Event-Triggered Adaptive Neural Network Tracking Control with Dynamic Gain and Prespeciﬁed Tracking Accuracy for a Class of Pure-Feedback Systems

: This paper studies the event-triggered adaptive tracking control problem of a class of pure-feedback systems. Via the backstepping method and the neural network approximation with the central symmetric distribution, an event-triggered adaptive neural network controller is designed. In particular, a dynamic gain driven by the tracking error is introduced into the event-triggering mechanism. Then, by using the Lyapunov stability theory, the boundedness of all the closed-loop signals is proved, and the tracking error falls into a prespeciﬁed (cid:101) -neighbourhood of zero. Meanwhile, the Zeno behaviour is avoided. Finally, two simulations verify the effectiveness of the proposed control scheme.


Introduction
In the last several years, the event-triggered control (ETC) of nonlinear systems [1][2][3][4][5][6] has received extensive attention from scholars. For example, the output-feedback eventtriggered (ET) controllers are designed for nonlinear plant models [1]. The Zeno behaviour is solved by combining ETC and the time-triggered control. In [2], the authors systematically co-design the adaptive controller and the event-triggering mechanism (ETM) and remove the input-to-state stability assumption on the measurement and requirement of the global Lipschitz continuousness of the nonlinearities. In [3], a periodic ETC policy is utilised to investigate the minimum bit rate conditions for stabilising a scalar nonlinear system under bounded network delay, where the sensor and the controller are updated asynchronously without acknowledgement. By adding different dwell times, two ETC strategies are proposed with static and dynamic ETMs, respectively [4]. Under a fixed/relative threshold strategy, the ETC problem is investigated for p-norm uncertain nonlinear systems to guarantee all the signals of the closed-loop systems converge to an arbitrarily small set [5]. A relative threshold event-triggered strategy is also investigated for large-scale high-order uncertain nonlinear systems [6]. Besides, an event-triggered (ET) stabilisation issue is investigated for IT2 fuzzy systems [7]. In [8], an ET strategy is proposed to save communication resources, and the triggering condition has an adaptive threshold. It is proved that the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB). In [9][10][11], the ETC problems of strict-feedback systems with external disturbances are studied. In [12], the authors rewrite the closed-loop error system as a linear system with nonlinear perturbation and novelly construct an event-based dynamic surface control law under convex optimisation. Dynamic gains are used in [13,14] to design ET controllers for strict-feedback systems. In [13], based on the dynamic gain, an adaptive ET controller is designed to achieve the desired tracking target and avoid the Zeno where Ω ρ i is a compact set; R denotes the set of real numbers; ideal weight vector W * i := arg min . . , s l i (ρ)] T is the basis function vector; l i is the number of neuron nodes and δ i (ρ) is the approximation error. In this paper, Gaussian functions s j (ρ) = exp − . . , l i ) are used as the basis functions, where ξ j ∈ Ω ρ and ψ j > 0 are the center and width of the Gaussian function, respectively.

Problem Statement
Consider the following pure-feedback system: where x i (i = 1, . . . , n) are the system states withx i = [x 1 , . . . , x i ] T ∈ R i ; u ∈ R and y ∈ R are the system input and output, respectively; f i (·)(i = 1, . . . , n) and g n (·) are unknown smooth nonlinear functions.
The purpose of this paper is to design an ET adaptive NN controller to meet the following objectives: (a) All the signals in the closed-loop systems are bounded on [0, +∞); (b) Tracking error y − y r falls into a prespecified −neighbourhood of zero; (c) The infinite ET phenomenon will not happen, that is, Zeno behaviour is avoided.

Adaptive Backstepping and ET NN Controller
In this section, an ET ANNC scheme is proposed through the adaptive backstepping design, where a dynamic gain is introduced into the adaptive NN controller.
In order to show the idea of this paper more intuitively, a frame diagram of the ETC architecture is shown in Figure 1, where ZOH means zero-order holder. It can clearly be seen from Figure 1 that both the dynamic gain in the sensor-controller channel and the ETM in the controller-actuator channel are considered comprehensively, where the ETM is integrated into the controller to determine whether the communication occurs in the controller-actuator channel. Step 1. Define tracking error z 1 = x 1 − y r , whose derivative is: Define: where c 1 is a positive constant. Obviously, ω 1 is just a function of y r and x 1 , so ∂ω 1 ∂x 2 = 0. Taking Assumption 3 into consideration, Thus, According to the process in [24], there is an ideal virtual control input α * 1 (x 1 , ω 1 ) for each x 1 and ω 1 such that: whereḡ 1 = g 1 (x 1 , χ 2 ) with χ 2 = µ 1 x 2 + (1 − µ 1 )α * 1 and 0 < µ 1 < 1. It can be deduced that Assumption 3 made for g 1 (x 1 , x 2 ) still applies toḡ 1 . Because α * 1 is a function of x 1 and y r , g 1 is a function of x 1 , x 2 and y r . Then one has:ḡ According to Assumption 4, |ġ 1 | ≤ g is still holds. Combining (2)-(4), one can obtain: RBF NN is used to approximate α * 1 , i.e., where γ 1 = (x 1 , y r ) T ∈ Ω 1 ⊂ R 2 . Define z 2 = x 2 − α 1 , and design virtual controller: whereŴ 1 is the estimate of W * 1 ; b 1 is a positive design constant and G is the dynamic gain defined as: with being a prespecified tracking accuracy. Then, Equation (6) becomes: Hereinafter, define( ·) =( ·) − (·) * . Select the Lyapunov function: where Γ 1 is a positive definite matrix with proper dimension and hereinafter, Γ i throughout the paper has the same meaning. Its derivative is: According to the Young's Inequality, it is obtained as: Design the adaptive law ofŴ 1 as:Ŵ where σ 1 is a small positive constant. By choosing b 1 = b 10 + b 11 with b 10 > 0 and b 11 > 0, Equation (12) becomes:V According to the Young's Inequality, the following inequality holds: )z 2 1 , by choosing b 10 large enough such that b * 10 = b 10 − g 1s 2g 2 1m > 0, Equation (15) becomes: Remark 1. According to (9), it can be seen that G will gradually increase untilĠ = 0, which implies that the control goal is achieved. Obviously, G is a non-decreasing function greater than 1.
where constantḠ is an upper bound of G and the boundedness of G will be proved later. Step Similar to Step 1, there exists µ i (0 < µ i < 1) satisfying: where, where whereŴ i is the estimate of W * i and b i is a positive constant. Then,ż i becomes:ż Choose Lyapunov function: Its derivative is:V Similar to Step 1, it holds: The adaptation law ofŴ i is designed: where σ i is a small positive constant. By Similar to Step 1, it holds: Step n. The derivative of z n = x n − α n−1 is: There exists the ideal virtual controller: where b n is a positive constant.
On account of that α n−1 is a function of y r ,x n−1 andŴ 1 , . . . ,Ŵ n−1 , where, The unknown part of α * n is approximated by using an RBF NN, i.e., where γ n = (x n , ∂α n−1 ∂x 1 , . . . , According to (37), design the adaptive NN controller and the weight update law as: where σ n is a small positive constant. Let b n = b n0 + b n1 with b n0 and b n1 > 0. In order to decrease the controller update frequency, an ETM is designed to decide the data transmission from controller to actuator. Now the ETM and the ET controller are given as follows: where k = 0, 1, . . .; t 0 = 0; A is a positive constant. Now select the Lyapunov function: and it is obtained as:V Combining (32) and (43) gets: Substituting (32), (36), (38) and (39) into (44), one obtains: The following inequalities hold: nm > 0, Equation (45) becomes:
(ii) Because G is a monotonically non-decreasing function, there exists finite time T 1 such that: Recalling (51), one obtains:V and thus, According to the definition of V n and (56), there is sufficiently large time T 2 such that: By (9), we haveĠ = 0, ∀t > T 2 , which implies sup t≥0 G(t) = G(T 2 ) < +∞. Therefore, the boundedness of G on [0, +∞) is obtained. Moreover, because y r and z i are bounded on [0, +∞) and z i = x i − α i−1 , it can be recursively derived that x i and α i are bounded on [0, +∞). Based on the forms of α n and u, the boundedness of u on [0, +∞) is obtained.
Thus, all the signals in the closed-loop system are bounded on [0, +∞).
Taking (38) into consideration, we have: The items in the right-hand side of (61) are continuous, soα n is continuous. Thus, ∃E > 0,s.t. |α n | ≤ E, ∀t ∈ [t k , t k+1 ]. Then one can have: Eventually, it is obtained that: Therefore, no Zeno behaviour occurs. The proof is completed.

Numerical Simulation
We give the following numerical example to illustrate the effectiveness of the proposed scheme.
    ẋ The reference trajectory is chosen as y d = 0.5 cos(0.5t) + 0.6 sin t. System output trajectory y and reference trajectory y d are shown in Figure 2. In Figure 3, it can be clearly seen that the tracking error trajectory is between −0.03 and 0.03 after 0.4vs, which implies that the control target is achieved. In Figure 4, the growth of dynamic gain G is clear at a glance, and it remains unchanged after 0.4vs, which corresponds to the fact that the tracking error in Figure 3 converges with the prespecified accuracy after 0.4vs. That is, the control target is achieved. The change of control u can be clearly seen in Figure 5. Figure 6 is the time interval diagram between the consecutive triggered moments.      Theoretically, the value of A is the fixed triggering threshold, which can be an arbitrary predetermined positive constant. On the premise of achieving the control objectives, different values of A are taken for simulation and the triggered times of events under different values of A are shown in Table 1. It can be seen from Table 1 that when A is 0.3, the number of the triggered events is the least. Table 1, it also can be seen that too small or too big A values can create a large number of events. Obviously, the smaller the triggering threshold, the more the events are triggered. That is, more network resources are wasted. Hence, it seems that the value of A should be chosen large enough. However, if A is chosen too large, the error between the controller and the actuator may deteriorate the control performance and cause big fluctuation of the controller; thus, more events may be triggered. In practical applications, the triggering threshold need to be selected eclectically to trade off the system performance and the communication frequency, which seems to be a dilemmatic choice. Therefore, in the future, the dynamic threshold method also should be considered in our research.

Simulation of One-Link Robot
In this part, the effectiveness of the proposed scheme is further verified by the practical application of the one-link robot. The equation of the one-link robot system is as follows [25]: where p,ṗ andp represent the position, velocity and acceleration of the link, respectively. φ andφ are the motor shaft angle and velocity. u denotes the control input of motor torque. Let x 1 = p, x 2 =ṗ, x 3 = φ R , y = x 1 , and then system (65) can be rewritten as: In the simulation, the system parameters are chosen as R = 1, F = 1, N = 10, L m = 10, Q = 0.5, M = 0.05 and reference trajectory y d is generated by the van der Pol oscillator system [25].
Here, 27, 243 and 2187 nodes were chosen for NNs: 1], respectively, with the same width ψ = 2. The simulation parameters are selected as follows: The trajectories of the system output y and reference signal y d are shown in Figure 7. In Figure 8, it can be clearly seen that the tracking error trajectory is between −0.05 and 0.05 after 0.5 s, which implies that the control target is achieved. In Figure 9, dynamic gain G grows at the beginning and remains unchanged after 0.5 s, which corresponds to the fact that the tracking error in Figure 8 converges with the prespecified accuracy after 0.4 s. That is, the control target has been achieved. The change of control u can be clearly seen in Figure 10. Figure 11 shows the time interval diagram between the consecutive triggered moments.    Time(sec) Figure 11. Time intervals between each two consecutive triggered instants for (66).
As mentioned in Remark 3, different values of A are also selected for simulation. The results are shown in Table 2. On the premise of achieving the control goal, when A = 0.4, the number of events is the least; that is, the saving of communication resources is the greatest.

Conclusions
In this paper, a new ETC scheme is proposed for a class of pure-feedback systems. The introduction of dynamic gain ensures that the output signal can track the reference signal effectively with the prespecified accuracy. Through theoretical analysis, it is proved that all signals are ultimately bounded, and Zeno behaviour is ruled out. Two simulations show the effectiveness of the designed scheme.
The threshold of the ETM in this paper is a given constant. In the future research, dynamic threshold [36] and controller fault-tolerance [37] will be further considered. The method proposed in this paper will also be expanded to the multi-agent systems with the symmetric communication topology.

Conflicts of Interest:
The authors declare no conflict of interest.