Event-triggered H∞ filtering for discrete-time Markov jump delayed neural networks with quantizations

ABSTRACT The problem of event-triggered filtering for discrete-time Markov jump delayed neural networks with quantizations is investigated in this paper. Firstly, an event-triggered communication scheme is proposed to determine whether or not the current sampled data can be transmitted to the quantizer. Secondly, a quantizer is used to quantify the sampled data, which can reduce the data transmission rate in the network. Next, through the analysis of network-induced delay's intervals, the discrete-time neural network, the event-triggered scheme and network-induced delay are unified into a discrete-time Markov jump delayed neural network. As a result, the sufficient conditions are obtained to guarantee the stability and performance of the augmented system and to present the filter design. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.


I. Introduction
In the past decades, neural networks have attracted more and more research attention due to their extensive applications in various areas such as signal processing, image processing and artificial intelligence. Recently, lots of significant subjects including stability analysis, feedback control and passivity analysis for delay recurrent neural networks have stirred a great deal of research interests (Jian & Zhao, 2015;Lin, He, Zhang, & Wu, 2018;Ma, Sun, Liu, & Xing, 2016;Yang, Li, & Huang, 2016).
In general, the neural networks display a characteristics of network modes jumps and such jumps are commonly considered to be determined by a time homogeneous Markov chain. With the aid of analysis and synthesis methodologies in the area of Markov jump linear systems (MJLSs) (Oliveira, Vargas, DoVal, & Peres, 2014;Xia, Sun, Teng, & Zhang, 2014), the resulting Markov jump neural networks (MJNNs) attract widely research interests, and a great number of literature are carried out for MJNNs, for more details, see Stoica and Yaesh (2008), Zhang, Zhu, Shi, and Zhao (2015), Zhuang, Ma, Xia, and Zhang (2016), Ren, Liu, Zhu, Zhong, and Shi (2017). Moreover, the filtering problems for neural networks are extensively studied by many researchers via various methodologies (Bao & Cao, 2011;Huang, Huang, & Chen, 2015). Nowadays, several methods are proposed to solve the problem of the H ∞ filter design (Liu, Liu, Cao, & Zhang, 2016;Wang, Shi, CONTACT Jinfeng Gao gaojf163@163.com Wang, Xue, Wang, & Lu, 2017). The authors in Wang et al. (2017) investigate the problem of eventbased H ∞ filtering for the discrete-time Markov jump system with network-induced delay. In , the problem of the adaptive event-triggered H ∞ filter design for a class of T-S fuzzy systems with time delay is studied. Overall, the H ∞ filter problem has received researchers' attention for a long time. Nevertheless, the study on MJNNs only has a short history and a few of vital results on this topic appear in the literature. Therefore, it is essential to pay attention to filter design in the various aspects of the MJNNs. As we all know, the control signals are transmitted in a shared communication network, and the bandwidth of this network is limited in networked control systems (NCSs). It is crucial to construct appropriate communication strategies to reduce the bandwidth occupation of the communication network. Compared with the traditional time-triggered scheme, the event-triggered scheme is utilized as an efficient way to reduce the burden of the communication network and improve the transmission efficiency. In the event-triggered communication scheme, the data is only transmitted if it meets certain conditions. Up to now, different event-triggered schemes have been proposed (Hu & Yue, 2012a;Liu, Tang, & Fei, 2016;Wang et al., 2017;Wang, Zhang, & Lu, 2018;Yuan, Wang, & Guo, 2017;Zha, Fang, Li, & Liu, 2017). The authors in Zha et al. (2017) are concerned with H ∞ output feedback control of event-triggered Markov jump systems (MJSs) with measured output quantizations. In Wang et al. (2018), the authors discuss the problem of the event-triggered H ∞ filter design for MJSs with output quantization. Hu Yue (2012a) investigate the problem of event-based H ∞ filtering for networked systems with communication delay. In Wang et al. (2017), the problem of event-based H ∞ filtering for discrete-time MJSs with network-induced delay is investigated. In , the problem of the H ∞ filter design for a class of neural network systems is studied with the event-triggered communication scheme and quantization. Motivated by the above results, it is necessary to design an event-triggered scheme to save the limited communication resources in the Markov jump delayed neural networks.
Signal quantization as another way to overcome this problem has been extensively researched. It is indispensable to quantize signal before it being transmitted through a communication channel with a limited bandwidth. In Hu and Yue (2012b), the authors are concerned with the control design problem of event-triggered networked systems with both state and control input quantizations. The H ∞ state estimation problem for a class of discrete-time neural networks with timevarying delays, randomly occurring quantizations and missing measurements is addressed in Zhang, Wang, Ding, and Liu (2015). The problem of robust H ∞ estimation for a class of MJNNs with transmission delay, measurement quantization and data packet dropout is studied in Zhuang et al. (2016). In Sasirekha, Rakkiyappan, Cao, Wan, and Alsaedi (2017), H ∞ state estimation of discrete-time Markov jump neural networks with general transition probabilities and output quantization is described. The effect of the quantization on the networked control systems is greater than the traditional control systems. To the best of the authors'knowledge, little work has been done in event-triggered H ∞ filtering for discrete-time Markov jump delayed neural networks with quantizations. This situation motivates our current investigation.
Inspired by the results mentioned above, we focus on the event-triggered H ∞ filtering for a class of discretetime Markov jump delayed neural networks with quantizations. The main work is as follows: (a) Event-triggered scheme for discrete-time Markov jump delayed neural networks, to reduce network resource wastage; (b) Quantization signals to save limited bandwidth and energy consumption; (c) Reduce system conservatism and obtain sufficient conditions by using the Jenson inequality, to guarantee the stability with the H ∞ performance index of the augmented system. Then, the filter parameters are designed.
The rest of the paper is organized as follows. In Section II, an H ∞ filter design is addressed for the discrete-time Markov jump delayed neural networks with the event-triggered communication scheme and quantization. H ∞ filter performance analysis and the method of filter design are presented in Section III. A numerical example to illustrate the effectiveness of the obtained results is proposed in Section IV. Section V is the conclusion.
Notation: R n and R n×m denote the n dimensional Euclidean space, and the set of n × m real matrices; superscript T and −1 represent the transposition of vectors or matrices and matrix inverse, respectively. The notation P > 0(≤ 0) means P is real symmetric positive definite. I and 0 represent identity matrix and zero matrix, respectively. In addition, * is used to denote the symmetry entries of symmetry matrices.

A. System description
Consider a discrete-time n-neuron network with timevarying delays as follows: where x(k) ∈ R n is the state vector of the neural network, y(k) ∈ R m is the measured output, z(k) ∈ R p denotes the neural signal to be estimated, ω(k) ∈ R q is the external disturbance with ω(k) ∈ L 2 [0, ∞). The positive integer d(k) ∈ [d m , d M ] describes the known time-varying delay, and d m and d M are the lower and upper bound of d(k), respectively. f (x(k)) ∈ R n and g(x(k)) ∈ R n are the neuron activation function. Respectively, A(r k ), B(r k ), C(r k ), D(r k ), E(r k ), L(r k ) are known real constant matrices with appropriate dimensions. r k is the discrete-time Markov jump process taking values in a finite space S = {1, 2, . . . , N}. The transition probability matrix = π ij (i, j ∈ S) is given by The probability transfer matrix is as follows: The following H ∞ filter will be adopted: Time-delay is inevitable during signals transmit through limited network bandwidth. It is supposed that the time-varying delay in network communication is τ k ∈ [0,τ ],τ is the maximal network-induced delay. The event generator sends out the signal at the time s l (l = 0, 1, 2, . . . , ∞) and reaches the controller at time instant s l + τ s l . Consider the effect of zero-order (ZOH), the actual measurements can be described as

B. Event-triggered scheme
To reduce the communication burden of network and guarantee the performance of system, an event generator is introduced between the sensor and the quantizer. Similar to Hu and Yue (2012a), where the current sample data y(k) is directly transmitted to the event-triggered mechanism, whether the latest information can be sent out and transmitted via the communication channels depends on the following condition: where i ∈ R m are symmetric positive matrices to be designed and σ i ∈ [0, 1) is a given scalar parameter. Only when the current sampled sensor measurement y(k + j), (j = 1, 2, . . . , ) and the latest transmitted sensor measurement y(k) satisfying the inequality (4), the current sampled sensor measurement y(k + j) will be sent out and transmitted to the quantizer q(·).
Remark II.1: The sensor measurement is sampled at time k ∈ N and the next sensor measurement is at time k+1.
For simplicity, the instant (k + j) is replaced by k j in follows.
According to Yue, Tian, and Han (2013), using the similar methods, τ s l is the delay at the instant s l , τ s l ∈ [0,τ ] Case A: If s l + 1 +τ ≥ s l+1 + τ s l+1 − 1, define a function Case B: If s l + 1 +τ < s l+1 + τ s l+1 − 1, consider the following intervals: where h ∈ Z + and satisfies h ≥ 1. It can be easily shown that there exists a positive integer m, such that and y(s l ), from (5) to (7), we can obtain Then it can be easily shown that From the definition of e(k) and the triggering algorithm (4), it can be seen that when k ∈ [s l + τ s l , s l+1 + τ s l+1 − 1],

C. Event-triggered quantized H ∞ filtering problem
In order to reduce the communication burden, the quantizer is employed. According to Hu and Yue (2012b), the quantizer q(·) is defined as q(y) = [q 1 (y 1 )q 2 (y 2 ) · · · q n (y n )] T , where q s (y s ) (s = 1, 2, . . . , n) are chosen as logarithmic quantizers given by where δ q s = (1 − ρ q s )/(1 + ρ q s ), 0 < ρ q s < 1, ρ q s is the quantization density with is a given constant. Moreover, similar to Peng and Tian (2007), the quantification level is defined as . . , n, then the actual input of filterỹ(k) can be expressed by the sector bound method as (Fu & Xie, 2005) Define the new state vectorx(k) = [x T (k), x T f (k)] T and the filtering error vectorz(k) = z(k) − z f (k), then the augmented system can be obtained as

wherē
Before ending this section, we recall the following definition and lemma, which will help us in deriving the main results.
(17) for all x, y ∈ R n , where U 1 , U 2 , V 1 and V 2 are constant real matrices of appropriate dimensions.
Definition II.1: The augmented system (16) with ω(k) = 0 is asymptotically stable in mean square, if for any initial conditions, such that Definition II.2: Given a scalar γ > 0, for all nonzero ω(k) ∈ L 2 [0, ∞), the filtering system (16) is asymptotically stable with an H ∞ performance index γ if it is asymptotically stable and the filtering errorz(k) satisfies
Then, continue to demonstrate the H ∞ performance of the system (16). When external disturbance ω(k) = 0 By using the Schur complement lemma, (20) implies that < 0. Hence, the augmented system (16) is stochastically stable with an H ∞ performance index γ if (20) and (21) are satisfied. This completes the proof.

Moreover, if the above conditions are feasible, the parameter matrices of the filter are given by
Therefore, LMIs (20) is equivalent to LMIs (37). Applying the Schur complement, 3i P T 2i > 0, P 2i and P 3i are nonsingular matrices. Note that P 2i and P 3i cannot be directly derived from the condition (37), the transfer function T fromỹ(k) to z f (k) can be described as So the parameter matrices (39) of the filter are readily obtained. This completes the proof.
In the following, we will deal with the nonlinear terms in Theorem III.2. Theorem III.3: For given scalars 0 ≤ σ i < 1, d m , d M , τ M , and appropriate scalar δ, the filtering error system (16) is stochastically stable with an H ∞ performance index γ under the event-triggered scheme (4), if there exist the real matri- andC fi with appropriate dimension and positive scalars α 1 > 0, α 1 > 0 satisfy P 1i − X i > 0 and the following LMIs: ⎡ Proof: The matrix (37) can be rewritten as the following: According to Lemma II.1, there exists > 0 such that and 2 q ≤ δ 2 I By the Schur complement, linear matrix inequality (41) can be easily obtained. This completes the proof.

IV. Numerical example
In this section, a numerical example of system (16) is provided as follows: Mode 1: The neuron activation functions are taken as It is easy to obtain U 1 , U 2 and V 1 , V 2 satisfying Assumption 1 Then, we consider two cases for this discrete-time Markov jump delayed neural networks. Case 1: σ 1 = σ 2 and Case 2: σ 1 = σ 2 . For Case 1, the event-triggered parameter σ 1 = σ 2 = 0.2, when d m = 1, d M = 4, the maximum delay in communication network τ M = 5, and = 1, γ = 3, by using the LMI toolbox of Matlab, it is easy to obtain the following matrices: For Case 2, the event-triggered parameter σ 1 = 0.2, σ 2 = 0.1, when d m = 1, d M = 4, the maximum delay in communication network τ M = 5, and = 1, γ = 3, by using the LMI toolbox of Matlab, it is easy to obtain the following matrices:  (Sasirekha et al., 2017;Zhuang et al., 2016), they are difficult to be applied directly to deal with the case that exist in the event-triggered scheme. Comparing the existing literature and simulation results, we can easily conclude that the event-triggered mechanism can reduce the use of network bandwidth effectively.       Remark IV.2: In , the authors designed a H ∞ filter for a class of neural network systems with quantization and event-triggered schemes. The simulation results of this article can further illustrate that the event-triggered mechanism scheme and quantization can reduce the use of network resources, but there are no relevant research on discrete-time Markov jump delayed neural networks. This situation motivates our current investigation.

V. Conclusion
The event-triggered H ∞ filter design problem for discrete-time Markov jump delayed neural networks with quantizations is studied in this paper. To reduce the communication bandwidth utilization, an eventtriggered communication scheme and a quantizer are introduced to the framework. Based on the analysis of network-induced delay, a unified discrete-time Markov jump filter error system with time-delay is constructed to describe the event-triggered scheme, network-induced delays, quantizations and the neural network system together. The sufficient conditions of stochastically H ∞ norm bound are obtained for the estimation error system with event-triggered scheme and quantization. Finally, a simulation example is presented to illustrate the effectiveness of the designed method.