Reliability Assessment and Performance Enhancement of Reset Control Systems

Abstract: This paper develops a frequency-domain reliability assessment framework for reset control systems. The closed-loop higher-order sinusoidal-input describing function formulation is extended to explicitly include the reset-triggering signal generated through a shaping filter. Based on this signal, two metrics are introduced: (σ_t), which quantifies reset-time deviation, and (σ_d), which evaluates the tendency toward additional zero crossings. These metrics provide design-oriented indicators for identifying potentially unreliable reset behavior. To improve reset-triggering reliability, a first-order shaping filter is proposed for a generalized first-order reset element, increasing the low-frequency attenuation slope of the nonzero higher-order harmonics. The proposed analysis is evaluated on an industrial motion stage. The results show that the proposed metrics capture reliability issues that are not evident from the first-order closed-loop response alone and can therefore support the design of reset controllers with more reliable reset-triggering behavior.

• The paper introduces a frequency-domain framework that utilizes higher-order sinusoidal-input describing functions (HOSIDFs) to assess reset control system reliability.
• A first-order shaping filter is proposed to reduce reset-time deviation (σt) and multi-reset events (σd), leading to smoother reset behavior validated on an industrial wire-bonder.
• The framework provides actionable diagnostic tools for robust nonlinear control design and paves the way for future research on advanced shaping filter techniques.

Reliability Assessment and Performance Enhancement of Reset Control Systems

Introduction

The paper "Reliability Assessment and Performance Enhancement of Reset Control Systems" (2606.21431) formulates a comprehensive frequency-domain reliability assessment framework for reset control systems, particularly addressing the challenges posed by their inherent nonlinearity. Central to this work are the higher-order sinusoidal-input describing functions (HOSIDFs), whose closed-loop formulation is extended to explicitly incorporate the reset-triggering signal generated via a shaping filter. The analysis characterizes the impact of higher-order harmonics in the reset-triggering signal with two metrics: $\sigma_t$, quantifying reset-time deviation, and $\sigma_d$, assessing proximity to additional zero crossings, thus providing design-guided diagnostics beyond standard linear sensitivity-based approaches.

Frequency-Domain Framework for Reset Controllers

Reset elements, nonlinear filters that modulate control performance limitations intrinsic to LTI systems, increasingly require frequency-domain analysis methodologies that are robust to their nonlinearity. The SIDF and HOSIDF frameworks serve this purpose by approximating the harmonic content of the reset element's response under sinusoidal excitation. The closed-loop analysis is generalized using a Lur’e-type representation, enabling coverage of broad interconnections of LTI and reset elements. The novelty here lies in extending these closed-loop HOSIDFs to account for the impact and structure of shaping filters within the reset-triggering signal path, an aspect previously excluded from analytical frameworks.

(Figure 1)

Figure 1: Reliability metrics for the reset-based controller with and without the shaping filter: (a) reset-time deviation metric $\sigma_t$, (b) zero-crossing reliability metric $\sigma_d$. Regions with $\sigma_d>100$ indicate $N_r>2$.

Reliability Metrics: Reset-Time Deviation and Zero-Crossing Behavior

While conventional frequency-domain predictions assume reset instants governed by the first harmonic, empirical reset behavior often diverges due to significant higher-order harmonic content in the reset-triggering signal. The metric $\sigma_t$ quantitatively measures the deviation between the predicted (first-harmonic) and actual reset instants, normalized by the nominal interval. This is critical for identifying the potential mismatch between theoretical performance predictions and actual system behavior.

Complementary to $\sigma_t$, $\sigma_d$ evaluates the tendency of the reset-triggering signal to generate additional zero crossings, indicating proximity to multi-reset phenomena that invalidate the assumptions underlying HOSIDF-based designs. Specifically, $\sigma_d$ incorporates the distance between critical stationary points and zero, revealing if the system is on the verge of violating the designed two-reset condition.

These metrics demonstrate the capability to detect latent reliability issues not apparent in first-order closed-loop response analysis, contributing diagnostic value for controller synthesis and tuning.

Shaping Filter Design for Enhanced Reliability

Motivated by the diagnostic insight from reliability metrics, the paper proposes a first-order shaping filter for generalized first-order reset elements (GFORE). Analytical derivation shows the shaping filter increases the low-frequency attenuation slope of the nonzero higher-order harmonics from $\sigma_d$0 to $\sigma_d$1. This substantially reduces the higher-order harmonic content in the reset-triggering path, directly improving both $\sigma_d$2 and $\sigma_d$3, i.e., enhancing reset-triggering reliability and regularity.

Case Study: Industrial Wire-Bonder Motion Stage

The practical utility of the analytical framework and reliability metrics is validated on an industrial wire-bonder motion stage. The baseline LTI controller (PID + notch) is band-limited by delay and resonance constraints. A reset-based controller (using GFORE) extends the bandwidth but suffers unreliable reset instants, revealed by elevated $\sigma_d$4 and $\sigma_d$5 over key frequencies, including frequent violation of the two-reset condition.

Upon equipping the GFORE element with the proposed shaping filter, the metrics demonstrate reduced reset-time deviation and elimination of multi-reset events across the operational frequency range. The first-order sensitivity response remains largely unchanged, but the higher-order closed-loop sensitivities are attenuated, ensuring the improved theoretical performance is matched in practice.

Experimental Validation

Experimental results on the motion stage corroborate the frequency-domain predictions. Both reset-based controllers achieve significant RMS tracking error reductions relative to the LTI baseline, with the shaped reset controller consistently producing a smoother, more regular reset-triggering signal. This experimentally validates the metrics’ diagnostic capability and the shaping filter's reliability enhancement.

Implications and Future Directions

This reliability-centric framework equips practitioners to go beyond mere performance enhancement and toward robust nonlinear controller design, where reset instants are predictable and consistent with theoretical assumptions. The methodology provides actionable diagnostic tools for design iterations and tuning of industrial motion systems, facilitating rigorous controller synthesis that addresses both nonlinear performance advantage and reset reliability.

From a theoretical perspective, the extension of closed-loop HOSIDF analysis with explicit shaping filter inclusion and metric-based diagnosis informs new controller architectures and design paradigms. Future research should investigate higher-order and fractional-order shaping filters, broader placement strategies, and joint optimization of performance and reliability under diverse nonlinearities.

Conclusion

The paper establishes a frequency-domain reliability assessment framework for reset controllers, introducing rigorous metrics ($\sigma_d$6, $\sigma_d$7) and a shaping filter mechanism for GFORE elements. The reliability metrics expose deficiencies not evident in classical sensitivity analysis and enable design of reset controllers with provable, reliable reset-triggering behavior. The approach is experimentally validated, marking a methodological advance in nonlinear precision control design and reliability assurance for industrial systems.