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Validation of Neural Network Controllers for Uncertain Systems Through Keep-Close Approach: Robustness Analysis and Safety Verification

Abdelhafid Zenati, Nabil Aouf

TL;DR

The paper tackles safety certification for neural-network controllers governing uncertain dynamical systems by introducing a keep-close approach that binds the NN-controlled output to a trusted reference model under bounded inputs. It combines the Differential Mean Value Theorem (DMV) with Integral Quadratic Constraints (IQCs) and Lyapunov-based dissipation to derive worst-case bounds on the Relative Integral Square Error ($RISE$) and Supreme Square Error ($SSE$) between the reference and NN-controlled systems, formulated via a robust LPV extended model. The authors provide two substantive validations—the Single-Link Robot Arm and the Apollo Lander guidance problem—using the IQClab toolbox to compute concrete bounds and demonstrate that the NN controller maintains performance within predicted envelopes despite unmodeled dynamics, delays, and nonlinearities. The work offers a practical, scalable verification framework for safety-critical ML-enabled controllers, enabling pre-deployment robustness certification and enabling designers to quantify worst-case dynamical errors under uncertainty. Overall, this framework advances confidence in deploying NN-based controllers by delivering explicit, interpretable performance envelopes grounded in rigorous control-theoretic guarantees.

Abstract

Among the major challenges in neural control system technology is the validation and certification of the safety and robustness of neural network (NN) controllers against various uncertainties including unmodelled dynamics, nonlinearities, and time delays. One way in providing such validation guarantees is to maintain the closed-loop system output with a NN controller when its input changes within a bounded set, close to the output of a robustly performing closed-loop reference model. This paper presents a novel approach to analysing the performance and robustness of uncertain feedback systems with NN controllers. Due to the complexity of analysing such systems, the problem is reformulated as the problem of dynamical tracking errors between the closed-loop system with a neural controller and an ideal closed-loop reference model. Then, the approximation of the controller error is characterised by adopting the differential mean value theorem (DMV) and the Integral Quadratic Constraints (IQCs) technique. Moreover, the Relative Integral Square Error (RISE) and the Supreme Square Error (SSE) bounded set are derived for the output of the error dynamical system. The analysis is then performed by integrating Lyapunov theory with the IQCs-based technique. The resulting worst-case analysis provides the user a prior knowledge about the worst case of RISE and SSE between the reference closed-loop model and the uncertain system controlled by the neural controller.

Validation of Neural Network Controllers for Uncertain Systems Through Keep-Close Approach: Robustness Analysis and Safety Verification

TL;DR

The paper tackles safety certification for neural-network controllers governing uncertain dynamical systems by introducing a keep-close approach that binds the NN-controlled output to a trusted reference model under bounded inputs. It combines the Differential Mean Value Theorem (DMV) with Integral Quadratic Constraints (IQCs) and Lyapunov-based dissipation to derive worst-case bounds on the Relative Integral Square Error () and Supreme Square Error () between the reference and NN-controlled systems, formulated via a robust LPV extended model. The authors provide two substantive validations—the Single-Link Robot Arm and the Apollo Lander guidance problem—using the IQClab toolbox to compute concrete bounds and demonstrate that the NN controller maintains performance within predicted envelopes despite unmodeled dynamics, delays, and nonlinearities. The work offers a practical, scalable verification framework for safety-critical ML-enabled controllers, enabling pre-deployment robustness certification and enabling designers to quantify worst-case dynamical errors under uncertainty. Overall, this framework advances confidence in deploying NN-based controllers by delivering explicit, interpretable performance envelopes grounded in rigorous control-theoretic guarantees.

Abstract

Among the major challenges in neural control system technology is the validation and certification of the safety and robustness of neural network (NN) controllers against various uncertainties including unmodelled dynamics, nonlinearities, and time delays. One way in providing such validation guarantees is to maintain the closed-loop system output with a NN controller when its input changes within a bounded set, close to the output of a robustly performing closed-loop reference model. This paper presents a novel approach to analysing the performance and robustness of uncertain feedback systems with NN controllers. Due to the complexity of analysing such systems, the problem is reformulated as the problem of dynamical tracking errors between the closed-loop system with a neural controller and an ideal closed-loop reference model. Then, the approximation of the controller error is characterised by adopting the differential mean value theorem (DMV) and the Integral Quadratic Constraints (IQCs) technique. Moreover, the Relative Integral Square Error (RISE) and the Supreme Square Error (SSE) bounded set are derived for the output of the error dynamical system. The analysis is then performed by integrating Lyapunov theory with the IQCs-based technique. The resulting worst-case analysis provides the user a prior knowledge about the worst case of RISE and SSE between the reference closed-loop model and the uncertain system controlled by the neural controller.
Paper Structure (30 sections, 5 theorems, 115 equations, 16 figures, 2 tables)

This paper contains 30 sections, 5 theorems, 115 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem formulation
  4. Keep-Close Approach
  5. Integral Quadratic Constraints (IQCs) Concept
  6. Controller Error Analysis
  7. IQCs Characteristics of Uncertainties
  8. Robust Tracking Analysis via Standard IQCs
  9. Relative Integral Square Error (RISE)
  10. Supreme Square Error (SSE)
  11. Validation and Numerical Simulation
  12. Single-Link Robot Arm Control Problem
  13. Step 1: Problem Setting
  14. Step 2: Linear Reference Model
  15. Step 3: Error Dynamical System
  16. ...and 15 more sections

Key Result

lemma 3

Consider the dynamics of the error as defined in (erroriqc) and Let $c$ the trajectory lies on the curve $\vartheta (\nu) = x + \nu(\hat{x}- x)$ for some $\nu \in [0, 1]$ called the convex domain $\mathbb{Co}(x,\hat{x})$. The controllers error $\mu(t)$, as given in (erroriqc-c), can be expressed as: where the trajecory $s(t)=[d(t), c(t)]^T$, the LPV matrix $\Lambda(s(t))=\nabla_x \pi(d(t),c(t))$ i

Figures (16)

  • Figure 1: Keep-Close Approach Illustration.
  • Figure 2: Graphical Interpretation of the IQCs with the Equivalent System of (\ref{['extendedsystem']}).
  • Figure 3: Single-link robot arm.
  • Figure 4: Neural controller $\pi^{*}$ architecture.
  • Figure 5: IQCs characterisation of the nonlinearity $\Delta_{\delta}$.
  • ...and 11 more figures

Theorems & Definitions (12)

  • Definition 1: Hard IQC ref30
  • lemma 3: Controller Error
  • proof
  • lemma 4
  • proof
  • corollary 1
  • Definition 2: RISE & SSE
  • Theorem 1: Worst-case RISE Performance
  • proof
  • Theorem 2: Worst-case SSE
  • ...and 2 more