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

TL;DR

The paper tackles certifying closed-loop stability when a gated-RNN-based virtual sensor sits in feedback with a nonlinear plant. It reveals that standard Hadamard gating in GRU/LSTM precludes a SNOF-based Lyapunov analysis and proposes the LP-GRNN, which uses a fixed mixing vector to achieve affine-like updates compatible with Luré-Postnikov LMIs. By unifying plant, controller, and LP-GRNN into a single SNOF, the authors derive tractable LMIs that certify global asymptotic stability and validate the approach on a CMAPSS benchmark and a linear boiler example. The work offers a practical route to formal stability guarantees for ML-enhanced sensing in control loops, while noting scalability and architectural flexibility as areas for future improvement.

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.

Figures (7)

• Figure 1: Illustration of the general problem of a discrete-time plant controlled by a discrete-time controller and a machine learning-based virtual sensor estimating a plant output signal 10394670.
• Figure 2: Architecture of the proposed LP-GRNN cell. The dynamic update gate $z_k$ of the standard GRU is replaced by a learnable time-invariant vector $\alpha$.
• Figure 3: Loop transformation of the sigmoid activation. The identity $\sigma(q) = \tfrac{1}{2}\tanh(\tfrac{1}{2}q) + \tfrac{1}{2}$ enables reformulation of the gating mechanism using only $\tanh(\cdot)$ nonlinearities, which are sector bounded in $[0,1]$ and slope restricted, thereby satisfying the requirements of Definitions \ref{['def:sb']} and \ref{['def:sr']}.
• Figure 4: Performance comparison of RNN architectures on the CMAPSS dataset. Bars represent mean RMSE with error bars showing the interquartile range (IQR).
• Figure 5: Training convergence analysis showing validation loss across epochs. Solid lines represent mean validation loss with shaded regions indicating the interquartile range (IQR).

Theorems & Definitions (20)

• Definition 3.1: Well‑posed SNOF Kim2018StandardRA
• Definition 3.2: Sector‑bounded non‑linearity Gupta1994SomePA
• Definition 3.3: Slope‑restricted non‑linearity Nguyen2021RobustCT
• Remark 3.1
• Theorem 3.1: Stability of a SNOF kim2009robust
• Proposition 4.1: GRU in SNOF‑like form
• proof
• Proposition 4.2: Hadamard Incompatibility with Luré--Postnikov
• proof
• Remark 4.1