Event-Triggered State Estimation for a Class of Delayed Recurrent Neural Networks with Sampled-Data Information

Abstract

The paper investigates the state estimation problem for a class of recurrent neural networks with sampled-data information and time-varying delays. The main purpose is to estimate the neuron states through output sampled measurement; a novel event-triggered scheme is proposed, which can lead to a significant reduction of the information communication burden in the network; the feature of this scheme is that whether or not the sampled data should be transmitted is determined by the current sampled data and the error between the current sampled data and the latest transmitted data. By using a delayed-input approach, the error dynamic system is equivalent to a dynamic system with two different time-varying delays. Based on the Lyapunov-krasovskii functional approach, a state estimator of the considered neural networks can be achieved by solving some linear matrix inequalities, which can be easily facilitated by using the standard numerical software. Finally, a numerical example is provided to show the effectiveness of the proposed event-triggered scheme.

1. Introduction

The research of neural networks has been paid much attention during the past few years, due to its potential application in various fields, such as image processing, pattern recognition, and associative memory [1–5]. As a special class of nonlinear dynamical systems, the dynamic behavior of recurrent neural networks has been one of the most important issues. In particular, the analysis problems of stability and synchronization of recurrent neural networks have received great attention and a number of profound results have been proposed [6–12].

In many application, such as signal processing and control engineering, for large-scale neural networks, it is quite common that only partial information can be accessible from the network outputs. Therefore, it is of great significance to estimate the neuron states through available output measurements of the networks and then utilizes the estimated neuron states to achieve certain design objectives; note that state estimation problem for neural networks has been hot reach topics that have drawn considerable attention, and many profound results have been available in the literature [13–25]. The authors in [13] studied the problem of state estimation for a class of delayed neural networks; the traditional monotonicity and smoothness assumption on the activation function had been removed. The design problem of state estimator for a class of neural networks with constant delays was investigated in [14], where a delay-dependent criterion for existence of the estimator was proposed. As an extension, The authors in [14, 15] further discussed state estimation for neural networks with time-varying delays. In practice, sometimes a neural network has finite state modes and modes may switch from one to another at different times. On the other hand, discrete-time neural networks could be more suitable to model digitally transmitted signals in dynamical way; based on the above reason, The authors in [16] investigated state estimation problem for a new class of discrete-time neural networks with Markovian jumping parameters and mode dependent mixed time-delays, where he discrete and distributed delays were mode-dependent. Different from the stuelies in [16, 17] which considered state estimation for Markovian jumping delayed continuous-time recurrent neural networks, where only matrix parameters were mode-dependent. Similar to [16], for continuous-time recurrent neural networks with discrete and distributed delays, state estimation was also investigated in [18]. In [19, 20], synchronization and state estimation had been studied for discrete-time complex networks with distributed delays; it was noticed that in [20], a novel notion of bounded 𝐻_∞ synchronization had been first defined to characterize the transient performance of synchronization. Some robust state estimation problems for uncertain neural networks with time-varying delays had been investigated in [21–23], where the parameter uncertainties are assumed to be norm bounded; some sufficient conditions were presented to guarantee the existence of the desired state estimator. Taking into account the stochastic properties of time-varying delays, the authors in [24] discussed state estimation problem for a class of discrete-time stochastic neural networks with random delays; sufficient delay-distribution-dependent conditions were established in terms of linear matrix inequalities (LMIs) that guarantee the existence of the state estimator.

The sampled-data control theory had attracted much attention due to its effectiveness in engineering applications. Especially, a new approach to deal with the sampled-data control problems had been proposed in [26], where the sampling period had been converted into time-varying delay. As its extension, the authors in[27] investigated the sampled-data state estimation problem for a class of recurrent neural networks with time-varying delays, where the sampled measurements had been used to estimate the neuron states. Using a similar approach, where the sampled-data synchronization control problem was investigated in [28] for a class of general delayed complex networks, the sampled-data feedback controllers were designed in terms of the solution to certain linear matrix inequalities. But in the above references, the sampling rate for each signal is the same; but in the actual system, it may be varying from sample to sample owing to unpredictable perturbations; this factor was considered in [29], the problem of robust 𝐻_∞ control was investigated for sampled-data systems with probabilistic sampling, where two different sampling periods were considered whose occurrence probabilities were given constants and satisfied Bernoulli distribution. In [30], stochastic sampled-data approach was used for studying the problem of distributed 𝐻_∞ filtering in sensor networks, by converting the sampling periods into bounded time-delays, the design problem of 𝐻_∞ filters amounted to solving the 𝐻_∞ filtering problem for a class of stochastic nonlinear systems with multiple bounded time delays. In [31], the sampled-data synchronization control problem was addressed, where the sampling period was time varying and switched between two different values in a random way. It is worth noting that most of the above results were involved the traditional approach of sampling at pre-specified time instances, which was called time-triggered sampling; this sampling method may lead to an inherently periodic transmission and produce many useless messages if the current sampled signal had not significantly changed in contrast to the previous sampled signal, which led to a conservative usage of the communication resources. Recently, event-triggered scheme provided an effective approach of determining; its main property was that the signal was sampled and only some functions of the system state or output measurement exceeded threshold. Compared with periodic sampling method, the event-triggered scheme could reduce the burden of the communication and also preserve the desired properties of the ideal continuous state feedback system, such as stability and convergence. The utilization on event-triggered scheme could be found in many literatures such as [32–37]. The event-triggered 𝐻_∞ control design was investigated in [32] for networked control systems with uncertainties and transmission delays, and a novel event-triggered scheme was proposed. The study in [33] was concerned with the control problem of event-triggered networked systems with both state and control input quantizations. In [34], the problems of exponential stability and 𝐿₂-gain analysis of event-triggered networked control systems were studied, where the event-triggered conditions were proposed in the sensor side and controller side. In [35–37], the consensus problems for multiagent systems were investigated by event-triggered control, where different trigger functions were proposed. Unfortunately, as far as we know, up to now, no theoretical results are given for state estimation of recurrent neural networks with time-varying delays based on event-triggered scheme. The purpose of our study is to fill the gap.

Motivated by the above discussion, the paper is concerned with the sampled-data state estimation problem for a class of recurrent neural networks with time-varying delays. The main purpose is to estimate the neuron states through output sampled measurement, and a novel event-triggered scheme is proposed, which can lead to a significant reduction of the information communication burden in the network. By using a delayed-input approach, the error dynamics system is equivalent to a dynamic system with two different time-varying delays. Based on constructing a Lyapunov-Krasovskii functional and employing some analysis techniques, a state estimator of the considered neural networks can be achieved by solving some linear matrix inequalities, which can be easily facilitated by using the standard numerical software. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

The main contributions of this paper are highlighted as follows.(1)The novel event-triggered scheme is proposed, compared with a time-triggered periodic communication scheme, since the proposed communication scheme only depends on the state at the sampled instant and the state error between the current sampled instant and the latest transmitted state. Therefore, the number of the transmitted state signals through the network could be reduced apparently. (2)Sufficient conditions obtained are in the form of linear matrix inequalities which can be readily solved by using the LMI toolbox in Matlab, and the solvability of derived conditions dependents on not only trigger parameters and sampling period but also the size of the delay.

Notation 1. The notation used here is fair standard except where otherwise stated. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space and ℝ^𝑛×𝑚 is the set of real 𝑛×𝑚 matrices. The superscript 𝑇 represents the transpose of the matrix (or vector). 𝐼 denotes the identity matrix of compatible dimensions. The asterisk represents the symmetric block in one symmetric matrix. diag{⋯} stands for a block-diagonal matrix. The notation 𝑋≥0 (𝑋>0) means that 𝑋 is positive semi-definite (positive definite). ‖⋅‖is the Euclidean norm in ℝ^𝑛. If they are not explicitly specified, arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.

2. Preliminaries

Consider a class of recurrent neural networks with time-varying delays as follows: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑊₀𝑔(𝑥(𝑡))+𝑊₁𝑔(𝑥((𝑡𝑦(𝑡)(𝑡),𝑡−𝜏)))=𝐶𝑥 (2.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector associated with 𝑛 neurons; 𝐴=diag{𝑎₁,𝑎₂,…,𝑎_𝑛} is a positive diagonal matrix; 𝑊₀∈ℝ^𝑛×𝑛 and 𝑊₁∈ℝ^𝑛×𝑛 are the connection weight matrix and the delayed connection weight matrix, respectively; 𝜏(𝑡)∈[0,𝜏] is the time-varying bound delay; 𝐶∈ℝ^𝑚×𝑛 is a constant matrix; 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑚(𝑡))^𝑇∈ℝ^𝑚 denotes the output vector; 𝑔(𝑥(𝑡))=[𝑔₁(𝑥₁(𝑡)),𝑔₂(𝑥₂(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇∈ℝ^𝑛 represents the neuron activation function with 𝑔(0)=0.

In this paper, the measurement output is sampled before it enters the estimator; based on the sampling technique and zero-order hold, the actual output can be described as 𝑦(𝑡)=𝑦𝑡_𝑘=𝐶𝑥𝑡_𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1, (2.2) where 𝑦(𝑡)∈ℝ^𝑚 is the actual output of the estimator, and 𝑡_𝑘 denotes the sampling instant satisfying lim_𝑘→∞𝑡_𝑘=∞.

Remark 2.1. In practical systems, periodic sampling mechanism may often lead to sending many unnecessary signals through the networks, which will increase the load of network transmission and waste the network bandwidth; therefore, it is significant to introduce a mechanism to determine which sampled signal should be sent out or not. As stated in [32, 33], the event-trigger sampling scheme is effective way because they can reduce the traffic and the power consumption.
The sampled data 𝑦(𝑡_𝑘+𝑗) is transmitted (or released) by the event generator only when the current sampled value 𝑦(𝑡_𝑘+𝑗) and the previously transmitted one 𝑦(𝑡_𝑘) satisfy the following judgement algorithm: 𝑦𝑡_𝑘+𝑗−𝑦𝑡_𝑘^𝑇𝑊𝑦𝑡_𝑘+𝑗−𝑦𝑡_𝑘<𝜎𝑦^𝑇𝑡_𝑘+𝑗𝑊𝑦𝑡_𝑘+𝑗, (2.3) where 𝑊∈ℝ^𝑚×𝑚 is a positive matrix, and 𝜎∈[0,1) is a positive scalar. The sampled state 𝑦(𝑡_𝑘+𝑗) satisfying the inequality (2.3) will not be transmitted; only the one that exceeds the threshold in (2.3) will be sent to the estimator. Specially, when 𝜎=0, the inequality (2.3) is not satisfied for almost all the sampled state 𝑦(𝑡_𝑘+𝑗), and the event-triggered scheme reduces to a periodic release scheme.

Remark 2.2. From event-triggered algorithm (2.3), it is easily seen that all the released signals are subsequences of the sampled data, that is, the set of the release instants {𝑡₀,𝑡₁,𝑡₂…}⊆{0,1,2,…}. The amount of {𝑡₀,𝑡₁,𝑡₂…} depends on not only the value of 𝜎 but also the variation of the system output.

Suppose that the time-varying delay in network communication is 𝑑_𝑘∈[0,𝑑](𝑘=1,2,…,+∞), the output 𝑦(𝑡) in (2.2) can be rewritten as 𝑦(𝑡)=𝑦𝑡_𝑘=𝐶𝑥𝑡_𝑘,𝑡∈𝑡_𝑘+𝑑_𝑘,𝑡_𝑘+1+𝑑_𝑘+1. (2.4)

Substituting the output (2.4) into the judgement algorithm (2.3), we can obtain 𝑥𝑡_𝑘+𝑗−𝑥𝑡_𝑘^𝑇𝐶^𝑇𝑊𝐶𝑥𝑡_𝑘+𝑗−𝑥𝑡_𝑘<𝜎𝑥^𝑇𝑡_𝑘+𝑗𝐶^𝑇𝑊𝐶𝑥𝑡_𝑘+𝑗. (2.5)

For technical convenience, similar to [32, 33], consider the following two intervals: 𝑡_𝑘+𝑑_𝑘,𝑡_𝑘+ℎ+𝑑,𝑡_𝑘+𝑙ℎ+𝑑,𝑡_𝑘+𝑙ℎ+ℎ+𝑑, (2.6) where 𝑙 is a positive integer and ℎ is a sampling period.(1)if 𝑡_𝑘+ℎ+𝑑>𝑡_𝑘+1+𝑑_𝑘+1, define a function 𝑑(𝑡) as follows: 𝑑(𝑡)=𝑡−𝑡_𝑘,𝑡∈𝑡_𝑘+𝑑_𝑘,𝑡_𝑘+1+𝑑_𝑘+1. (2.7) It can easily be obtained that the following inequality holds: 𝑑_𝑘≤𝑑(𝑡)≤𝑡_𝑘+1−𝑡_𝑘+𝑑_𝑘+1≤ℎ+𝑑. (2.8)(2)if 𝑡_𝑘+ℎ+𝑑<𝑡_𝑘+1+𝑑_𝑘+1, there exists a positive integer 𝑚, such that 𝑡_𝑘+𝑚ℎ+𝑑<𝑡_𝑘+1+𝑑_𝑘+1<𝑡_𝑘+𝑚ℎ+ℎ+𝑑. (2.9) It can be easily shown that 𝑡_𝑘+𝑑_𝑘,𝑡_𝑘+1+𝑑_𝑘+1=𝐼₁∪𝐼₂∪𝐼₃, (2.10)where 𝐼₁=𝑡_𝑘+𝑑_𝑘,𝑡_𝑘𝐼+ℎ+𝑑₂=^𝑙−1_𝑚=1𝐼^𝑚₂𝐼^𝑚₂=𝑡_𝑘+𝑚ℎ+𝑑,𝑡_𝑘𝐼+𝑚ℎ+ℎ+𝑑₃=𝑡_𝑘+𝑙ℎ+𝑑,𝑡_𝑘+1+𝑑_𝑘+1. (2.11)

Define a function 𝑑(𝑡) as 𝑑(𝑡)=𝑡−𝑡_𝑘𝑡∈𝐼₁𝑡−𝑡_𝑘−𝑚ℎ𝑡∈𝐼^𝑚₂(𝑚=1,2,…,𝑙−1)𝑡−𝑡_𝑘−𝑙ℎ𝑡∈𝐼₃. (2.12)

From the definition of 𝑑(𝑡) defined in (2.12), we can derive 0≤𝑑_𝑘≤𝑑(𝑡)<ℎ+𝑑,𝑡∈𝐼₁0≤𝑑_𝑘≤𝑑≤𝑑(𝑡)<ℎ+𝑑,𝑡∈𝐼^𝑚₂(𝑚=1,2,…,𝑙−1)0≤𝑑_𝑘≤𝑑≤𝑑(𝑡)<ℎ+𝑑,𝑡∈𝐼₃. (2.13) From (2.13), it can be derived that 0≤𝑑(𝑡)<𝑑_𝑀, where 𝑑_𝑀=ℎ+𝑑. For 𝑡∈[𝑡_𝑘+𝑑_𝑘,𝑡_𝑘+1+𝑑_𝑘+1), we define 𝑒_𝑘(𝑡)=0𝑡∈𝐼₁𝑥𝑡_𝑘+𝑚ℎ−𝑥𝑡_𝑘𝑡∈𝐼^𝑚₂()𝑥𝑡𝑚=1,2,…,𝑙−1_𝑘+𝑙ℎ−𝑥𝑡_𝑘𝑡∈𝐼₃. (2.14)

Combining the above definitions of 𝑑(𝑡) and 𝑒_𝑘(𝑡), the algorithm (2.5) can be rewritten as 𝑒^𝑇_𝑘(𝑡)𝐶^𝑇𝑊𝐶𝑒_𝑘(𝑡)<𝜎𝑥^𝑇(𝑡−𝑑(𝑡))𝐶^𝑇𝑊𝐶𝑥(𝑡−𝑑(𝑡)). (2.15)

Based on the available sampled measurement 𝑦(𝑡), the following state estimator is adopted: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝐾𝑦(𝑡)−𝐶𝑥(𝑡), (2.16) where 𝐾 is feedback gain matrix to be designed, and 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 is estimator state vector.

Setting 𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡), the estimation error dynamics can be obtained from (2.1) and (2.16), and it follows that 𝑒̇(𝑡)=−(𝐴+𝐾𝐶)𝑒(𝑡)+𝐾𝐶𝑥(𝑡)−𝐾𝐶𝑥(𝑡−𝑑(𝑡))+𝐾𝐶𝑒_𝑘(𝑡)+𝑊₀𝑔(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−𝜏(𝑡))). (2.17)

Let 𝑥(𝑡)=(𝑥^𝑇(𝑡),𝑒^𝑇(𝑡))^𝑇, we can get the following augmented system from (2.1) and (2.17) 𝑥̇‌(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝑑(𝑡))+𝑊₀𝑔(𝐻𝑥(𝑡))+𝑊₁𝑔(𝐻𝑥(𝑡−𝜏(𝑡)))+𝐶𝑒_𝑘(𝑡), (2.18) where 𝐴=−𝐴0𝐾𝐶−𝐴−𝐾𝐶𝐵=00−𝐾𝐶0𝑊₀=𝑊₀𝑊₀𝑊₁=𝑊₁𝑊₁𝐻^𝑇=𝐼00𝐶=.𝐾𝐶 (2.19)

Before giving the main results, the following assumption, definition, and lemma are essential in establishing our main results.

Assumption 2.3 (see, [27]). The activation function 𝑔(⋅) satisfies the following sector-bounded condition: 𝑔(𝑥)−𝑈₁𝑥^𝑇𝑔(𝑥)−𝑈₂𝑥≤0, (2.20) where 𝑈₁ and 𝑈₂ are two real constant matrices with 𝑈₂−𝑈₁≥0.

Definition 2.4 (see, [27]). The augmented system (2.18) is exponentially stable, if there exist two constants 𝛼>0 and 𝛽>0, such that ‖𝑥(𝑡)‖²≤𝛼𝑒^−𝛽𝑡sup_{−𝑟≤𝜃≤0}𝜙(𝜃)², (2.21) where 𝜙(⋅) is in the initial function system (2.18) as 𝜙(𝑡)=𝑥(𝑡), 𝑡∈[−𝑟,0].

Lemma 2.5 (see, [38, 39]). Suppose 𝜏(𝑡)∈[𝜏_𝑚,𝜏_𝑀], 𝑄_𝑖(𝑖=1,2,3) are some constant matrices with appropriate dimensions, then 𝑄₁+𝜏_𝑀−𝜏(𝑡)𝑄₂+𝜏(𝑡)−𝜏_𝑚𝑄₃<0, (2.22) if the following inequalities hold 𝑄₁+𝜏_𝑀−𝜏_𝑚𝑄₂𝑄<0₁+𝜏_𝑀−𝜏_𝑚𝑄₃<0. (2.23)

Lemma 2.6 (see, [40]). For any constant positive matrix 𝑇∈ℝ^𝑛×𝑛, scalar 𝜏₁≤𝜏(𝑡)<𝜏₂ and vector function 𝑥̇(𝑡)∶[−𝜏₂,𝜏₁]→ℝ^𝑛 such that the following integration is well defined, then it holds that −𝜏₂−𝜏₁^𝑡−𝜏1_𝑡−𝜏2𝑥̇^𝑇(𝑣)𝑇𝑥̇(𝑣)𝑥𝑑𝑣≤𝑡−𝜏₁𝑥((𝑡𝑥𝑡−𝜏))𝑡−𝜏₂^𝑇−𝑇𝑇0𝑥∗−2𝑇𝑇∗∗−𝑇𝑡−𝜏₁𝑥((𝑡𝑥𝑡−𝜏))𝑡−𝜏₂. (2.24)

3. Main Results

In this section, we design a sampled-date estimator with form (2.18) for recurrent neural networks with time-varying delay based event-triggered control.

The system (2.18) can be rewritten as 𝑥̇‌(𝑡)=𝒜𝜉(𝑡), (3.1) where 𝜉(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝑑(𝑡)),𝑥^𝑇𝑡−𝑑_𝑀,𝑥^𝑇(𝑡−𝜏(𝑡)),𝑥^𝑇(𝑡−𝜏),𝑔^𝑇(𝐻𝑥(𝑡)),𝑔^𝑇(𝐻𝑥(𝑡−𝜏(𝑡))),𝑒^𝑇_𝑘^𝑇𝒜=𝐴,𝐵,0,0,0,𝑊₀,𝑊₁,𝐶. (𝑡)(3.2)

Theorem 3.1. Suppose that Assumption 2.3 holds, for given estimator gain matrix 𝐾, the augmented system (3.1) is exponentially stable, if there exist some positive definite matrices 𝑃>0, 𝑄_𝑖>0, 𝑅_𝑖>0 and 𝑆_𝑖, 𝑇_𝑖(𝑖=1,2) with appropriate dimension, and two positive scalars 𝛼>0, 𝛽>0, such that the following linear matrix inequalities hold: Π_𝑖=ΠΦ₁Φ₂Φ₃^(𝑖)∗−𝑑_𝑀𝑅₁00∗∗−𝜏𝑅₂0∗∗∗−𝜏𝑅₂<0(𝑖=1,2), (3.3) where ΠΠ=¹¹Π₁₂0Π₁₃0Π₁₄𝑃𝑊₁𝑃𝐶∗Π₂₂1𝑑_𝑀𝑅₁00000∗∗Π₃₃00000∗∗∗Π₄₄Π₄₅0−𝛽𝑈₂0∗∗∗∗Π₅₅000∗∗∗∗∗−𝛼𝐼00∗∗∗∗∗∗−𝛽𝐼0∗∗∗∗∗∗∗−𝐶^𝑇Φ𝑊𝐶₁=𝑑_𝑀𝑅₁𝐴𝑑_𝑀𝑅₁𝐵000𝑑_𝑀𝑅₁𝑊₀𝑑_𝑀𝑅₁𝑊₁𝑑_𝑀𝑅₁𝐶^𝑇Φ₂=𝜏𝑅₂𝐴𝜏𝑅₂𝐵000𝜏𝑅₂𝑊₀𝜏𝑅₂𝑊₁𝜏𝑅₂𝐶^𝑇Φ₃⁽¹⁾=𝜏𝑆^𝑇₁00𝜏𝑆^𝑇₂0000^𝑇Φ₃⁽²⁾=000𝜏𝑇^𝑇₁𝜏𝑇^𝑇₂000^𝑇Π₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝑄₁+𝑄₂−1𝑑_𝑀𝑅₁+𝑆₁+𝑆^𝑇₁−𝛼𝑈₁Π₁₂=𝑃1𝐵+𝑑_𝑀𝑅₁Π₁₃=𝑆^𝑇₂−𝑆₁Π₁₄=𝑃𝑊₀−𝛼𝑈₂Π₂₂2=−𝑑_𝑀𝑅₁Π+𝜎Ω₃₃=−𝑄₁−1𝑑_𝑀𝑅₁Π₄₄=−𝑆₂−𝑆^𝑇₂+𝑇₁+𝑇^𝑇₁−𝛽𝑈₁Π₄₅=−𝑇₁+𝑇^𝑇₂Π₅₅=−𝑄₂−𝑇₂−𝑇^𝑇₂𝐶Ω=^𝑇.𝑊𝐶000 (3.4)

Proof. Construct the following Lyapunov-Krasovskii functional candidate: 𝑉(𝑡,𝑥(𝑡))=𝑉₁(𝑡,𝑥(𝑡))+𝑉₂(𝑡,𝑥(𝑡))+𝑉₃(𝑡,𝑥(𝑡))+𝑉₄(𝑡,𝑥(𝑡)), (3.5) where 𝑉₁(𝑡,𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)𝑉₂(𝑡,𝑥(𝑡))=^𝑡_{𝑡−𝑑𝑀}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑉𝑑𝑠₃(𝑡,𝑥(𝑡))=^𝑡_{𝑡−𝑑𝑀}^𝑡_𝜃𝑥̇‌^𝑇(𝑠)𝑅₁𝑥̇‌(𝑠)𝑉𝑑𝑠𝑑𝜃₄(𝑡,𝑥(𝑡))=^𝑡_𝑡−𝜏^𝑡_𝜃𝑥̇‌^𝑇(𝑠)𝑅₂𝑥̇‌(𝑠)𝑑𝑠𝑑𝜃, (3.6) and 𝑃>0, 𝑄_𝑖>0 and 𝑅_𝑖>0(𝑖=1,2) are matrices to be determined.
The derivative of 𝑉_𝑖(𝑡,𝑥(𝑡))(𝑖=1,2,3,4) along the trajectory of system (3.1) can be shown as follows: 𝑉̇₁(𝑡,𝑥(𝑡))=2𝑥^𝑇(𝑡)(𝑡)𝑉̇𝑃𝒜𝜉₂(𝑡,𝑥(𝑡))=𝑥^𝑇(𝑡)𝑄₁+𝑄₂𝑥(𝑡)−𝑥^𝑇𝑡−𝑑_𝑀𝑄₁𝑥𝑡−𝑑_𝑀−𝑥^𝑇(𝑡−𝜏)𝑄₂𝑥()𝑉̇𝑡−𝜏₃(𝑡,𝑥(𝑡))=𝑑_𝑀𝑥̇‌^𝑇(𝑡)𝑅₁𝑥̇‌(𝑡)−^𝑡_{𝑡−𝑑𝑀}𝑥̇‌^𝑇(𝑠)𝑅₁𝑥̇‌(𝑠)𝑑𝑠=𝑑_𝑀𝜉^𝑇(𝑡)𝒜^𝑇𝑅₁𝒜𝜉(𝑡)−^𝑡_{𝑡−𝑑𝑀}𝑥̇‌^𝑇(𝑠)𝑅₁𝑥̇‌(𝑠)𝑉̇𝑑𝑠₄(𝑡,𝑥(𝑡))=𝜏𝑥̇^𝑇(𝑡)𝑅₂𝑥̇‌(𝑡)−^𝑡_𝑡−𝜏𝑥̇‌^𝑇(𝑠)𝑅₂𝑥̇‌(𝑠)𝑑𝑠=𝜏𝜉^𝑇(𝑡)𝒜^𝑇𝑅₂𝒜𝜉(𝑡)−^𝑡_𝑡−𝜏𝑥̇‌^𝑇(𝑠)𝑅₂𝑥̇‌(𝑠)𝑑𝑠. (3.7)(3.8)(3.9)(3.10)
Noting that (3.9), it follows from Lemma 2.6 that −^𝑡_{𝑡−𝑑𝑀}𝑥̇‌^𝑇(𝑠)𝑅₁𝑥̇‌(𝑠)1𝑑𝑠≤𝑑_𝑀𝑥(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥𝑡−𝑑_𝑀^𝑇−𝑅₁𝑅₁0∗−2𝑅₁𝑅₁∗∗−𝑅₁𝑥(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥𝑡−𝑑_𝑀. (3.11)
Employing the free matrix method [38, 39], it is easily derived that 2𝜉^𝑇(𝑡)𝑆𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑥̇‌(𝑠)𝑑𝑠=0,2𝜉^𝑇(𝑡)𝑇𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−𝜏)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑥̇‌(𝑠)𝑑𝑠=0, (3.12) where 𝑆=𝑆^𝑇₁00𝑆^𝑇₂0000^𝑇,𝑇=000𝑇^𝑇₁𝑇^𝑇₂000^𝑇. (3.13)
It follows that from (3.12) that −2𝜉^𝑇(𝑡)𝑆^𝑡_{𝑡−𝜏(𝑡)}𝑥̇‌(𝑠)𝑑𝑠≤𝜏(𝑡)𝜉^𝑇(𝑡)𝑆𝑅₂⁻¹𝑆^𝑇𝜉(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑥̇‌^𝑇(𝑠)𝑅₂𝑥̇‌(𝑠)𝑑𝑠−2𝜉^𝑇(𝑡)𝑇^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑥̇‌(𝑠)𝑑𝑠≤(𝜏−𝜏(𝑡))𝜉^𝑇(𝑡)𝑇𝑅₂⁻¹𝑇^𝑇𝜉(𝑡)+^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑥̇‌^𝑇(𝑠)𝑅₂𝑥̇‌(𝑠)𝑑𝑠. (3.14)
By Assumption 2.3, the following inequality holds: 𝑥(𝑡)𝑔(𝐻𝑥(𝑡))^𝑇𝑈₁𝑈₂𝑈^𝑇₂𝐼𝑥(𝑡)𝑔(𝐻𝑥(𝑡))≤0, (3.15) where 𝑈₁=𝐻^𝑇𝑈₁𝐻,𝑈₂=𝐻^𝑇𝑈₂𝑈₁=𝑈^𝑇₁𝑈₂+𝑈^𝑇₂𝑈₁2,𝑈₂=𝑈^𝑇₁+𝑈^𝑇₂2. (3.16)
For all 𝛼,𝛽>0, it can be derived from (3.15) that −𝛼𝑥(𝑡)𝑔(𝐻𝑥(𝑡))^𝑇𝑈₁𝑈₂𝑈^𝑇₂𝐼𝑥(𝑡)𝑔(𝐻𝑥(𝑡))≥0−𝛽𝑥((𝑡𝑔(𝐻𝑡−𝜏))𝑥(𝑡−𝜏(𝑡)))^𝑇𝑈₁𝑈₂𝑈^𝑇₂𝐼𝑥((𝑡𝑔(𝐻𝑡−𝜏))𝑥(𝑡−𝜏(𝑡)))≥0. (3.17)
Then, (2.15) can be rewritten as 𝜎𝑥^𝑇(𝑡−𝑑(𝑡))Ω𝑥((𝑡−𝑒𝑡−𝑑))^𝑇𝑘(𝑡)𝐶^𝑇𝑊𝐶𝑒_𝑘(𝑡)000>0, (3.18) where 𝐶Ω=^𝑇𝑊𝐶000. (3.19)
It follows from (3.7)–(3.18) that 𝑉̇(𝑡,𝑥(𝑡))≤2𝑥^𝑇(𝑡)𝑃𝒜𝜉(𝑡)+𝑥^𝑇(𝑡)𝑄₁+𝑄₂𝑥(𝑡)−𝑥^𝑇𝑡−𝑑_𝑀𝑄₁𝑥𝑡−𝑑_𝑀−𝑥^𝑇(𝑡−𝜏)𝑄₂𝑥(𝑡−𝜏)+𝑑_𝑀𝜉^𝑇(𝑡)𝒜^𝑇𝑅₁𝒜𝜉(𝑡)+𝜏𝜉^𝑇(𝑡)𝒜^𝑇𝑅₂𝒜𝜉(𝑡)+2𝜉^𝑇(𝑡)𝑆(𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡)))+2𝜉^𝑇(𝑡)𝑇(𝑥(𝑡−𝜏(𝑡))−𝑥(+1𝑡−𝜏))𝑑_𝑀𝑥(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥(𝑡−𝑑_𝑀)^𝑇−𝑅₁𝑅₁0∗−2𝑅₁𝑅₁∗∗−𝑅₁𝑥(𝑡)𝑥(𝑡−𝑑(𝑡))𝑥𝑡−𝑑_𝑀−𝛼𝑥(𝑡)𝑔(𝐻𝑥(𝑡))^𝑇𝑈₁𝑈₂𝑈^𝑇₂𝐼𝑥(𝑡)𝑔(𝐻𝑥(𝑡))−𝛽𝑥(𝑡−𝜏(𝑡))𝑔(𝐻𝑥(𝑡−𝜏(𝑡)))^𝑇𝑈₁𝑈₂𝑈^𝑇₂𝐼𝑥((𝑡𝑔(𝐻𝑡−𝜏))𝑥(𝑡−𝜏(𝑡)))+𝜎𝑥^𝑇(𝑡−𝑑(𝑡))Ω𝑥((𝑡−𝑒𝑡−𝑑))^𝑇𝑘(𝑡)𝐶^𝑇𝑊𝐶𝑒_𝑘(𝑡)0(𝑡)𝜉00+𝜏^𝑇(𝑡)𝑆𝑅₂⁻¹𝑆^𝑇𝜉(𝑡)+(𝜏−𝜏(𝑡))𝜉^𝑇(𝑡)𝑇𝑅₂⁻¹𝑇^𝑇𝜉(𝑡)=𝜉^𝑇(𝑡)Π+𝑑_𝑀𝒜^𝑇𝑅₁𝒜+𝜏𝒜^𝑇𝑅₂𝒜𝜉(𝑡)+𝜏(𝑡)𝜉^𝑇(𝑡)𝑆𝑅₂⁻¹𝑆^𝑇𝜉(𝑡)+(𝜏−𝜏(𝑡))𝜉^𝑇(𝑡)𝑇𝑅₂⁻¹𝑇^𝑇𝜉(𝑡). (3.20)
By using Schur complement and Lemma 2.5, it can be seen that (3.3) is equivalent to Π+𝑑_𝑀𝒜^𝑇𝑅₁𝒜+𝜏𝒜^𝑇𝑅₂𝒜+𝜏(𝑡)𝑆𝑅₂⁻¹𝑆^𝑇+(𝜏−𝜏(𝑡))𝑇𝑅₂⁻¹𝑇^𝑇<0 (3.21) which implies 𝑉̇(𝑡,𝑥(𝑡))<−𝜀‖𝑥(𝑡)‖²; then similar to [41], we can obtain the exponential stability of system (3.1). The proof is completed.

Remark 3.2. From Theorem 3.1, it can be seen that the trigger parameters 𝜎, 𝑊 and the upper bound of time delay 𝜏 are involved in (3.3); for given 𝜎, the corresponding trigger parameter 𝑊 and the upper bound of 𝜏 can be obtained by using LMI toolbox in Matlab. From the simulation example, it can be derived that the larger the 𝜎, the small the 𝜏; the larger average release period, which means the load of network transmission will be reduced.

Remark 3.3. When the estimator gain matrix 𝐾 is given, the conditions (3.3) are in the form of linear matrix inequalities, which can be readily solved by using the standard numerical software. The conditions (3.3) are not linear matrix inequalities when the estimator gain matrix 𝐾 is a matrix variable to be designed, and thus Theorem 3.1 cannot be used to design 𝐾 directly, a design method will be provided in the following Theorem.

After establishing analysis results in Theorem 3.1, the design problem of state estimator is to be considered and the following results can be readily derived from Theorem 3.1.

Theorem 3.4. Suppose that Assumption 2.3 holds, the augmented system (3.1) is exponentially stable, if there exist 𝑃=diag{𝐏₁,𝐏₂}>0, 𝑄_𝑖=diag{𝐐_𝑖,𝐐_𝑖}>0, 𝑅_𝑖=diag{𝐑_𝑖,𝐑_𝑖}>0 and 𝑆_𝑖=diag{𝐒_𝑖,𝐒_𝑖}, 𝑇_𝑖=diag{𝐓_𝑖,𝐓_𝑖}(𝑖=1,2) and 𝑉 with appropriate dimension, and two positive scalars 𝛼>0, 𝛽>0, such that the following linear matrix inequalities hold: Π_𝑖=ΠΦ₁Φ₂Φ₃^(𝑖)∗Φ₄00∗∗Φ₅0∗∗∗−𝜏𝑅₂<0(𝑖=1,2), (3.22) where Π=Π₁₁Π₁₂0Π₁₃0Π₁₄Π₁₅Π₁₆∗Π₂₂Π₂₃00000∗∗Π₃₃00000∗∗∗Π₄₄Π₄₅0Π₄₆0∗∗∗∗Π₅₅000∗∗∗∗∗Π₆₆00∗∗∗∗∗∗Π₇₇0∗∗∗∗∗∗∗Π₈₈Φ₁=Π^𝑇₁₇Π^𝑇₂₄000Π^𝑇₆₇Π^𝑇₇₈Π^𝑇₈₉^𝑇Φ₂=Π^𝑇₁₈Π^𝑇₂₅000Π^𝑇₆₈Π^𝑇₇₉Π^𝑇_8,10^𝑇Φ₃⁽¹⁾=Π^𝑇₁₉00Π^𝑇₄₇0000^𝑇Φ₃⁽²⁾=000Π^𝑇₁₉Π^𝑇₄₇000^𝑇Φ₄=2𝑑_𝑀𝐏₁+𝑑_𝑀𝐑₁002𝑑_𝑀𝐏₂+𝑑_𝑀𝐑₁Φ₅=2𝜏𝐏₁+𝜏𝐑₂002𝜏𝐏₂+𝜏𝐑₂Π₁₁=−𝐏₁𝐴−𝐴^𝑇𝐏₁−𝛼𝑈₁𝐶^𝑇𝑉^𝑇𝑉𝐶−𝐏₂𝐴−𝐴^𝑇𝐏₂−𝑉𝐶−𝐶^𝑇𝑉^𝑇+𝐐₁+𝐐₂−1𝑑_𝑀𝐑₁+𝐒₁+𝐒^𝑇₁00𝐐₁+𝐐₂−1𝑑_𝑀𝐑₁+𝐒₁+𝐒^𝑇₁Π₁₂=1𝑑_𝑀𝐑₁01𝑉𝐶𝑑_𝑀𝐑₁,Π₁₃=𝐒^𝑇₂−𝐒₁00𝐒^𝑇₂−𝐒₁Π₁₄=𝐏₁𝑊₀−𝛼𝑈₂𝐏₂𝑊₀,Π₁₅=𝐏₁𝑊₁𝐏₂𝑊₁Π₁₆=0,𝑉𝐶Π₁₇=−𝑑_𝑀𝐴^𝑇𝐏₁𝑑_𝑀𝐶^𝑇𝑉^𝑇0−𝑑_𝑀𝐴^𝑇𝐏₂−𝑑_𝑀𝐶^𝑇𝑉^𝑇Π₁₈=−𝜏𝐴^𝑇𝐏₁𝜏𝐶^𝑇𝑉^𝑇0−𝜏𝐴^𝑇𝐏₂−𝜏𝐶^𝑇𝑉^𝑇,Π₁₉=𝜏𝐒₁00𝜏𝐒₁Π₁₉=𝜏𝐓₁00𝜏𝐓₁,Π₂₂=−2𝑑_𝑀𝐑₁+𝜎𝐶^𝑇2𝑊𝐶00−𝑑_𝑀𝐑₁Π₂₃=1𝑑_𝑀𝐑₁001𝑑_𝑀𝐑₁,Π₂₄=0−𝑑_𝑀𝐶^𝑇𝑉^𝑇00Π₂₅=0𝜏𝐶^𝑇𝑉^𝑇,00Π₃₃=−𝐐₁−1𝑑_𝑀𝐑₁00−𝐐₁−1𝑑_𝑀𝐑₁Π₄₄=−𝐒₂−𝐒^𝑇₂+𝐓₁+𝐓^𝑇₁−𝛽𝑈₁00−𝐒₂−𝐒^𝑇₂+𝐓₁+𝐓^𝑇₁,Π₄₅=𝐓^𝑇₂−𝐓₁00𝐓^𝑇₂−𝐓₁Π₄₆=−𝛽𝑈₂0,Π₄₇=𝜏𝐒₂00𝜏𝐒₂,Π₄₇=𝜏𝐓₂00𝜏𝐓₂Π₅₅=−𝐐₂−𝐓₂−𝐓^𝑇₂00−𝐐₂−𝐓₂−𝐓^𝑇₂,Π₆₆=−𝛼𝐼Π₆₇=𝑑_𝑀𝑊^𝑇₀𝐏₁𝑑_𝑀𝑊^𝑇₀𝐏₂,Π₆₈=𝜏𝑊^𝑇₀𝐏₁𝜏𝑊^𝑇₀𝐏₂,Π₇₇=−𝛽𝐼Π₇₈=𝑑_𝑀𝑊^𝑇₁𝐏₁𝑑_𝑀𝑊^𝑇₁𝐏₂,Π₇₉=𝜏𝑊^𝑇₁𝐏₁𝜏𝑊^𝑇₁𝐏₂,Π₈₈=−𝐶^𝑇𝑊𝐶Π₈₉=0−𝑑_𝑀𝐶^𝑇𝑉^𝑇,Π_8,10=0−𝜏𝐶^𝑇𝑉^𝑇, (3.23) then the desired estimator gain matrix is given as 𝐾=𝐏₂⁻¹𝑉.

Proof. By using Schur complement in Theorem 3.1, Π_𝑖<0  (𝑖=1,2) can be rewritten as Π+𝑑_𝑀𝒜^𝑇𝑅₁𝒜+𝜏𝒜^𝑇𝑅₂𝒜+𝜏𝑆𝑅₂⁻¹𝑆^𝑇<0Π+𝑑_𝑀𝒜^𝑇𝑅₁𝒜+𝜏𝒜^𝑇𝑅₂𝒜+𝜏𝑇𝑅₂⁻¹𝑇^𝑇<0. (3.24)
By using Lemma 2.5, (3.24) are equivalent to the following matrix inequalities ΠΦ₁Φ₂Φ₃^(𝑖)∗−𝑑_𝑀𝑅₁⁻¹00∗∗−𝜏𝑅₂⁻¹0∗∗∗−𝜏𝑅₂<0(𝑖=1,2), (3.25) where Π=Π₁₁Π₁₂0Π₁₃0Π₁₄Π₁₅Π₁₆∗Π₂₂Π₂₃00000∗∗Π₃₃00000∗∗∗Π₄₄Π₄₅0Π₄₆0∗∗∗∗Π₅₅000∗∗∗∗∗Π₆₆00∗∗∗∗∗∗Π₇₇0∗∗∗∗∗∗∗Π₈₈Φ₁=𝑑_𝑀𝐴𝑑_𝑀𝐵000𝑑_𝑀𝑊₀𝑑_𝑀𝑊₁𝑑_𝑀𝐶^𝑇Φ₂=𝜏𝐴𝜏𝐵000𝜏𝑊₀𝜏𝑊₁𝜏𝐶^𝑇. (3.26)
Then performing a congruence transformation of diag{𝐼,𝑃,𝑃,𝐼} to (3.25), it can be derived that ΠΦ₁Φ₂Φ₃^(𝑖)∗−𝑑_𝑀𝑃𝑅₁⁻¹𝑃00∗∗−𝜏𝑃𝑅₂⁻¹𝑃0∗∗∗−𝜏𝑅₂<0(𝑖=1,2), (3.27) where Φ₁=𝑑_𝑀𝑃𝐴𝑑_𝑀𝑃𝐵000𝑑_𝑀𝑃𝑊₀𝑑_𝑀𝑃𝑊₁𝑑_𝑀𝑃𝐶^𝑇Φ₂=𝜏𝑃𝐴𝜏𝑃𝐵000𝜏𝑃𝑊₀𝜏𝑃𝑊₁𝜏𝑃𝐶^𝑇. (3.28)
Setting 𝑃₂𝐾=𝑉 in (3.27) and considering the following inequality: −𝑃𝑅_𝑖⁻¹𝑃≤−2𝑃+𝑅_𝑖(𝑖=1,2). (3.29)
By using (3.29), we can obtain ΠΦ₁Φ₂Φ₃^(𝑖)∗−𝑑_𝑀𝑃𝑅₁⁻¹𝑃00∗∗−𝜏𝑃𝑅₂⁻¹𝑃0∗∗∗−𝜏𝑅₂<ΠΦ₁Φ₂Φ₃^(𝑖)∗−2𝑑_𝑀𝑃+𝑑_𝑀𝑅₁00∗∗−2𝜏𝑃+𝜏𝑅₂0∗∗∗−𝜏𝑅₂. (3.30)
Substitute 𝐴, 𝐵, 𝑊₀, 𝐻,𝑊₁, 𝐶, 𝑃, 𝑄_𝑖, 𝑅_𝑖, 𝑆_𝑖, 𝑇_𝑖 (𝑖=1,2) into the right of (3.30), combining (3.22), we can obtain ΠΦ₁Φ₂Φ₃^(𝑖)∗−2𝑑_𝑀𝑃+𝑑_𝑀𝑅₁00∗∗−2𝜏𝑃+𝜏𝑅₂0∗∗∗−𝜏𝑅₂<0(𝑖=1,2). (3.31) The rest of the proof follows directly from Theorem 3.1.

Remark 3.5. When the estimator gain matrix 𝐾 is a matrix variable to be designed, in order to transform the conditions (3.3) to linear matrix inequalities, and meanwhile reduce the computational complexity (i.e., reduce the number of matrix variables), in Theorem 3.4, matrix variables in Theorem 3.1 are replaced by some diagonal matrices. Then setting 𝑃₂𝐾=𝑉, we can obtain (3.22), which is in the form of linear matrix inequalities, which are easy to be verified by LMI toolbox.

Remark 3.6. It is noticed that 𝑑_𝑀=ℎ+𝑑, if 𝑑_𝑀 is solved, we can select a sampling period ℎ<𝑑_𝑀. For given 𝑑, the maximal allowable sampling period ℎ_max can be obtained by the following two-step procedure.(1)For given 𝜏 and 𝑑, setting ℎ_max=ℎ₀ and step size STEP=STEP₀, where ℎ₀ and STEP₀ are two specified positive constants.(2)If LMIs (3.22) are feasible, set ℎ_max=ℎ₀+STEP₀ and return to step (2): otherwise, ℎ is the maximal allowable sampling period.

4. Numerical Results

In this section, a numerical example is given to verify the effectiveness of the proposed control techniques for estimation of recurrent neural networks with time-varying delays.

Example 4.1. Consider recurrent neural networks (2.1) with the following parameters 𝐴=1.5002,𝑊₀=0.3−0.4−0.40.3,𝑊₁=0.30.30.30.3,𝐶=0.90.80.70.5. (4.1)
The neuron activation function is described as follows: 𝑔(𝑥)=0.5𝑥₁(𝑡)−tanh0.2𝑥₁(𝑡)+0.2𝑥₂(𝑡)0.95𝑥₂(𝑡)−tanh0.75𝑥₂(𝑡). (4.2)
It is easy to verify that the nonlinear function 𝑓(⋅) satisfies Assumption 2.3; by some simple calculations, we can obtain 𝑈₁=𝑈0.30.200.2²=0.50.200.95. (4.3)
Setting 𝑑_𝑀=0.01 and 𝜎=0.1, by applying Theorem 3.4, it can be obtained the maximum allowable delay 𝜏=1.1572. More detailed calculation results for different values of 𝜎 are given in Table 1. It can be shown that the larger 𝜎, the smaller 𝜏. For given 𝜏=1 and 𝑑=0.01, based on Remark 3.6, we can obtain the maximal allowable sampling period ℎ_max, which are shown in Table 2. For given 𝜏=1, 𝜎=0.1 and 𝑑_𝑀=0.01, by using LMI Toolbox in LMIs (3.22), the feasible solution can be obtained as follows: 𝐏₁=5.4676−0.1329−0.13295.4000,𝐏₂=5.0204−0.1155−0.11554.2212,𝐐₁=𝐐2.7863−0.0955−0.09552.0982₂=3.2539−0.0133−0.01332.8910,𝐑₁=0.02220.00100.00100.0234,𝐑₂=𝐒2.2237−0.0246−0.02461.4752₁=−0.55030.01950.0478−0.4156,𝐒₂=1.01850.06740.13781.1920,𝐓₁=𝐓−1.0844−0.0634−0.0902−1.2378₂=0.3511−0.0221−0.01950.3384,𝑉=−0.0792−0.1282−0.1755−0.0863,𝛼=6.4604,𝛽=5.7723. (4.4)
Then the triggered matrix and the desired estimator can be obtained as follows: 𝑊=4.6153−2.7354−2.73546.5131,𝐾=−0.0165−0.0308−0.0354−0.0214. (4.5)
For giving the sampling period ℎ=0.005, Table 3 gives the relation of the trigger parameter 𝜎, trigger times, the average release period, and the percentage of data transmissions; it can be seen that the larger the 𝜎, the smaller trigger times; the larger average release period, the smaller percentage of data transmission, which are reasonable results. In the following, we provide some simulation results: when 𝜎=0, the time varying delay 𝜏(𝑡) obeys uniform distribution over [0,1], and the curves of the error dynamics of the neural networks 𝑒_𝑖(𝑡)(𝑖=1,2) are depicted in Figure 1, from which we can see the errors converge to zero asymptotically. If setting 𝜎=0.1, The response of the error dynamics for the delayed neural networks (2.17) which converge to zero asymptotically in the mean square is given in Figure 2. Figure 3 shows the event-triggered release instants and intervals. It can be seen from Figures 1 and 2 that the simulation results are almost the same, but the percentage of data transmission under even-triggered scheme used much small number than time-triggering scheme. To make this clear, seen the computation results lists in Table 2, from which we can see that data transmission rate with even-triggered scheme (𝜎=0.1) is only 3.71% of sampled measurement output with time-triggered scheme (𝜎=0); from these results, we can draw a conclusion that event-triggered scheme has advantage over the time-triggered scheme in improving the resource utilization.

Table 1 

𝑑_𝑀=0.01.

Table 2 

𝜏=1, 𝑑=0.01.

Table 3 

ℎ=0.005, 𝑑_𝑀=0.01, 𝜏=0.1, 𝑡=10.

Figure 1 

The error curves 𝑒𝑖(𝑡)(𝑖=1,2) with trigger parameter 𝜎=0 (time-triggered scheme).

Figure 2 

The error curves 𝑒𝑖(𝑡)(𝑖=1,2) with trigger parameter 𝜎=0.1 (event-triggered scheme).

Figure 3 

Release instants and release interval by event-triggered scheme.

5. Conclusions

This paper has provided a novel event-triggered scheme to investigate the sampled-data state estimation problem for a class of recurrent neural networks with time-varying delays. This scheme can lead to a significant reduction of the information communication burden in the network. By using a delayed-input approach, the error dynamics system is equivalently to a dynamic system with two different time-varying delays. Based on the Lyapunov-krasovskii functional approach, a state estimator of the considered neural networks can be achieved by solving some linear matrix inequalities, which can be readily solved by using the standard numerical software. Finally, an illustrative example is exploited to show the effectiveness of the event-triggered scheme.

Acknowledgments

This work was in part jointly supported by the National Science Foundation of China (under Grant 61074024, 60874113), the Natural Science Foundation of Jiangsu Province of China (under Grant BK2010543), the Education Department Research Project of Zhejiang Province of China (under Grant Y201019013), and the Outstanding Young Teacher Project of Zhejiang Province.