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On System Operators with Variation Bounding Properties

Christian Grussler, Chaim Roth, Kang Tong

TL;DR

This work develops tractable algebraic certificates for variation bounding (VB_k) and variation diminishing (VD_k) properties of discrete-time LTI observability operators by leveraging total-positivity concepts and SSC/SC criteria. By showing that VB_k/VD_k can be certified through SSC_k of the t-step observability operators or via strictly externally positive subsystems, the authors relax traditional rank/dimension restrictions on A and connect these properties to the impulse response and Hankel operator structure. The approach yields practical, computable certificates even in infinite-dimensional settings, and it demonstrates that OVD/VB properties can be inferred without requiring k-positivity of A, including a case where no k-positive realization exists. The results provide bounds on impulse-response sign changes and Bangs in bounded control, with implications for model-order reduction and robust design of systems preserving monotonicity-like variation properties.

Abstract

The property of linear discrete-time time-invariant system operators mapping inputs with at most $k-1$ sign changes to outputs with at most$k-1$ sign changes is investigated. We show that this property is tractable via the notion of $k$-sign consistency in case of the observability/controllability operator, which as such can also be used as a sufficient condition for the Hankel operator. Our results complement the mathematical literature by providing an algebraic characterization, independent of rank and dimension for variation bounding and diminishing matrices as well as by discussing their computational tractability. Based on these, we conduct our studies of variation bounding system operators beyond existing studies on order-preserving $k$-variation diminishment. Our findings are applied to the open problem of bounding the number of sign changes in a system's impulse response, which appears, e.g., when bounding the number of over- and undershoots in a step response or the number of bangs in bounded optimal control problems.

On System Operators with Variation Bounding Properties

TL;DR

This work develops tractable algebraic certificates for variation bounding (VB_k) and variation diminishing (VD_k) properties of discrete-time LTI observability operators by leveraging total-positivity concepts and SSC/SC criteria. By showing that VB_k/VD_k can be certified through SSC_k of the t-step observability operators or via strictly externally positive subsystems, the authors relax traditional rank/dimension restrictions on A and connect these properties to the impulse response and Hankel operator structure. The approach yields practical, computable certificates even in infinite-dimensional settings, and it demonstrates that OVD/VB properties can be inferred without requiring k-positivity of A, including a case where no k-positive realization exists. The results provide bounds on impulse-response sign changes and Bangs in bounded control, with implications for model-order reduction and robust design of systems preserving monotonicity-like variation properties.

Abstract

The property of linear discrete-time time-invariant system operators mapping inputs with at most sign changes to outputs with at most sign changes is investigated. We show that this property is tractable via the notion of -sign consistency in case of the observability/controllability operator, which as such can also be used as a sufficient condition for the Hankel operator. Our results complement the mathematical literature by providing an algebraic characterization, independent of rank and dimension for variation bounding and diminishing matrices as well as by discussing their computational tractability. Based on these, we conduct our studies of variation bounding system operators beyond existing studies on order-preserving -variation diminishment. Our findings are applied to the open problem of bounding the number of sign changes in a system's impulse response, which appears, e.g., when bounding the number of over- and undershoots in a step response or the number of bangs in bounded optimal control problems.
Paper Structure (29 sections, 26 theorems, 63 equations, 4 figures)

This paper contains 29 sections, 26 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Variation diminishing maps
  5. Total Positivity Theory
  6. Variation diminishing observability operators
  7. Main Results
  8. The $\text{SVB}_{k-1}$ case
  9. The $\text{VB}_{k-1}$ case
  10. The $\text{OVD}_{k}\slash \text{VD}_{k}$ case
  11. Discussion
  12. Illustrative Examples
  13. Example 1: $\text{OVD}_{1}$ ${\mathcal{O}^{}}$
  14. Example 2: $\text{SVB}_{1}$ ${\mathcal{O}^{}}$
  15. Example 3: $\text{SVB}_{1}$ ${\mathcal{O}^{}}$ without $k$-positive $A$ representation
  16. ...and 14 more sections

Key Result

Lemma 1

Let $X \in \mathds{R}^{n \times p}$ and $Y \in \mathds{R}^{p \times m}$.

Figures (4)

  • Figure 1: Impulse responses $\tilde{g}_{1,1,\beta}(t)$ of $(\tilde{A}_1,b_{1,1,\beta},\tilde{c}_1)$ in Theorem \ref{['thm:main2']}: \ref{['symbol:r1beta12_g3']}$\tilde{g}_{1,1,\{1\}}(t)$, \ref{['symbol:r1beta13_g3']}$\tilde{g}_{1,1,\{2\}}(t)$ and \ref{['symbol:r1beta23_g3']}$\tilde{g}_{1,1,\{3\}}(t)$ are strictly positive.
  • Figure 2: Impulse responses $\tilde{g}_{2,2,\beta}(t)$ of $(\tilde{A}_2,b_{2,2,\beta},\tilde{c}_2)$ in Theorem \ref{['thm:main2']}: \ref{['symbol:r2beta12_g3']}$\tilde{g}_{2,2,\{1,2\}}(t)$, \ref{['symbol:r2beta13_g3']}$\tilde{g}_{2,2,\{1,3\}}(t)$ and \ref{['symbol:r2beta23_g3']}$\tilde{g}_{2,2,\{2,3\}}(t)$ are strictly positive.
  • Figure 3: Impulse responses $\tilde{g}_{2,1,\beta}(t)$ of $(\tilde{A}_1,b_{2,1,\beta},\tilde{c}_1)$ in Theorem \ref{['thm:main2']}: \ref{['symbol:r1beta12_g2']}$\tilde{g}_{2,1,\{1,2\}}(t)$, \ref{['symbol:r1beta13_g2']}$\tilde{g}_{2,1,\{1,3\}}(t)$ and \ref{['symbol:r1beta23_g2']}$\tilde{g}_{2,1,\{2,3\}}(t)$ are strictly positive.
  • Figure 4: Impulse responses $\tilde{g}_{2,2,\beta}(t)$ of $(\tilde{A}_2,b_{2,2,\beta},\tilde{c}_2)$ in Theorem \ref{['thm:main2']}: \ref{['symbol:r2beta12_g2']}$\tilde{g}_{2,2,\{1,2\}}(t)$, \ref{['symbol:r2beta13_g2']}$\tilde{g}_{2,2,\{1,3\}}(t)$ and \ref{['symbol:r2beta23_g2']}$\tilde{g}_{2,2,\{2,3\}}(t)$ are strictly positive.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 2
  • Proposition 4
  • Proposition 5
  • ...and 19 more