Event-triggered H∞ state estimation for time-varying neural networks with variance-constraint and fading measurements

This paper addresses the event-triggered H∞ state estimation problem for a class of discrete recurrent neural networks subject to variance-constraint and fading measurements. The phenomena of fading measurements are described by introducing a set of mutually independent random variables, which reflect that each sensor has individual missing probability. In addition, for the purpose of energy saving, an eventtriggered H∞ state estimation scheme is used for time-varying neural networks to determine whether the measurement output is transmitted to the estimator or not. Some sufficient conditions are obtained to guarantee that the estimation error system satisfies both estimation error variance constraint and prescribed H∞ performance requirement. Finally, the feasibility of the proposed event-triggered H∞ state estimation method is verified by a numerical example.


Introduction
In the past decades, the analysis and design of recursive neural networks (RNNs) have attracted great attention due to their powerful advantages, such as showing dynamic time behaviour and using internal memory to process arbitrary input sequences. Accordingly, the RNNs have been successfully applied to broad areas such as speech recognition, pattern recognition and associate memory. Hence, more and more researchers have paid attention to the state estimation problem for time varying system. In [1], an event-based recursive input and state estimator has been designed to ensure that the covariance of the estimation error has an upper bound at any time for the linear discrete time-varying systems.
Since the perfect measurements cannot be always available, especially in the unreliable network circumstances, the state estimation problems with fading measurements have aroused the extensive research. To be specific, the stochastic stability of a modified unscented Kalman filter problem has been analyzed in [2] for a class of nonlinear systems with stochastic nonlinearities and multiple fading measurements. In addition, a novel envelope-constrained performance criterion has been proposed in [3] to better quantify the transient dynamics of the filtering error process over the finite horizon. Moreover, it is worth noting that few scholars have studied the state estimation of time-varying neural networks with fading measurements. Different from the optimal estimation of minimum error covariance, the varianceconstrained estimation method can provide a more loose technique, where the upper bound constraints are introduced to reflect the estimation accuracy. For example, in [4], the finitehorizon state estimator has been designed to provide the estimation scheme for timevarying complex networks, where both the H∞ performance requirements and prescribed variance constraints on the estimation error can be guaranteed simultaneously. On the other hand, compared with the time-triggered control scheme, the advantage of the event triggered scheme is that it can effectively realize the sharing of communication resources. For example, a new event-triggered distributed state estimation strategy has been presented in [5], which can ensure the existence of the desired estimator gains and the exponentially stability in the mean square of the estimation error dynamics simultaneously.
Motivated by the aforementioned discussion, we handle the event-triggered H∞ state estimation problem for discrete time-varying RNNs subject to variance-constraint and fading measurements. The main work conducted can be listed: 1) the event-triggered H∞ state estimation problem is, for the first time, investigated for a class of discrete timevarying stochastic RNNs subject to variance-constraint and fading measurements; 2) a new solvable method is given for addressed variance-constrained state estimation problem based on the recursive linear matrix inequalities (RLMIs); and 3) the usual literature considers the augmented system satisfying both the prescribed H∞ performance requirement and the estimation error variance constraints, but we analyse the estimation error system directly with same order of original system, which may reduce the computational complexity.

Problem formulation and preliminaries
In this paper, the addressed discrete time-varying neural networks are described by 1 1 are Gaussian white noises with zero mean values and covariances 1 0 V > and 2 0 V > , respectively. k δ is zero mean Gaussian white noise with unity covariance. and δ is the triggering threshold. The execution is triggered as long as the condition ( , ) 0 k f ξ δ > is satisfied. Therefore, the sequence of event-triggered instants In order to estimate the states of neurons, the following state estimator is constructed: where ˆk x is the estimation of neural state k x , k K is the estimator gain matrix to be determined. Let the estimation error be  X e e . The main purpose of this paper is to design a time-varying state estimator (2), the following two requirements are satisfied simultaneously: (i) Let the scalar 0 γ > , the positive define matrices W ϕ and W φ be given. For the initial state 0 e , the estimation error k z satisfies: (ii) The estimation error covariance satisfies the bounded constraint 2 is a set of pre-defined known matrix.     22  22  33  44  55  66  77  88  99   =diag{ ,  ,  ,  ,  ,  , , . In addition, the simulation results are shown in Fig. 1, which illustrate that the proposed event-triggered H∞ state estimation method is effective.

Conclusion
In this paper, we have discussed the event-triggered H∞ state estimation problem for a class of discrete RNNs subject to variance-constraint and fading measurements. The phenomena of fading measurements have been described by introducing a set of mutually independent random variables. Applying the RLMI technique, some criteria have been established to guarantee the prescribed H∞ performance and the estimation error covariance constraints of the estimation error system. Finally, the feasibility of the proposed method has been verified by a numerical example.
This work was supported in part by the National Natural Science Foundation of China under Grants 12071102.