Luré-Postnikov Stability Analysis of Closed-Loop Control Systems with Gated Recurrent Neural Network-based Virtual Sensors

2025/05/14 by Eric Hilgert, Hilgert, Eric, Andreas Schwung +1
Computer Science · Engineering · #Adaptive Control of Nonlinear Systems #Adaptive Dynamic Programming Control #FOS: Electrical engineering #Stability and Control of Uncertain Systems #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · pdf · doi:10.48550/arxiv.2505.11539

openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02

Abstract

This article addresses certification of closed-loop stability when a virtual-sensor based on a gated recurrent neural network operates in the feedback path of a nonlinear control system. The Hadamard gating used in standard GRU/LSTM cells is shown to violate the Luré-Postnikov Lyapunov conditions of absolute-stability theory, leading to conservative analysis. To overcome this limitation, a modified architecture-termed the Luré-Postnikov gated recurrent neural network (LP-GRNN)-is proposed; its affine update law is compatible with the Luré-Postnikov framework while matching the prediction accuracy of vanilla GRU/LSTM models on the NASA CMAPSS benchmark. Embedding the LP-GRNN, the plant, and a saturated PI controller in a unified standard nonlinear operator form (SNOF) reduces the stability problem to a compact set of tractable linear matrix inequalities (LMIs) whose feasibility certifies global asymptotic stability. A linearized boiler case study illustrates the workflow and validates the closed-loop performance, thereby bridging modern virtual-sensor design with formal stability guarantees.

Citations

• SigGate: Enhancing Recurrent Neural Networks with Signature-Based Gating Mechanisms
• Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systems
• Chordal Sparsity for SDP-based Neural Network Verification
• Refined Gate: A Simple and Effective Gating Mechanism for Recurrent Units
• LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification
• Block-Diagonal Solutions to Lyapunov Inequalities and Generalisations of\n Diagonal Dominance
• Minimal Gated Unit for Recurrent Neural Networks
• CVXPY: A Python-Embedded Modeling Language for Convex Optimization
• Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation
• Long Short-Term Memory

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