Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hopfield Neural Networks for Online Constrained Parameter Estimation with Time-Varying Dynamics and Disturbances

Miguel Pedro Silva

TL;DR

The paper tackles online parameter estimation under linear constraints, disturbances, and slow parameter drift by mapping constrained LS-like problems onto projector-based Hopfield networks. It introduces two estimators, CA-HNN and CA$^2$-HNN, that enforce equalities/inequalities via slack variables and absorb disturbances with compensation channels, all within a unified energy framework. Global uniform ultimate boundedness with explicit rates and radii is established, and practical tuning rules connect gains to bandwidth and steady-state error, complemented by an online identifiability monitor. Through 2-DOF MSD simulations, the proposed methods demonstrate competitive accuracy, zero constraint violations, and favorable disturbance handling with lower online computational burden than MHE, highlighting their potential for real-time, parallelizable implementations.

Abstract

This paper proposes two projector-based Hopfield neural network (HNN) estimators for online, constrained parameter estimation under time-varying data, additive disturbances, and slowly drifting physical parameters. The first is a constraint-aware HNN that enforces linear equalities and inequalities (via slack neurons) and continuously tracks the constrained least-squares target. The second augments the state with compensation neurons and a concatenated regressor to absorb bias-like disturbance components within the same energy function. For both estimators we establish global uniform ultimate boundedness with explicit convergence rate and ultimate bound, and we derive practical tuning rules that link the three design gains to closed-loop bandwidth and steady-state accuracy. We also introduce an online identifiability monitor that adapts the constraint weight and time step, and, when needed, projects updates onto identifiable subspaces to prevent drift in poorly excited directions...

Hopfield Neural Networks for Online Constrained Parameter Estimation with Time-Varying Dynamics and Disturbances

TL;DR

The paper tackles online parameter estimation under linear constraints, disturbances, and slow parameter drift by mapping constrained LS-like problems onto projector-based Hopfield networks. It introduces two estimators, CA-HNN and CA-HNN, that enforce equalities/inequalities via slack variables and absorb disturbances with compensation channels, all within a unified energy framework. Global uniform ultimate boundedness with explicit rates and radii is established, and practical tuning rules connect gains to bandwidth and steady-state error, complemented by an online identifiability monitor. Through 2-DOF MSD simulations, the proposed methods demonstrate competitive accuracy, zero constraint violations, and favorable disturbance handling with lower online computational burden than MHE, highlighting their potential for real-time, parallelizable implementations.

Abstract

This paper proposes two projector-based Hopfield neural network (HNN) estimators for online, constrained parameter estimation under time-varying data, additive disturbances, and slowly drifting physical parameters. The first is a constraint-aware HNN that enforces linear equalities and inequalities (via slack neurons) and continuously tracks the constrained least-squares target. The second augments the state with compensation neurons and a concatenated regressor to absorb bias-like disturbance components within the same energy function. For both estimators we establish global uniform ultimate boundedness with explicit convergence rate and ultimate bound, and we derive practical tuning rules that link the three design gains to closed-loop bandwidth and steady-state accuracy. We also introduce an online identifiability monitor that adapts the constraint weight and time step, and, when needed, projects updates onto identifiable subspaces to prevent drift in poorly excited directions...

Paper Structure

This paper contains 34 sections, 4 theorems, 88 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction and Background
  2. Problem setting and motivation
  3. Hopfield neural networks
  4. Contributions
  5. Paper organization
  6. Mapping Online Constrained Estimation with Disturbances onto a Hopfield Neural Network
  7. Mapping the unconstrained parameter estimation problem
  8. Mapping the constrained (equalities and inequalities) parameter estimation problem
  9. Mapping the constrained parameter estimation problem subject to additive disturbances
  10. Stability and Tracking Analysis of the HNN Estimator
  11. Global bounds
  12. Tuning rules
  13. Choose the saturation level $\alpha$ (avoid saturation, keep sensitivity)
  14. Set the bandwidth via $\beta$ (rate is independent of $P$)
  15. Tune the constraint weight $\eta$ (secure curvature without over-stiffening)
  16. ...and 19 more sections

Key Result

Lemma 1

Let $U(t)=\mathcal{R}(W^\top(t))\subset\mathbb{R}^p$ and $V=\mathcal{R}(A_\theta^\top)\subset\mathbb{R}^p$. Assume $\mathrm{rank}\![]=p$ for all $t$ and $\eta>0$. Let $P_W$ and $P_{A,\theta}$ be the orthogonal projectors onto $U(t)$ and $V$, respectively. Then $\Pi_\eta=P_W+\eta P_{A,\theta}\succ 0$ In particular, with $c(t)=\lambda_{\min}(\Pi_\eta(t))$ we have $c(t)>0$.

Figures (9)

  • Figure 1: Model of the 2-DOF Mass–Spring–Damper (MSD) system.
  • Figure 2: Standard LS–HNN (no constraints). Parameter trajectories: $v_1{=}\hat{k}_1$ (red), $v_2{=}\hat{b}_1$ (magenta), $v_3{=}\hat{k}_2$ (blue), $v_4{=}\hat{b}_2$ (cyan); Asterisks mark true values. LS-HNN settings: $h=10^{-4}$, $\beta=1$, $\alpha=6$.
  • Figure 3: Standard LS–HNN: energy function $E(t)$ for $\beta=1$. Non-monotonic due to time-varying $T,b$. HNN settings: $h=10^{-4}$, $\beta=1$, $\alpha=6$
  • Figure 4: Comparison between proposed CA-HNN estimator and PB-RLS algorithm. MSD parameter estimates with constraints (constant parameters, no disturbance). CA-HNN settings: $h=10^{-5}$, $\alpha=10$, $\eta=50$ and $\beta=250$. PB–RLS settings: forgetting factor $\lambda=0.995$, initial covariance $P_0=10^6I$ and sampling period $h=10^{-5}$. CA-HNN converges smoothly and remains within bounds; PB--RLS reaches the same steady state but shows a transient overshoot for $b_2$ that briefly hits the upper bound.
  • Figure 5: Simulation of CA-HNN estimator. Estimated parameters $v_i(t)$ under a disturbance $d(t) \sim \mathcal{N}(1,\,1)$ on $m_2$ (without compensation neurons). Asterisks mark true values. $\hat{k}_1$ and $\hat{b}_1$ converge near their true values; $\hat{k}_2$ and $\hat{b}_2$ show a steady bias and a small ripple at the $d\!\to\!x_2$ resonance (period $\approx 13.7$ s). CA-HNN settings: $h=10^{-5}$, $\alpha=10$, $\eta=50$ and $\beta=200$. All constraints are satisfied at all times.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Remark 1: Scalar low–pass tracking and bandwidth
  • Lemma 1: Coercivity of $P_W+\eta P_{A,\theta}$
  • Theorem 1: GUUB for constraint-aware HNN (CA-HNN) estimator under disturbances and parameter drift (no compensation)
  • Lemma 2: Coercivity of the augmented projector
  • Theorem 2: GUUB for constraint-aware compensation-augmented HNN (CA$^2$-HNN) estimator under disturbances and parameter drift
  • Remark 2: Effects of compensation neurons