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Robust Hybrid Finite-Time Parameter Estimation Without Persistence of Excitation

Adnane Saoud, Ryan S. Johnson, Ricardo G. Sanfelice

TL;DR

A hybrid algorithm is proposed allowing the estimate to converge to the exact value of the unknown parameters in predetermined finite time, and it is shown that for the case of constant parameters, the convergence property of the hybrid algorithm holds while only requiring the regressor to be exciting on a given interval.

Abstract

In this paper, we consider the problem of estimating parameters of a linear regression model. Using a hybrid systems framework, a hybrid algorithm is proposed allowing the estimate to converge to the exact value of the unknown parameters in predetermined finite time. Interestingly, we show that for the case of constant parameters, the convergence property of the hybrid algorithm holds while only requiring the regressor to be exciting on a given interval. For the case of piecewise constant parameters, the classical persistency of excitation condition is required to guarantee the convergence. Robustness of the proposed algorithm with respect to measurements noise is analysed. Finally, illustrative examples are provided showing the merits of the proposed approach in terms of scalability and the applicability for the general class of time-varying unknown parameters

Robust Hybrid Finite-Time Parameter Estimation Without Persistence of Excitation

TL;DR

A hybrid algorithm is proposed allowing the estimate to converge to the exact value of the unknown parameters in predetermined finite time, and it is shown that for the case of constant parameters, the convergence property of the hybrid algorithm holds while only requiring the regressor to be exciting on a given interval.

Abstract

In this paper, we consider the problem of estimating parameters of a linear regression model. Using a hybrid systems framework, a hybrid algorithm is proposed allowing the estimate to converge to the exact value of the unknown parameters in predetermined finite time. Interestingly, we show that for the case of constant parameters, the convergence property of the hybrid algorithm holds while only requiring the regressor to be exciting on a given interval. For the case of piecewise constant parameters, the classical persistency of excitation condition is required to guarantee the convergence. Robustness of the proposed algorithm with respect to measurements noise is analysed. Finally, illustrative examples are provided showing the merits of the proposed approach in terms of scalability and the applicability for the general class of time-varying unknown parameters
Paper Structure (15 sections, 9 theorems, 52 equations, 2 figures, 1 table)

This paper contains 15 sections, 9 theorems, 52 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Linear Regression
  4. Preliminaries on Hybrid Systems
  5. Hybrid finite time convergent parameter estimation algorithm
  6. Excitation Conditions
  7. Constant Unknown Parameter
  8. Piecewise Constant Unknown Parameter
  9. Robustness to measurement noise
  10. Examples
  11. Constant Unknown Parameters
  12. Time-varying Unknown Parameter
  13. Scalability of the Hybrid Parameter Estimator
  14. Conclusion
  15. Appendix: Auxiliary results

Key Result

Proposition 1

Suppose that $\gamma_1\neq \gamma_2$, the regressor $t \mapsto \phi(t)$ is exciting over $[0,\delta]$ (i.e, $\int_{0}^{\delta}\phi(s)\phi^{\top}(s)ds\geq \eta I$ for $\eta>0$), there exists $\phi_M \geq 0$ such that $|\phi(s)| \leq \phi_M$ for all $s\in [0,\delta]$, and the constants $\delta,\gamma_ Then, the gains $K_1$ and $K_2$ in (eqn:functionals) are well defined at hybrid time $(\delta,0)$,

Figures (2)

  • Figure 1: Results of simulation for unknown constant parameters
  • Figure 2: Results of simulation for unknown time-varying parameter

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 15 more