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Analysis and Design of Reset Control Systems via Base Linear Scaled Graphs

T. de Groot, W. P. M. H. Heemels, S. J. A. M. van den Eijnden

Abstract

In this letter, we prove that under mild conditions, the scaled graph of a reset control system is bounded by the scaled graph of its underlying base linear system, i.e., the system without resets. Building on this new insight, we establish that the negative feedback interconnection of a linear time-invariant plant and a reset controller is stable, if the scaled graphs of the underlying base linear components are strictly separated. This result simplifies reset system analysis, as stability conditions reduce to verifying properties of linear time-invariant systems. We exploit this result to develop a systematic approach for reset control system design. Our framework also accommodates reset systems with time-regularization, which were not addressed in the context of scaled graphs before.

Analysis and Design of Reset Control Systems via Base Linear Scaled Graphs

Abstract

In this letter, we prove that under mild conditions, the scaled graph of a reset control system is bounded by the scaled graph of its underlying base linear system, i.e., the system without resets. Building on this new insight, we establish that the negative feedback interconnection of a linear time-invariant plant and a reset controller is stable, if the scaled graphs of the underlying base linear components are strictly separated. This result simplifies reset system analysis, as stability conditions reduce to verifying properties of linear time-invariant systems. We exploit this result to develop a systematic approach for reset control system design. Our framework also accommodates reset systems with time-regularization, which were not addressed in the context of scaled graphs before.

Paper Structure

This paper contains 13 sections, 5 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Signal spaces, systems, and stability
  4. Scaled graphs and feedback analysis
  5. SGs and reset systems
  6. Reset systems
  7. SGs for LTI systems
  8. Properties inherited from base linear SGs
  9. Stability analysis and design
  10. Conditions for stability
  11. Design for stability
  12. Example
  13. Conclusions

Key Result

Theorem 1

Let $H_1, H_2 : \mathcal{L}_{2e}^n \rightrightarrows \mathcal{L}_{2e}^n$ be causal, stable systems in negative feedback interconnection. Suppose the interconnection of $H_1$ and $\mu H_2$ is well-posed for all $\mu \in (0,1]$. If there exists $r > 0$ such that for all $\mu \in (0,1]$, then $\Sigma$ is stable. If, in addition, one or both of $\textup{SG}(H_1)$ and $\textup{SG}(H_2)$ satisfy the ch

Figures (3)

  • Figure 1: Feedback interconnection $\Sigma$.
  • Figure 2: Left: $\textup{SG}$ (blue) and $\textup{patch}(\textup{SG})$ (hatched region). Right: $(\textup{SG}^c)_{\infty}$ (yellow) and $\left(\textup{SG}^c \setminus (\textup{SG}^c)_{\infty}\right)$ (red).
  • Figure 3: Left: $\textup{SG}_e^\dagger(-H_1)$ (grey) and ${\textup{SG}}(\mathcal{R}_{\textup{BLS}})$ (blue). Right: step-response $y_1$ of BLS (black) and reset system (red).

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Theorem 1: Chen25
  • Lemma 1: Groot25
  • Definition 3
  • Theorem 2
  • Corollary 1
  • proof
  • Theorem 3
  • proof