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Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers

Hamidreza Montazeri Hedesh, Milad Siami

TL;DR

This work addresses stability of positive linear systems in feedback with a fully connected FFNN controller by formulating the closed-loop as a positive Lur'e system. It introduces a sector bound for a fully connected FFNN without biases, parameterized by activation sector and weight magnitudes, and proves a global exponential stability result under a positive Aizerman-like framework: if $A+B\Gamma_1C$ is Metzler and $A+B\Gamma_2C$ is Hurwitz, then the NN-controlled system is globally exponentially stable. The key contributions are (i) a concrete sector bound for general FFNNs without biases, (ii) a stability theorem leveraging positive system theory and Aizerman conjecture, and (iii) a practical example demonstrating stability and favorable computational efficiency versus IQC-based methods. The approach offers a scalable, Lyapunov-free route for verifying NN-controlled positive systems, with potential extensions to local sectors and biased networks. This has practical impact for robust NN-based control in safety-critical domains where positivity and stability must be guaranteed.

Abstract

This paper introduces a novel method for the stability analysis of positive feedback systems with a class of fully connected feedforward neural networks (FFNN) controllers. By establishing sector bounds for fully connected FFNNs without biases, we present a stability theorem that demonstrates the global exponential stability of linear systems under fully connected FFNN control. Utilizing principles from positive Lur'e systems and the positive Aizerman conjecture, our approach effectively addresses the challenge of ensuring stability in highly nonlinear systems. The crux of our method lies in maintaining sector bounds that preserve the positivity and Hurwitz property of the overall Lur'e system. We showcase the practical applicability of our methodology through its implementation in a linear system managed by a FFNN trained on output feedback controller data, highlighting its potential for enhancing stability in dynamic systems.

Ensuring Both Positivity and Stability Using Sector-Bounded Nonlinearity for Systems with Neural Network Controllers

TL;DR

This work addresses stability of positive linear systems in feedback with a fully connected FFNN controller by formulating the closed-loop as a positive Lur'e system. It introduces a sector bound for a fully connected FFNN without biases, parameterized by activation sector and weight magnitudes, and proves a global exponential stability result under a positive Aizerman-like framework: if is Metzler and is Hurwitz, then the NN-controlled system is globally exponentially stable. The key contributions are (i) a concrete sector bound for general FFNNs without biases, (ii) a stability theorem leveraging positive system theory and Aizerman conjecture, and (iii) a practical example demonstrating stability and favorable computational efficiency versus IQC-based methods. The approach offers a scalable, Lyapunov-free route for verifying NN-controlled positive systems, with potential extensions to local sectors and biased networks. This has practical impact for robust NN-based control in safety-critical domains where positivity and stability must be guaranteed.

Abstract

This paper introduces a novel method for the stability analysis of positive feedback systems with a class of fully connected feedforward neural networks (FFNN) controllers. By establishing sector bounds for fully connected FFNNs without biases, we present a stability theorem that demonstrates the global exponential stability of linear systems under fully connected FFNN control. Utilizing principles from positive Lur'e systems and the positive Aizerman conjecture, our approach effectively addresses the challenge of ensuring stability in highly nonlinear systems. The crux of our method lies in maintaining sector bounds that preserve the positivity and Hurwitz property of the overall Lur'e system. We showcase the practical applicability of our methodology through its implementation in a linear system managed by a FFNN trained on output feedback controller data, highlighting its potential for enhancing stability in dynamic systems.
Paper Structure (11 sections, 5 theorems, 25 equations, 4 figures, 1 table)

This paper contains 11 sections, 5 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Positive Systems
  5. Lur'e Systems and Aizerman's Conjecture
  6. Positive Lur’e Systems
  7. Problem Formulation
  8. Main Results
  9. Neural Network controlled system as Lur'e system
  10. Examples
  11. Discussion & Conclusion

Key Result

Proposition 1

Given a system described by eq:generallti and considering the definition of a positive system as per Definition def:positive system, the system is positive if and only if matrix $A$ is Metzler, and matrices $B \in \mathbb R^{n\times m}_+$ and $C\in \mathbb R^{p\times n}_+$ebihara2016analysis.

Figures (4)

  • Figure 1: Lur'e system with plant $G$ and nonlinear controller $\Phi$.
  • Figure 2: $\mathop{\mathrm{ReLU}}\nolimits$ and $\tanh$ sector-bounded in $\left[0,1\right]$.
  • Figure 3: The sector bounds vs NN output for two different NNs.
  • Figure 4: Trajectory of system for $50$ random initial conditions.

Theorems & Definitions (7)

  • Definition 1: Positive Linear System
  • Definition 2: Metzler Matrix
  • Proposition 1
  • Lemma 1
  • Theorem 1: Positive Aizermandrummond2022aizerman
  • Theorem 2
  • Theorem 3