controller.The authors report that Theorem 16 appears satisfied along all 100 trajectories.Source: S1.5 Results · p. 13
  • The NN controller’s computation time was significantly lower than the robust MPC’s across the 100 simulated trajectories.Source: S1.5 Results · p. 13, Table S1 · p. 14
  • Approximation errors in l2-norm across the 100 trajectories were all below the allowed bound ϵ = 3.The reported minimum is 0, but the first observed digit is at the seventh decimal place and is not clearly distinguishable from numerical noise.Source: S1.5 Results · p. 13, Table S2 · p. 14

S2 Weight-projected neural network training

The network’s Lipschitz constant is bounded analytically from its layer properties, and projected training keeps each completed update within the prescribed bound.

S3 Existence of a neural network with prescribed finite-dataset accuracy

For any finite dataset with distinct inputs, a feed-forward neural network can achieve any prescribed positive pointwise approximation tolerance. The proof constructs a continuous interpolant and approximates it on a compact set.

  • This existence result guarantees finite-dataset accuracy only; it does not establish that a particular training algorithm finds the network or that accuracy holds outside the dataset.Source: S3 Existence of a neural network with… · p. 14
  • For any finite dataset with pairwise distinct inputs and any εD > 0, a feed-forward neural network with a continuous non-polynomial activation can approximate every data point within εD.Source: S3 Existence of a neural network with… · p. 14
  • The proof first extends the finite data to a continuous interpolant, then uses universal approximation on a compact set containing the training inputs.Source: S3 Existence of a neural network with… · pp. 14–15
  • Figure S2 depicts 100 closed-loop state trajectories for robust MPC and NN controllers from initial states in ˆ XN, with enlarged views of selected trajectories.Source: Figure S2 · p. 15

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