Unimodal self-oscillations and their sign-symmetry for discrete-time relay feedback systems with dead zone

Abstract

This paper characterizes self-oscillations in discrete-time linear time-invariant (LTI) relay feedback systems with nonnegative dead zone. Specifically, we aim to establish existence criteria for unimodal self-oscillations, defined as periodic solutions where the output exhibits a single-peaked period. Assuming that the linear part of system is stable, with a strictly monotonically decreasing impulse response on its infinite support, we propose a novel analytical framework based on the theory of total positivity to address this problem. We demonstrate that unimodal self-oscillations subject to mild variation-based constraints exist only if the number of positive and negative values of the system's loop gain coincides within a given strictly positive period, i.e., the self-oscillation is sign-symmetric. Building upon these findings, we derive conditions for the existence of such self-oscillations, establish tight bounds on their periods, and address the question of their uniqueness.

Figures (7)

• Figure 1: Feedback system in discrete-time consisting of a linear time-invariant system with transfer function $G(z)$, $z\in\mathds{C}$ and a relay with a symmetric deadzone of width $2 \chi_0$. The system here exhibits a unimodal (i.e, single-peaked) self-oscillation $u(t) = -y(t)$.
• Figure 2: A single period of periodically unimodal sequence $x \in \ell_\infty(13)$, where $k_1 = 0$ and $k_2 = 4$.
• Figure 3: The relay function with symmetric dead zone of width $2 \chi_0 \geq 0$.
• Figure 4: Illustration of \ref{['prop:PMP_three_props']}: a single period of a periodically unimodal sequence $u \in \ell_\infty(13)$ and its cyclic forward difference \ref{['DeltauP']}$\Delta_c u^P$. Periodic unimodality is characterized by $S_c^-(\Delta u^P)=2$, or equivalently, by $S_c^-(u^P - \gamma \boldsymbol{1}_P) \leq 2$ for all $\gamma \in \mathds{R}$.
• Figure 5: A DT Relay Feedback System with dead zone and delay module.

Theorems & Definitions (15)

• Definition 1: Variation
• Definition 2: Self-oscillation
• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Corollary 1