Adaptive fuzzy output-feedback event-triggered control for fractional-order nonlinear system

: This paper studies the issue of adaptive fuzzy output-feedback event-triggered control (ETC) for a fractional-order nonlinear system (FONS). The considered fractional-order system is subject to unmeasurable states. Fuzzy-logic systems (FLSs) are used to approximate unknown nonlinear functions, and a fuzzy state observer is founded to estimate the unmeasurable states. By constructing appropriate Lyapunov functions and utilizing the backstepping dynamic surface control (DSC) design technique, an adaptive fuzzy output-feedback ETC scheme is developed to reduce the usage of communication resources. It is proved that the controlled fractional-order system is stable, the tracking and observer errors are able to converge to a neighborhood of zero, and the Zeno phenomenon is excluded. A simulation example is given to verify the availability of the proposed ETC algorithm.


Introduction
Fractional-order nonlinear systems (FONSs) are a class of complex systems modeled by fractional calculus. In recent years, the application of fractional calculus has attracted wide-ranging attention. In practice, examples such as the blood ethanol concentration system by Qureshi et al. [1], the dynamics of the TB virus by Ullah et al. [2], the fractional Brusselator reaction-diffusion system by Jena et al. [3], etc. can be modeled as fractional order systems. This kind of system can well describe the genetic effect and long memory effect of a real physical system. Nevertheless, it is noteworthy that conventional control scheme is no longer applicable to fractional-order systems due to some general rules for integer-order calculation, such as Chain rules and Leibniz rules, not being well built with regard to fractional derivatives. Hence, how to solve the stability analysis or controller of fractional order systems has attracted extensive attention of scholars. For example, in [4], a fractional order controller designed by Liu et al. to ensure that the synchronization errors of the fractional order chaotic proposed control algorithm erases the restrictive condition in [18][19][20], with which the state of the system must be completely measured.
2) Due to the DSC technique being used to control design, the put forward control method settles the computational complexity issue in current works [18][19][20].
3) In this paper, the adaptive control law and the event-triggered mechanism are designed together. The stability of the controlled system can be guaranteed by using the fractional-order Lyapunov criterion. Unlike [9][10][11][12], the control signal needs to be sampled and updated regularly. The system drive will be generated only when the preset conditions are met in this paper, which greatly reduces the consumption of network resources.

System statements
Consider the following FONS: ., x n ] T ∈ R n are the system state vectors, and y ∈ R and u ∈ R denote the output variable and control input of the system. f i (·) ∈ R, i = 1, . . . , n, denotes an unknown smooth nonlinear function. This paper assumes that only the output variable y is measurable. Assumption 1 [9][10][11][12][13][14]: The given reference signals y d , C 0 D α t y d and C 0 D α t ( C 0 D α t y d ) are smooth and bounded. Furthermore, assumed that there exists known constant Z 0 > 0 satisfying that Control Objectives: In this article, a fuzzy adaptive event-triggered controller is designed for System (1) such that all signals in the considered system are bounded, and the tracking error converges to the compact set of the origin.

Fuzzy state observer
Write the FONS (2.1) as follows: , C = 1 0 · · · 0 T , K is chosen such that (A + CK) is a Hurwitz matrix. Q = Q T > 0 is a positive definite matrix, and exist a positive definite matrix P = P T > 0 such that It is worth noting that f i (x i ) in (3.1) is an unknown continuous function, so it is necessary to approximate f i (x i ) with the help of an FLSf i (x i |ξ i ) = ξ T i ϕ i (x i ),. In the bounded sets Ω, the definition of ideal parameter vectors ξ * i are described as: The definition of the optimal approximation errorε i is described as where ε = [ε 1 , ε 2 , . . . , ε n ] T , ξ i are the estimations of ideal parameters ξ * i , andξ = ξ * − ξ. Construct the Lyapunov function candidate as V 0 = 1 2 e T Pe, and then the following Theorem can be obtained.
Remark 1: Theorem 1 shows that ifξ T iξ i is bounded, smaller observation errors e i can be obtained by selecting a large enough λ 0 . It is further concluded that the constructed fuzzy state observer (3.5) can better estimate the unknown states.

ETC controller design and stability analysis
This part will use the adaptive fuzzy backstepping control algorithm to provide the observer-based adaptive ETC control design program and give its stability analysis.

Design procedure
Make the coordinate transforms as where S 1 is tracking error, S i are dynamic surface errors, υ i are filter variables, and η i are filter output errors, and τ i−1 are the virtual control functions.
Step1: Via (3.3), (4.1), and x 2 = e 2 +x 2 , one has Choose the Lyapunov function as where γ 1 > 0 is a known constant. The virtual controller τ 1 and the adaptive law C 0 D α t ξ 1 are designed as where c 1 > 0 and κ 1 > 0 are known constants. Introduce dynamic surface filter in [15] as where σ 1 is a constant. By using (4.2) and (4.6), one has where W 1 (·) is a continuous function. Remark 2: In the backstepping ETC design of FONSs, it is difficult to obtain the mathematical analytical expression of the fractional derivative of the virtual controllers. To solve this problem, some authors used the packaged approximation technology in [18][19][20] to repeatedly approximate the virtual controllers. Because this method takes all the signals of the closed-loop system as the input variables of the NN or FLS, it will increase the dimensions of the adjusted parameter vector, resulting in the problem of computational complexity. Therefore, this paper adopts the DSC technology to effectively avoid this problem.
Step i: From (3.5) and (4.1), one has The Lyapunov function candidate is chosen as where γ i > 0 is a known constant. The virtual controller τ i and the adaptive law C 0 D α t ξ i are designed as where c i > 0 and κ i > 0 are known constants. Introduce dynamic surface filter as where σ i is a constant. By using (4.8) and (4.12), one can obtain where W i (·) is a continuous function.
Step n: We first devise an event-triggered controller as where c n > 0 is a known constant, and t k (k ∈ z + ) defines input updating time. Thus, in order to get a lower communication rate, the event-triggered condition can be designed as whereδ ∈ (0, 1), m > 0 andm > [m (1 −δ)] are given as the known parameters, and ν(t) = ω(t) − u(t) is called as the measurement error. When (4.17) is triggered, the time will be marked as t k+1 , and the controller u(t k+1 ) will be utilized to the system. At the time t ∈ [t k , t k+1 ) the control signal is always unchanging.
If v(t k ) > 0, the measured error can be rewritten as If v(t k ) < 0, we can transform the event-triggered condition (4.17) as where λ 2 (t) ∈ [−1, 1]. From (4.18) and (4.19), one obtains where |λ i | ≤ 1, i = 1, 2 are time-varying variables. Then, one has Remark 3: The event-triggered parametersδ and m in (4.17) are determined according to the required communication rate. Therefore, in practical applications, while ensuring satisfactory tracking performance, we should try to reduce the communication burden.
The adaptive law C 0 D α t ξ n is designed as: where κ n > 0 is a known constant. Remark 4: A backstepping control algorithm is indicated for FONSs in [17]. It applies the stability analysis of integer-order Lyapunov methods to known fractional-order systems. However, in this article, the system model may be completely unknown. In addition, the stability of the control algorithm is analyzed by the fractional order adaptive stability criterion.

Closed-loop systems stability analysis
Theorem 2: Consider System (2.1), under Assumptions 1-2, and then the put forward adaptive fuzzy output feedback event-triggered controller (4.21) with the event-triggered mechanism (4.17) can keep that controlled fractional-order system is stable, and the tracking error is able to regulate to a small residual set of the origin. Meanwhile, Zeno behavior is removed effectively.
Ultimately, Zeno behavior is removed through the following proof. By recalling the measurement error From (4.20) that ω(t) is a differentiable signal of order α, and C 0 D α t ω is a bounded function. Thus, the existence of ρ > 0 makes C 0 D α t ω ≤ ρ hold. According to ν(t k ) = 0 and lim t→t k+1 v(t) = m, one can have t k+1 − t k ≥ m/ρ. So, Zeno behavior does not occur.
Remark 6: It is worth noting that the control schemes designed by the authors in [7][8][9][10][11][12][13][14] are based on time triggered control. Because the control signal is sampled and updated periodically, it leads to a waste of communication resources. In order to solve this issue, an event-triggered mechanism is introduced in the backstepping technology. This mechanism enables the control signal to be sampled and updated only when the given conditions cannot be met, thus decreasing the communication load.
Remark 7: The repeated differentiation of virtual control function will lead to complexity explosion, so filter is introduced to solve this problem. By using event-triggered mechanism and dynamic surface filter at the same time, this paper greatly saves the waste of computing resources.
Remark 8: In [15,16], it is important to study the adaptive ETC for integer-order nonlinear systems. However, we have designed an event-triggered rule for FONSs with unmeasurable states. Within the framework of event-triggered DSC scheme, we solved the computing explosion caused by duplicate derivation of virtual controllers. Unlike event-triggered rule in Wang et al. [17], threshold is a function of system state or tracking error. This paper determines the improved event-triggered mechanism based on the size of the control signal itself.

Simulation example
This part gives a simulation example to verify the availability of theoretical results. Example: The fractional-order strict-feedback system is described as follows: The membership functions can be chosen as The given reference signal is y d = cos(2t). The observer gain is selected as K 1 = [k 1 , k 2 ] T = [10, 220] T . Then, the observer (3.3) can be written as The control law can be given as The event-triggered controller is devised as The event-triggered condition can be designed as whereδ ∈ (0, 1), m > 0 andm > [m (1 −δ)]. With the following adaptive laws:  Figure 1 shows the curves of reference signal y d and the system output y. Figure 2 shows the curves of reference signal y d and the system output y without ETC. From Figures 1 and 2, we can see that ETC can ensure satisfactory system performance. The trajectories of the tracking error are shown in Figure 3. Figures 4 and 5 response of x i andx i , i = 1, 2. Figure 6 responses of u. Figure 7 shows the trigger time intervals with the event-triggered control. However, by calculating, we know that the controller executes 2000 times without eventtriggered control; under the event-triggered control method, the number of samples is only 1007, which greatly reduces the waste of communication resources. From Figures 1-7, it is concluded that the proposed event-triggered controller can achieve the stability of the controlled system and effectively decrease the communication load.

Conclusions
The observer-based adaptive ETC algorithm for FONS was investigated in this study. By employing FLS to model the unknown dynamics, a fuzzy state observer is constructed for the unmeasurable state vectors. Using an adaptive backstepping control algorithm, an observer-based adaptive fuzzy DSC method is proposed. The put forward control algorithm has avoided computational complexity problem resulted in the repeated iteration of virtual controllers in the inherent backstepping method. Additionally, it has reduced the burden of communication and removed the Zeno behavior. The simulation results testify the validity of the controller. The further research will focus on the intelligent adaptive ETC problem of fractional-order nonlinear impulsive systems based on this study and literature [23,24].