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Discrete Homogeneity and Quantizer Design for Nonlinear Homogeneous Control Systems

Yu Zhou, Andrey Polyakov, Gang Zheng, Masaaki Nagahara

TL;DR

The paper addresses stabilization of nonlinear homogeneous control systems under state quantization by introducing a discrete-dilation (𝔡) notion of homogeneity and a corresponding discrete Lyapunov framework that yields finite-/fixed-time convergence rates determined by the homogeneity degree $\mu$. It then develops homogeneous sector-boundedness in a homogeneous vector space to translate quantization errors into stability guarantees, and designs a geometry-aware homogeneous polar-spherical quantizer that preserves discrete homogeneity and bounds the quantization error via $\|\frak{q}(x)-x\|_d$. A constructive quantizer based on homogeneous coordinates $\Phi_{\boldsymbol{d}}(x)$ and a logarithmic radial quantizer, together with a spherical quantizer on the unit sphere, yields $\frak{q}_h$ that satisfies $\frak{q}_h(\frak{d}(s)x)=\frak{d}(s)\frak{q}_h(x)$ and a provable error bound $\|\frak{q}_h(x) \tilde{-} x\|_d \le \tilde{\epsilon}\|x\|_d$. Numerical validation demonstrates preserved stability under quantization, with insights into convergence rates and quantization efficiency. Overall, the framework offers a principled route to robust nonlinear control with discrete measurements and quantization by leveraging discrete homogeneity and geometry-aware quantizers.

Abstract

This paper proposes a framework for analysis of generalized homogeneous control systems under state quantization. In particular, it addresses the challenge of maintaining finite/fixed-time stability of nonlinear systems in the presence of quantized measurements. To analyze the behavior of quantized control system, we introduce a new type of discrete homogeneity, where the dilation is defined by a discrete group. The converse Lyapunov function theorem is established for homogeneous systems with respect to discrete dilations. By extending the notion of sector-boundedness to a homogeneous vector space, we derive a generalized homogeneous sector-boundedness condition that guarantees finite/fixed-time stability of nonlinear control system under quantized measurements. A geometry-aware homogeneous static vector quantizer is then designed using generalized homogeneous coordinates, enabling an efficient quantization scheme. The resulting homogeneous control system with the proposed quantizer is proven to be homogeneous with respect to discrete dilation and globally finite-time, nearly fixed-time, or exponentially stable, depending on the homogeneity degree. Numerical examples validate the effectiveness of the proposed approach.

Discrete Homogeneity and Quantizer Design for Nonlinear Homogeneous Control Systems

TL;DR

The paper addresses stabilization of nonlinear homogeneous control systems under state quantization by introducing a discrete-dilation (𝔡) notion of homogeneity and a corresponding discrete Lyapunov framework that yields finite-/fixed-time convergence rates determined by the homogeneity degree . It then develops homogeneous sector-boundedness in a homogeneous vector space to translate quantization errors into stability guarantees, and designs a geometry-aware homogeneous polar-spherical quantizer that preserves discrete homogeneity and bounds the quantization error via . A constructive quantizer based on homogeneous coordinates and a logarithmic radial quantizer, together with a spherical quantizer on the unit sphere, yields that satisfies and a provable error bound . Numerical validation demonstrates preserved stability under quantization, with insights into convergence rates and quantization efficiency. Overall, the framework offers a principled route to robust nonlinear control with discrete measurements and quantization by leveraging discrete homogeneity and geometry-aware quantizers.

Abstract

This paper proposes a framework for analysis of generalized homogeneous control systems under state quantization. In particular, it addresses the challenge of maintaining finite/fixed-time stability of nonlinear systems in the presence of quantized measurements. To analyze the behavior of quantized control system, we introduce a new type of discrete homogeneity, where the dilation is defined by a discrete group. The converse Lyapunov function theorem is established for homogeneous systems with respect to discrete dilations. By extending the notion of sector-boundedness to a homogeneous vector space, we derive a generalized homogeneous sector-boundedness condition that guarantees finite/fixed-time stability of nonlinear control system under quantized measurements. A geometry-aware homogeneous static vector quantizer is then designed using generalized homogeneous coordinates, enabling an efficient quantization scheme. The resulting homogeneous control system with the proposed quantizer is proven to be homogeneous with respect to discrete dilation and globally finite-time, nearly fixed-time, or exponentially stable, depending on the homogeneity degree. Numerical examples validate the effectiveness of the proposed approach.
Paper Structure (15 sections, 19 theorems, 133 equations, 5 figures)

This paper contains 15 sections, 19 theorems, 133 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on homogeneity
  3. Problem Formulation
  4. Homogeneity with discrete dilation
  5. Homogeneous sector-boundedness
  6. Homogeneous stabilization with quantization
  7. generalized homogeneous quantizer design
  8. Homogeneous vector quantizer
  9. Quantization error estimate
  10. Numerical example
  11. Conclusion
  12. Proof of Lemma \ref{['lem:discrate_partial_d']}
  13. Proof of Theorem \ref{['thm:exist_LF_discrete']}
  14. Proof of Lemma \ref{['lem:distance']}
  15. Proof of Theorem \ref{['thm:main']}

Key Result

Proposition 1

polyakov2019IJRNC A linear continuous dilation $\mathbf{d}$ is strictly monotone with respect to the weighted Euclidean norm $\|z\|=\sqrt{z^\top P z}$ if and only if the following linear matrix inequality holds $P\succ 0$, $P G_\mathbf{d}+G_\mathbf{d}^{\top} P \succ 0$, where $G_\mathbf{d} \in \math

Figures (5)

  • Figure 1: Comparison between element-wise logarithmic (left) and polar-spherical (right) quantizers, where the black points represent quantization seeds.
  • Figure 2: Homogeneous curves for linear continuous and discrete dilations generated by $e^{G_{\mathbf{d}_i} s}$, $i\!=\!1,2,3$, $G_{\mathbf{d}_1}\!=\!I_2$, $G_{\mathbf{d}_2}\!=\!\left[2001\right]$, $G_{\mathbf{d}_3}\!=\!\left[2-1.511\right],$$x_0\!=\!\left[11\right],$ where $s\!\in\! \mathbb{R}$ (left) or $s \!\in\!\mathcal{S}$ (right).
  • Figure 3: Quantization seeds under different generators: $G_{\mathbf{d}_1}$ (top) and $G_{\mathbf{d}_2}$ (bottom) with $P = I_2$. Colored points indicate quantization seeds, and lines outline the boundaries of quantization cells. The color bar represents the value of the homogeneous norm $\|\mathfrak{q}_h(x)\|_{\mathbf{d}_i}$, $i=1,2$.
  • Figure 4: States of the system under control with and without quantized data.
  • Figure 5: Comparison of state norms. The upper subfigure shows the norm of the system states under control with and without quantization. The lower subfigure illustrates the logarithmic norm of the states and the quantized states for the system with quantized state feedback.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Proposition 2
  • Definition 4
  • Proposition 3
  • Theorem 1
  • Proposition 4
  • Definition 5
  • ...and 29 more