Finite-time non-fragile state estimation for discrete neural networks with sensor failures, time-varying delays and randomly occurring sensor nonlinearity

https://doi.org/10.1016/j.jfranklin.2018.10.032Get rights and content

Highlights

  • The considered system is modeled as a kind of nonlinear singular neural network.

  • A finite-time state estimation algorithm is provided for the singular system.

  • A finite-time non-fragile state estimation algorithm is given to avoid the estimator’s parameter perturbation.

Abstract

A finite-time non-fragile state estimation algorithm is discussed in this article for discrete delayed neural networks with sensor failures and randomly occurring sensor nonlinearity. First, by using augmented technology, such system is modeled as a kind of nonlinear stochastic singular delayed system. Then, a finite-time state estimator algorithm is provided to ensure that the singular error dynamic is regular, causal and stochastic finite-time stable. Moreover, the states and sensor failures can be estimated simultaneously. Next, in order to avoid the affection of estimator’s parameter perturbation, a finite-time non-fragile state estimation algorithm is given, and a simulation result demonstrates the usefulness of the proposed approach.

Introduction

In the past several years, the theory of neural networks have received considerable attractions that the systems have obtained wide application in a large number of fields, such as pattern recognition, process of static image and signal, combinatorial optimization and associative memory, the corresponding results can be found in [1], [2], [3], [4], [5], [6], [7], [8], [9], [17], [18] and the references therein. In most of algorithms above, a prerequisite is necessary that the factual information of the neuron states should be known in advance. However, due to the characteristic of neural networks and some physical limits, it is doubtless that only a part of the neuron states information can be obtained, thus, state estimation problems for neural networks have attracted an increasing research interest [31]. The state estimation is an important area of the modern control theory since the sensor is used to measure the information of outputs, and in such approach, a filter/estimator is designed to reconstruct the unknown signals [13]. For instance, an extended dissipativity-based state estimation algorithm was provided in [14] for discrete Markov jump neural networks, and a state estimator was provided in [19] when the transition probabilities of the mode jumps were partially unknown. For linear or nonlinear uncertain systems, a sliding mode observer was designed to estimate the states in [15]. By dividing the bounding of the activation function into two subintervals, a state estimator in [16] was designed with stochastic sampling for delayed neural networks.

Besides, it is worth mentioning that time delay, considered as an unavoidable characteristic of signals transmission between different neurons, has been investigated as a major causes of instability and unsatisfactory performance. Recently, the dynamic of delayed neural networks have widely studied and some excellent results are provided in [10], [11], [12], [24], [26], [27], [28], [29], [30], [36]. However, in practical, sensors are often placed under harsh environment, which may caused nonlinear characteristic of sensors, and the corresponding results based on linear measurement may not provide a reliable solution [32], [33]. Moreover, due to the limitation of physical sensor device, failures are inevitable, thus, it is important and necessary to pay attention on the state estimation problem for systems in the presence of sensor nonlinearities and sensor failures [34].

Furthermore, it is well known that, most of the state estimation (or control) algorithms are implemented with accurate estimator’s (or controller’s) gain. In practice, the corresponding gain is not an ideal value because of the complex and changeable environments. Therefore, some researchers have pay attention on non-fragile problem in the past few years, see [20] (for controller), [25], [35] (for state estimator) and the references therein. In classical control theory, the considered system’s asymptotic stability is mainly based on the Lyapunov stability over an infinite interval. In the application point of view, the estimation problem should be realized in finite-time. Thus, it is necessary to analyze the finite-time estimation problem for nonlinear neural networks [21]. Best to the author’s knowledge, the finite-time non-fragile state estimation problem for discrete-time neural networks with sensor failures, time-varying delays and randomly occurring sensor nonlinearity has not been fully investigated.

In this article, considering the characteristic of time-varying delays, finite-time condition and sensor constraints, some Lyapunov–Krasovskii functions are proposed, then, by using augmented technology, a finite-time non-fragile estimator is constructed to ensure that the singular error dynamic is regular, casual and stochastic finite-time stable. Moreover, the main contributions of this research can be described as following: (a) by using augmented technology, a kind of nonlinear singular neural network is modeled to describe the delayed system with sensor failures and randomly occurring sensor nonlinearity. (b) A finite-time state estimation algorithm is provided for the singular system to ensure that the singular error dynamics is regular, casual and stochastic finite-time stable, besides, the states and sensor failures can be estimated simultaneously. (c) A finite-time non-fragile state estimation algorithm is provided to avoid the estimator’s parameter perturbation.

Notation: Throughout the paper, Rn and Rn×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. The notation X ≥ Y (or X > Y) means that X and Y are symmetric matrices and XY is positive semi-definite (or positive definite). The superscript “T” denotes the matrix transposition. ‖ · ‖ is the Euclidean norm in Rn.

Section snippets

Model formulation and preliminaries

In the present paper, a kind of discrete-time neural networks with sensor failures, time-varying delays and randomly occurring sensor nonlinearities are described as follows:x(k+1)=Ax(k)+Adx(kd1(k))+Wg(x(k))+Wdg(x(kd2(k)))y(k)=α(k)ψ(Cx(k))+(1α(k))Cx(k)+xf(k)z(k)=Cx(k)x(k)=ϕ,kZwhere x(k)=[x1(k),x2(k),,xn(k)]Tis the neural state vector; d1(k) and d2(k) denote the discrete-time varying delays, and sastifying d̲idi(k)d¯i(k),i=1,2, A=diag{a1,a2,,an} and Ad=diag{ad1,ad2,, adn} stand for the

Main results

In this section, we first provide the criteria on stochastic finite-time state estimation of the nominal error dynamic system (10) (ΔL=0).

Theorem 1

The nominal system (10) (ΔL=0) is stochastically finite-time stable with respect to (δ, ϵ, Z, N) if there exists μ ≥ 1, ϵ > 0, γ1 > 1, γ3 > 0, γ5 > 0, J1 > 0, J2 > 0, sets of positive scalars κ1, κ2 and γ¯, symmetric positive-definite matrices P, Q1, Q2, Q31, Q32, Q41, Q42 and R such thatΦ=[Φ1Φ2Φ3Φ4Φ5Φ6*P0000**P000***P00****P0*****2P]<0,Z<P<γ1ZQn<γ3Z,Qn{

Numerical example

In this section, a numerical example of networks (1) and (2) with following parameters:A=[0.50000.30000.4],Ad=[0.050000.010000.04],W=[0.20.20.100.40.30.300.2],Wd=[0.200.10.10.200.100.1],C=[0.30.20.60.10.80.2000.8]

Take the activation functions as follows:g1(x1(k))=0.2tanh(x1(k))g2(x2(k))=0.3tanh(x2(k))g3(x3(k))=0.1tanh(x3(k))ψ(·)=[tanh(x1(k))+0.2x1(k)+0.1x2(k)+0.1x3(k)0.1x1(k)tanh(x2(k))+0.2x2(k)0.1x1(k)+0.2x3(k)tanh(x3(k))]and we can conclude thatB=[0.20000.30000.1],J1=[0.80.10.10.1

Conclusion

In this paper, we have discussed the finite-time non-fragile state estimation problem for discrete neural networks with sensor failures, time-varying delays and randomly happening sensor nonlinearity. Considering the characteristics of time-varying delays and sensor conditions, a Bernoulli distribution based stochastic process is used to model the random sensor nonlinearities. In order to estimate the system’s states and sensor failures simultaneously, the considered system is modeled as a kind

Acknowledgment

This work was supported in part by Zhejiang Provincial Natural Science Foundation of China (No. LY19F030020), in part by the National Natural Science Foundation of China (No. 61733009), in part by the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (No. U1509203)

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