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Robust Finite-time Stabilization of Linear Systems with Limited State Quantization

Yu Zhou, Andrey Polyakov, Gang Zheng

Abstract

This paper investigates the robust asymptotic stabilization of a linear time-invariant (LTI) system by a static feedback with a static state quantization. It is shown that the controllable LTI system can be stabilized to zero in a finite time by means of a nonlinear feedback with a quantizer having a limited (finite) number of values (quantization seeds) even when all parameters of the controller and the quantizer are time-invariant. The control design is based on generalized homogeneity. A homogeneous spherical quantizer is introduced. The static homogeneous feedback is shown to be local (or global) finite-time stabilizer for the linear system (dependently of the system matrix). The tuning rules for both the quantizer and the feedback law are obtained in the form of Linear Matrix Inequalities (LMIs). The closed-loop system is proven to be robust with respect to some bounded matched and vanishing mismatched perturbations. Theoretical results are supported by numerical simulations. \

Robust Finite-time Stabilization of Linear Systems with Limited State Quantization

Abstract

This paper investigates the robust asymptotic stabilization of a linear time-invariant (LTI) system by a static feedback with a static state quantization. It is shown that the controllable LTI system can be stabilized to zero in a finite time by means of a nonlinear feedback with a quantizer having a limited (finite) number of values (quantization seeds) even when all parameters of the controller and the quantizer are time-invariant. The control design is based on generalized homogeneity. A homogeneous spherical quantizer is introduced. The static homogeneous feedback is shown to be local (or global) finite-time stabilizer for the linear system (dependently of the system matrix). The tuning rules for both the quantizer and the feedback law are obtained in the form of Linear Matrix Inequalities (LMIs). The closed-loop system is proven to be robust with respect to some bounded matched and vanishing mismatched perturbations. Theoretical results are supported by numerical simulations. \
Paper Structure (10 sections, 9 theorems, 44 equations, 4 figures)

This paper contains 10 sections, 9 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Preliminary
  4. Homogeneity and homogeneous function, system
  5. Main results
  6. Generalized homogeneous spherical quantizer
  7. Finite-time stabilization with homogeneous spherical quantization
  8. A possible spherical quantizer design
  9. Numerical example
  10. Conclusion

Key Result

Proposition 1

A linear continuous dilation $\mathbf{d}$ in $\mathbb{R}^n$ is strictly monotone with respect to the norm $\|z\|=\sqrt{z^\top P z}$ if and only if the following linear matrix inequality holds where $G_\mathbf{d} \in \mathbb{R}^{n\times n}$ is the generator of $\mathbf{d}$.

Figures (4)

  • Figure 1: Illustration of quantization in $\mathbb{R}^2$. (a) Uniform quantizer per axis. (b) Logarithmic quantizer = per axis. (c) Homogeneous spherical quantizer. ($G_\mathbf{d}=I_2$). (d) Homogeneous spherical quantizer ($G_\mathbf{d}=\left[2001\right]$). The dots in (c) and (d) represent quantization seeds on the sphere.
  • Figure 2: Quantization seeds (red markers) with $N = 512$ are placed on the unit sphere $x^\top P x=1$. The lines represent the intersections of quantization cells with the weighted sphere.
  • Figure 3: The system state under finite static quantization feedback.
  • Figure 4: The quantization value $\mathfrak{q}_{\pi_\mathbf{d}}(x)$.

Theorems & Definitions (15)

  • Definition 1: kawski1991
  • Definition 2: polyakov2019IJRNC
  • Proposition 1
  • Definition 3: kawski1991
  • Definition 4: polyakov2020book
  • Definition 5: polyakov2019IJRNC
  • Lemma 1
  • Theorem 1
  • Theorem 2: filippov2013differential, page 152
  • Definition 6
  • ...and 5 more