Event-triggered compound learning tracking control of nonstrict-feedback nonlinear systems in sensor-to-controller channel

This paper investigates the event-triggered tracking control of the nonstrict-feedback nonlinear system with time-varying disturbances. While the fuzzy logic systems (FLSs) approximate the unknown dynamics, an event-triggered compound learning algorithm is originally developed to accurately estimate the total uncertainties. By referring to an event-triggered adaptive model, the control laws are derived without provoking the problem of “algebraic loop,” seeing Remark 3. The command filters are employed to generate the continuous substitutes for both the virtual control laws and their derivatives, so as to solve the recently proposed problem of “jumps of virtual control laws” arising in the backstepping-based event-triggered control (ETC). The triggering condition is constructed to guarantee the similarity between the adaptive model and the original system. Estimation of optimal fuzzy weights and compound disturbances follows from the event-triggered update laws. While the satisfactory learning performance is achieved, the proposed control scheme can guarantee the semi-globally uniformly ultimate boundedness (SGUUB) of all the tracking errors. Finally, a numerical experiment verifies the effectiveness of the proposed control scheme.


Introduction
Plenty of industrial plants can be described by nonstrictfeedback nonlinear systems, such as the ball and beam system, the spring damper system, the remote manipulator, and the stirred tank reactor [1]. Tracking control is the most common subject of the nonlinear systems, where the backstepping method is frequently used. As the strict-feedback system is attributed to the special case of the nonstrict-feedback system, the control strategies for the nonstrict-feedback system can easily extend to the strict-feedback one, but conversely, the conclusion is not true. According to the backstepping method, the ith step virtual control law of the strictfeedback system only utilizes the states from x 1 to x i , whereas that of the nonstrict-feedback system requires all the states such that the "algebraic loop" problem emerges [2]. Based on the monotonously increasing property of the bounding function of unknown dynamics, [3,4] developed a so-called variable separation method, in which the unknown dynamics can be transformed to the polynomials of tracking errors. The back-stepping method is available. This method was further combined with the observer design in [5], the smallgain theorem to deal with the unmodeled dynamics in [6], and extended to the stochastic nonstrict-feedback system in [7][8][9]. To exempt the restrictive assumptions of the unknown dynamics, [2] fabricated the backstepping-based control laws by using the bounded property of fuzzy basis functions. This method was applied to the switched nonstrict-feedback systems in [1,10], and combined with the finite-time control in [11]. By using the structural property of the radial basis function (RBF) NN, the "algebraic loop" problem was also solved in [12]. By using the semirecurrent neural networks (NNs) as the one-step predictors, [13] designed the backstepping-based control laws for the discrete nonstrict-feedback system, which also solved the "noncausality" problem [14].
There exist two drawbacks to control the nonstrictfeedback nonlinear systems in the existing researches. First, most of them followed the principle of "direct adaptive control" [15]. Although the closed-loop stability was ensured, the interpretability for approximation was missing in the FLS or NN. Second, the time-varying disturbances were seldom addressed. The "composite learning" technique provides a powerful solution to the first drawback, which combines "direct adaptive control" with "indirect adaptive control." By referring to a serial-parallel model [16][17][18][19][20][21][22][23] or a predictor [24,25], the identification components (namely the prediction errors) are utilized to estimate the fuzzy or neural weights, so as to ensure both the tracking and the approximation performance. It should be noted that the update laws in [16][17][18][19][20][21][22][23][24][25] only utilized the current information of the states, such that the parameters were not convergent if the persistent excitation (PE) condition was not satisfied. To release this constraint, [26][27][28][29][30] utilized the online-recorded data to construct the prediction errors, which led to a more relaxed condition of the interval excitation (IE) and the faster parameter convergence. For the second drawback, [20] found that disturbances will degrade the learning performance of the NNs or the FLSs, and neither the unknown dynamics nor the disturbances can be accurately approximated separately, whereas the sum of them can be estimated with a satisfactory precision by combining the composite learning with disturbance observers. This method was applied in the canonical nonlinear system in [20,21], and the strict-feedback systems in [30]. To be distinguished from the composite learning, this thought to identify both the unknown dynamics and the time-varying disturbances was named as "compound learning" in [30]. While reviewing these literature, we found that the compound learning was never applied to the nonstrict-feedback system. Although [22] applied the composite learning to the nonstrict-feedback system without the disturbances, it followed the primitive predictor-based composite learning. Moreover, it is noted that neither the compound learning nor the composite learning has ever been achieved in the ETC.
Due to the convincible merit to save communication load, ETC has attracted compelling interests in recent years, which has been widely employed in the strictfeedback systems [31][32][33][34][35][36][37][38][39][40], the nonstrict-feedback systems [41][42][43][44][45][46], and the other industrial plants [47][48][49][50][51][52][53][54]. It can be found that [31][32][33][34][35][41][42][43][44][45]47,[50][51][52] only addressed the ETC in the controller-to-actuator channel, namely the real control inputs kept unchanged by the zero-order holder (ZOH) during the inter-event time and were replaced by the command control signals at the triggering instants. As the communication congestion is more apt to occur in the sensor-to-controller channel, the ETC in the sensor-to-controller channel is indeed more preferred in control practice. For this goal, we notice the problem of "jumps of virtual control laws" in the backstepping design. A brief description of this problem is: while the states are involved in the virtual control law, the renewal of states at the triggering instants will precipitate the virtual control law into a sudden jump. This problem was first put forward in our previous researches of [40,53], and the formulaic definition will be presented in Sect. 2 of this paper. Although [47][48][49] addressed the backstepping-based ETC in the sensor-to-controller channel, this problem did not emerge as the backstepping therein was only used in one step. In fact, this problem is inevitable in the strict-feedback and the nonstrict-feedback systems, where the backstepping is iteratively used. [37][38][39]46] neglected this problem and acquiesced in the continuous virtual control laws at the triggering instants. Although the backstepping-based ETC was derived therein, it cannot procure a strict analysis on the closedloop stability. To our knowledge, the problem of "jumps of virtual control laws" was only solved in [36,40,53]. In [36], the final control input was constructed as the composition of each virtual control law without the involvement of event-triggered samples, and approximated by a NN. Although this paper meritoriously solved the problem of "jumps of virtual control laws" and circumvented the recursive appearance of virtual control laws in the triggering condition, the control performance therein totally relied on the approximation of the NN in the final control input. Because the direct adaptive control was adopted, the learning performance of the NN was not well. [40,53] substituted the virtual control laws with the continuous variables generated by the first-order filters. This setup can solve both the problems of "jumps of virtual control laws" and "complexity explosion." Nevertheless, the continuous variables were not smooth, and the parameter selection is fussy for the high-order nonlinear systems.
Motivated by above challenges, this paper develops an event-triggered compound learning scheme of the nonstrict-feedback nonlinear system. Based on an event-triggered adaptive fuzzy model, the backsteppingbased control laws are derived. The prediction errors are constructed according to the adaptive model and the original system, which are utilized to estimate the fuzzy weights and the compound disturbances. By using the online-recorded data collected during the inter-event time, estimation is renewed at the triggering instants and kept by the ZOH during the inter-event time. By referring to the second-order command filter, the smooth substitute for the virtual control law and the continuous substitute for its derivative are involved in the backstepping design. The adaptive triggering condition is fabricated to ensure the similarity between the adaptive model and the original system. Compared with the precedents, the contributions of this paper are mainly threefold.
1. By constructing an event-triggered adaptive model, the "algebraic loop" problem is solved. Compared with [3][4][5], the restriction on nonlinear functions is relaxed; Compared with [2,10], the fuzzy basis functions are no longer placed in the denominator, which refrains from the overshoot of adaptive laws. 2. By using the online-recorded-data-based prediction errors, the event-triggered compound learning technique is originally developed in this paper. Compared with [20,21,30], the communication resources are largely reduced without compromising learning performance of the FLS and the disturbance observer. 3. By referring to the second-order command filters, the problem of "jumps of virtual control laws" is solved. Compared with [37,38], the logicality of the stability analysis is ensured; Compared with the [40,53], the complicated analysis on selection of filter parameters is no longer required.
The remainder of this paper is organized as follows. Section 2 provides the preliminaries. Section 3 presents the control design. Section 4 designs the triggering condition and analyzes the closed-loop stability of the system. A practical example is provided to exemplify the effectiveness of the proposed scheme in Sect. 5. Section 6 concludes the entire work. Appendices prove the bounded estimation errors and Lemma 2.
Uniformly in this paper, · denotes the Euclidean norm of the vector, λ max (·) denotes the largest eigenvalue of the matrix,ŝ denotes the estimate of s and s = s −ŝ denotes the estimation error. t j implies the triggering instant with j = 0, 1, . . . , +∞, in which t 0 is also deemed as the initial time.

Problem formulation
Consider the single-input-single-output (SISO) nonstrict-feedback nonlinear system as . , x n ] T denotes the n-dimensional vector of states, y is the single output, u denotes the single control input. In the ith step of (1), f i (x) denotes the smooth unknown dynamics and d i denotes the unknown time-varying disturbance. To facilitate the following design and analysis, an assumption is preset as Assumption 1 [2,20,30,36] The time-varying disturbance d i and its first-order derivativeḋ i are bounded by |d i | ≤ b d i and |ḋ i | ≤ bḋ i , respectively, where b d i and bḋ i are two unknown positive constants.
Assumption 2 [12] The linear part x i+1 is unique in (1), namely f i (x) with i = 1, . . . , n − 1 cannot be reexpressed as Remark 1 Assumption 2 guarantees the feasibility of backstepping in the control of (1). As x i+1 is deemed as the virtual control input of ith subsystem, uniqueness of x i+1 excludes its failure to control the subsystem, for example, f i (x) = g i (x) + b i x i+1 exists and b i = −1.
The control objective of this paper is concluded as: design the control law of u such that the output y can track its smooth reference of y d , which has the bounded first-order derivative.
Next, we will discuss the problem of "jumps of virtual control laws." Following the backstepping design, the virtual control laws for (1) at the ith step usually has the form of is a smooth function, seeing [1][2][3]5]. If the ETC is designed in the controller-to-actuator channel like [41][42][43][44][45] is the continuous control command. Denote the tracking error in the ith step as z i+1 = x i+1 − α i . It can be inferred that the update of u(t) at t j will not affect the continuity of α i and z i . Nevertheless, if the ETC is designed in the sensor-to-controller channel, things are totally different. The definition of the problem of "jumps of virtual control laws" is provided as follows.
Definition 1 For the backstepping-based ETC in the sensor-to-controller channel, the states are sampled by the controller in an event-triggered way and at t j , which causes Δα i = 0. This is the so-called jumps of virtual control laws problem. As In [37][38][39]46], it presumed Δα i = 0 at t j . This disposal is not reasonable according to the above analysis. [36] fabricated a NN-based controller without α i involved, so as to solve the problem of "jumps of virtual control laws." In [40,53], this problem was solved by finding a continuous substitute for α i .

Approximation of FLSs
The FLSs are employed to approximate the unknown dynamics in (1). A FLS works mainly through three steps, namely fuzzification, fuzzy reasoning, and defuzzification. The fuzzy rule base is consulted to conduct the entire procedure, and it is composed of the IF-THEN rules as x n ] T is the input vector, y is the output, F l i and G l are the fuzzy sets with the membership functions of μ F l i (x i ) and μ G l (y), respectively. Through the singleton fuzzification, the product reasoning and the center average defuzzification, the FLS can be rewritten as whereȳ l = max y∈R μ G l (y). For simplicity, denote the fuzzy basis function in (2) as and let ϕ( The well-known universal approximation theorem of the FLS is concluded as follows.

Lemma 1 For any multiple-inputs-single-output function f (x) defined in a compact set x ∈ Ω, there always exists the FLS satisfying
where ε can be deemed as a slow time-varying variable bounded by |ε| ≤ b ε , where b ε is a positive constant. In this paper, W is determined as the optimal weight vector satisfying W = arg min W ∈R N {b ε } As the universal approximation theorem of the NN can also be described by Lemma 1, the FLS in this paper can be replaced by the NN.

Control design
The framework of the proposed control scheme is illuminated in Fig. 1. It adopts the second configuration of adaptive models in [49], in which the event-triggered adaptive model and the update laws are co-located with the controller. This configuration allows for the distributed layout of sensors, and is more practical than the first configuration in [49], namely the sensor-located configuration. The control design is divided into three steps.

Event-triggered adaptive model
According to Lemma 1, the unknown dynamics in (1) is where k i and k n are tuning parameters to be designed later. It is observed that x(t j ) in the right side of (4) is renewed at the triggering instant andx i keeps continuous. Denote the measurement error as e i = x i (t j ) − x i and e n = x n (t j ) − x n . By subtracting (4) from (1), it renders the estimation error system in t ∈ (t j , t j+1 ] as It can be inferred from (5)

Event-triggered compound learning
The update laws ofŴ i andσ i will be fabricated to guarantee the satisfactory learning performance ofŴ T i ϕ(x) andσ i . By using (5), the prediction error is constructed as It is clear in (6) that the prediction errors s i and s n synthesize both the recorded data from (1) and (4) during the previous inter-event time. Although the prediction errors can be constructed by solely using (1), this design of (6) can reduce the dependence on model precision. AsŴ i andσ i are only updated at the triggering instant, (6) can be further transformed to It can be inferred from (7) that the recorded data are only transmitted to the controller at the triggering instant through the integrators in Fig. 1. By substituting (5) to (6), it further renders where According to (8), the update laws ofŴ i are designed as where γ wi > 0 denotes the learning rate, l wiŴi (t j ) with l wi > 0 is used to prevent the parameter drifting. The selection principle of γ wi and l wi is discussed in Appendix A. By using (1), the prediction error is designed as Similar with (7), it can be transformed to (11) illustrates that all the data are achievable for p i and p n at the triggering instant. By substituting (1) to (10), it renders According to (12), the update laws ofσ i are designed as where γ σ i > 0 and l σ i > 0 have the same definitions with those in (9). The proposed update laws of (9) and (12) can guarantee the boundedness of W i and |σ i | all along the control process. The proof is provided in Appendix A.

Control laws
The control laws herein are designed by referring to the event-triggered adaptive model (4) instead of the original system (1). Denote the tracking errors as z 1 = x 1 − y d and z i =x i − β i−1 with i = 1, . . . , n, where β i is the filtered variable of the virtual control law α i and q i = β i − α i . The control design is divided into three parts.
Step 1: According to (4), the backstepping method is employed to eliminate the tracking error of z 1 , and the first virtual control law in t ∈ (t j , t j+1 ] is designed as where k a1 is the positive tuning parameter to be discussed later. Differentiating z 1 along (4) and (14), it renderṡ Step i: Similarly to eliminate z i with i = 2, lcdots, n − 1, the virtual control law in the ith step is designed as where k a2 is the positive tuning parameter to be discussed later. β i andβ i with i = 1, . . . , n − 1 are generated by the following second-order command filter [55].
where ζ > 0 is the damping coefficient and ω > 0 is the natural frequency, and it hasβ i = η i . According to the definition of q i , (16) and (17), Δq i at t j is expressed as By invoking (3), the following relationship holdŝ By invoking Assumption 1 and Lemma 1, (18) can be further expressed as As is proved in Appendix A, W i and |σ i | are bounded all along. It is reasonable to assume a positive constant b q i satisfying |Δq i | ≤ b q i . Then, it is proved in Appendix B that (17) has the following property.

Lemma 2
There exists a positive constantb q i such that |q i | ≤b q i always holds. By increasing ω in (17), it has b q i → b q i .
Remark 2 Becausex i is continuous in (4) and α i is filtered by (17), the components of −k ai z i and −z i−1 in (14) and (16) are continuous at t j . Different with the first-order DSC filter in [53], the second-order command filter of (17) can also guarantee the continuity ofβ i−1 in (16), such that the sophisticated analysis of Δβ i is avoided.
Differentiating z i along (4) and (16), it renderṡ Step n: Similarly to eliminate z n , the real control law of u in the final step is designed as u = −k an z n − z n−1 −Ŵ T n ϕ n (x(t j )) −σ n +β n−1 .
By differentiating z n along (4) and (16), it renderṡ z n = −k an z n − z n−1 + k n e n + k nxn .
Remark 3 Following from (1), the essence of "algebraic loop" problem implies that the x i cannot converge to the α i−1 (x) in the backstepping method, which is synchronously changed with x i . By using the eventtriggered adaptive model of (4) in the proposed scheme, the convergence ofx i to β i−1 will not cause the synchronous change of α i−1 (x(t j ),x i−1 ). As x(t j ) is fixed in t ∈ (t j , t j+1 ], (4) has the similar structure with the strict-feedback system in t ∈ (t j , t j+1 ]. Thus, the "algebraic loop" problem will never occur.

Design of triggering condition
Select the Lyapunov function as Select the Lyapunov function as V 2 = n i=1 z 2 i /2. Differentiating V 2 along (15), (21) and (23), it renderṡ Define V = V 1 +V 2 . By adding up (24) and (25) and invoking Lemma 2, the differential of V is obtained aṡ where b l i = sup{ W i 2 /2 +σ 2 i /2}. Because W i and |σ i | are bounded all along according to the proof in Appendix A, b l i is an unknown constant.
According to (26), the triggering condition is designed as where μ is the positive triggering threshold adjusting triggering frequency of ETC. The next triggering instant is activated while (27) is violated. Then, the proof of the existence of minimum inter-event time δ t (namely to avoid the "Zeno" behavior) is given as follows.
Proof of δ t : Denote the left side of (27) as . It is observed that equates to 0 at the triggering instant.
As W i is bounded all along the control process and W i = W i −W i , it is known that Ŵ i is also bounded. Furthermore, according to (9),Ŵ i is a constant vector in t ∈ (t j , t j+1 ]. As is mentioned in the subsequent theorem, all the x are defined in a compact set. It is clear thatẋ i is bounded according to (1). The membership function μ F l i is adopted as a smooth bounded function. According to (9), φ i is also bounded. Then, it is clear that˙ can be limited by a positive constant l . According to (27), δ t ≥ μ/l can be proved. The proof is completed.  (7) and calculate p i (t j ) from (11) end if updateŴ i by (9) andσ i by (13) calculatex i from (4) calculate α i from (14) and (16) calculate β i andβ i from (17) calculate u from (22) execute u in the nonstrict-feedback system of (1) end for

Stability analysis
The main result of this paper is concluded as the following theorem.
Theorem 1 Assume all the states of (1) are defined in a compact set, namely If the update laws of (9) and (13), the real control law of (22), and the triggering condition of (27) are used, all the tracking errors are proved to satisfy SGUUB. Moreover, by tuning the parameters appropriately, the output y can track the reference y d in a satisfactory precision.
The application of the proposed scheme can be described by the following pseudocode. As the processor runs in a digital way, the computing times are discrete with a fixed period t s , where the k = 1, . . . , N denotes the kth computation after t 0 .
Proof of Theorem 1 By substituting (27) to (26), and selecting a = 2 min{k 1 − 3, (26) can be simplified aṡ According to the comparison theorem, it can be inferred from (30) that for i = 1, . . . , n Thus, the SGUUB of z i is proved. Because y − y d = z 1 +x 1 and |x i | ≤ √ 2V too, it is inferred that |y−y d | ≤ 2 √ 2V . According to (31), we can infer that |y − y d | ≤ 2 √ 2V → 2 √ 2b/a while t → +∞. Thus, we can only prove the convergence of y to y d in 2 √ 2b/a. By selecting the large a, the steady tracking error 2 √ 2b/a will be small. The selection principle of parameters is provided as follows.

Remark 4
The parameter selection follows two steps: 1) select k 1 > 3, k i > 7/2, k n > 3 and μ > 0 to ensure the acceptable estimation performance of (4); 2) select k ai > 1 + k 2 i /2 to ensure the satisfactory tracking performance. It is observed in (30) that the larger a will lead to the smaller steady tracking and estimation errors as well as the faster convergence rate. Meanwhile, the larger a will also result in the undesired control transient and the energy waste.
Remark 5 According to the definition of z 1 , it is inferred that z 1 → 0 ⇒ x 1 → y d if x i →x i . Becausẽ x i is affected by the approximation errors and the measurement errors according to (5), the tracking performance is positively associated with the learning performance and decreasing the triggering threshold μ in (27). Besides, as μ is involved in b of (30), decreasing μ can also reduce the steady error of z i . Nevertheless, this operation will increase triggering frequency and lengthen inter-event time.

Numerical experiment
In this section, the proposed event-triggered compound learning scheme is applied to the ball and beam system in [56], and its setup is shown in Fig. 2.
In Fig. 2, o 1 , o 2 and o 3 are three fixed pivots. By regulating the voltage of the DC motor (namely u), the position of the ball in beam 2 (namely x 1 ) can be regulated as needed. This system possesses the mathe- where  Table 2. To correspond with the structure of (1), let f 2 (x) = Ax 1 x 2 4 − Agx 3 − x 3 and f 4 (x) = −B cos(x 3 )(Dx 4 cos(l b x 3 /d) + E + H x 1 ) − BGx 1 x 2 x 4 express the unknown dynamics in (32), and the unknown dynamics and the disturbances in the first and the third steps are omitted. Reconstruct the control input as U = BC cos( The time-varying disturbances are described by the following second-order Markov process.
where ω 2 and ω 4 are two independent Gauss white noises with the variance of 1. The initial values of d 2 , d 2 , d 4 andḋ 4 are assigned as 0. The FLS is set with 7 fuzzy rules, and the membership function for each In this simulation, f 2 (x) and f 4 (x) in (32) employ the same set of fuzzy membership functions. The reference signal is selected as a sinusoidal function of time, namely y d = 0.2 + 0.05 sin(0.1257(t − t 0 )). The initial values of states are assigned as x 1 (t 0 ) = 0.3m, x 2 (t 0 ) = 0m/s, x 3 (t 0 ) = 0 • and x 4 = 0 • /s. The initial values of the estimation states in (4) and the estimations in (9) and (13) are assigned as zeros.
The numerical calculation period is set as 0.01s in the simulation, which will not deteriorate the algorithm effectiveness. To demonstrate the differences, two comparative schemes are set in this simulation. The first one is the time-triggered compound learning, marked as "TT-CL." This scheme can be achieved by setting μ = 0 in (27). According to [30], the continuous update law ofŴ i and the disturbance observer ofσ i are presented aṡ  (35) is the prediction error using the data collected in the latest 0.1s. The second scheme adopts the predictor-based learning in [23], which is marked as "ET-PL." In this scheme, the compound disturbance σ i is not addressed, and the update law ofŴ i is designed as where the event-triggered adaptive model (4) functions as the predictor andx i is deemed as the prediction error.
Simulation results are exhibited as follows. Figure 3 shows the evolution of disturbances. The magnitudes of them correspond with the magnitudes of total uncertainties. Fig. 4 demonstrates the outputs of three schemes. It is clear that TT-CL and ET-CL have better tracking performance than ET-PL due to their better learning performance. This point is verified in J z (IAE) and J z (ITAE) of Table 3. Because the ETC is employed, the outputs of ET-CL and ET-PL inevitably suffer from fluctuation. Figure 5 shows the evolution ofx i in (4) of ET-CL, in which a fast and accurate convergence to x i is observed. Fig. 6 demonstrates the control inputs of three schemes. The discontinuity of ET-CL and ET-PL is observed in the local enlarged view.
Figs. 7-9 compare the leaning performance of three schemes. TT-CL has the best learning performance among three, and ET-CL has much better learning performance than ET-PL. This is also verified in J c (IAE) and J c (ITAE) of Table 3. Fig. 10 shows the evolution of performance indexes. It is observed that TT-CL has the slowest growing rate of indexes in three schemes, which means the best tracking and learning performance. The indexes of ET-CL grow slower than ET-PL, and attest to the better tracking and learning performance. Figure 11 demonstrates the variation of update laws in ET-CL, in which the event-triggered renewal is observed. Figure 12 compares the accumulation of sampling times in three schemes. A considerable communication saving is observed in the event-triggered schemes. The statistics of total sampling times are provided in Table 3. Figure 13 shows the evolution of inter-event time in ET-CL. The minimum inter-event time is observed at the initial stage and equates to 0.01s. The maximum inter-event time is recorded as 0.26s. Periodic inter-event time is  Fig. 13, which is caused by periodic fluctuation of u in Fig. 6. This phenomenon is caused by lack of state information while t ∈ (t j , t j+1 ]. As shown in TT-CL of Fig. 6, the fluctuation can be averted with sufficient state information.

Conclusion
This paper proposed an ETC scheme of the nonstrictfeedback nonlinear system with the event-triggered compound learning technique. An event-triggered adaptive fuzzy model was fabricated to direct the backstepping design of control laws, such that the "algebraic loop" problem was solved. Two prediction errors were constructed by using the estimation error system (5) and the original system (1), respectively, which can collect the data during the previous inter-event time.
With the prediction errors involved, the estimates of the fuzzy weights and the compounding disturbances were updated in an event-triggered manner. The secondorder command filters were employed to substitute for the virtual control laws and their derivatives, such that the problem of "jumps of virtual control laws" was solved. An adaptive triggering condition was constructed to guarantee the closed-loop stability. Finally, the effectiveness of the proposed scheme was exemplified in the ball and beam system via numerical simulation. In the future, we plan to incorporate the presented event-triggered compound learning technique into some industrial plants. of Hebei Province (No. E2020203174). We'd like to thank Professor Chen Bing in Institute of Complexity Science, Qingdao University for his suggestions on the nonstrict-nonlinear system.

Data Availability Statement
The data for supporting the findings will be made available upon the reasonable request for academic use by contacting the corresponding author.

Compliance with ethical standards
Conflict of interest There is no potential conflict of interest in this paper.

A Proof of bounded estimation errors
Select the Lyapunov function as V wi =W T iW i . By invoking (8), (9) and the Young's inequality, ΔV wi is derived as Because the maximum inter-event time T max is finite and ϕ i and σ i are bounded, it is tenable that the positive constants of n φ i and b D i exist.
Select the Lyapunov function as V σ i =σ 2 i . By invoking (12), (13) and the Young's inequality, ΔV σ i is derived as By adding up (A-1) and (A-2), it renders By selecting l σ i < 1/2, 3γ 2 wi n 2 φ i + 4γ 2 σ i n φ i < 1 and Becauseσ i is changing during the inter-event time, one cannot infer the boundedness of |σ i | from ΔV i ≤ −a i V i + b i . A further processing is needed.
Assume |σ i | grows with the fastest speed during the inter-vent time. According to Assumption 1 and (13), it is tenable to assume |σ i | = |σ i | ≤ bσ i in t ∈ (t j , t j+1 ], where bσ i is an unknown constant. Thus, the following inequality holds.
Thus, a new interconnected system is obtained as If V i (t j ) > b i /a i , it has ΔV i < 0 for the first inequality in (A-5). In this case, the second inequality in (A-5) is transformed to is bounded all along and is ultimately bounded by max{b i /a i , (bσ i T max + b 2 σ i T 2 max (1 + a i ) + a i b i ) 2 /a 2 i }. The proof is completed.
It is obvious that the larger T max (namely by increasing μ) will lead to the smaller a i and the larger ultimate bound of V i , which suggests the slower learning rate and the worse identification accuracy.

B Proof of Lemma 2
For convenience, set ζ = 1 in (17). By subtracting [α i , 0] T from (17), it renders Thus, a 0 (t) = (1 + ωt)e −ωt and a 1 (t) = te −ωt are obtained. Ulteriorly, the state transformation matrix of (B-1) in t ∈ (t j , t j+1 ] is obtained as Φ(t) = a 1 (t − t j )A +a 0 (t −t j ). Then, the analytical solution of (B-1) is derived as Because all the states are defined in a compact set, it is tenable to assume |α i (t)| ≤ bα i , where bα i is an unknown constant. Then, the following inequality can be derived from (B-3).
For (B-4), an assumption is made at first that |q i (t + j )| ≤ μ q i + b q i and |η i (t j )| ≤ μ η i + bα i , where μ q i and μ η i can two arbitrarily small positive constants. For the first inequality in (B-4), it is inferred that |q i (t)| → 0 with ω → +∞. Thus, there must exist ω = ω q i such that |q i (t j +δ t )| ≤ μ q i holds, where δ t is the minimum inter-event time proved in Section IV. Invoking |Δq i | ≤ b q i , |q i (t + j )| ≤ μ q i + b q i holds. For the second inequality in (B-4), it is inferred that |η i (t)| ≤ bα i with ω → +∞. Thus, there must exist ω = ω η i such that |η i (t j + δ t )| ≤ μ η i + bα i holds. By selecting ω = max{ω q i , ω η i }, this assumption holds.