Frequency Domain Stability and Convergence Analysis for General Reset Control Systems Architecture

TL;DR

This work furnishes a rigorous FRF-based $H_eta$ stability framework for general reset control systems, including parallel reset structures and nonzero after-reset values. By deriving a transfer-function correspondence between the matrix-based $H_eta$ and an FRF representation, it enables stability assessment using measured FRFs rather than full parametric models, while still guaranteeing quadratic stability under SPR. The method is extended to account for time delay (via Padé approximations) and is validated through mass-spring-damper and industrial precision-positioning examples, illustrating model-free stability checks and UBIBS/convergence properties. The results highlight that delay challenges CI configurations but leaves FORE/GFORE viable, offering practical guidance for industrial reset-control design and a foundation for future robustness and conservativeness analyses.

Abstract

A key factor that generates significant interest in reset control systems, especially within industrial contexts, is their potential to be designed using a frequency-domain loop-shaping procedure. On the other hand, formulating and assessing stability analysis for these nonlinear elements often depends on access to parametric models and numerically solving linear matrix inequalities. These specific factors could present challenges to the successful implementation of reset control within industrial settings. Moreover, one of the most effective structures for implementing reset elements is to use them in parallel with a linear element. Therefore, this article presents the development of the frequency domain-based $H_β$ stability method from a series to a more general structure of reset control systems. Additionally, it investigates the behavior of different reset elements in terms of the feasibility of stability in the presence of time delay. To illustrate the research findings, two examples are provided, including one from an industrial application.

Figures (5)

• Figure 1: The closed-loop architecture of a general reset control system.
• Figure 2: NSV plot ($\stackrel{\rightarrow}{\mathcal{N}}(\omega)$) in the presence of delay at high frequencies ($\omega \rightarrow \infty$), for (a) $\omega_r\neq 0$, and (b) $\omega_r$= 0.
• Figure 3: $\theta_{\mathcal{N}}(\omega)$ for the MSD system.
• Figure 4: Frequency response data from the x-axis motion platform of the ASMPT's wire-bonding machine.
• Figure 5: $\theta_{\mathcal{N}}(\omega)$ for the precision positioning control system.

Theorems & Definitions (28)

• Theorem 1
• Lemma 1
• Definition 1
• Definition 2
• Remark 1
• Lemma 2
• Definition 3
• Lemma 3
• Remark 2
• Lemma 4: Partitioned matrix inversion