Variance-constrained resilient H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\infty }$\end{document} state estimation for time-varying neural networks with randomly varying nonlinearities and missing measurements

This paper addresses the resilient H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\infty }$\end{document} state estimation problem under variance constraint for discrete uncertain time-varying recurrent neural networks with randomly varying nonlinearities and missing measurements. The phenomena of missing measurements and randomly varying nonlinearities are described by introducing some Bernoulli distributed random variables, in which the occurrence probabilities are known a priori. Besides, the multiplicative noise is employed to characterize the estimator gain perturbation. Our main purpose is to design a time-varying state estimator such that, for all missing measurements, randomly varying nonlinearities and estimator gain perturbation, both the estimation error variance constraint and the prescribed H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\infty }$\end{document} performance requirement are met simultaneously by providing some sufficient criteria. Finally, the feasibility of the proposed variance-constrained resilient H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{\infty }$\end{document} state estimation method is verified by some simulations.


Introduction
Over the past few decades, the study of neural networks (NNs) has become an attractive research topic [1][2][3]. As it is well recognized, the state of neurons is not always available in reality, so effective estimation methods need to be used to estimate them [4][5][6]. Accordingly, the corresponding state estimation (SE) algorithms under different performance indices have been successfully applied in wide fields including associative memory, pattern recognition, combinatorial optimization and image processing [7][8][9][10]. Up to now, there have been many studies on the SE for different types of NNs [11][12][13][14]. For example, some effective SE methods have been established in [15][16][17] for Markov jump NNs with different time delays. Nevertheless, it is worthwhile to notice that many existing approaches are only applicable to deal with the time-invariant cases, which may lead to the application limitations [18][19][20]. Recently, many scholars have paid attention to the study of dynamical systems with time-varying parameters and proposed many methods to analyze the behavior of dynamical systems [21][22][23][24][25]. For example, a novel SE scheme has been developed in [19] for time-varying systems, where a suboptimal algorithm has been presented to tackle the effect induced by redundant channels. Nevertheless, it is worth noting that there is little research on the SE problem of discrete time-varying uncertain recurrent neural networks (DTVURNNs), which deserves the further investigations.

Related Work
In the process of the network communication or transmission, many scholars have done a lot of research work regarding the SE problems subject to the saturated phenomena [26][27][28]. In fact, the sensor saturation is one of the common nonlinear problems encountered in the actual system. It will reduce the performance of the estimation error (EE) system and even lead to the instability of the EE system if this phenomenon is ignored when tackling the design problem of the SE algorithm. Up to now, many results have been obtained to cope with the SE problems for dynamical networks with sensor saturation [29,30]. Specifically, in [29], a new SE strategy has been proposed for two-dimensional stochastic delayed systems, and a state estimator has been constructed to ensure that the EE dynamical system is globally robustly asymptotical stable. Nevertheless, it is worth mentioning that there are few research results on SE problem, where the sensor saturation might exist in a random way under uncertain occurrence probability. As far as the authors know, the variance-constrained resilient H ∞ SE problem for time-varying NNs with random saturation observation under uncertain occurrence probability has not been fully considered in the literature. Therefore, we address the SE problem for DTVURNNs with random saturation observation, where major attention is to handle the random saturation observation and the effect of the uncertain occurrence probability.
At present, the problem of non-fragile/resilient SE becomes one of the hot research topics [31][32][33]. In the actual systems, because of the limited communication bandwidth and the limited number of independent power supply in each node, it is desirable to provide the efficient estimation methods to meet increasing performance requirements especially in the networked environment [34][35][36][37]. For instance, the event-triggered resilient SE problem has been investigated in [38] for discrete Markovian jumping NNs, where a new SE method has been proposed to weaken the effect induced by randomly occurring nonlinearities. In addition, a novel resilient SE method has been presented in [39] for discrete Markovian jumping NNs, and sufficient conditions have been obtained to guarantee that the EE dynamical system is exponentially stable. On the other hand, it is necessary to use the upper bounds of EE covariance to characterize the estimation performance, which has important theoretical value and practical significance. In fact, the variance-constrained SE strategy is different from the traditional minimum variance estimation. Accordingly, a specific upper bound constraint is introduced to describe the allowable accuracy of the proposed estimation method, which provides a more relaxed estimation method. Recently, in [40] and [41], the novel varianceconstrained H ∞ SE algorithms have been presented for time-varying dynamical systems, and sufficient conditions have been obtained to ensure the desired H ∞ performance constraint and the error variance boundedness. However, it should be noted that few SE method can be available for coping with the effects of random saturation observation under uncertain occurrence probability, not to mention the corresponding problem for DTVURNNs subject to two performance indices with respect to the estimation accuracy and disturbance attenuation level.

Methodology
Inspired by the aforementioned discussions, the main purpose is to study the varianceconstrained resilient H ∞ SE problem for DTVURNNs with random saturation observation under uncertain occurrence probability. Overall, the essential difference between the methods for dealing with the SE problems of time-invariant or time-varying NNs are how to present proper estimation ways, where the effects from the time-varying parameters are needed to be handled and the estimation performance is examined by utilizing the proper index. In particular, the recursive linear matrix inequalities (RLMIs) method is given to tackle the SE problem for NNs with time-varying parameters in this paper and the combination of H ∞ performance and variance constraint indices is taken into account to evaluate the estimation performance. Accordingly, a new variance-constrained resilient H ∞ SE algorithm is presented by using the stochastic analysis technique and matrix theory, which can guarantee both the upper bound of error variance boundedness and the pre-defined H ∞ performance constraint simultaneously. The main contributions of this paper can be summarized as follows: (1) the variance-constrained resilient H ∞ SE issue is investigated for DTVURNNs with random saturation observation under uncertain occurrence probability, thereby reflecting engineering practice more closely; (2) a new RLMIs method without resorting to the augmentation method is proposed for the EE dynamical system satisfying the H ∞ performance constraint and the error variance boundedness, which might reduce the computational burden; (3) the proposed variance-constrained SE method has time-varying characteristics which is suitable for online applications; and (4) the information of addressed network-induced phenomena is fully taken into account in order to achieve desired estimation performance, and then a new SE algorithm is obtained to examine the effects from the random saturation observation under uncertain occurrence probabilities on the whole estimation performance. Notations: In this paper, the superscript T depicts the transpose of the matrix. N + , R r , E{x} and I denote, respectively, the sets of positive integers, the r -dimensional Euclidean space, the mathematical expectation of x and the identity matrix of proper dimension. * represents the ellipsis of the term resulting from symmetry in a symmetric block matrix, and diag{…} is defined as the block diagonal matrices. The full names and abbreviations are given as follows:

Problem Formulation
As illustrated in Fig. 1, the basic framework of resilient H ∞ SE is given, where the random saturation observation phenomenon is characterized by the random variable to describe the real transmission process of neural signals. The major methodology is the RLMIs-based technique to fulfill both the H ∞ performance and the error variance boundedness by proposing new sufficient condition.
Consider the class of DTVURNNs with random saturation observation as follows: where x k ∈ R n , y k ∈ R m and z k ∈ R r stand for the state vector of the NNs, the measurement output and the controlled output. A k is the self-feedback diagonal matrix, B k is the connection weight matrix, E k , C k and M k are known matrices with compatible dimensions, σ (·) is the saturation function, v 1k and v 2k are white noises with zero mean values and covariances V 1k > 0 and V 2k > 0, respectively. f (x k ) is the neuron activation function. A k characterizes the parameter uncertainty and satisfies where H k and N k are known proper dimension matrices, and the unknown matrix F k satisfies the following form The phenomenon of random saturation observation is described by random variable, which obeys whereβ + β ∈ [0, 1] and | β| ≤ ι 0 withβ and ι 0 being known scalars.

Remark 1
In this paper, the main reasons for using uncertain occurrence probability to describe the random saturation observation phenomenon are as follows: (1) it is difficult to obtain accurate probability information for random saturation observation due to the inaccuracy of statistical tests or other reasons; (2) the measuring method might undergo failure resulting in the inaccurate statistic result; and (3) the occurrence of the random saturation observation changes frequently as the external environment changes. Here, we attempt to characterize the case when the system in reality is unavoidably influenced by random saturation observation phenomena and the occurrence probabilities could be uncertain due to some reasons. To be specific, the Bernoulli distributed random variable β k is introduced to characterize the random saturation observation, and it is of great significance to propose an effective and admissible method to attenuate the impact of uncertain probability information on the estimation performance.
The activation function f (s) obeys the following sector-bounded condition where U 1k and U 2k are known proper compatible matrices and U k = U 2k −U 1k is a symmetric positive-definite real-value matrix (PDRVM). The saturation function σ (·) is defined as follows: where σ i (ξ i ) = sign(ξ i )min{ξ i,max , |ξ i |} for each i ∈ {1, 2, · · · , m}, where ξ i,max is the i-th element of the vector ξ max (i.e., the saturation level). Moreover, we assume that σ (ξ) satisfies the following condition whereḡ is a positive constant with 0 <ḡ < 1. Remark 2 Due to the physical properties and technical limitations of the sensor itself, the sensor can only measure signals within a certain strength, which is called the sensor saturation. The main reason for reducing the estimation effect: (1) the phenomenon of sensor saturation may cause the data transmission not to be the ideal value, thus affecting the estimation effect; and (2) the nonlinear characteristic of sensor saturation may also lead to EE system oscillation and even instability, which reduces the estimation performance.
In this paper, we construct the following finite-horizon resilient state estimator (FHRSE): wherex k ,K k , K k and ξ k denote, respectively, the estimation of x k , the known matrix with compatible dimension, the estimator gain parameter (EGP) to be determined and the white noise with zero mean and unity covariance. Remark 3 In this paper, we consider the DTVURNNs, in which the H ∞ performance requirement under the finite-horizon case is considered. It should be noted that the influence of initial value is also considered. Accordingly, the transient performance of the system is discussed and the H ∞ performance under the finite-horizon case is examined, where k ∈ [0, N ] describes the finite-horizon. Moreover, notice that the data received by the estimator are slightly different from the expected ones owing to various reasons such as numerical round off errors, limited word length of the computer and the imprecision in analog-digital conversion. Tiny variations of estimator parameters could degrade the whole estimation performance. Therefore, we aim to propose a resilient state estimator with admissible adjustment ability for handling the estimation problem of time-varying NNs, where the estimator gain perturbation is taken into account as in (6) during the design of the state estimator. In particular, the estimator's execution error is allowed within an acceptable range of accuracy domain. It will be shown later that the newly proposed resilient estimation method has acceptable estimation accuracy, which has the recursive computation feature with online implementation potential and can be used to handle the problem of resilient SE problem for time-varying NNs.
Subsequently, let the controlled output EE bez k = z k −ẑ k and the EE be e k = x k −x k . It follows from (1) and (6) that withf To proceed, the state covariance matrix X k is defined as Our purpose is to design a variance-constrained resilient H ∞ SE algorithm and achieve the following two conditions simultaneously.
(R1) For given positive scalar γ , the PDRVMs U ν and U φ , and the initial state e 0 , the controlled output EEz k obeys the following constraint: where v k The EE covariance satisfies the following performance criterion where ϒ k (0 ≤ k < N ) is a series of admissible estimation precision requirements corresponding to the actual situation.
To end this section, in order to facilitate subsequent derivations, we provide the following lemmas. where Proof It is easy to obtain this lemma based on [42]. (4), then we can derive

Lemma 2 If the activation function f (s) satisfies
where U 1k and U 2k are real matrices of compatible dimensions.

Lemma 3 The saturation function σ (τ ) satisfies
where τ = C k (x k + e k ) and 0 < < 1 1+ḡ . Proof It is easy to obtain this proof, and it is omitted for brevity.

Remark 4
The main problems we faced are: (1) How to choose appropriate conditions to deal with the nonlinear activation function and saturation observation for DTVURNNs? (2) How to guarantee the satisfactory estimation performance by utilizing the proper indices? (3) How to propose an effective method to deal with the recursive SE problem for time-varying NNs with random saturation observation under uncertain occurrence probabilities? The corresponding solutions are given as follows: i) Efficient techniques are given in Lemmas 1-3. ii) During the design of the estimation method, we consider these two performance constraints (i.e., H ∞ performance and the error variance boundedness) simultaneously. iii) Some sufficient criteria are obtained to achieve that the EE system satisfies the preset H ∞ performance requirement within the finite-horizon and the error variance boundedness, that is, both the disturbance attenuation ability and flexible estimation accuracy are ensured via the RLMIs technique.

Main Results
In this section, several sufficient conditions are given for EE dynamical system (7), where both the H ∞ performance constraint and the error variance boundedness are guaranteed by using the RLMIs method.
Firstly, sufficient conditions are obtained for the EE dynamical system to meet the H ∞ performance constraint. (1) with random saturation observation under uncertain occurrence probability. Suppose that the PDRVMs U ν and U φ , the scalars γ > 0, λ > 0 andβ ≥ 0, and the EGP K k in (6) are given. If Q 0 ≤ γ 2 U φ and there exist a set of PDRVMs 12 13

Theorem 1 Consider the DTVURNNs
then the H ∞ performance requirement in (9) is ensured.
Taking the EE dynamical system (7) into account, it can be concluded that Using the inequality 2x T Py ≤ x T Px + y T Py (P > 0) and together with (15), it is easy to derive the following form where i (i = 1, 2, · · · , 11) are mentioned below (13). where with 55 and 66 defined in (13). Based on Lemma 1 and (5), it can be obtained that where is defined in (13). Summarizing both sides of (18) from 0 to N − 1 on k, the following form can be obtained Therefore, it is straightforward to see that Note that Q 0 ≤ γ 2 U φ , < 0 and Q N > 0, it follows that J 1 < 0. The proof is complete now.
To sum up, the sufficient criterion has been proposed to guarantee that the EE dynamical system satisfies H ∞ performance. Next, we obtain the sufficient condition to guarantee the requirement of error variance boundedness.
In this subsection, a sufficient criterion is obtained such that the error variance boundedness is guaranteed. (1) with random saturation observation under uncertain occurrence probability. Let the scalarβ ≥ 0 and the EGP K k in (6) be given. Under the initial condition P 0 = X 0 , if there exist a series of PDRVMs {P k } 1≤k≤N +1 satisfying the matrix inequality as follows:
Proof Based on (8), the state covariance matrix X k can be calculated by By using the inequality x y T + yx T ≤ x x T + yy T and together with (23), the following form can be derived where i (i = 1, 2, · · · , 11) are mentioned below (13). Subsequently, according to Lemma 2 and Lemma 3, it is obvious to obtain where Y , κ 1 and κ 2 are defined in (22). Therefore, it can be obtained that According to the feature of the trace, the following conditions can be derived: Furthermore, combining (24) with (25) results in It is easy to get that P 0 ≥ X 0 . Letting P k ≥ X k , we obtain Then, from (21) and (26), we can derive the following inequality The proof is complete.
According to the above analysis, the sufficient conditions have been presented, which can ensure that the EE dynamical system satisfies prescribed H ∞ performance constraint and the error variance boundedness by solving the RLMIs. (1) with random saturation observation under uncertain occurrence probability. Assume that the EGP K k in (6) is given. For given PDRVMs U ν and U φ , scalars γ > 0, λ > 0 andβ ≥ 0, under the initial conditions Q 0 ≤ γ 2 U φ and P 0 = X 0 , if there are two sets of PDRVMs {Q k } 1≤k≤N +1 and {P k } 1≤k≤N +1 obeying two RLMIs 11 12  Proof Based on Theorems 1-2, it is not difficult to testify that (28) implies (13) and (29) implies (21). Thus, we can conclude that both the prescribed H ∞ performance constraint and the error variance boundedness can be guaranteed, which ends the proof of Theorem 3.

Design of Finite-Horizon Resilient State Estimator
In this section, a sufficient criterion is given to handle the design problem of FHRSE by using the stochastic analysis technique, which provides the solution of the FHRSE gain.
Theorem 4 For a given scalarβ ≥ 0, the attenuation level γ > 0 and the PDRVMs U ν and U φ , a series of prescribed variance upper bound matrices {ϒ k } 0≤k≤N +1 , under the initial conditions with the updating rule as follows:Q and other elements are defined in Theorems 1-3, then it is concluded that the design problem of FHRSE is solvable.

Proof
In the first place, in order to deal with parameter uncertainty, (28) can be rewritten as Subsequently, it is not difficult to see that Similarly, we rewrite (29) as follows: where F 0 12 =   12  0  13  14  15 16 , 0 13 = 0 0 Consider the variance-constrained resilient SE problem for DTVURNNs, where the corresponding parameters are given as follows: Furthermore, the activation function is taken as follows: where x k = x 1,k x 2,k T represents the neuron state vector of the NNs. Let the upper bound matrices be {ϒ k } 1≤k≤N = diag{0.3, 0.3}, the weighted matrices be U ν = 1, the disturbance attenuation level be γ = 0.9 and N = 90, and covariances be V 1k = V 2k = 1. In addition, set the saturation level ξ max = 0.8, the initial states x 0 = −2.5 0.2 T andx 0 = −1.5 1.3 T .
Then, according to (31)- (33) in Theorem 4, the parameter matrices K k , Q k and P k can be designed in TABLE 2 and TABLE 3 Fig. 4 describes the trajectories of the controlled output EEsz 1,k , and the second figure in Fig. 4 describes the trajectories of the controlled output EEsz 2,k . The upper bound of error variance and actual error variance are shown in Fig. 5. It can be seen from Fig. 5 that the actual error variance is lower than the upper bound of error variance. In addition, we calculate the sum of the norm of the output EEs under the above two cases and make a simple comparison. From TABLE 4, it is easy to observe that the biggerβ, the smaller the error is. Therefore, the simulation results have illustrated the necessity of designing the variance-constrained resilient H ∞ SE scheme for the sake of mitigating the effects of random saturation observation on the estimation results, which further show the feasibility of the new variance-constrained resilient H ∞ SE algorithm.  Table 4 The sum of the norm of the output EEs

Conclusion
In this paper, the resilient H ∞ SE method subject to variance constraint has been proposed for a class of DTVURNNs with random saturation observation under uncertain occurrence probability. The phenomenon of random saturation observation has been described by a random variable obeying the Bernoulli distribution. The FHRSE has been designed such that, in the presence of estimator gain perturbation and random saturation observation under uncertain occurrence probability, some sufficient criteria have been obtained ensuring the pre-defined H ∞ performance constraint and error variance boundedness simultaneously. The major feature of the constructed state estimator lies in that both the random saturation observation under uncertain occurrence probability and estimator gain perturbations have been taken into account in the same framework. Accordingly, a novel variance-constrained resilient H ∞ SE algorithm has been proposed, and the EGP has been obtained by solving RLMIs. Finally, a simulation example has been given to illustrate the validity of the proposed variance-constrained resilient H ∞ SE algorithm.