Hopfield Neural Networks for Online Constrained Parameter Estimation With Time‐Varying Dynamics and Disturbances

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. A two‐degree‐of‐freedom mass‐spring‐damper study with Monte Carlo trials compares the proposed HNN estimators against projector‐based recursive least squares, disturbance‐aware projector‐based Kalman filtering, and disturbance‐aware projector‐based moving‐horizon estimation. The HNN estimators achieve competitive or superior accuracy with zero constraint violations, reduced disturbance‐induced bias (especially with compensation), and low per‐step computational cost suitable for high‐rate deployment.