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The Analysis and the Performance of the Parallel-Partial Reset Control System

Xinxin Zhang, S. Hassan HosseinNia

Abstract

Reset controllers have demonstrated their effectiveness in enhancing performance in precision motion systems. To further exploiting the potential of reset controllers, this study introduces a parallel-partial reset control structure. Frequency response analysis is effective for the design and fine-tuning of controllers in industries. However, conducting frequency response analysis for reset control systems poses challenges due to their nonlinearities. We develop frequency response analysis methods for both the open-loop and closed-loop parallel-partial reset systems. Simulation results validate the accuracy of the analysis methods, showcasing precision enhancements exceeding 100% compared to the traditional describing function method. Furthermore, we design a parallel-partial reset controller within the Proportional-Integral-Derivative (PID) control structure for a mass-spring-damper system. The frequency response analysis of the designed system indicates that, while maintaining the same bandwidth and phase margin of the first-order harmonics, the new system exhibits lower magnitudes of higher-order harmonics, compared to the traditional reset system. Moreover, simulation results demonstrate that the new system achieves lower overshoot and quicker settling time compared to both the traditional reset and linear systems.

The Analysis and the Performance of the Parallel-Partial Reset Control System

Abstract

Reset controllers have demonstrated their effectiveness in enhancing performance in precision motion systems. To further exploiting the potential of reset controllers, this study introduces a parallel-partial reset control structure. Frequency response analysis is effective for the design and fine-tuning of controllers in industries. However, conducting frequency response analysis for reset control systems poses challenges due to their nonlinearities. We develop frequency response analysis methods for both the open-loop and closed-loop parallel-partial reset systems. Simulation results validate the accuracy of the analysis methods, showcasing precision enhancements exceeding 100% compared to the traditional describing function method. Furthermore, we design a parallel-partial reset controller within the Proportional-Integral-Derivative (PID) control structure for a mass-spring-damper system. The frequency response analysis of the designed system indicates that, while maintaining the same bandwidth and phase margin of the first-order harmonics, the new system exhibits lower magnitudes of higher-order harmonics, compared to the traditional reset system. Moreover, simulation results demonstrate that the new system achieves lower overshoot and quicker settling time compared to both the traditional reset and linear systems.

Paper Structure

This paper contains 14 sections, 2 theorems, 25 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Definition of the Reset Controller
  4. Frequency Response Analysis for the Reset Controller
  5. Problem Statement
  6. Frequency Response Analysis for the Parallel-Partial Reset Control System
  7. Frequency Response Analysis for the Open-loop and Closed-loop PP-RCS
  8. Illustrative Example: The Accuracy of the Analysis Method for the PP-RCS
  9. The Performance of the PP-RCS
  10. The PP-CgLp-PI$^2$D control system
  11. Open-loop Analysis for Systems $C_1$-$C_3$
  12. Closed-loop Analysis for Systems $C_1$-$C_3$
  13. Transient Responses for Systems $C_1$-$C_3$ and Future Work
  14. Conclusion

Key Result

Theorem 1

(Frequency Response Analysis for the Open-loop PP-RCS) The $n$-th Higher-order sinusoidal input describing function (HOSIDF) for $\mathcal{H}_p$, defined as $H_{p}^n(\omega)$, represents the transfer function from $E_{1n}(\omega) = \mathscr{F}[e_{1n}(t)]$ to $V_{1n}(\omega) = \mathscr{F}[v_{1n}(t)]$ where $\mathcal{C}_{bl}(\omega)$ is given in eq: RL and $\mathcal{C}_{nl}(n\omega)$ is given in rcs

Figures (9)

  • Figure 1: The block diagram of the RCS, where the blue lines represent the reset action.
  • Figure 2: The block diagram of the PP-RCS, where $r(t)$, $e(t)$, $v(t)$, $u(t)$, and $y(t)$ are the reference input, error, reset output, control input, and system output signals respectively. The blue lines represent the reset action.
  • Figure 3: The block diagram of the parallel-partial reset controller (PP-RC) $\mathcal{H}_p$.
  • Figure 4: The PP-PCI-PID control system structure.
  • Figure 5: The steady-state error $e(t)$ of the PP-PCI-PID system at input frequencies of (a) $f = 100$ Hz, (b) $f = 1000$ Hz, and (c) $f=20$ Hz, obtained from simulation, HOSIDF analysis \ref{['hosidfeq: y,e,u']}, and DF analysis \ref{['dfeq: y,e,u']}.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2