Passivity-based control for Hopfield neural networks using convex representation
Introduction
Artificial neural network has attracted much attention because it can be applied for many practical research areas, e.g., signal processing, pattern recognition, parallel computation, fixed-point computation, optimization, associative memories, and so on. The Hopfield neural network is one of the most popular artificial neural networks and is extensively studied during the last decade. Since neural networks need a global equilibrium point in order to be applied to practical and theoretical applications, there have been lots of analysis and control schemes to deal with global stability in the literature. For example, the researchers in [1], [2], [3], [4], [5] proposed global stability analysis for neural networks. Also, in [6], [7], [8], [9], [10], [11], [12], it has been considered the effect of time delays which are an inevitable factor in real world. By considering uncertainties in the model dynamics, robust stability criteria have been derived in [13] and references therein in recent years. The stability of stochastic Hopfield neural networks with time delays was also studied in [14] and the global asymptotic stability of delayed cellular neural networks of neutral-type was investigated in [15].
On the other hand, the concept of passivity for nonlinear systems is attracting new interest recently. The passive theory considers power flow into the system so that passive properties can guarantee stability of the system. Since the passivity theory plays an important role in designing stabilizing controllers for asymptotic stability of nonlinear systems, the passivity scheme has been used in various applications, e.g., the control design for chaotic Lü systems [16], a controller for a unified chaotic system [17], and so on. The passivity theory was also devoted to analyze the stability of neural networks [18], [19], [20], [21]. However, there does not exist passive-based control schemes for neural networks, especially, Hopfield neural networks.
In this paper, we address a design problem for passivity-based controllers of Hopfield neural networks. The nonlinearities in the Hopfield neural networks is represented in convex combination. Then, a sufficient condition for asymptotic stability is derived using Lyapunov method and passivity theory. The derived condition is a criterion for existence of such controller, which is represented in terms of LMIs so that it can be easily solved by a convex optimization problem. Finally, a numerical simulation is given to show the effectiveness of the proposed method.
The paper is organized as follows: a problem statement and a convex representation are described in Section 2. In Section 3, a stabilizing controller based on passivity theory is designed. Section 4 contains a numerical example and Conclusions are given in Section 5. Notation In this presentation, the following notation will be used. denotes the n-dimensional Euclidean space and is the set of all m × n real matrices. ★ denotes the symmetric part. X > 0 (X ⩾ 0) means that X is a real, symmetric and positive-definite matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. ∥·∥ refers to the induced matrix 2-norm and diag(⋯) denotes the block diagonal matrix.
Section snippets
Problem statement
Consider the following Hopfield neural networkwhere is the state vector, is the output vector, is the control input, , and are constant matrices, and f(x(t)) is a nonlinear function satisfying following sector bounded condition:where σi(t) is the ith element of σ(·), αi and βi are lower and upper bound of the sector, respectively.
The nonlinear function f(·) can be represented by a
Main results
In this section, we consider the control problem for passivity of Hopfield neural networks given in Eq. (1). Based on the system description and definition of passivity in Section 2, we have the following main results. Theorem 1 If there exist matrices , with appropriate dimensions and a positive scalar γ satisfying following LMI:then the Hopfield neural network system (1) is output strictly passive and the origin is the unique equilibrium point which is
Numerical example
In this section, an example is given to verify the effectiveness of the proposed method in Theorem 1. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.01.
Consider the Hopfield neural networks as shown in [22]:and the nonlinear function iswhich belongs to sector [0, 1].
The initial condition
Conclusions
This paper has been proposed a passivity-based controller design method for Hopfield neural networks. Nonlinearities in the system are represented in convex combination so that a sufficient condition for asymptotic stability can be formulated in terns of LMI. The designed state feedback controller renders the Hopfield neural network passive from the external input and the output. A numerical example is given to show the effectiveness of the proposed method.
Acknowledgements
The authors thank the editor and reviewers for their valuable comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009373).
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