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Discrete-Time Periodic Monotonicity Preserving Systems

Christian Grussler

TL;DR

The paper tackles predicting self-sustained oscillations in Lur'e-type feedback by identifying discrete-time LTI kernels that preserve invariants of $T$-periodic signals. It develops tractable certificates for two nested signal-sets—periodic monotonicity PM$(T)$ and cyclic 2-variation bounds CVD$_2$(T)—via total positivity theory and a geometric interpretation of sequential convex contours. Key contributions include a complete characterization of PMP$(T)$ and a practical certificate for CVD$_2$(T) using cyclic minors and compound LTI representations, with extensions to Lur'e loop gains and relay feedback systems. The results provide a rigorous, checkable framework for predicting self-sustained oscillations and lay the groundwork for signal-based fixed-point theorems in relay-feedback contexts, as demonstrated by example systems of orders three and four.

Abstract

Two nested classes of discrete-time linear time-invariant systems, which differ by the set of periodic signals that they leave invariant, are studied. The first class preserves the property of periodic monotonicity (period-wise unimodality). The second class is invariant to signals with at most two sign changes per period, and requires that periodic signals with zero sign changes are mapped to the same kind. Tractable characterizations for each system class are derived by the use and extension of total positivity theory via geometric interpretations. Central to our results is the characterization of sequentially convex contours via consecutive minors. Our characterizations also extend to the loop gain of Lur'e feedback systems as the considered signals sets are invariant under common static non-linearities, e.g., ideal relay, saturation, sigmoid function, quantizer, etc. The presented developments aim to form a base for future signal-based fixed-point theorems towards the prediction of self-sustained oscillations. Our examples on relay feedback systems indicate how periodic monotonicity preservation gives rise to useful insights towards this goal.

Discrete-Time Periodic Monotonicity Preserving Systems

TL;DR

The paper tackles predicting self-sustained oscillations in Lur'e-type feedback by identifying discrete-time LTI kernels that preserve invariants of -periodic signals. It develops tractable certificates for two nested signal-sets—periodic monotonicity PM and cyclic 2-variation bounds CVD(T)—via total positivity theory and a geometric interpretation of sequential convex contours. Key contributions include a complete characterization of PMP and a practical certificate for CVD(T) using cyclic minors and compound LTI representations, with extensions to Lur'e loop gains and relay feedback systems. The results provide a rigorous, checkable framework for predicting self-sustained oscillations and lay the groundwork for signal-based fixed-point theorems in relay-feedback contexts, as demonstrated by example systems of orders three and four.

Abstract

Two nested classes of discrete-time linear time-invariant systems, which differ by the set of periodic signals that they leave invariant, are studied. The first class preserves the property of periodic monotonicity (period-wise unimodality). The second class is invariant to signals with at most two sign changes per period, and requires that periodic signals with zero sign changes are mapped to the same kind. Tractable characterizations for each system class are derived by the use and extension of total positivity theory via geometric interpretations. Central to our results is the characterization of sequentially convex contours via consecutive minors. Our characterizations also extend to the loop gain of Lur'e feedback systems as the considered signals sets are invariant under common static non-linearities, e.g., ideal relay, saturation, sigmoid function, quantizer, etc. The presented developments aim to form a base for future signal-based fixed-point theorems towards the prediction of self-sustained oscillations. Our examples on relay feedback systems indicate how periodic monotonicity preservation gives rise to useful insights towards this goal.

Paper Structure

This paper contains 31 sections, 19 theorems, 70 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Sets
  5. Sequences
  6. Matrices
  7. Functions
  8. Linear Dynamic Systems
  9. Cyclic Variation Bounding Properties
  10. Variation Bounding
  11. Variation Diminishing & Bounding
  12. Periodic Monotonicity Preservation
  13. Total Positivity
  14. Convex Contours
  15. Monotonicity & Convexity
  16. ...and 16 more sections

Key Result

Lemma 1

Let $X \in \mathds{R}^{n \times p}$, $Y \in \mathds{R}^{p \times m}$ and $1 \leq r \leq \min \{n,m,p \}$.

Figures (7)

  • Figure 1: Lur'e feedback systems with linear time-invariant system $G$ and static non-linearity $\psi$.
  • Figure 2: Example of a periodically monotone (single-peaked) $u \in \ell_\infty(T)$: $\textnormal{S}^{-}_c[u - \gamma \mathbf{1}] \leq 2$ for all $\gamma \in \mathds{R}$, or equivalently, there exist $t_1 \le t_2 \le t_1+T-1$ such that \ref{['eq:unimod_perido']} is fulfilled. In other words, over the period $(t_1:t_1+T-1)$, every local maximum $u(t)$ is a global maximum (single peak), or simply $\textnormal{S}^{-}_c[\Delta u] \leq 2$. In particular, $u$ is also strictly periodically monotone, because $\textnormal{S}^{+}_c[\Delta u] \leq 2$.
  • Figure 3: Visualization of a strictly convex contours $\gamma$ with period $T=10$: $\gamma \in \textnormal{CC}(T)$, since the polygon resulting from connecting $\gamma(0),\dots,\gamma(T_1),\gamma(0)$ sequentially is non-intersecting and the boundary of a convex set. Moreover, $\gamma \in \textnormal{SCC}(T)$, since no three points $\gamma(t-1),\gamma(t),\gamma(t+1)$ are co-linear.
  • Figure 4: Illustration for the need of $\tilde{\gamma}$ in \ref{['prop:convc_ct']}: for the contour $\gamma$ with period $T=7$ all consecutive $3$-minors of $M^\gamma$ in \ref{['eq:M_mat']} are zero and $\gamma_1, \gamma \in \textnormal{PM}(T)$. However, the curve resulting from connecting $\gamma(0),\dots,\gamma(T_1),\gamma(0)$ sequentially is intersecting and not the boundary of a convex set, which is why $\gamma \not \in \textnormal{CC}(T)$.
  • Figure 5: Visualization of the curve $(\Delta g_T,g_T)$ belonging to $G_2(s)$ in \ref{['eq:second_order']} (left) and $G_3(s)$ in \ref{['eq:third_order']} (right) for \ref{['line:T_5_1']}$T=5$, \ref{['line:T_10_1']}$T=10$ and \ref{['line:T_100_1']}$T=100$. All curves are strictly convex contours, which by \ref{['thm:pmp_main']} implies that $\mathcal{S}_{g} \in \textnormal{S}\textnormal{PMP}(T)$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Lemma 1
  • Lemma 2
  • Definition 1: Variation
  • Lemma 3
  • Definition 2: Cyclic Variation
  • Definition 3: $k$-variation bounding/diminishing
  • Lemma 4
  • Definition 4: Periodic Monotonicity
  • Lemma 5
  • Lemma 6
  • ...and 21 more