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Robust Stability Analysis of Positive Lure System with Neural Network Feedback

Hamidreza Montazeri Hedesh, Moh. Kamalul Wafi, Bahram Shafai, Milad Siami

TL;DR

The paper addresses robustness of positive Lur'e systems under structured uncertainties and unknown nonlinear sector bounds, using positivity-based linear-system tools to obtain explicit stability-radius formulas. It develops a positive Aizerman-based argument to relate nonlinear stability to linear counterparts and extends the theory to NN feedback loops by bounding NN nonlinearities within sectors. The main results provide closed-form radii for both Lur'e and NN-controlled systems, and introduce a refinement procedure to tighten NN sector bounds when exact bounds are unknown. The illustrated examples demonstrate accurate prediction of stability thresholds and show how sector-bound refinement can improve robustness guarantees for NN controllers in feedback loops.

Abstract

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and robustness properties of Metzler matrices. Specifically, we consider a control system of Lur'e type in which not only the linear part includes parametric uncertainty but also the nonlinear sector bound is unknown. We investigate tools from positive linear systems to effectively solve the problems in complicated and uncertain nonlinear systems. By leveraging the positivity characteristic of the system, we derive an explicit formula for the stability radius of Lur'e systems. Furthermore, we extend our analysis to systems with neural network (NN) feedback loops. Building on this approach, we also propose a refinement method for sector bounds of NNs. This study introduces a scalable and efficient approach for robustness analysis of both Lur'e and NN-controlled systems. Finally, the proposed results are supported by illustrative examples.

Robust Stability Analysis of Positive Lure System with Neural Network Feedback

TL;DR

The paper addresses robustness of positive Lur'e systems under structured uncertainties and unknown nonlinear sector bounds, using positivity-based linear-system tools to obtain explicit stability-radius formulas. It develops a positive Aizerman-based argument to relate nonlinear stability to linear counterparts and extends the theory to NN feedback loops by bounding NN nonlinearities within sectors. The main results provide closed-form radii for both Lur'e and NN-controlled systems, and introduce a refinement procedure to tighten NN sector bounds when exact bounds are unknown. The illustrated examples demonstrate accurate prediction of stability thresholds and show how sector-bound refinement can improve robustness guarantees for NN controllers in feedback loops.

Abstract

This paper investigates the robustness of the Lur'e problem under positivity constraints, drawing on results from the positive Aizerman conjecture and robustness properties of Metzler matrices. Specifically, we consider a control system of Lur'e type in which not only the linear part includes parametric uncertainty but also the nonlinear sector bound is unknown. We investigate tools from positive linear systems to effectively solve the problems in complicated and uncertain nonlinear systems. By leveraging the positivity characteristic of the system, we derive an explicit formula for the stability radius of Lur'e systems. Furthermore, we extend our analysis to systems with neural network (NN) feedback loops. Building on this approach, we also propose a refinement method for sector bounds of NNs. This study introduces a scalable and efficient approach for robustness analysis of both Lur'e and NN-controlled systems. Finally, the proposed results are supported by illustrative examples.

Paper Structure

This paper contains 12 sections, 9 theorems, 29 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Proposed Theoretical Framework
  4. Positive Lur'e Systems
  5. Positive Aizerman Conjecture
  6. Stability Radius of Positive Systems
  7. Stability Radius of Positive Lur'e Systems
  8. Stability Radius of Lur'e Based Neural Network Feedback Loops
  9. Examples
  10. Lur'e System with Structured Perturbations
  11. Neural Network-Controlled System
  12. Discussion and Conclusion

Key Result

Lemma 1

The system eq:generallti is positive if and only if the matrix $A$ is a Metzler matrix, $B \in \mathbb R_+^{n\times m}$, and $C \in \mathbb R_+^{p\times n}$.

Figures (6)

  • Figure 1: Lur'e system with plant $G$ and nonlinear controller $\Phi$.
  • Figure 2: A schematic of the static nonlinearity tailored to lie within the sector bound $[\Sigma_1,\Sigma_2]$.
  • Figure 3: Random output trajectories of the system for various $\Delta$ values. The system preserves stability for $\Delta \leq 0.26$.
  • Figure 4: Random output trajectories of the system for various $\Delta$ values. The system is stable for $\Delta\leq2.04$ and transfers to instability with larger amounts of perturbations. This observation indicates that the NN upper sector bound $\Gamma_2$ might not be tight.
  • Figure 5: Random output trajectories of the system for various $\Delta$ values. As illustrated, the system becomes unstable at $\Delta = 3.15$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1: Positive Aizermandrummond2022aizerman
  • Remark 1
  • Theorem 2: Metzler Stability Radiusshafai1997explicit
  • Lemma 3: Monotonicity of Stability Radius
  • proof
  • Remark 2
  • Theorem 3
  • ...and 5 more