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

1. Problem Statement

The paper investigates the existence, characterization, and properties of unimodal self-oscillations in discrete-time linear time-invariant (LTI) relay feedback systems that include a nonnegative dead zone.

• System Configuration: The system consists of a discrete-time LTI plant $G(z)$ in a negative feedback loop with a relay function $rel_{\chi_0}(u)$ that has a symmetric dead zone of width $2\chi_0$.
• Target Phenomenon: The authors focus on unimodal self-oscillations, defined as periodic solutions where the output signal exhibits a single peak within one period (i.e., the signal increases to a maximum and then decreases).
• Gap in Literature: While extensive tools exist for continuous-time relay systems (e.g., describing functions, Tsypkin loci, Poincaré maps), discrete-time counterparts remain largely unexplored. Existing discrete-time methods often rely on differentiability assumptions or yield conservative bounds, failing to address unimodality and specific period constraints rigorously.

2. Methodology

The authors propose a novel analytical framework based on the theory of total positivity and variation theory, avoiding the need for differentiability assumptions or approximation techniques common in frequency-domain methods.

• Variation and Sign Changes: The core tool is the counting of sign changes in sequences.
  • $S_-(v)$: The number of sign changes in a vector $v$ (ignoring zeros).
  • $S_+(v)$: The maximum possible sign changes if zeros are replaced by real numbers to maximize variation.
  • Cyclic Variation ($S_c^-$): The variation of a sequence when viewed cyclically.
• Unimodality Characterization: A periodic sequence is unimodal if and only if its cyclic forward difference has exactly two sign changes ($S_c^-(\Delta_c u_P) = 2$).
• Total Positivity: The authors utilize the property that if the impulse response $g(t)$ of the LTI system is strictly monotonically decreasing on its infinite support, the convolution operator maps unimodal inputs to unimodal outputs. This creates an invariant set of signals amenable to fixed-point analysis.
• Sign-Symmetry: A critical concept introduced is sign-symmetry, defined as a periodic signal having an equal number of positive and negative elements within one period ($P_p = P_n$).

3. Key Contributions and Results

A. Necessary Conditions for Existence (Sign-Symmetry)

The paper establishes that for a system with a strictly monotonically decreasing impulse response, a unimodal self-oscillation can only exist if the relay output is sign-symmetric.

• Theorem 2: If a unimodal self-oscillation exists, the number of positive and negative elements in the relay output over one period must be equal ($P_p = P_n$).
• This generalizes the concept of "half-wave symmetry" (where $u(t + P/2) = -u(t)$), showing that sign-symmetry is a fundamental prerequisite for unimodality in this class of systems.

B. Absence of Oscillations for Zero Relative Degree

• Theorem 3: If the relative degree of the system is zero (i.e., $g(0) > 0$, meaning the system has no inherent time delay), no unimodal self-oscillations exist.
• This result mirrors findings in continuous-time passive systems, proving that a delay is necessary to sustain these specific oscillatory patterns.

C. Period Bounds and Existence with Time Delays

When the system has a time delay (relative degree $d > 0$, so $g(0)=0$ and $g(t)=0$ for $t < d$), the authors derive tight bounds on the oscillation period $P$:

• Theorem 4: For a system with delay $P_d$ and a strictly decreasing impulse response, any unimodal self-oscillation with period $P \ge P_d$ must satisfy:
  $$2P_d \le P \le 2(P_d + P_s)$$
  where $P_s$ is a specific index determined by the impulse response $g_0$ (the part of the system after the delay).
• Corollary 2 (Convexity): If the impulse response is also convex, the upper bound tightens significantly to:
  $$P \le 4P_d + 2$$
• Uniqueness and Multiplicity: The paper provides conditions under which multiple self-oscillations with different periods can coexist. Specifically, if a period $2P_d$ oscillation exists, oscillations with periods $P = \frac{2P_d}{2n+1}$ may also exist under specific conditions on the dead zone size.

D. Numerical Validation

The theoretical findings are validated through numerical examples (Example 1 and 2), demonstrating:

• The existence of oscillations with periods $2, 6, 18$ for a specific delay $P_d=9$.
• The effect of the dead zone $\chi_0$: A non-zero dead zone can preserve oscillations from the ideal relay case or introduce new ones, but the period bounds derived in Theorem 4 remain valid.

4. Significance

• Rigorous Discrete-Time Analysis: This work fills a critical gap by providing the first rigorous algebraic framework for analyzing unimodal self-oscillations in discrete-time relay systems, moving beyond approximate frequency-domain methods.
• Fundamental Insight: The discovery that sign-symmetry is a necessary condition for unimodality provides a powerful filter for predicting system behavior without simulation.
• Design Guidelines: The derived bounds on the oscillation period ($2P_d \le P \le 4P_d + 2$) offer concrete design constraints for engineers tuning relay-based controllers (e.g., PID autotuning) or analyzing biological oscillators modeled in discrete time.
• Extension of Total Positivity: The paper successfully adapts total positivity theory, traditionally used in continuous-time monotone systems, to discrete-time non-smooth systems, opening new avenues for analyzing hybrid and switched systems.

In summary, the paper proves that for discrete-time relay systems with dead zones and strictly decreasing impulse responses, unimodal self-oscillations are strictly constrained by the system's time delay and must exhibit sign-symmetry, with their periods bounded by simple linear functions of the delay.