Outlier-resistant l 2-l ∞ state estimation for discrete-time memristive neural networks with time-delays

In this paper, the outlier-resistant - state estimation issue is investigated for a class of discrete-time memristive neural networks (DMNNs) with time-delays. Measurement outputs could occur unpredictable abnormal data due possibly to outliers from abnormal interferences, cyber-attacks as well as vibration of equipment. Obviously, the estimation performance could be degraded if these abnormal measurements were directly taken into the innovation to drive the estimation dynamics. As such, a novel outlier-resistant estimator for DMNNs with time-delays is developed to diminish the adverse effects from predictable abnormal data. By resorting to the robust analysis theory and the Lyapunov stability theory, some sufficient conditions are established to ensure a prescribed - performance index while achieving the stochastic stability of the estimation error dynamics. Furthermore, the desired estimator gains are derived by solving a convex optimization problem. Finally, a simulation example is provided to demonstrate the feasibility of the proposed design algorithm of outlier-resistant state estimators.


I. Introduction
For decades, a recurring research interest has been paid to the recurrent neural networks (RNNs) because of their broad applications in various areas such as handwriting recognition (Liu et al., 2003), speech recognition (Qian et al., 2019), natural language processing (Tarantino et al., 2018), object tracking (Feng et al., 2018) as well as computer vision (Kurtulmus & Kavdir, 2014). These successes can be attributed to the dynamic behaviour of the RNNs. In particular, the global stability of neural networks is great importance due to the applications in approximation and optimization issue, and an increasing number of results have been available in the literature, e.g. Lam et al. (2012), Liang et al. (2016), Liu et al. (2006Liu et al. ( , 2008, Kwon et al. (2013), Singh (2004), Shen et al. (2017), Wang et al. (2005), Zhang et al. (2017), Zhang, Tang, et al. (2015) and the references therein.
In practice, the RNNs can be realized via very large scale integration circuits where the connection weights are executed via resistors (Zhang, Shen, et al., 2013). Nonetheless, the resistor itself has inherent weakness regardless of its usefulness in realizing some specific functions. Taking the neural circuit as an example, the volatility of the resistor renders the state information disappear CONTACT Hongjian Liu hjliu1980@gmail.com in the absence of voltage and, moreover, huge amount of resistors leads inevitably to a substantial reduction of the integration degree of the neural circuit (Strukov et al., 2008). As we know, the memristor firstly proposed in Chua (1971) and materialized by Williams's team (Strukov et al., 2008). The memristors can 'memorize' a state of internal resistance based on the previous accumulation of applied voltage and current even if the system is in the power-off state. As such, the memristors have been introduced to take place of the resistors in implementing NNs, and the memristive neural networks (MNNs) have been considered as the most promising device to realize brain-like NN computers. Accordingly, the analysis problems (e.g. stability, convergence and synchronization) of many kinds of the MNNs have aroused much research attention, and some pioneering work has been available (see e.g. Feinberg et al., 2018;Pedretti et al., 2017;Wang et al., 2018). Nevertheless, little associated literature puts their emphasis on the stochastic stability issue for MNNs. Based on the practical instances, the state information of MNNs plays a key role for accomplishing optimization and approximation. However, it is a great challenge for us to obtain full neuronal states. In this case, information of the neurone states should be accessed from output information of the neurone states, and the state estimation issue (SEI) for MNNs becomes both practically significant and theoretically important . It is, therefore, the main motivation of this paper to establish a state estimation framework to solve such an appealing problem. So far, much research effort has recently been devoted to SEIs for MNNs, see e.g. Bao et al. (2018), Liu et al. (2018), Sakthivel et al. (2015), Zeng and Sheng (2018) and the reference therein.
It is worthy noting that, the SE algorithm of MNNs is sometimes required to be realized at a remote location in a networked environment, and this gives rise to the remote SEI. On the other hand, the measurement outliers are often happened because of the signal transmission along the communication channel of limited bandwidth. The measurement outliers are generally understood as contaminated measurements that deviate significantly from the normal measurements (Alessamdri & Zaccarian, 2018;Ding et al., 2017;Nakahira & Mo, 2018;Shen et al., 2020). Such kind of measurement outliers seriously degrades the estimation performance of the SEI for MNNs. To handle these contaminated measurements, some initial research has conducted for SE or filtering, see Akkaya and Tiku (2008), Alessamdri and Zaccarian (2018) and Gandhi and Mili (2010). For example, in Akkaya and Tiku (2008), a modified maximum likelihood estimator has been designed which is robust to the possible outliers, and a new prewhitening method has been applied to the filtering issue for discrete linear system in Gandhi and Mili (2010). Recently, in Alessamdri and Zaccarian (2018), by utilizing a saturated output, a novel observer has been constructed for SEIs. Nevertheless, to our best knowledge, very few results have been acquired so far on the remote SSI for delayed MNNs, not to mention the case where the contaminated measurements and the l 2 -l ∞ performance are both embraced, which is mainly due to the following two identified challenges: (1) how to design a suitable structure to remove the abnormal data? (2) how to propose a feasible approach to obtain the estimator gain. It is, therefore, our main purpose of this paper to deal with these two challenges and shorten such a gap. As such, we will make a dedicated effort in this article to deal with the listed two challenges coupled with the complexities resulting from the state-dependent parameters.
Inspired by the above discussions, in this paper, we aim at developing a outlier-resistant l 2 -l ∞ estimator for delayed MNNs. The primary contributions we deliver in this paper are outlined in threefold. (1) A artificial saturation term is introduced into the estimator of discrete-time MNNs with time-delays so as to attenuate the adverse effects from contaminated measurements. (2) Sufficient conditions are derived to achieve the prescribed l 2 -l ∞ performance while satisfying the stochastic stability of estimation error dynamics. (3) The desired estimator gains are derived by solving a linear matrix inequality. This paper is organized as follows. In Section II, a class of discrete-time MNNs with time-delays is presented. In Section III, by utilizing the robust analysis theory and the Lyapunov stability, some delay-dependent sufficient conditions are established in the form of matrix inequalities, and then the estimator gain is obtained by solving a convex optimization problem. In Section IV, a simulation example is provided. The paper is finally concluded in Section V.
Notations: The notation used here is fairly standard except where otherwise stated. N is used to be the set {1, 2, . . . , n}. R n and R n×m represent, respectively, the ndimensional Euclidean space and the set of all n × m real matrices. mod(a, b) represents the unique nonnegative remainder on division of the integer a by the positive integer b. If A is a matrix, A T represents the transpose of A, λ min (A) (λ max (A)) denote the smallest (largest) eigenvalue of A. diag{· · · } stands for a block-diagonal matrix. E{x} stands for the expectation of the stochastic variable x. x describes the Euclidean norm of a vector If not explicitly specified, matrices are assumed to have compatible dimensions.

II. Problem formulation and preliminaries
For the purpose of simple presentation, let us denotè In this paper, consider the following discrete-time memristive neural networks with time-delays: wherex(s) is the neurone state,ỳ(s) is the ideal measurement, andz(s) is the interested output. D(x(s)) is the selffeedback matrix, A(x(s)) = (a ij (x i (s))) nx×nx stands for the connection weight matrix, B(x(s)) = (b ij (x i (s))) nx×nx is the discretely delayed connection weight matrix, and C, M and N are known, real matrices with appropriate dimensions. The external disturbancev(s) ∈ R n p belongs to l 2 ([0, +∞); R p ), the nonlinear functionf (x(s)) means the neurone activation function, and the positive scalarτ represents the time delay. The initial condition of neural network (1) has the formx(s) = ϕ(s) for s ∈ [−τ , 0].
Benefiting from our previous research (Liu et al., 2020, the state-dependent matrices in (1) The uncertain matrices D(s), A(s) and B(s) can be described as the following structures whereH andÈ i (i = 1, 2, 3) are known real constant matrices, and unknown matrices Remark II.1: As the simplified models in Liu et al. (2020Liu et al. ( , 2018, all elements in D(x(s)), A(x(s)) and B(x(s)) are state-dependent and takes one of two values. Taking Under this kind of transformation, matricesH,È 1 ,È 2 andÈ 3 can be easily obtained, and their specific structures can be found in our previous research.
To reduce the effect of measurement outliers, we introduce the following outlier-resistant state estimator wherex(s) ∈ R nx is the estimate of the statex(s),ẑ(s) ∈ R nz represent the estimate of the outputz(s), and K is the estimator gain to be designed. sat(·) is a symmetric vector saturation function, which defined as follows.
The saturation function sat(·) : R nỳ −→ R nỳ is given as below Similar to the approach used in Ding, Wang, Shen, et al. (2012), the saturation function sat(ὴ(s)) can be divided into a linear and a nonlinear part like: where R 1 and R 2 are diagonal matrices satisfying 0 ≤ R 1 < I ≤ R 2 , the matrix R = R 2 − R 1 , and (ὴ(s)) is a nonlinear vector-valued function.
Remark II.2: In this paper, such a state estimator is named as outlier-resistant state estimator. Different from traditional saturation from the physical limitation, the stubborn term in (5) is an artificial constraint on the innovation 'ỳ − Mx(s)'. The advantage of outlier-resistant state estimator is that possible outliers in the measurements are suitably cancelled and their effect on the estimation error is reduced by the limiting effect of saturation. As such, the reliability of obtained estimation will be increased. Defineè Combining (1) and (5), we obtain the dynamics of the estimation error: It should be remarkable thatx(s) andè(s) coexist in (8), which bring some difficulties in solving the addressed problem. For this reason, introduce the new vectors The following augmented system can be readily presented: Before presenting further, we first introduce the following definition.
Our aim in this paper is to design a outlier-resistant state estimator of the form (5) for the system (1) such that the following requirements are met simultaneously.

III. Main result
Let us start with giving the following lemmas that will be used in the proof of our main result in this paper.

holds for all F(s) satisfying F T (s)F(s) ≤ I, if and only if there exists a scalar ε > 0, such that the following inequality
holds.

Lemma III.2 (S-procedure): Let N = N T , H and E be real matrices with appropriate dimensions, and F T F ≤ I. Then, the inequality N + HFE + (HFE) T < 0 holds if and only if there exists a scalar μ > 0 such that N
Theorem III.1: Consider DMNNs (1) and let the estimator gain K and the scalar γ > 0 be given. The augmented system (9) is stochastically stable while achieving the l 2 -l ∞ performance constraint if there exist a symmetric positive definite matrix P > 0, and two positive constant scalars λ 1 and λ 2 satisfying Proof: Construct the following Lyapunov function: Along the trajectory of the augmented system (9), one can calculate that

s)D T (s)PD(s)ὴ(s)
where Considering Assumption II.1 and (7), (14) is satisfied when the following inequality is true whereξ Now, let us discuss the stochastic stability of the augmented system (9) withζ (s) = 0.
From the inequality (16), for N > 0, it follows According to Definition II.1, the augmented estimation error system is stochastically stable. Finally, let us disclose the l 2 -l ∞ performance. To this end, define the cost In terms of (7), (15) and the zero initial condition, it follows According to Lemma III.1, we have that 1 < 0, which means J < 0, that is Moreover, we can obtain from (11) that which means This completes the proof.
In the above theorem, all uncertain matrices should be removed and an analysis form should be provided to carry out the design of the desired gain.
Theorem III.2: Consider DMNNs (1) and let the scalar γ > 0 be given. If there exist a symmetric positive definite matrix P = diag{P 1 , P 2 } > 0, a matrix X, and three positive scalars λ 1 , λ 2 and ς such that following linear matrix inequality holds: then the augmented estimation error system (9) is stochastically stable while achieving the l 2 -l ∞ performance constraint. Further, the desired estimator parameter is determined by K = P −1 2 X.

Proof:
To eliminate the uncertainties of in the matrix (12), using the well-known Schur lemma results in that 1 < 0 if and only if By considering K = P −1 2 X, it follows from (24) that: It follows from Lemma III.2 that the inequality (25) holds if and only if the inequality (22) is true, and the proof of Theorem III.2 is now complete.
Remark III.1: At this moment, we have solved the outlier-resistant SSI for a class of discrete-time MNNs subject to time-delays. Compared to existing literature, our main results has two notable features: (1) the outlierresistant SSI is new where a confidence-dependent saturation function is introduced into the estimator structure to reduce the adverse effects from the abnormal measurement outputs on the estimation error dynamics; and (2) a delay-dependent criterion is derived to verifies the stochastic stability of the corresponding estimation error dynamics with a guaranteed l 2 -l ∞ performance constraint.

V. Conclusions
In this paper, a outlier-resistant state estimator has been designed for a series of discrete-time MNNs subject to time-delays. By introducing a saturation function, the outlier-resistant state estimator can reduce the effect abnormal measurement outputs on the estimation results. First, addressed discrete-time MNNs (1) has been turned into a system involving uncertain parameters. According to this uncertain system, the robust analysis and Lyapunov stability theory can be utilized, and some conditions are established under which the estimation error system is stochastically stable and the prescribed l 2 − l ∞ performance is satisfied. Then, by using Matlab to solve a certain LMI to get the estimator gain matrix. At the last of this paper, a simulation example has been provided to show the effectiveness of the main results. The present outlier-resistant estimator design approach can be extended to in more complicated systems including sensor networks, genetic regulatory networks and social networks.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported in part by the National Natural Science