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Adaptive observer and control of spatiotemporal delayed neural fields

Lucas Brivadis, Antoine Chaillet, Jean Auriol

Abstract

An adaptive observer is proposed to estimate the synaptic distribution between neurons asymptotically from the measurement of a part of the neuronal activity and a delayed neural field evolution model. The convergence of the observer is proved under a persistency of excitation condition. Then, the observer is used to derive a feedback law ensuring asymptotic stabilization of the neural fields. Finally, the feedback law is modified to ensure simultaneously practical stabilization of the neural fields and asymptotic convergence of the observer under additional restrictions on the system. Numerical simulations confirm the relevance of the approach.

Adaptive observer and control of spatiotemporal delayed neural fields

Abstract

An adaptive observer is proposed to estimate the synaptic distribution between neurons asymptotically from the measurement of a part of the neuronal activity and a delayed neural field evolution model. The convergence of the observer is proved under a persistency of excitation condition. Then, the observer is used to derive a feedback law ensuring asymptotic stabilization of the neural fields. Finally, the feedback law is modified to ensure simultaneously practical stabilization of the neural fields and asymptotic convergence of the observer under additional restrictions on the system. Numerical simulations confirm the relevance of the approach.
Paper Structure (19 sections, 9 theorems, 63 equations, 6 figures, 1 table)

This paper contains 19 sections, 9 theorems, 63 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement and mathematical preliminaries
  3. Delayed neural fields
  4. Problem statement
  5. Definitions and notations
  6. Preliminaries on Hilbert-Schmidt operators
  7. Properties of the system
  8. Adaptive observer
  9. Observer design
  10. Observer convergence
  11. Proof of Theorem \ref{['th:obs']} (observer convergence)
  12. Step 1: Proof that $\lim_{t\to+\infty}\|\tilde{z}_1(t)\|_{\mathcal{X}_{z_1}}= \lim_{t\to+\infty} \|\tilde{z}_2(t)\|_{\mathcal{X}_{z_2}}=0$
  13. Step 2: Proof that $\lim_{t\to+\infty} \|\tilde{w}_{11}(t)P_X\|_{\mathcal{X}_{w_{11}}}= \lim_{t\to+\infty} \|\tilde{w}_{12}(t)P_Y\|_{\mathcal{X}_{w_{12}}}=0$
  14. Adaptive control
  15. Exact stabilization
  16. ...and 4 more sections

Key Result

Proposition 2.1

Suppose that Assumption ass:wp is satisfied. Then, for any initial condition $(z_{1,0}, z_{2,0})\in C^0([-\mathrm{\textnormal{d}}, 0], \mathcal{X}_{z_1})\times C^0([-\mathrm{\textnormal{d}}, 0], \mathcal{X}_{z_2})$, the open-loop system eq:wcij admits a unique corresponding solution $(z_1, z_2)\in C

Figures (6)

  • Figure 1: Evolution of the kernel estimation $\hat{w}_{11}(t, r, r')$ when running the observer \ref{['eq:obs']}.
  • Figure 2: Evolution of the estimation errors $\|\tilde{w}_{1i}\|_{\mathcal{X}_{w_{1i}}}$ and $\|\tilde{z}_i\|_{\mathcal{X}_{z_i}}$ for $i\in\{1,2\}$ of the observer \ref{['eq:obs']}.
  • Figure 3: Evolution of the norm of the state $z_i$, of the estimation errors $\tilde{w}_{1i}$ and $\tilde{z}_i$ for $i\in\{1,2\}$, and of input $u_1$, for the control law \ref{['eq:cont']}.
  • Figure 4: Evolution of the norm of the state $\|z\|_{\mathcal{X}_{z}}$ and of the estimation errors $\|\tilde{w}\|_{\mathcal{X}_{w}}$ and $\|\tilde{z}\|_{\mathcal{X}_{z}}$ for the control law \ref{['eq:cont_sim']} with $v(t, r)=100\sin(\lambda_1 tr)$.
  • Figure 5: Steady-state behavior of $\|z_1(t)\|_{L^2}$, in terms of its maximal (blue dots) and average (red dots) values for the control law \ref{['eq:cont']}, as a function of the value of the applied additive disturbance.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1: Open-loop well-posedness and BIBS
  • proof
  • Proposition 3.1: Observer well-posedness
  • proof
  • Definition 3.2: Persistence of excitation
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 22 more