Adaptive memory-event-triggered H ∞ control for network-based T-S fuzzy systems with asynchronous premise constraints

This paper presents a novel adaptive memory-event-triggered scheme (A-METS) for network-based T-S fuzzy systems with asynchronous premise constraints. Different from the event-triggered scheme (ETS), the proposed A-METS has two characters. First, some recent released packets are applied in deciding whether the current packet is supposed to be released. Second, the triggering parameter can adjust itself according to the variation of the state. It should be pointed out that the proposed adaptive memory-event-triggered scheme can save network recourses and improve the system performance simultaneously when compared with ETS or adaptive ETS. Furthermore, considering the network envi-ronment, the asynchronous premise between the fuzzy plant and controller are considered. By resorting to the Lyapunov functional approach, sufﬁcient conditions are derived for the stability and H ∞ performance of the network-based T-S fuzzy systems. For the sake of illustrating the usefulness of the proposed adaptive memory-event-triggered scheme, a tunnel diode circuit example is

in ETS usually contains a triggering threshold, the most recent released packet, the current packets and/or a positive matrix variable to be designed. In most of the studies, a static triggering threshold is used [12][13][14][15][16][17][18], which is pre-selected and cannot be adjusted according to the system and network conditions. For example, in [18], fuzzy bipartite tracking control problem for stochastic non-linear multi-agent systems is investigated, where ETS has been applied to reduce the buren of communication. More recently, in order to make the ETS more "smart," an adaptive ETS (AETS) is introduced [19][20][21][22], wherein the threshold can adjust itself according to the system dynamics. To name a few, in [21], an AETS was proposed to mitigate the burden of network-bandwidth for a class of the non-linear systems, where a new adaptive law was presented to achieve the threshold of event-triggering condition on-line. In [22], an AETS was well used in reducing the load frequency of the power system and ensuring resources utilisation. Compared with static ETS, AETS can transmit fewer packets because its threshold is a parameter that can be dynamically adjusted. Although the AETS can further reduce the packets transmission, the system's dynamic may have a degradation when compared with the static ETS. For other kinds of ETS methods, when considering specific media access protocols, a dynamic ETS was proposed in an outputbased or decentralised form in [23]. It should be noted that in most of the ETS mentioned above, only the current packets and the newly released packets are used in the event-triggered conditions. As such, some newly released packets were considered, and a memory event-triggered communication scheme (METS) was constructed in [24,25]. By using the METS, when the system fluctuates wildly, more packets will be released to stabilise the system quickly. However, more packets are probably to be released when compared with static ETS. Therefore, how to design a triggering scheme to give consideration to data transmission and system performance still needs further investigation.
As we all know, T-S fuzzy models have been well paid to characterising non-linear systems, and the common used fuzzy controller design method is parallel distribution compensation (PDC) [26][27][28][29][30][31]. The T-S fuzzy system is represented by numerous If-Then rules, which can treat the non-linear system as a fuzzy approximation of multiple local linear models. Therefore, much research enthusiasm has recently been attracted towards the T-S fuzzy systems because of the wide applications in automobile suspension system, the internal combustion engine systems and other fields. [32,33]. When considering the networkbased T-S fuzzy systems, some event-triggered control/filter methods have been concerned in [34][35][36][37]. For example, in [37], a new discrete ETS was proposed for T-S fuzzy system. In [38], the problem of tracking control of T-S fuzzy systems is studied, where a novel AETS is employed to save the limited network bandwidth. In these literature, the premise variables of the system and the fuzzy controller are synchronous. Actually, it is worth noting that the network exists all the time in the process of data transmission, and the asynchronous premise variables will reflect the real situation more truly. Considering this, in [39][40][41], methods of asynchronous premise reconstruction are designed in terms of the fuzzy systems with ETS. Moreover, for some non-PDC methods, the asynchronous problem is handled by the membership functions with upper and lower bounds [42,43]. In [44], dynamic output feedback control problem with ETS was investigated to deal with the problem of asynchronous constraints by designing two independent membership functions.
Inspired by the aforementioned discussion, combining the advantages of the AETS and METS, a novel adaptive METS is proposed for network-based T-S fuzzy system with asynchronous premise variables. The main contributions of this paper are summarised as follows: 1) An adaptive memory event-triggered communication scheme (A-METS) is, for the first time, introduced to reduce the number of transmissions effectively and improve the system performance for network-based T-S fuzzy systems.
2) For the sake of ensuring the stability and H ∞ performance of the considered systems, an asynchronous premise reconstruct method for T-S fuzzy systems is considered. Correspondingly, the memory fuzzy controller is designed while taking both the effects of the asynchronous premise and A-METS into consideration.
The organisation of this paper is as follows: Section 2 presents an A-METS and asynchronous premise variables for networked T-S fuzzy systems. In Section 3, two theorems are given for the stability and H ∞ controller design for the studied systems. A simulation example is presented in Section 4 and Section 5 states conclusions.

Network-based T-S fuzzy systems
Considering a non-linear system which is denoted as where x(t ) ∈ ℝ n and u(t ) ∈ ℝ m are the state and input vector, respectively. (t ) ∈  2 [0, ∞] denotes the external disturbance signal, z(t ) ∈ ℝ p represents the system output. i = 1, 2, … , r, r is the number of IF-THEN rules, ℰ i j are fuzzy sets and j (x(t ))( j = 1, 2, … , g) represent premise variables and define (x(t )) = [ 1 (x(t )), … , g (x(t ))] T . A i , B i , C i , B i and D i are known constant matrices with appropriate dimensions. The global fuzzy system can be derived by means of the weighted average of multiple local linear models Considering the impact of the network environment, the premise variables at the controller should be (x(t k h)), the ith controller rule is where K j are controller gain to be determined later.

Network-based T-S fuzzy controller with asynchronous premise constraints
Similar to the analysis of [14], we define t k as the communication delay of the sampled packet x(t k h) and (t ) = t − (t k h + lh), h is the sampling period. The control input will remain the same in the interval [t k h + t k , t k+1 h + t k+1 ) △ = ℛ. Using virtual partition method, the interval is divided into the following sub-intervals Then, the fuzzy controller is easily obtained Similar to [40], supposing the membership functions satisfy where ϝ j are known positive constants. From (3), j ( (x(t ))) and j ( (x(t k h))) can be obtained, we assume j ( (x(t k h))) = j j ( (x(t ))), where j is a positive constant. From (8)-(9), one obtains Combining (7) and (9) together, for j = 1, 2, … , r, we conclude that

The adaptive memory event-triggered scheme
Firstly, let us review the static ETS proposed in [14] is expressed as where e k,l = x(t k h) − x(t k h + lh), l ∈ ℕ, Φ is a triggered parameters and is a given arbitrary scalar.
Remark 1. In the ETS (14), it is obvious that the triggering threshold and the error e k,l both determine whether a newly sampled packet is transmitted. For example, a smaller and a larger e k,l will give rise to more triggering instants, that is, the more packets are probably to be transmitted.
In this paper, combining the advantages of AETS and METS, an A-METS is proposed, which is described as (14), (t k h) in (15) can be adaptively adjusted based on the following rule: wherē> 0, and the parameter is determined by where we set > 0. The initial of (t k h) is given as (0) = . From (16), we can see that̄is the lower bound of (t k h) and a larger will result a larger change rate of (t k h). Remark 2. Obviously, the function arctan(⋅) in (16) is a bounded function, that is, arctan(⋅) ∈ (− 2 , 2 ). The system can adaptively adjust the triggered threshold parameter (t k h) based on this feature. For example, if ||x(t k h)|| < ||x(t k+1 h)||, one can derive that (t k h) > (t k+1 h). Consequently, a smaller (t k+1 h) will lead to a higher communication efficiency. In reverse, a larger value of (t k+1 h) can obtain a lower transmission efficiency.
Remark 3. In some existing work, AETS and METS have been investigated yet. AETS has the advantage of reducing the transmission of packets effectively and saving limited bandwidth [22]. METS has the advantage of utilising some historic signals to improve the performance of the system [24,25]. In this paper, considering the advantages of both AETS and METS, an A-METS is proposed. In the A-METS (15), m is view as the number of the historic released packets to reflect the existence of memory. It is clear that the A-METS becomes the traditional AETS when m = 1. The weighting parameters p can illustrate the significance of the released packets. Usually, we view the newly released packets as more important. Therefore, it is assumed to be p > p+1 , (p = 1, 2, … , m − 1).
For convenience, i ( (x(t ))) is written as i , according to (15), the expression of memory fuzzy controller is obtained as Substituting (18) into (2) leads to the following fuzzy system: where By using the proposed A-METS, the purpose of this paper is to design the fuzzy memory feedback controller, such that the following two requirements are satisfied.

MAIN RESULTS
We aim to exploit a method for the sake of ensuring stability and designing controller for the studied systems. First of all, the following three lemmas are given for obtaining the desired results. Then, the sufficient condition that ensures the H ∞ performance is established. Lemma 1. [45] For arbitrary matrices Υ 1 , Υ 2 and Δ with compatible dimensions, (t ) is the network-induced delay and satisfies (t ) ∈ [0, M ], the following inequality: holds if and only if Lemma 2. [46] For arbitrary vector a, b ∈ ℝ n×n , positive definite matrix F ∈ ℝ n×n , we have the inequality Lemma 3. [46] For arbitrary variable , and H , D, P are some real matrices with compatible dimensions and ||P|| ≤ 1, the following inequality is true Theorem 1. For given scalars M , ,̄, p , 1 , 2 , m, and matrix K jp , the system (19) is asymptotically stable if there exist matrices P > 0, S 1 > 0, S 2 > 0, L 1 > 0, L 2 > 0 and some free matrices U, G, W with compatible dimensions such that the following matrix inequalities hold for l, i, j = 1, 2, … , r, i < j. [ where Ω i j Proof. Considering the following Lyapunov function where and P, S 1 , S 2 , L 1 , L 2 are positive definite matrices, s k = t k h + lh + t k . From the proposed A-METS in (15), we can obtain easily that, for t ∈ ℛ l , ] .
(29) Taking derivation on V (t, x t ) and using d dt x(s k )=0, which yieldṡ For the convenience of study, we define .. e T m ]. Then, for some free matrices G , U , W with compatible dimensions, we define , by using the Newton-Leibnitz formula, we have  1 = 0,  2 = 0,  3 = 0. By using Lemma 2, there exist positive definite matrices L 2 > 0 and S 2 > 0, such that From (28)-(31), we can easily derivė where .
From the inequalities (24)- (27), it can be concluded that when (t ) = 0, there exists a positive scalar > 0 such thaṫ  22 11 =mΦ Proof. First, we define a set of new matrices: X = P −1 , Using Lemma 3 and defining = L −1 1 , we obtain that Then, pre and post multiplying (24) and (25) and (26) and (27) by diag( 1 ,  2 ) and diag( 1 ,  2 ,  2 ), respectively, where  1 = diag(X X X X I X … X ),  2 = diag(I I X I ), we obtain (47)-(50) from (24)-(27) by resorting to the Schur complement and (51). The proof has been completed. □ Remark 4. Noticing that the existence of non-linear items −XS −1 2 X , −XL −1 1 X , −XL −1 2 X will result (47)-(50) cannot be solved straightly by LMI toolbox. Therefore, we substitute non-linear items XS −1 2 X , XL −1 2 X , XL −1 1 X with 2 1S 2 − 2 1 X , 2 2L 2 − 2 2 X and 2 3L 1 − 2 3 X , where i > 0 is a scalar. Generally, there are two methods in the literature to deal with non-linear items. One method is the CCL algorithm, which transfers the feasible solutions of non-linear matrix inequalities to a non-linear optimisation problem involving LMI conditions. Another method is the linearisation of non-linear method. In this paper, we have used the second method and enlarge some items, therefore, and that has created some conservatism. However, the less conservative results based on CCL generally than those based on linearisation of non-linear method. It should be pointed out that although CCL can effectively reduce the conservativeness of this paper, the main work of this article is to handle adaptive memory-event-triggered H ∞ control for network-based T-S fuzzy systems with asynchronous premise constraints. Therefore, CCL approach has not been considered.
Remark 5. For a preselect threshold̄, the performance index , triggered parameter Φ and memory fuzzy controller gains K jp can be obtained from Theorem 2. Too largēwill result fewer packets need to be transmitted. However, if̄is too large, (47)-(50) maybe infeasible, that is, the performance of the system cannot be ensured.

AN ILLUSTRATIVE EXAMPLE
A case study on tunnel diode circuit [44] as shown in Figure 1 is carried out in this section to demonstrate the efficacy of proposed method, the diode is described as i D (t ) = 0.002v D (t ) + 0.01v 3 D (t ), based on Kirchhoff Laws and let x 1 (t ) = v c (t ) and x 2 (t ) = i L (t ), and set C = 20mF , L = 1000mH and R = 10Ω, we can obtain where z(t ) and (t ) are the controlled output and disturbance noise, respectively. In this paper, two fuzzy rules are used, for example, as shown in Figure 2. Supposing that −3 ≤ x 1 (t ) ≤ 3, the system (52) can be approximately expressed by the system (19), where For the simulation purpose, under the initial condition With the asynchronous premise variables, the numbers of transmitted packets among AETS, METS and the proposed A-METS are listed in Table 1. By using the proposed A-METS, the state responses, release intervals, the trajectory of adaptive threshold (t ) are shown in Figures 3-5, respectively. Figure 6 depicts the curve of the circuit control input, we can see that u(t ) tends to initial state after a short oscillation, which indicates a good control performance. Obviously, the threshold (t ) is dynamically adjusted in terms of the variation of the state, while ensuring the control performance and stability of the system. The state responses, release instants and intervals under METS and AETS are shown in Figures 7-10, respectively.   Figures 3 and 7, it can be found that the overshoot and settling time are almost the same. Further, we can see that the proposed A-METS can transmit less packets than METS from Table 1. Therefore, the proposed A-METS can Release intervals under AETS with asynchronous premise achieve similar system performance by using less packets, thus more communication resource can be saved. From Table 1 and Figures 3 and 9, we can see that although the proposed A-METS can transmit more packets than AETS, the overshoot and settling time outperform the AETS. It is worth to point out there are properly more packets released at the beginning time t ∈ [0, 2s] when the states have a drastic change (see Figure 4). That is, the proposed A-METS can improve the performance of the system.
From Table 1 and the figures, we can conclude that (i) the packets transmitted by proposed A-METS are less than the METS and larger than AETS ; (ii) the proposed A-METS by transmitting fewer packets can achieve the similar system performance to METS. Both the system performance of the A-METS and METS are better than that of AETS. Further, compared with the AETS, the proposed A-METS will release more packets when the state of the system fluctuates severely, which makes the responses of the system reach the stable more quickly. That is, the proposed A-METS can give consideration to the data transmission reduction and system performance improving, simultaneously.

CONCLUSION
An adaptive METS has been investigated for networked T-S fuzzy systems with asynchronous premise variables. An adaptive METS has been employed to save the communication resources and effectively reduce data packets transmission through utilising historical packets. The memory fuzzy controller has been designed by constructing the asynchronous premise variables. Sufficient conditions of the system have been derived by using the Lyapunov theory. Simulation results have been presented to illustrate the efficacy of the proposed method. It should be noted that H ∞ control problem is considered in this paper; however, filtering problem has also been given much attention recently. Therefore, how to consider an adaptive memory-eventtriggered H ∞ filtering problem for networked T-S fuzzy systems with asynchronous constraints is one of our future work.