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Self-triggered Stabilization of Contracting Systems under Quantization

Masashi Wakaiki

TL;DR

This work tackles stabilizing contracting nonlinear systems when measurements are quantized and control and triggering are driven by self-triggered sampling. It develops two self-triggered schemes: (i) a logarithmic-quantization approach with a threshold-based STM, and (ii) a joint zooming-quantization design where the quantization range adapts with inter-sampling times. Using contraction theory and a trajectory-based analysis, the authors establish conditions under which inter-sampling times are bounded away from zero and the closed-loop state decays exponentially, thereby avoiding Zeno behavior. The results are illustrated on a two-tank system, showcasing how quantization and sampling parameters influence convergence and transient behavior, and they extend to Lur’e systems via a structured norm-based contraction verification. The work advances quantized networked control by providing rigorous, executable guidelines for self-triggered stabilization under nonlinear dynamics.

Abstract

We propose self-triggered control schemes for nonlinear systems with quantized state measurements. Our focus lies on scenarios where both the controller and the self-triggering mechanism receive only the quantized state at each sampling time. We assume that the ideal closed-loop system without quantization or self-triggered sampling is contracting. Moreover, an upper bound on the growth rate of the open-loop system is assumed to be known. We present two control schemes that achieve closed-loop stability without Zeno behavior. The first scheme is implemented under logarithmic quantization and uses the quantized state for the threshold in the triggering condition. The second one is a joint design of zooming quantization and self-triggered sampling, where the adjustable zoom parameter for quantization changes based on inter-sampling times and is also used for the threshold of self-triggered sampling. In both schemes, the self-triggering mechanism predicts the future state from the quantized data for the computation of the next sampling time. We employ a trajectory-based approach for stability analysis, where contraction theory plays a key role.

Self-triggered Stabilization of Contracting Systems under Quantization

TL;DR

This work tackles stabilizing contracting nonlinear systems when measurements are quantized and control and triggering are driven by self-triggered sampling. It develops two self-triggered schemes: (i) a logarithmic-quantization approach with a threshold-based STM, and (ii) a joint zooming-quantization design where the quantization range adapts with inter-sampling times. Using contraction theory and a trajectory-based analysis, the authors establish conditions under which inter-sampling times are bounded away from zero and the closed-loop state decays exponentially, thereby avoiding Zeno behavior. The results are illustrated on a two-tank system, showcasing how quantization and sampling parameters influence convergence and transient behavior, and they extend to Lur’e systems via a structured norm-based contraction verification. The work advances quantized networked control by providing rigorous, executable guidelines for self-triggered stabilization under nonlinear dynamics.

Abstract

We propose self-triggered control schemes for nonlinear systems with quantized state measurements. Our focus lies on scenarios where both the controller and the self-triggering mechanism receive only the quantized state at each sampling time. We assume that the ideal closed-loop system without quantization or self-triggered sampling is contracting. Moreover, an upper bound on the growth rate of the open-loop system is assumed to be known. We present two control schemes that achieve closed-loop stability without Zeno behavior. The first scheme is implemented under logarithmic quantization and uses the quantized state for the threshold in the triggering condition. The second one is a joint design of zooming quantization and self-triggered sampling, where the adjustable zoom parameter for quantization changes based on inter-sampling times and is also used for the threshold of self-triggered sampling. In both schemes, the self-triggering mechanism predicts the future state from the quantized data for the computation of the next sampling time. We employ a trajectory-based approach for stability analysis, where contraction theory plays a key role.
Paper Structure (26 sections, 14 theorems, 151 equations, 11 figures, 3 tables)

This paper contains 26 sections, 14 theorems, 151 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic assumptions and system properties
  3. Assumptions on nonlinear systems
  4. Growth and decay properties
  5. Preliminary results on open-loop dynamics
  6. Self-triggered stabilization under logarithmic quantization
  7. System model
  8. Logarithmic quantizer
  9. Self-triggered control system
  10. Stability analysis
  11. Main result in logarithmic quantization case
  12. Proof of main result
  13. Self-triggered stabilization under zooming quantization
  14. System model
  15. Zooming quantization
  16. ...and 11 more sections

Key Result

Theorem 2.4

Suppose that Assumptions assump:closed and assump:control_part hold. Let $\tau \in \mathbb{R}_{>0}$, and let $e\colon [0,\tau) \to \mathop{\mathbf B}_{\mathop{\mathrm{op}}\nolimits}\nolimits(\sigma_0 R)$ be continuous. If the ODE $\dot x = F(x,e)$ with $x(0) = x_0\in \mathop{\mathbf B}_{\mathop{\ma for all $t \in [0,\tau]$.

Figures (11)

  • Figure 1: Closed-loop system.
  • Figure 2: Closed-loop system in networked control framework. Encoding and decoding of $\ell_k$ are omitted for brevity.
  • Figure 3: Two-tank system.
  • Figure 4: Stabilizable region of $(\rho, \sigma)$.
  • Figure 5: State under logarithmic quantization.
  • ...and 6 more figures

Theorems & Definitions (28)

  • Remark 2.1: Selection of norms
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.2
  • ...and 18 more