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Enhancing the Reliability of Closed-Loop Describing Function Analysis for Reset Control Applied to Precision Motion Systems

Xinxin Zhang, S. Hassan HosseinNia

TL;DR

The paper tackles the reliability of SIDF analysis for closed-loop reset control in precision motion systems, showing that the common two-reset assumption can fail when multiple resets occur per cycle, which degrades SIDF accuracy and amplifies high-order harmonics. It presents a principled method to identify frequency regions where multiple resets occur via piecewise steady-state expressions and a Delta theorem, and proposes a shaped reset strategy with a PID-shaped shaping filter to suppress higher harmonics while preserving the benefits of the first-order harmonic. The approach is validated through six CI-based reset controllers on a precision motion stage, demonstrating improved SIDF accuracy, reduced steady-state errors, better disturbance/noise rejection, and elimination of limit cycles in step responses. The work provides a practical, frequency-domain design framework that enhances the reliability and performance of reset-control systems in high-precision applications, with potential applicability to other reset structures. Overall, the shaping strategy yields tangible gains in tracking precision and robustness, while enabling more faithful SIDF-based design in complex reset control contexts.

Abstract

The Sinusoidal Input Describing Function (SIDF) is an effective tool for control system analysis and design, with its reliability directly impacting the performance of the designed control systems. This study enhances the reliability of SIDF analysis and the performance of closed-loop reset feedback control systems, presenting two main contributions. First, it introduces a method to identify frequency ranges where SIDF analysis becomes inaccurate. Second, these identified ranges correlate with high-magnitude, high-order harmonics that can degrade system performance. To address this, a shaped reset control strategy is proposed, which incorporates a shaping filter to tune reset actions and reduce high-order harmonics. Then, a frequency-domain design procedure of a PID shaping filter in a reset control system is outlined as a case study. The PID filter effectively reduces high-order harmonics and resolves limit-cycle issues under step inputs. Finally, simulations and experimental results on a precision motion stage validate the efficacy of the proposed shaped reset control, showing enhanced SIDF analysis accuracy, improved steady-state precision over linear and reset controllers, and elimination of limit cycles under step inputs.

Enhancing the Reliability of Closed-Loop Describing Function Analysis for Reset Control Applied to Precision Motion Systems

TL;DR

The paper tackles the reliability of SIDF analysis for closed-loop reset control in precision motion systems, showing that the common two-reset assumption can fail when multiple resets occur per cycle, which degrades SIDF accuracy and amplifies high-order harmonics. It presents a principled method to identify frequency regions where multiple resets occur via piecewise steady-state expressions and a Delta theorem, and proposes a shaped reset strategy with a PID-shaped shaping filter to suppress higher harmonics while preserving the benefits of the first-order harmonic. The approach is validated through six CI-based reset controllers on a precision motion stage, demonstrating improved SIDF accuracy, reduced steady-state errors, better disturbance/noise rejection, and elimination of limit cycles in step responses. The work provides a practical, frequency-domain design framework that enhances the reliability and performance of reset-control systems in high-precision applications, with potential applicability to other reset structures. Overall, the shaping strategy yields tangible gains in tracking precision and robustness, while enabling more faithful SIDF-based design in complex reset control contexts.

Abstract

The Sinusoidal Input Describing Function (SIDF) is an effective tool for control system analysis and design, with its reliability directly impacting the performance of the designed control systems. This study enhances the reliability of SIDF analysis and the performance of closed-loop reset feedback control systems, presenting two main contributions. First, it introduces a method to identify frequency ranges where SIDF analysis becomes inaccurate. Second, these identified ranges correlate with high-magnitude, high-order harmonics that can degrade system performance. To address this, a shaped reset control strategy is proposed, which incorporates a shaping filter to tune reset actions and reduce high-order harmonics. Then, a frequency-domain design procedure of a PID shaping filter in a reset control system is outlined as a case study. The PID filter effectively reduces high-order harmonics and resolves limit-cycle issues under step inputs. Finally, simulations and experimental results on a precision motion stage validate the efficacy of the proposed shaped reset control, showing enhanced SIDF analysis accuracy, improved steady-state precision over linear and reset controllers, and elimination of limit cycles under step inputs.

Paper Structure

This paper contains 28 sections, 4 theorems, 87 equations, 22 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reset Control System
  4. Stability and Convergence Conditions for Reset Control Systems
  5. SIDF Analysis for Open-Loop Reset Control Systems
  6. Precision Motion Stage
  7. Motivation and Problems Statement via Illustrative Examples
  8. Problem 1: Multiple-Reset Actions Leading to Inaccuracies in Closed-Loop SIDF Analysis
  9. Problem 2: Multiple-Reset Actions Relating to High-Order Harmonics that Need to be Reduced
  10. Identifying Two-Reset Conditions in SIDF Analysis and Experimental Validation
  11. Piecewise Expressions for Steady-State Trajectories in Sinusoidal-Input Closed-Loop Reset Control Systems
  12. Identifying Multiple-Reset and Two-Reset Actions in Sinusoidal-Input Closed-Loop Reset Control Systems
  13. Simulations and Experimental Validation of Theorem \ref{['thm: Delta']}
  14. Analysis and Design of Shaped Reset Control Systems
  15. Illustrative Example: Designing a PID Shaping Filter in CI-Based Reset Control Systems
  16. ...and 13 more sections

Key Result

Lemma 1

Consider a closed-loop reset control system as shown in Fig. fig: RCS_d_n_r_n_n, with a sinusoidal reference input $r(t) = |R| \sin (\omega t)$, and satisfying Assumptions assum: stable. Within one steady-state period $(0, 2\pi/\omega]$, the reset instant $t_i$, at which $z_s(t_i) = 0$, divides the where

Figures (22)

  • Figure 1: Block diagram of the closed-loop reset feedback control system.
  • Figure 2: Experimental precision positioning setup.
  • Figure 3: FRF data from actuator $A_1$ to attached mass $M_1$.
  • Figure 4: Block diagram of the reset PID control system.
  • Figure 5: Bode plots of the PID and the first-order harmonic of the PCID control systems.
  • ...and 17 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • ...and 8 more