Finite-time non-fragile state estimation for discrete neural networks with sensor failures, time-varying delays and randomly occurring sensor nonlinearity
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, and 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 is positive semi-definite (or positive definite). The superscript “T” denotes the matrix transposition. ‖ · ‖ is the Euclidean norm in .
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:where is the neural state vector; d1(k) and d2(k) denote the discrete-time varying delays, and sastifying and 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) (). Theorem 1 The nominal system (10) () 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 symmetric positive-definite matrices P, Q1, Q2, Q31, Q32, Q41, Q42 and R such that
Numerical example
In this section, a numerical example of networks (1) and (2) with following parameters:
Take the activation functions as follows:and we can conclude that
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)
References (39)
- et al.
Mean-square exponential stability for stochastic discrete-time recurrent neural networks with mixed time delays
Neurocomputing
(2015) - et al.
Exponential synchronization of discrete-time mixed delay neural networks with actuator constraints and stochastic missing data
Neurocomputing
(2016) - et al.
A new approach to non-fragile state estimation for contiuous neural networks with time-delays
Neurocomputing
(2016) - et al.
State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties
Automatica
(2015) - et al.
Stochastic sampled-data control for state estimation of time-varying delayed nerual networks
Neural Networks
(2013) - et al.
State estimation for discrete Markovian jumping neural networks with time delay
Neurocomputing
(2010) - et al.
Stochastic finite-time state estimation for discrete time-delay neural networks with Markovian jumps
Neurocomputing
(2015) - et al.
Non-fragile state estimation for discrete Markovian jump neural networks
Neurocomputing
(2016) - et al.
Event-triggered generalized dissipativity filtering for neural networks with time-varying delays
IEEE Trans. Neural Netw. Learn. Syst.
(2016) - et al.
Asynchronoous filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities
Automacia
(2014)