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Non-overshooting output shaping for switched linear systems under arbitrary switching using eigenstructure assignment

Kai Wulff, Maria Christine Honecker, Robert Schmid, Johann Reger

TL;DR

The paper tackles robust set-point tracking for switched MIMO systems under arbitrary switching by designing a switched state-feedback law via eigenstructure assignment that yields global stability and controlled transient behavior. It extends LTI eigenstructure methods to the switched setting, enforcing simultaneous eigenvector rectification across subsystems and distributing closed-loop modes to outputs through a feasible partitioning $(d_0,\dots,d_p)$. A structural feasibility condition and constructive algorithms are provided to compute $F_q$ and $G_q$ that guarantee $GUAS$ while achieving either non-overshooting or monotonic step-tracking, with the latter requiring stronger partitioning requirements. Two numerical examples illustrate both design paradigms, highlighting the method's ability to shape output dynamics under arbitrary switching and illustrating practical limits. Overall, the framework offers a rigorous, constructive route to stable, predictable tracking in switched MIMO systems with stringent transient specifications.

Abstract

We consider the analytical control design for a pair of switched linear multiple-input multiple-output (MIMO) systems that are subject to arbitrary switching signals. A state feedback controller design method is proposed to obtain an eigenstructure assignment that ensures that the closed-loop switched system is globally asymptotically stable, and the outputs achieve the non-overshooting tracking of a step reference. Our analysis indicates whether non-overshooting or even monotonic tracking is achievable for the given system and considered outputs and provides a choice of possible eigenstructures to be assigned to the constituent subsystems. We derive a structural condition that verifies the feasibility of the chosen assignment. A constructive algorithm to obtain suitable feedback matrices is provided, and the method is illustrated with numerical examples.

Non-overshooting output shaping for switched linear systems under arbitrary switching using eigenstructure assignment

TL;DR

The paper tackles robust set-point tracking for switched MIMO systems under arbitrary switching by designing a switched state-feedback law via eigenstructure assignment that yields global stability and controlled transient behavior. It extends LTI eigenstructure methods to the switched setting, enforcing simultaneous eigenvector rectification across subsystems and distributing closed-loop modes to outputs through a feasible partitioning . A structural feasibility condition and constructive algorithms are provided to compute and that guarantee while achieving either non-overshooting or monotonic step-tracking, with the latter requiring stronger partitioning requirements. Two numerical examples illustrate both design paradigms, highlighting the method's ability to shape output dynamics under arbitrary switching and illustrating practical limits. Overall, the framework offers a rigorous, constructive route to stable, predictable tracking in switched MIMO systems with stringent transient specifications.

Abstract

We consider the analytical control design for a pair of switched linear multiple-input multiple-output (MIMO) systems that are subject to arbitrary switching signals. A state feedback controller design method is proposed to obtain an eigenstructure assignment that ensures that the closed-loop switched system is globally asymptotically stable, and the outputs achieve the non-overshooting tracking of a step reference. Our analysis indicates whether non-overshooting or even monotonic tracking is achievable for the given system and considered outputs and provides a choice of possible eigenstructures to be assigned to the constituent subsystems. We derive a structural condition that verifies the feasibility of the chosen assignment. A constructive algorithm to obtain suitable feedback matrices is provided, and the method is illustrated with numerical examples.

Paper Structure

This paper contains 14 sections, 13 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Switched linear system
  4. Stabilising state-feedback for switched linear systems
  5. Set-point tracking for switched linear systems
  6. Common steady state and design of the feedforward control
  7. Eigenstructure assignment for LTI MIMO systems
  8. Eigenstructure assignment for switched linear systems
  9. Main Results: Shaping of the tracking response
  10. Examples
  11. Non-overshooting design
  12. Monotonic design
  13. Conclusion
  14. Appendix

Key Result

Theorem 13

Given the LTI system eq:sysLTI, let the sets $\mathcal{L}_k = \{\lambda_{k,1}, \ldots, \lambda_{k,d_k} \} \subset \mathbb{R}$, and $\mathcal{V}_k = \{v_{k,1}, \ldots, v_{k,d_k} \} \subset \mathbb{R}^n$, $k \in\{0,1,\ldots,p\}$, both be compatible with the $(p,n)$-partitioning $(d_0, d_1,\dots, d_p) for each $\lambda_{k,i} \in \mathcal{L}_k$ and $v_{k,i} \in \mathcal{V}_k$. Let where $V \in \math

Figures (2)

  • Figure 1: Step reference tracking for the rectified eigenstructure design with non-overshooting outputs.
  • Figure 2: Step reference tracking for the rectified eigenstructure design with monotonic outputs.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 7
  • Definition 8
  • Remark 9
  • Remark 10
  • Definition 11
  • Remark 12
  • Theorem 13
  • ...and 34 more