A contribution to the use of Hopfield neural networks for parameter estimation Hugo Alonso, Teresa Mendonça, Paula Rocha Abstract— This paper presents a contribution to the use of Hopfield neural networks (HNNs) for parameter estimation. Our focus is on time-invariant systems that are linear in the parameters. We introduce a suitable HNN and present a weaker condition than the currently existing ones that guarantees the convergence of the parameterization estimated by the network to the actual parameterization. The application of our results is illustrated in a parameter estimation problem for a two carts system. Keywords Parameter estimation, Hopfield neural net- works, Lyapunov stability theory. I. I NTRODUCTION System identification plays undoubtedly an important role in different areas such as biomedicine, robotics and fluid dynamics applied to aircraft and motor vehicle industries. In fact, obtaining an accurate model for a process is important not only for the study of the process itself, but very often for the design of a suitable control strategy. In this con- text, consider the example of a patient undergoing general anesthesia, where a control action is used for drug delivery optimization [7]: the safety of the patient obviously depends on the reliability of the control strategy, but the latter is based on the identification of the patient dynamics. Many approaches to system identification have been pro- posed, in particular the use of neural networks as black-box models (see, for instance, [8]). However, there are problems where gray-box models are preferred, since they can be ap- plied not only for prediction but also for description purposes, offering in this case some insight into the underlying system dynamics. This motivated us to propose instead the use of neural networks as a tool designed to estimate the parameters of a gray-box model intended to fit the system data. In this paper, we consider Abe’s formulation of a Hopfield neural network (HNN) [1]. Compared to the original Hopfield’s formulation [4], this formulation is simpler, computationally less demanding, and hence more suitable for parameter estimation. These advantages have been pointed out in [2], where the use of such formulation in that context has been proposed. The authors proved that the parameterization estimated by their network asymptotically converges to the Hugo Alonso (corresponding author) and Teresa Mendonça are with the Departamento de Matemática Aplicada, Faculdade de Ciências da Universi- dade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (e-mail: hugo.alonso@fc.up.pt,tmendo@fc.up.pt). Paula Rocha is with the Departamento de Matemática, Universidade de Aveiro, Campo de Santiago, 3810-193 Aveiro, Portugal (e-mail: procha@mat.ua.pt). actual parameterization for a system assumed to be time- invariant and linear in the parameters. However, this result was obtained under very restrictive assumptions on the matrix (W ij ). In fact, these assumptions are so restrictive that they do not hold for the case study presented in that work, in spite of the good performance of the corresponding estimation process. This suggested us that the conditions of [2] could be relaxed, motivating this contribution. The paper is organized as follows. In Section II, the prob- lem of applying a HNN to parameter estimation is formulated and our contributions clarified. Section III presents both the way how we define the network estimator and its stability analysis. In Section IV, the application of our results is illustrated in a parameter estimation problem for a two carts system. Finally, we present the conclusions and future work. II. PROBLEM FORMULATION Our focus is on time-invariant systems that are linear in the parameters, i.e., systems that can be represented in the form y(t)= A(t)θ, (1) for some y :[t 0 , +∞[→ R m×1 , A :[t 0 , +∞[→ R m×n , being θ ∈ R n×1 the vector of the unknown parameters to es- timate. It is assumed that y, A are continuously differentiable and bounded functions, and that y(t), A(t) are available at each time t, although y, A are possibly explicitly unknown. Furthermore, it is reasonable to assume the knowledge about some c> 0 for which θ ∈] − c, c[ n . Our problem is that of defining a Hopfield neural network (HNN) that is able to generate a trajectory ˆ θ(·) such that ∀t ≥ t 0 ˆ θ(t) ∈] − c, c[ n and lim t→+∞ ˆ θ(t) = θ under mild assumptions. In [2], the proposed network generates a trajectory confined to ˆ θ(t 0 )+] − 1, 1[ n and asymptotically convergent to θ under the assumption that ∀t ≥ t 0 ker(A(t)) = {0}. But if the latter condition holds, then the solution to the estimation problem given by θ = ( A T (t)A(t) ) -1 A T (t)y(t) can be obtained at any time t ≥ t 0 , even if A T (t)A(t) is ill- conditioned, in which case simpler standard methods can be applied to compute θ. In addition, a necessary condition for ∀t ≥ t 0 ker(A(t)) = {0} to hold is that m ≥ n, i.e., the system should not be overparameterized. However, a successful application of a HNN to an estimation problem where m<n is given in [2], which motivated us to find more general conditions. Here, we start by introducing a different HNN that has the advantage of requiring no prior knowledge on the choice of ˆ θ(t 0 ), the initial estimate of θ, and which Proceedings of the European Control Conference 2007 Kos, Greece, July 2-5, 2007 WeD04.2 ISBN: 978-960-89028-5-5 3844