Frequency Response Analysis for Reset Control Systems: Application to Predict Precision of Motion Systems

TL;DR

This paper tackles the challenge of frequency-domain analysis for reset control systems, where nonlinear resets break linear Bode relations. It proposes a pulse-based decomposition of reset outputs into base-linear and nonlinear components for open-loop analysis and derives a closed-loop higher-order sinusoidal-input describing function (HOSIDF); further, it develops a closed-loop HOSIDF by introducing a two-reset model and a Gamma(ω) factor linking open- and closed-loop harmonics. The approach is validated through simulations and experiments on PCI-PID precision-motion systems, including a two-reset PCI-PID (T-PCI-PID) design that demonstrates superior low-frequency tracking. The results enable loop-shaping and predictive design of reset controllers in precision motion applications and highlight the importance of considering higher-order harmonics and reset timing in design.

Abstract

The frequency response analysis describes the steady-state responses of a system to sinusoidal inputs at different frequencies, providing control engineers with an effective tool for designing control systems in the frequency domain. However, conducting this analysis for closed-loop reset systems is challenging due to system nonlinearity. This paper addresses this challenge through two key contributions. First, it introduces novel analysis methods for both open-loop and closed-loop reset control systems at steady states. These methods decompose the frequency responses of reset systems into base-linear and nonlinear components. Second, building upon this analysis, the paper develops closed-loop higher-order sinusoidal-input describing functions for reset control systems at steady states. These functions facilitate the analysis of frequency-domain properties, establish a connection between open-loop and closed-loop analysis, and enable the prediction of time-domain performance for reset systems. The accuracy and effectiveness of the proposed methods are successfully validated through simulations and experiments conducted on a reset Proportional-Integral-Derivative (PID) controlled precision motion system.

Figures (15)

• Figure 1: The block diagram of the reset control system, where $r(t)$, $e(t)$, $v(t)$, $u(t)$, and $y(t)$ denote the reference input signal, the error signal, the reset output signal, the control input signal, and the output signal, respectively. The blue lines indicate the reset-triggered actions, with $e(t)$ serving as the reset-triggered signal in this system.
• Figure 2: $u_{ci}(t)$ (solid line), $u_i(t)$ (dotted line), and $q_i(t)$ (dashed line) of open-loop CI.
• Figure 3: The new block diagram of an open-loop reset controller $\mathcal{C}$.
• Figure 4: The comparison between the simulated and the Theorem \ref{['thm: Open loop model for RC']}-predicted output signals in a reset control system.
• Figure 5: Block diagrams for the closed-loop RCS, wherein (a) the resetting actions are indicated by the blue lines. In (b), the reset controller is decomposed into two components: the linear part $\mathcal{C}_{bl}$ within the grey box and the nonlinear part $\mathcal{C}_{nl}$ contained within the blue box.

Theorems & Definitions (22)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Remark 1
• Theorem 3