Adaptive event-triggered dynamic output feedback H ∞ control for networked T-S fuzzy systems

This paper investigates the design problem of fuzzy dynamic output feedback controller for nonlinear networked systems via mismatched membership functions and adaptive event-triggered (AET) mechanism. Firstly, an AET mechanism is introduced to save communication resources, which causes the controller and the original system's premise variables to be asynchronous. Then, considering the influence of AET and mismatched membership functions, a model of fuzzy control system is established. In addition, utilizing the Lyapunov-Krasovskii(L-K) functional, sufficient conditions for the global exponential stability of the closed-loop system with performance are derived. Besides, the controller parameters and event-triggered (ET) weight matrix are solved by a set of linear matrix inequalities (LMIs). Finally, an example is given to demonstrate the effectiveness of the proposed control method.


Introduction
Compared with traditional point-to-point control systems, networked control systems (NCSs) have become a research hotspot in the field of control due to their strong adaptability and low cost, and many important literatures have been published (Qiu et al., 2016;X. Zhang et al., 2016). However, the limited bandwidth makes the NCSs inevitably have problems such as network delay and blocked transmission when sharing communication information, and the system performance is greatly reduced. An ET mechanism is proposed to replace the traditional communication scheme in Tabuada (2007). You et al. (2019) and Hua et al. (2019) study the related issues of multi-agent systems through ET mechanisms. Gu et al. (2018) propose an AET mechanism to further save communication resources and apply it to the comprehensive problem of fuzzy networked systems. In K. Zhang et al. (2020), Ran et al. (2018), and Liu et al. (2020), the introduction of the ET mechanism designs dynamic output feedback controller for nonlinear networked systems. However, how to design a fuzzy dynamic output feedback controller utilizing AET mechanism is worthy of further investigation.
The well-known Takagi-Sugeno (T-S) fuzzy model (Takagi & Sugeno, 1985) is a powerful tool to approximate the nonlinear systems through linear subsystems described by IF-THEN fuzzy rules. In Guerra and Vermeiren (2004) and Kim and Kim (2002), the classical linear system theory is successfully extended to nonlinear systems. The L-K functional method is used to analyze the stability of time-varying delay systems in C. , C. , and X. . It is worth mentioning that this paper fully considers that after the introduction of the ET mechanism, the membership functions of the fuzzy system and the fuzzy controller may be different. Firstly, a fuzzy dynamic output feedback H ∞ controller with unmatched premise variables is designed by using AET mechanism. Then, based on L-K functional method, the stability and stabilization conditions of the closed-loop system with network delays are provided. Finally, the effectiveness of the proposed method is verified by an example.
The rest of the structure of this paper is organized as follows: Section 2 introduces the problem under consideration, AET mechanism and establishes a closed-loop system under dynamic output feedback H ∞ control. The main results of controller design, system stability and qualitative analysis are given in Section 3. An example is shown in Section 4 to describe the effectiveness of the control method. Section 5 summarizes the conclusion. the matrix transpose and inverse, respectively; * stands for the transposed elements of the symmetric matrix; He(X) denotes the expression X T + X; diag{. . .} stands for the the block-diagonal matrix; · stands for the Euclidean norm; L 2 [0, ∞) denotes the space of square integrable vector functions.

System description
Consider a nonlinear networked system represented by the T-S fuzzy model, in which the entire structure of dynamic output feedback controller is shown in Figure 1. The system is described as follows: where x(t) ∈ R n , z(t) ∈ R n 1 , y(t) ∈ R n 2 , u(t) ∈ R m 1 and w(t) ∈ R m 2 stand for the state vector, control output, measurement output, control input and the noise disturbance input which belongs to L 2 ∈ [0, ∞), respectively.
. . , r, j = 1, 2, . . . , p) represents the fuzzy set, r and p are the number of the IF-THEN rules and the prerequisite variables, respectively. For simplicity, By using central average defuzzifier and single-case fuzzer to generate fuzzy inference, system (1) can be rewritten aṡ (2) For subsequent development, the following assumptions are required.

Event-triggered mechanism
As seen in Figure 1, considering the bandwidth limitation of the communication network, an ET device is introduced between the sensor and the controller to determine whether the sampling output is transmitted instantaneously, in order to save communication resources. Inspired by W. Li et al. (2020), define the error between the latest released data y(t k h) and the current sampling data y(t k h + ch) as follows: Then the next transmission instant through the AET mechanism can be expressed as where h is the sampling period, y(t k h) and y(t k h + ch) denote the latest released signal and the current sampled signal, respectively; > 0 is the weight matrix to be designed; α(t) and β(t) are two independent ET thresholds, which are two adaptive functions and satisfy the following constrainṡ whereα(t) andβ(t) are adaptive law, μ 1 , μ 2 , κ 1 , κ 2 are given constants greater than zero, (5) and (6) are given to ensure that α(t) and β(t) satisfy the following formula : Remark 2.1: It is worth noting that through using AET mechanism, which can further improve transmission efficiency and save communication resources compared with Ran et al. (2018) and K. Zhang et al. (2020). And ET thresholds α(t) and β(t) are no longer a constant that is related to error e y (t), the latest released data y(t k h) and the current sampling data y(t k h + ch).

Networked dynamic output feedback controller design with asynchronous premise variables
Define the following subintervals on the time interval Define and 12) Then, according to (10)-(12), the actual inputŷ(t) of the dynamic output feedback controller can be given bŷ There is an AET mechanism between the sensor and the controller, which makes the premise variables of the controller and the system model asynchronous, so the IF-THEN rules of the controller are designed as Control Rule j: where N j d (j = 1, 2, . . . , r, d = 1, 2, . . . , q) represents the fuzzy set. x c (t) ∈ R n is the state vector of the controller,ŷ(t) denotes measured output through the eventtriggered communication network, A cj , B cj , C cj are the controller matrices to be designed.
By using central average defuzzifier and single-case fuzzer to generate fuzzy inference, system (12) can be rewritten aṡ Remark 2.3: It is worth mentioning that the membership functions of the controller are different from the original system, in other words, ϕ i (f (x)) are not necessarily the same as ψ j (g(x c )). In addition, unlike (Ning et al., 2018), asynchronous constraints are embodied in the mismatched premise variables in this paper.
, the method presented in this paper can be reduced to the case in Z. Zhang et al. (2015), the controller and the system use membership functions of the same structure, then by using the PDC method, the parameters of the fuzzy controller can be obtained. (13) and (15), the closed-loop fuzzy system is constructed as follows: +B 2ij e y (t) +B ωij ω(t) The goal of this paper is to design a dynamic feedback output controller for a fuzzy networked system based on an AET mechanism, such that: (1) Under the condition ω = 0, the closed-loop system (16) is exponentially stable; (2) Under zero initial condition, the closed-loop system satisfies z(t) 2 ≤ γ ω(t) 2 for any nonzero ω(t) ∈ L 2 [0, ∞), where γ > 0 is H ∞ performance target.
In order to obtain the main results, a useful lemma is first given.

H ∞ performance analysis
In this section, sufficient conditions for the asymptotic stability of the closed-loop system (16) with H ∞ performance are proposed.
Theorem 3.1: For given positive scalars τ 1 , τ 2 , μ 1 , μ 2 , κ 1 , κ 2 and γ , the membership functions satisfying ψ j (x c ) − ρ j ϕ j (x c ) ≥ 0(0 < ρ j < 1), the closed-loop system (16) is exponentially stable and meets the H ∞ performance target if there exist matrices P > 0, Q 1 > 0, Q 2 > 0, R 1 > 0, R 2 > 0, > 0, A cj , B cj , C cj and i = T i with suitable dimensions such that the following matrix inequities hold with i, j = 1, 2, . . . , r Proof: Consider the following Lyapunov-Krasovskii functional Then, seeking the time derivative of V(t) with respect to t getṡ By use of Lemma 2.1, we obtain where Combining (20) to (23), one haṡ Introduce a relaxation matrix i = T i and consider Under ψ j (x c ) − ρ j ϕ j (x c ) ≥ 0 for any j, according to the conditions (17)-(19), it yields thaṫ When the initial condition is zero, integrating the left and right sides of (22) from 0 to ∞ gives Therefore, the closed-loop system (16) is asymptotically stable with the H ∞ performance index. In addition, it can be seen from (17)-(19) thatV(t) < 0 is satisfied when ω = 0. This completes the proof.
Remark 3.1: Theorem 3.1 is obtained by the combination of Wirtinger inequality and relaxation matrix. Different from some existing works (H. Li et al., 2014;Z. Zhang et al., 2015), this paper fully considers the effects of network-induced delay and AET, the membership functions of the system (2) and the controller (15) are asynchronous and make use of the membership function information with designing the controller.

Fuzzy H ∞ controller design
On the basis of Theorem 3.1, the following Theorem 3.2 is given to obtain the ET weight matrix and fuzzy controller parameters by matrix decomposition.
Then, the parameters of the dynamic output feedback controller, the ET weight matrix, and the performance target can be obtained by the following algorithm.  (27) and (28), hold for i, j = 1, 2, . . . , r.
(32) Remark 3.2: Notice that the major differences between the present paper and Ran et al. (2018) are as follows: (1) A more general adaptive event-triggered mechanism is adopted, which saves communication resources; (2) By constructing a slightly different L-K functional and using the improved Wirtinger inequality, the results and the performance target have been improved, which will be illustrated by subsequent examples.

Conclusion
This paper has investigated the problem of dynamic output feedback H ∞ controller design for networked T-S fuzzy system under AET mechanism. The membership function of the designed controller and the original system is asynchronous. Based on L-K functional method, the stability and stabilization conditions of the control system are obtained. Finally, an example is used to verify the feasibility of the design method. Future research includes extending the method to static output feedback control.