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Predictor-Feedback Stabilization of Globally Lipschitz Nonlinear Systems with State and Input Quantization

Florent Koudohode, Nikolaos Bekiaris-Liberis

TL;DR

This work tackles global stabilization of nonlinear systems with arbitrarily long input delay in the presence of state and input quantization. It introduces a switched nonlinear predictor-feedback controller that uses dynamic, zooming quantizers and a backstepping-based predictor to compensate for delays and quantization, under global Lipschitz and ISS-type conditions. The main contributions are a constructive two-phase zooming strategy (open-loop zooming out followed by closed-loop zooming in), explicit stability bounds via small-gain arguments, and an extension to input quantization with corresponding bounds. The results address practical digital-control constraints for predictor-based stabilization and lay groundwork for relaxing stringent assumptions in future work.

Abstract

We develop a switched nonlinear predictor-feedback control law to achieve global asymptotic stabilization for nonlinear systems with arbitrarily long input delay, under state quantization. The proposed design generalizes the nonlinear predictor-feedback framework by incorporating quantized measurements of both the plant and actuator states into the predictor state formulation. Due to the mismatch between the (inapplicable) exact predictor state and the predictor state constructed in the presence of state quantization, a global stabilization result is possible under a global Lipschitzness assumption on the vector field, as well as under the assumption of existence of a globally Lipschitz, nominal feedback law that achieves global exponential stability of the delay and quantization-free system. To address the constraints imposed by quantization, a dynamic switching strategy is constructed, adjusting the quantizer's tunable parameter in a piecewise constant manner-initially increasing the quantization range, to capture potentially large system states and subsequently refining the precision to reduce quantization error. The global asymptotic stability of the closed-loop system is established through solutions estimates derived using backstepping transformations, combined with small-gain and input-to-state stability arguments. We also extend our approach to the case of input quantization.

Predictor-Feedback Stabilization of Globally Lipschitz Nonlinear Systems with State and Input Quantization

TL;DR

This work tackles global stabilization of nonlinear systems with arbitrarily long input delay in the presence of state and input quantization. It introduces a switched nonlinear predictor-feedback controller that uses dynamic, zooming quantizers and a backstepping-based predictor to compensate for delays and quantization, under global Lipschitz and ISS-type conditions. The main contributions are a constructive two-phase zooming strategy (open-loop zooming out followed by closed-loop zooming in), explicit stability bounds via small-gain arguments, and an extension to input quantization with corresponding bounds. The results address practical digital-control constraints for predictor-based stabilization and lay groundwork for relaxing stringent assumptions in future work.

Abstract

We develop a switched nonlinear predictor-feedback control law to achieve global asymptotic stabilization for nonlinear systems with arbitrarily long input delay, under state quantization. The proposed design generalizes the nonlinear predictor-feedback framework by incorporating quantized measurements of both the plant and actuator states into the predictor state formulation. Due to the mismatch between the (inapplicable) exact predictor state and the predictor state constructed in the presence of state quantization, a global stabilization result is possible under a global Lipschitzness assumption on the vector field, as well as under the assumption of existence of a globally Lipschitz, nominal feedback law that achieves global exponential stability of the delay and quantization-free system. To address the constraints imposed by quantization, a dynamic switching strategy is constructed, adjusting the quantizer's tunable parameter in a piecewise constant manner-initially increasing the quantization range, to capture potentially large system states and subsequently refining the precision to reduce quantization error. The global asymptotic stability of the closed-loop system is established through solutions estimates derived using backstepping transformations, combined with small-gain and input-to-state stability arguments. We also extend our approach to the case of input quantization.
Paper Structure (8 sections, 6 theorems, 37 equations)

This paper contains 8 sections, 6 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation and Control Design
  3. Nonlinear Systems With State Quantization
  4. Properties of the Quantizer
  5. Quantized Predictor-Feedback Law
  6. Stability Under State Quantization
  7. Extension to Input Quantization
  8. Conclusions and current work

Key Result

Theorem 1

Consider the closed-loop system consisting of the plant pde_representation--pde_representation1 and the switched predictor-feedback law control_quantizer--switching_parameter. Under Assumptions globalLipschitz and issODE, if $\Delta$ and $M$ satisfy then for all $X_{0} \in \mathbb{R}^{n}$, $u_{0} \in \mathcal{C}_{\rm rpw}([0, D], \mathbb{R})$, there exists a unique solution such that $X(t) \in AC

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Lemma 3
  • Lemma 4