Design of state estimator for BAM fuzzy cellular neural networks with leakage and unbounded distributed delays

doi:10.1016/j.ins.2017.02.056

Information Sciences

Volumes 397–398, August 2017, Pages 91-109

https://doi.org/10.1016/j.ins.2017.02.056 Get rights and content

Abstract

In this paper, the state estimation problem is studied for the bidirectional associative memory (BAM) fuzzy cellular neural networks (FCNNs) with delays, where the delays include leakage and unbounded distributed delays. The problem addressed is to estimate the neuron states, through available output measurements, such that for all admissible leakage and unbounded distributed delays, the dynamics of the estimation error is globally asymptotically stable. A delay-dependent linear matrix inequality (LMI) criterion for the existence of the estimator is proposed by using the Lyapunov–Krasovskii functional method. In addition, the unknown gain matrix is determined by solving a delay-dependent LMI. Finally, numerical examples and simulations are provided to illustrate the effectiveness of the theoretical results.

Introduction

In recent years, neural networks (NNs) have been successfully applied to various fields such as signal processing, associative memories, pattern recognition among others [5], [22]. Various NN models, such as cellular neural networks (CNNs), Hopfield-type NNs and bidirectional associative memory (BAM) NNs, have been extensively investigated [9], [11], [15], [36]. Concurrently, the state estimation problem for NNs with delays has also received much attention [23], [29], [34], [48], [50]. Recently, the state estimation problem for NNs with multiple time delays has been investigated by using the Lyapunov–Krasovskii functional, a convex polyhedron method and with a linear matrix inequality (LMI) technique [41]. Li et al. [33] have also analyzed the state estimation for uncertain Markovian jump NNs with mixed delays. In a relatively large-scale NNs, only partial information about the neuron states are available in the network outputs. However, in order to achieve certain practical performances, such as system modeling, signal processing and control engineering, one often has to estimate the relevant neuron state through available measurements.

In many practical problems, a typical time delay called as Leakage delay, may exist in the negative feedback terms of the system and it has a great impact on the dynamic behaviors of the NNs, such as on the instability or poor system performance of NNs (see [20], [26], [32]). Li et al. [31] have shown that time delays in the negative feedback terms have a tendency to destabilize the performance of NNs. Hence, understanding leakage delay in dynamical NNs are vital in the field of stability analysis of NNs. Li and Cao [28] have analyzed the delay-dependent stability of NNs of neutral-type with time delays in the leakage terms. In BAM NNs, Gopalsamy [21] had studied the stability with constant delays in the leakage terms. Peng [38] had also considered the BAM NNs with continuously distributed delays in the leakage terms via the continuation theorem with Lyapunov functional. Further, Li and Fu [30] also investigated the leakage time-varying delay effects on the stability of nonlinear differential systems. The result of their study implied that the effects of leakage delay on NNs cannot be ignored.

There are two types of cellular neural networks (CNNs) structures currently studied. The first is the traditional CNNs [6], [7], [8], [17], while the other is the fuzzy cellular neural networks (FCNNs) [46], [49] which integrates fuzzy logic into the structure of a CNN. Unlike the traditional CNNs structure, FCNNs implements fuzzy logic between its template input and/or output besides the sum of product operations. Initially, Gupta and Knopf [19] analyzed the uncertainty in image processing by using fuzzy logic due to its capability of representing and processing imprecise information. Moreover, the best known applications of fuzzy logic that relate to image processing in a broad sense are fuzzy clustering and fuzzy pattern recognition [5], [18]. Studies have shown that FCNNs has its potential in image processing, pattern recognition [22], [27], [44] and even in white blood cell detection [43]. Similar to the traditional CNNs, the stability of the system is very important in the design of FCNNs. We note that several stability conditions for FCNNs have been proposed [45], [46]. As with relevance to this paper, we also highlight that the state estimation problem for FCNNs, with various time delays have been studied [3], [24], [40].

Kosko [25] studied the stability of BAM NNs by modeling a system of ordinary differential equations. Due to its wide applications in various fields, BAM NNs have attracted the attention of many researchers [10], [12], [13], [14]. Other studies on stability criteria of BAM NNs with leakage delay and the state estimation of BAM NNs have also been reported [1], [4], [21], [39]. Recently, the problem of state estimator for BAM NNs with leakage delay [42] and the stability of BAM FCNNs with leakage delay [2] have been reported. Vadivel et al. [47] studied the problem of robust state estimation for uncertain fuzzy BAM networks with time-varying delays by using Takagi–Sugeno fuzzy BAM NNs model. To the best of the authors’ knowledge, the state estimation problem for BAM FCNNs with leakage and unbounded distributed delays has yet to be addressed. Thus, in this paper we investigate the problem of state estimation for BAM FCNNs with leakage and unbounded distributed delays by using the suitable Lyapunov–Krasovskii functional and the LMI technique. Moreover, the Lyapunov–Krasovskii functional approach is one of the most powerful tools to deal with the control problem of uncertain systems and no one may find some utility in addressing the aforementioned issue. This approach is in practice necessary when dealing with nonlinear systems with time-varying parameters, and we propose a technique which is effective and insightful for both important classes of problems and special classes of functions. In addition, that theory is supported by efficient numerical tools, such as those based on linear matrix inequalities (LMIs).

The main contributions of this paper are: 1) The problem of state estimation on the BAM FCNNs with leakage and unbounded distributed delays are considered. 2) The sufficient conditions are derived to guarantee the state estimation for the considered system based on the Lyapunov–Krasovskii functional and LMIs.

Notations: Throughout this paper, the superscript ``T'' stands for the transpose of a vector or a matrix. Let $R, R^{n}$ and $R^{n \times m}$ denote the set of real numbers, n-dimensional Euclidean space and the set of all n × m real matrices, respectively. I is the identity matrix with appropriate dimension. * represents the elements below the main diagonal of a symmetric block matrix. For a real square matrix X, the notation X > 0 (X ≥ 0, X < 0, X ≤ 0) means that X is symmetric and positive definite (positive semi-definite, negative, negative semi-definite, respectively). If A is a matrix, its operator norm is denoted by ‖A‖. That is, $∥ A ∥ = s u p {| A x | : | x | = 1} = \sqrt{λ_{\max} (A^{T} A)}$ where $λ_{\max} (\cdot) (r e s p e c t i v e l y, λ_{\min} (\cdot))$ means the largest (respectively, smallest) eigenvalue of A. $Δ = {1, 2, \dots, L} . I = {1, 2, \dots, n}$ and $J = {1, 2, \dots, m}$ .

Section snippets

Model description and preliminaries

In general, BAM FCNNs with leakage and unbounded distributed delays are described by the following differential equations: $\begin{matrix} {\begin{matrix} {\dot{x}}_{i} (t) & = - a_{i} x_{i} (t - σ_{1}) + \sum_{j = 1}^{m} w_{1_{i j}} {\bar{f}}_{j} (y_{j} (t)) + \sum_{j = 1}^{m} w_{2_{i j}} {\bar{f}}_{j} (y_{j} (t - τ (t))) \\ + ⋀_{j = 1}^{m} α_{i j} \circ \int_{- \infty}^{t} k_{2 j} (t - s) {\bar{f}}_{j} (y_{j} (s)) d s + ⋁_{j = 1}^{m} β_{i j} \circ \int_{- \infty}^{t} k_{2 j} (t - s) {\bar{f}}_{j} (y_{j} (s)) d s, \\ {\dot{y}}_{j} (t) & = - {\bar{a}}_{j} y_{j} (t - σ_{2}) + \sum_{i = 1}^{n} {\bar{w}}_{1_{j i}} {\bar{g}}_{i} (x_{i} (t)) + \sum_{i = 1}^{n} {\bar{w}}_{2_{j i}} {\bar{g}}_{i} (x_{i} (t - ρ (t))) \\ + ⋀_{i = 1}^{n} {\bar{α}}_{j i} \circ \int_{- \infty}^{t} k_{1 i} (t - s) {\bar{g}}_{i} (x_{i} (s)) d s + ⋁_{i = 1}^{n} {\bar{β}}_{j i} \circ \int_{- \infty}^{t} k_{1 i} (t - s) {\bar{g}}_{i} (x_{i} (s)) d s, \\ x_{i} (s) & = ζ_{i} (s), s \in (- \infty, 0], i \in I, \\ y_{j} (s) & = ψ_{j} (s), s \in (- \infty, 0], j \in J, \end{matrix} \end{matrix}$ $\begin{matrix} {\begin{matrix} z_{1_{j}} (t) = b_{j i} x_{i} (t) + ς_{1_{j}} (t, x_{i} (t)), \\ z_{2_{i}} (t) = c_{i j} y_{j} (t) + ς_{2_{i}} (t, y_{j} (t) \end{matrix} \end{matrix}$

Existence and uniqueness of the equilibrium point

In order to study the existence and uniqueness of the equilibrium point, we consider the following equation associated with system (6): $\begin{matrix} {\begin{matrix} - a_{i} {\bar{e}}_{1 i} + \sum_{j = 1}^{m} w_{1_{i j}} {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) + \sum_{j = 1}^{m} w_{2_{i j}} {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) + ⋀_{j = 1}^{m} α_{i j} \circ {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) + ⋁_{j = 1}^{m} β_{i j} \circ {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) \\ - h_{1_{i j}} b_{j i} {\bar{e}}_{1 i} - h_{1_{i j}} {\bar{ϕ}}_{1_{j}} ({\bar{e}}_{1 i}) = 0, \\ - {\bar{a}}_{j} {\bar{e}}_{2 j} + \sum_{i = 1}^{n} {\bar{w}}_{1_{j i}} {\bar{\bar{g}}}_{i} ({\bar{e}}_{1 i}) + \sum_{i = 1}^{n} {\bar{w}}_{2_{j i}} {\bar{\bar{g}}}_{i} ({\bar{e}}_{1 i}) + ⋀_{i = 1}^{n} {\bar{α}}_{j i} \circ {\bar{\bar{g}}}_{i} ({\bar{e}}_{1 i}) + ⋁_{i = 1}^{n} {\bar{β}}_{j i} \circ {\bar{\bar{g}}}_{i} ({\bar{e}}_{1 i}) \\ - h_{2_{j i}} c_{i j} {\bar{e}}_{2 j} - h_{2_{j i}} {\bar{ϕ}}_{2_{i}} ({\bar{e}}_{2 j}) = 0 . \end{matrix} \end{matrix}$ Define the maps H and H^⋆ respectively as follows $\begin{matrix} {\begin{matrix} H_{i} ({\bar{e}}_{1}) = - a_{i} {\bar{e}}_{1 i} + \sum_{j = 1}^{m} w_{1_{i j}} {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) + \sum_{j = 1}^{m} w_{2_{i j}} {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j}) + ⋀_{j = 1}^{m} α_{i j} \circ {\bar{\bar{f}}}_{j} ({\bar{e}}_{2 j} \end{matrix} \end{matrix}$

Main results

Theorem 4.1

The unique equilibrium point of BAM FCNNs (6) is globally asymptotically stable provided the conditions of Theorem 3.1 are satisfied.

Proof

Consider the following Lyapunov–Krasovskii functional $\begin{matrix} V (t) & = & \sum_{i = 1}^{5} V_{i} (t), \end{matrix}$ where $\begin{matrix} V_{1} (t) & = & {[e_{1} (t) - A \int_{t - σ_{1}}^{t} e_{1} (s) d s]}^{T} P_{1} [e_{1} (t) - A \int_{t - σ_{1}}^{t} e_{1} (s) d s] \\ + {[e_{2} (t) - \bar{A} \int_{t - σ_{2}}^{t} e_{2} (s) d s]}^{T} P_{2} [e_{2} (t) - \bar{A} \int_{t - σ_{2}}^{t} e_{2} (s) d s], \\ V_{2} (t) & = & \int_{t - τ (t)}^{t} f^{T} (s) Q_{1} f (s) d s + \int_{t - ρ (t)}^{t} g^{T} (s) Q_{2} g (s) d s \\ + \int_{t - σ_{1}}^{t} e_{1}^{T} (s) Q_{3} e_{1} (s) d s + \int_{t - σ_{2}}^{t} e_{2}^{T} (s) Q_{4} e_{2} (s) d s, \\ V_{3} (t) & = & ρ \int_{t - ρ (t)}^{t} e_{1}^{T} (s) R_{1} e_{1} (s) d s + τ \int_{t - τ (t)}^{t} e_{2}^{T} (s) R_{2} e_{2} (s) d s, \\ V_{4} (t) & = & σ_{1} \int_{- σ_{1}}^{0} \int_{t + θ}^{t} e_{1}^{T} (s) T_{1} e_{1} (s) d s d θ + \end{matrix}$

Numerical examples and simulations

Two numerical examples are presented to demonstrate the effectiveness of the developed LMI approach to the state estimator design for the BAM FCNNs.

Example 5.1

Consider the error-state system (8) or (10) with the following parameters: $\begin{matrix} A & = & [\begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{matrix}], W_{1} = [\begin{matrix} 1.25 & 0.1 & 1 \\ - 0.3 & 0.1 & - 2.1 \\ - 1.2 & 2.3 & 1 \end{matrix}], W_{2} = [\begin{matrix} 2.4 & - 1 & - 1 \\ - 3 & 0.2 & - 2.5 \\ 0.2 & 1.1 & - 2.3 \end{matrix}], \\ α & = & β = [\begin{matrix} 0.05 & 0.05 & 0.05 \\ 0.05 & 0.05 & 0.05 \\ 0.05 & 0.05 & 0.05 \end{matrix}], B = C = [\begin{matrix} 0.5 & 0 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 0.5 \end{matrix}], \bar{A} = [\begin{matrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{matrix}], \\ {\bar{W}}_{1} & = & [\begin{matrix} 0.2 & 0.2 & - 1.2 \\ - 2.1 & 1.1 & - 2 \\ - 1 & 2.4 & 1 \end{matrix}], {\bar{W}}_{2} = [\begin{matrix} 0.2 & - 1.5 & - 2 \\ - 4 & 1 & - 2 \\ - 0.2 & 1.5 & - 2.3 \end{matrix}], \bar{α} = \bar{β} = [\begin{matrix} 0.03 & 0.03 & 0.03 \\ 0.03 & 0.03 & 0.03 \\ 0.03 & 0.03 & 0.03 \end{matrix}] . \end{matrix}$ The

Conclusion

In this paper, the state estimation problem has been investigated for the BAM FCNNs with leakage and unbounded distributed delays. By applying the Lyapunov–Krasovskii functional, we have utilized a LMI approach to derive the sufficient conditions which guarantee the existence of the state estimators. The estimator gain matrices are found by solving a set of LMIs, which could guarantee the global asymptotic stability of the error dynamical system. Further, the stability results were derived by

Acknowledgement

This work was supported by the University of Malaya HIR grant UM.C/625/1/HIR/MOHE/SC/13. Authors are very much thankful to Prof M.J. Brunger of Flinders University for his careful reading of this paper and his valuable suggestions and comments to improve the manuscript. Further authors are thankful to the Editor-in-Chief, Associate Editor and anonymous reviewers for their constructive comments and fruitful suggestions for improving this manuscript.

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Design of state estimator for BAM fuzzy cellular neural networks with leakage and unbounded distributed delays

Abstract

Introduction

Section snippets

Model description and preliminaries

Existence and uniqueness of the equilibrium point

Main results

Numerical examples and simulations

Conclusion

Acknowledgement

Neurocomputing

Math. Comput. Model

Comput. Math. Appli.

Phys. Lett. A

Nonlinear Anal.

Physica D

Nonlinearity

Neural Netw.

Nonlinear Anal.

IEEE Trans. Circ. Syst.

Neurocomputing

IEEE Trans. Neural Netw.

IEEE Trans. Syst. Man Cybern.

Nonlinear Anal.

J. Franklin Inst.

Neurocomputing

Phys. Lett. A

Appl. Math Comput.

Nonlinear Anal.

Adv. Diff. Eq.

Neurocomputing

Inf. Sci. (Ny)

IEEE Trans. Inf. Technol. Biomed.

Int. J. Comput. Intell. Syst.

Robust state estimation for uncertain stochastic bidirectional associative memory networks with time-varying delays

Phys. Scr.