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Regional stability conditions for recurrent neural network-based control systems

Alessio La Bella, Marcello Farina, William D'Amico, Luca Zaccarian

TL;DR

This work develops linear matrix inequality (LMI) based global and regional stability conditions for discrete-time recurrent neural networks (RNNs) with sigmoidal nonlinearities, enabling effective $H_2$-based state-feedback control. It introduces two regional stability frameworks: an auxiliary-function approach and a sector-narrowing approach, each yielding LMIs that certify local stability and provide estimates of the basin of attraction; the region definitions rely on ellipsoids and sector bounds tied to the nonlinearities. The control design combines these stability tools with an $\ ext{H}_2$ objective, recasting the problem as quasi-convex LMIs to compute stabilizing gains $K$ (via $K=J S^{-1}$) while balancing performance and robustness to nonlinearities. A data-driven ESN benchmark (pH neutralization) demonstrates feasibility and highlights trade-offs: the auxiliary-function method often yields larger attraction regions but can be numerically challenging for high-dimensional systems, whereas the sector-narrowing method scales better to large orders at the cost of smaller basins. Overall, the paper provides scalable stability certificates and practical design procedures for RNN-based control, with potential extensions to more complex architectures and observers.

Abstract

In this paper we propose novel global and regional stability analysis conditions based on linear matrix inequalities for a general class of recurrent neural networks. These conditions can be also used for state-feedback control design and a suitable optimization problem enforcing H2 norm minimization properties is defined. The theoretical results are corroborated by numerical simulations, showing the advantages and limitations of the methods presented herein.

Regional stability conditions for recurrent neural network-based control systems

TL;DR

This work develops linear matrix inequality (LMI) based global and regional stability conditions for discrete-time recurrent neural networks (RNNs) with sigmoidal nonlinearities, enabling effective -based state-feedback control. It introduces two regional stability frameworks: an auxiliary-function approach and a sector-narrowing approach, each yielding LMIs that certify local stability and provide estimates of the basin of attraction; the region definitions rely on ellipsoids and sector bounds tied to the nonlinearities. The control design combines these stability tools with an objective, recasting the problem as quasi-convex LMIs to compute stabilizing gains (via ) while balancing performance and robustness to nonlinearities. A data-driven ESN benchmark (pH neutralization) demonstrates feasibility and highlights trade-offs: the auxiliary-function method often yields larger attraction regions but can be numerically challenging for high-dimensional systems, whereas the sector-narrowing method scales better to large orders at the cost of smaller basins. Overall, the paper provides scalable stability certificates and practical design procedures for RNN-based control, with potential extensions to more complex architectures and observers.

Abstract

In this paper we propose novel global and regional stability analysis conditions based on linear matrix inequalities for a general class of recurrent neural networks. These conditions can be also used for state-feedback control design and a suitable optimization problem enforcing H2 norm minimization properties is defined. The theoretical results are corroborated by numerical simulations, showing the advantages and limitations of the methods presented herein.
Paper Structure (12 sections, 13 theorems, 67 equations, 6 figures, 1 algorithm)

This paper contains 12 sections, 13 theorems, 67 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Plant model
  4. Global exponential stability analysis
  5. Regional stability analysis
  6. Regional conditions with standard saturation
  7. Regional stability analysis via auxiliary function
  8. Regional stability analysis via sector narrowing
  9. Choice of the regional stability condition
  10. Controller design
  11. Numerical results
  12. Conclusions

Key Result

Lemma 1

Let function $\sigma:\mathbb{R}^\nu\to\mathbb{R}^\nu$ satisfy Assumption ass:1 and consider $q:\mathbb{R}^\nu\to\mathbb{R}^\nu$ defined as $y\mapsto q(y)= y- \sigma(y)$. Then, condition eq:sec_bound holds for any $y\in \mathbb{R}^\nu$ and $W\in \mathbb{D}_{\succ 0}^{\nu}$.

Figures (6)

  • Figure 1: Left plot: Examples of functions $\sigma_i$ satisfying Assumption 1: $\sigma_i(y_i)=\text{sat}(y_i)$ (solid black line), $\sigma_i(y_i)=\text{tanh}(y_i)$ (dotted blue line) and $\sigma_i(y_i)=y_i/(1+|y_i|)$ (dashed red line); Right plot: corresponding functions $q_i(y_i)$.
  • Figure 2: Function $\psi_i(y_i)$ (solid line) and the sector bound $\theta_i\,y_i$ (dashed black line). Left plot: $\sigma_i=\text{tanh}(y_i)$, with $\theta_i=0.2384$; Right plot: $\sigma_i(y_i)=y_i/(1+|y_i|)$, with $\theta_i=0.52$.
  • Figure 3: Function $\sigma_i(y_i)=\text{tanh}(y_i)$ (solid black) and the ensuing function $h_iq_i(y_i)$ for $h_i=0.5$ (dashed blue), $h_i=1$ (dashed-dotted green), $h_i=2$ (dotted red).
  • Figure 4: State-feedback control scheme with integral action.
  • Figure 5: Radius of the ellipsoid $\mathcal{E}(S)$, i.e., $\gamma$, for varying $\mathcal{H}_2$ norm $\bar{\delta} \in [5,20]$. Left plot: solution obtained from \ref{['eq:H2_th1']}, right plot: solution obtained from \ref{['eq:H2_th2']}.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Remark 1
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Lemma 3
  • Theorem 1
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • ...and 8 more