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Adaptive tracking control for non-periodic reference signals under quantized observations

Chuiliu Kong, Ying Wang

TL;DR

This work addresses adaptive tracking for a second-order stochastic regression system with multi-threshold quantized observations and non-periodic reference signals. It introduces two backward-shifted polynomials with time-varying parameters and a projection structure to ensure uniform boundedness and persistent excitation, enabling reliable online parameter estimation and tracking. The main contributions include almost sure and mean-square convergence of the parameter estimates with a $O\left(\frac{1}{k}\right)$ rate, and an asymptotically optimal tracking law where $\lim_{k\to\infty}\mathbb{E}[ (y(k)-y^{*}(k))^{2} ] = \mathbb{E}[ w^{2}(1) ]$. The approach extends adaptive tracking with quantized observations to non-periodic references and provides a rigorous rate analysis, supported by simulations.

Abstract

This paper considers an adaptive tracking control problem for stochastic regression systems with multi-threshold quantized observations. Different from the existing studies for periodic reference signals, the reference signal in this paper is non-periodic. Its main difficulty is how to ensure that the designed controller satisfies the uniformly bounded and excitation conditions that guarantee the convergence of the estimation in the controller under non-periodic signal conditions. This paper designs two backward-shifted polynomials with time-varying parameters and a special projection structure, which break through periodic limitations and establish the convergence and tracking properties. To be specific, the adaptive tracking control law can achieve asymptotically optimal tracking for the non-periodic reference signal; Besides, the proposed estimation algorithm is proved to converge to the true values in almost sure and mean square sense, and the convergence speed can reach $O\left(\frac{1}{k}\right)$ under suitable conditions. Finally, the effectiveness of the proposed adaptive tracking control scheme is verified through a simulation.

Adaptive tracking control for non-periodic reference signals under quantized observations

TL;DR

This work addresses adaptive tracking for a second-order stochastic regression system with multi-threshold quantized observations and non-periodic reference signals. It introduces two backward-shifted polynomials with time-varying parameters and a projection structure to ensure uniform boundedness and persistent excitation, enabling reliable online parameter estimation and tracking. The main contributions include almost sure and mean-square convergence of the parameter estimates with a rate, and an asymptotically optimal tracking law where . The approach extends adaptive tracking with quantized observations to non-periodic references and provides a rigorous rate analysis, supported by simulations.

Abstract

This paper considers an adaptive tracking control problem for stochastic regression systems with multi-threshold quantized observations. Different from the existing studies for periodic reference signals, the reference signal in this paper is non-periodic. Its main difficulty is how to ensure that the designed controller satisfies the uniformly bounded and excitation conditions that guarantee the convergence of the estimation in the controller under non-periodic signal conditions. This paper designs two backward-shifted polynomials with time-varying parameters and a special projection structure, which break through periodic limitations and establish the convergence and tracking properties. To be specific, the adaptive tracking control law can achieve asymptotically optimal tracking for the non-periodic reference signal; Besides, the proposed estimation algorithm is proved to converge to the true values in almost sure and mean square sense, and the convergence speed can reach under suitable conditions. Finally, the effectiveness of the proposed adaptive tracking control scheme is verified through a simulation.
Paper Structure (6 sections, 6 theorems, 58 equations, 2 figures)

This paper contains 6 sections, 6 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Adaptive control algorithm
  4. Main results
  5. Simulations
  6. Conclusion

Key Result

proposition 1

The projection operator $\Pi_{\Omega}(\cdot)$ satisfies

Figures (2)

  • Figure 1: The performance of estimations
  • Figure 2: Tracking effects

Theorems & Definitions (13)

  • remark 1
  • remark 2
  • remark 3
  • proposition 1: Lemma 2.1 in Calamai-More
  • proposition 2
  • proof
  • lemma 1: Zhao-Wang-Bi-1
  • theorem 1
  • proof
  • theorem 2
  • ...and 3 more