Event-Triggered Compound Learning TrackingControl of Nonstrict-Feedback Nonlinear Systems inSensor-to-Controller ChannelYingjie Deng ( dyj_dl@126.com )

Yanshan University, School of Mechanical Engineering https://orcid.org/0000-0003-0819-9619Tao Ni

Yanshan University, School of Vehicle and EnergyJiantao Wang

Yanshan University, School of Vehicle and Energy

Research Article

Keywords: Event-triggered control, Fuzzy logic system (FLS), Compound learning, Jumps of virtual controllaws, Command lter backstepping

Posted Date: April 6th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-372665/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

Version of Record: A version of this preprint was published at Nonlinear Dynamics on October 4th, 2021.See the published version at https://doi.org/10.1007/s11071-021-06870-w.

Nonlinear Dyn No. xx(will be inserted by the editor)

Event-Triggered Compound Learning Tracking Control ofNonstrict-Feedback Nonlinear Systems in Sensor-to-ControllerChannel

Yingjie Deng · Tao Ni · Jiantao Wang

Received: xx March 2021 / Accepted: xx xx 2021

Abstract This paper investigates the event-triggered

tracking control of the nonstrict-feedback nonlinear sys-

tem with the time-varying disturbances. While the fuzzy

logic systems (FLSs) serve as the approximators to the

unknown dynamics, the compound disturbance is com-

prised of the time-varying disturbance and the approxi-

mation error of the FLS. An event-triggered compound

learning algorithm is originally developed to accurate-

ly 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 em-

ployed 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) that functions in the channelof sensor to controller. The triggering condition is con-structed to guarantee the similarity between the adap-

tive model and the original system. While the satis-

factory learning performance of the FLSs and the com-

pound disturbances estimation are maintained, the pro-

posed control scheme can guarantee the semi-globallyuniformly ultimate boundedness (SGUUB) of all thetracking errors. Finally, a numerical experiment is car-ried out to exemplify the effectiveness of the proposed

control scheme.

Yingjie DengSchool of Mechanical Engineering, Yanshan University,Qinhuangdao 066044, ChinaE-mail: dyj@ysu.edu.cn

Tao Ni () · Jiantao WangSchool of Vehicle and Energy, Yanshan University,Qinhuangdao 066044, ChinaE-mail: nitao@jlu.edu.cn, wjt@ysu.edu.cn

Keywords Event-triggered control · Fuzzy logic

system (FLS) · Compound learning · Jumps of virtual

control laws · Command filter backstepping

1 Introduction

Plenty of industrial plants can be described by nonstrict-

feedback nonlinear systems, such as the ball and beam

system, the spring damper system, the remote manipu-

lator, and the stirred tank reactor [1]. 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

usually not true. It is because in the backstepping frame-

work to control the strict-feedback system, only the s-

tates before the current step are required to constructthe virtual control laws, whereas all the states are re-quired for this purpose to control the nonstrict-feedbacksystem such that the “algebraic loop” problem emerges

[2]. Based on the monotonously increasing property of

the bounding function of unknown dynamics, [3, 4] de-

veloped a so-called variable separation method, in which

the unknown dynamics can be bounded by the sum oftracking errors and the backstepping design was avail-able. This method was further combined with the ob-

server design in [5], the small-gain theorem to deal with

the unmodeled dynamics in [6], and extended to the s-

tochastic nonstrict-feedback system in [7–9]. To exempt

the restrictive assumptions of the unknown dynamics in

these literature, [2] fabricated the novel backstepping-

based control laws and parameter adaptation laws, in

which the bounded property of fuzzy basis functions

was utilized. This method was applied to the switched

nonstrict-feedback systems in [1,10], and combined with

2 Yingjie Deng et al.

the finite-time control in [11]. Morever, the problem

of “complexity explosion” to differentiate the virtual

control laws was avoided by introducing the filters in

[10, 11]. By using the semirecurrent neural network-

s (NNs) as the one-step predictors, [12] designed the

backstepping-based control laws for the discrete nonstrict-

feedback system, which solved both the “algebraic loop”

problem and the “noncausality” problem [13]. Also basedon the property of bounded neural basis functions, [14]integrated the reinforcement learning to the backstep-

ping approach of the discrete nonstrict-feedback sys-

tem, such that the suboptimal control performance can

be achieved, and yet the “noncausality” problem re-

mained to be unsolved.

Two drawbacks can be found in the above attempt-

s to control the nonstrict-feedback systems. First, al-

though the FLSs and the NNs were employed as ap-

proximators, the update laws of their weights were fab-

ricated by following the principle of “ direct adaptivecontrol” [15], such that the closed-loop stability wasensured but the interpretability for approximation was

missed. Second, the time-varying disturbances were sel-

dom addressed in these literature. Although [1] con-

sidered the disturbances in the system, they were not

handled in the control design and the stability anal-

ysis utilized the bounded property of them. To over-come the first drawback, we can get enlightened fromthe “composite learning” technique. To our comprehen-

sion, the composite learning adopts the thought to com-

bine “direct adaptive control” with “indirect adaptive

control”. By referring to a serial-parallel model [16–23]

or a predictor [24, 25], the identification components

(namely the prediction errors) are added to the update

laws following “direct adaptive control”, such that both

the tracking performance and the approximation per-

formance are ensured through the entire control frame.

It should be noted that the update laws in [16–25] 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, [26–30] utilized the online-recorded da-

ta to construct the prediction errors, which leaded to

a more relaxed condition of the interval excitation (IE)

and the faster parameter convergence. For the second

drawback, [20] found that the existence of disturbances

in the nonlinear system will degrade the learning per-formance of the NNs or the FLSs, and neither the un-known 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 the disturbance observers.

This method was applied in the canonical nonlinear sys-

tem in [20, 21], and the strict-feedback systems in [30].

To be distinguished from the composite learning, the

thought to identify both the unknown dynamics and thetime-varying disturbances was named as “compoundlearning” 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 sys-

tem without the disturbances, it followed the primitive

predictor-based composite learning. Moreover, it was

observed from the above literature that both the com-

pound learning and the composite learning were never

achieved in the event-triggered manner.

Another concern of this paper rests with the ETC.As one of the hotspots in the control field, ETC has at-

tracted compelling interests in recent years. Due to its

convincible merit to save the communication load, ETC

has been widely employed in the strict-feedback sys-

tems [31–39], the nonstrict-feedback systems [40–45],

and the other industrial plants [46–53]. It can be found

that [31–35, 40–44, 46, 49–51] only addressed the ETCin the channel of controller to actuator, namely the realcontrol inputs kept unchanged by the zero order hold-

er (ZOH) during the inter-event time and replaced by

the command control signals at the triggering instants.

As the communication congestion is more apt to occur

in the channel of sensor to controller, the ETC in the

channel of sensor to controller is indeed more preferred

in control practice. For this goal, we notice that the

problem of “jumps of virtual control laws” arises while

the backstepping design is employed. A brief descrip-

tion of this problem is: while the states are involved in

the virtual control law, the renewal of states at the trig-

gering instants will precipitate the virtual control law

into a sudden jump. This problem was first put forward

in [52], and the formulaic definition will be presented

in Section II of this paper. Although [46–48] addressed

the backstepping-based ETC in the channel of sensor

to controller, this problem did not emerge as the back-

stepping therein was only used in one step. Thus, it is

more meaningful to discuss this problem in the strict-feedback and the nonstrict-feedback systems, in whichthe backstepping is iteratively used. In [37–39,45], this

problem was neglected and the virtual control laws were

deemed to be continuous at the triggering instants. Al-

though the backstepping-based ETC in the channel of

sensor to controller was derived therein, it cannot pro-

cure a strict analysis on the closed-loop stability. Toour knowledge, the problem of “jumps of virtual con-trol laws” was only solved in [36, 52, 53]. In [36], the

final control input was constructed as the composition

of each virtual control laws without the involvement of

event-triggered samples, and approximated by a NN.

Although this paper meritoriously solve the problem of

Title Suppressed Due to Excessive Length 3

“jumps of virtual control laws” and circumvented the

recursive appearance of virtual control laws in the trig-

gering condtion, the control performance therein total-

ly 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. [52, 53] substituted the virtual control laws with

the continuous variables generated by the first-order fil-ters. This setup can solve both the problems of “jumpsof virtual control laws” and “complexity explosion”. N-

evertheless, the continuous variables were not smooth,

and a complicated analysis on the parameter selection

is required if these schemes are applied in the high-order

nonlinear systems.

Motivated by the above challenge, this paper devel-

ops an ETC compound learning scheme of the nonstrict-

feedback nonlinear system. Based on an event-triggered

adaptive fuzzy model, the backstepping-based control

laws are derived. The prediction errors are construct-

ed according to the adaptive model and the original

system, which are involved in the update laws of the

estimates of the weights of FLSs and the compound dis-

turbances. By using the online-recorded data collected

during the inter-event time, the update laws are re-

newed 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 vir-

tual control law and the continuous substitute for itsderivative are involved in the backstepping design. Theadaptive triggering condition is fabricated to ensure thesimilarity between the adaptive model and the original

system. Compared with the precedents, the contribu-

tions of this paper are mainly threefold.

1. By referring to the event-triggered adaptive mod-

el, the “algebraic loop” problem is avoided in the

proposed scheme. Both the control design and the

analysis are less complicated than the existing tech-niques.

2. By using the online-recorded-data-based prediction

errors, the event-triggered compound learning tech-

nique is originally developed in this paper. Com-

pared with the continuous composite learning schemes,

it ensures the better understanding of total uncer-

tainties and less communication resources.3. By referring to the second-order command filters,

the problem of “jumps of virtual control laws” is

solved. Moreover, the problem of “complexity ex-

plosion” is avoided. Compared with the [52, 53] us-

ing the first-order filters, the proposed scheme can

address the high-order nonlinear system with less

complicated analysis.

The remainder of this paper are organized as fol-

lows. Section II provides the preliminaries. Section III

presents the control design. Section IV designs the trig-

gering condition and analyzes the closed-loop stabilityof the system. A practical example is provided to exem-plify the effectiveness of the proposed scheme in SectionV. Section VI concludes the entire work. Appendices

proves the bounded estimation errors and Lemma 2.

Uniformly in this paper, ‖ · ‖ denotes the Euclidean

norm of the vector, λmax(·) denotes the largest eigen-

value of the matrix, s denotes the estimate of s and s =s − s denotes the estimation error. tj implies the trig-

gering instant with j = 0, 1, · · · ,+∞, in which t0 is also

deemed as the initial time. V +(tj) denotes limt→tj V (t)

from the right side, ∆V = V +(tj) − V (tj) and ΘV =

V (tj+1)− V (tj).

2 Preliminaries

2.1 Problem Formulation

Consider the single-input-single-output (SISO) nonstrict-

feedback nonlinear system as

xi = fi(x) + xi+1 + di, i = 1, · · · , n− 1

xn = fn(x) + u+ dn

y = x1

(1)

where x = [x1, · · · , xn]T denotes the n-dimensional vec-

tor of states, y is the single output, u denotes the singlecontrol input, fi(x) denotes the smooth unknown dy-

namics in each hierarchy and di denotes the unknown

time-varying disturbance. To facilitate the following de-

sign and analysis, an assumption is preset as

Assumption 1 [2,20,30,36] The time-varying distur-

bance di and its first-order derivative di are deemed to

be bounded by |di| ≤ bdiand |di| ≤ bdi

respectively,

where bdiand bdi

are two unknown positive constants.

Remark 1 The nonstrict-feedback nonlinear system of

(1) has the same structure with that in [1]. Without

the disturbances, it has the same structure with those

in [5, 11, 45]. By setting the gain function as gn(·) = 1and reexpressing fi(·)+gi(·)−xi as fi(x), the nonstrict-

feedback nonlinear sytems in [2–4,10,22,42–44] and all

the referenced strict-feedback systems in this paper are

equivalent to (1).

Then, the control objective of this paper is conclude

as: design the control law of u such that the output

y can track its smooth reference of yd, which has the

bounded first-order derivative.

Next, we will discuss the problem of “jumps of vir-

tual control laws”. Following the backstepping design,

the virtual control laws for (1) at the ith step usually

4 Yingjie Deng et al.

has the form of αi = hi(xi, t), where xi = [x1, · · · , xi]T

and hi(xi, t) is a smooth function, seeing [1–3,5]. If the

ETC is designed in the channel of controller to actuator

like [40–44], it has u(t) = v(tj) for the inter-event time

t ∈ (tj , tj+1], where v(t) = hn(x, t) is the command

control signal. Denote the tracking error in the ith stepas zi+1 = xi+1 − αi. It can be inferred that the update

of u(t) at tj will not affect the continuity of αi and

zi. Nevertheless, if the ETC is designed in the channel

of sensor to controller, 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

channel of sensor to controller, the states are event-

triggered sampled in the controller, such that αi(t) =

hi(xi(tj), t) holds in t ∈ (tj , tj+1]. Because xi(tj) 6=xi(tj−1), there is αi(tj) = hi(xi(tj−1), tj) and α+

i (tj) =hi(xi(tj), tj) at tj , such that ∆αi 6= 0 and the jump

happens. Although x+i (tj) = xi(tj), z

+i (tj) 6= zi(tj) due

to ∆αi 6= 0 and the Lyapunov function of V (zi) is also

discontinuous at tj .

In [37–39, 45], it presumed ∆αi = 0 at tj . This dis-

posal is not reasonable according to the above analysis.

In [36], αi was not fabricated in the NN-based con-

troller, such that this problem was solved. In our pre-

vious work [52], this problem was solved by finding a

continuous substitute for αi.

2.2 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 defuzzi-

fication. The fuzzy rule base is consulted to conduct the

entire procedure, and it is composed of the IF-THEN

rules asRl: If x1 is F l

1, x2 is F l2, · · · and xn is F l

n, then y is

Gl.

where l = 1, · · · , N , x = [x1, · · · , xn]T is the input

vector, y is the output, F li andGl are the fuzzy sets with

the membership functions of µF li(xi) and µGl(y) respec-

tively. Through the singleton fuzzification, the product

reasoning and the center average defuzzification, theFLS can be rewritten as

y =

∑N

l=1 yl∏n

i=1 µF li(xi)

∑N

l=1

∏n

i=1 µF li(xi)

(2)

where yl = maxy∈R µGl(y). For simplicity, denote thefuzzy basis function in (2) as

ϕfl(x) =

∏n

i=1 µF li(xi)

∑N

l=1

∏n

i=1 µF li(xi)

(3)

and let ϕ(x) = [ϕf1, · · · , ϕfN ]T andW = [y1, · · · , yN ]T.

Then, (2) can be simplified as y = WTϕ(x). The well-known approximation property of the FLS is concluded

as follows.

Lemma 1 For any multiple-inputs-single-output func-tion f(x) defined in a compact set x ∈ Ω, there always

exists the FLS satisfying

f(x) = WTϕ(x) + ε

where ε can be deemed as a slow time-varying variable

bounded by |ε| ≤ bε, where bε is a positive constant.

In this paper, W is determined as the optimal weightvector satisfying W = argminW∈RN bε

3 Control Design

The framework of the proposed control scheme is illu-minated in Fig. 1. It belongs to the second configura-

tion in [48], in which the event-triggered adaptive modeland the update laws are co-located with the controller.This configuration allows for the distributed layout ofsensors, and is more practical than the first configu-

ration in [48], namely the sensor-located configuration.

The control design is divided into three steps.

Nonstrict-feedback

nonlinear system

Event-triggered

adaptive modelControl laws

Triggering condition

Update lawsIntegrators

ZOHZOH

u

ˆi

x

ˆ ˆ,i i

W

ix

ix

11dt,

j

j

t

i it

x

Fig. 1 Framework of proposed ETC

3.1 Event-triggered Adaptive Model

According to Lemma 1, the unknown dynamics in (1)

is expressed as fi(x) = Wiϕi(x)+ εi. According to [30],

it is impractical to distinguish εi from di. Thus, the

compound disturbance is constructed with the form of

Title Suppressed Due to Excessive Length 5

σi = εi+di. According to (1), the event-triggered adap-

tive model in t ∈ (tj , tj+1] is designed as

˙xi = WTi ϕi(x(tj)) + xi+1 + σi + ki(xi(tj)− xi),

i = 1, · · · , n− 1

˙xn = WTn ϕn(x(tj)) + u+ σn + kn(xn(tj)− xn)

(4)

where ki and kn are tuning parameters to be designedlater. It is observed that x(tj) in the right side of (4) is

renewed at the triggering instant and xi keeps continu-

ous. Denote the measurement error as ei = xi(tj) − xi

and en = xn(tj) − xn. By subtracting (4) from (1), it

renders the estimation error system in t ∈ (tj , tj+1] as

˙xi =WTi ϕi(x) + WT

i (ϕi(x)− ϕi(x(tj))) + xi+1

+ σi − kiei − kixi, i = 1, · · · , n− 1

˙xn =WTn ϕn(x) + WT

n (ϕn(x)− ϕn(x(tj))) + σn

− knen − knxn

(5)

It can be inferred from (5) that if the inter-event time

is small enough (namely kiei → 0 and WTi (ϕi(x) −

ϕi(x(tj))) → 0) and the identification is accurate e-

nough (namely WTi ϕi(x) → 0 and σi → 0), the fast

convergence of xi to xi is ensured.

3.2 Event-triggered Compound learning

The update laws of Wi and σi will be fabricated to guar-

antee the satisfactory learning performance of WTi ϕ(x)

and σi. According to (5), the prediction error is con-

structed as

si(tj) =

∫ tj

tj−1

˙xi − WTi (ϕi − ϕi(tj−1))− xi+1 + σi

+ ki(xi(tj−1)− xi) dt, i = 1, · · · , n− 1

sn(tj) =

∫ tj

tj−1

˙xn − WTn (ϕn − ϕn(tj−1)) + σn

+ kn(xn(tj−1)− xn) dt

(6)

It is clear in (6) that the prediction errors si and snutilize the recorded data during the previous inter-event

time. As Wi and σi are only updated at the triggering

instant, (6) can be further transformed to

si(tj) =xi(tj)− xi(tj−1)− W+i (tj−1)

Tφi(tj)

+ W+i (tj−1)

Tϕi(tj−1)(tj − tj−1)

−∫ tj

tj−1

xi+1 dt+

∫ tj

tj−1

xi+1 dt

+ σ+i (tj−1)(tj − tj−1) + kixi(tj−1)(tj − tj−1)

− ki

∫ tj

tj−1

xi dt, i = 1, · · · , n− 1

sn(tj) =xn(tj)− xn(tj−1)− W+n (tj−1)

Tφn(tj)

+ W+n (tj−1)

Tϕn(tj−1)(tj − tj−1)

+ σ+n (tj−1)(tj − tj−1)− kn

∫ tj

tj−1

xn dt

+ knxn(tj−1)(tj − tj−1)

(7)

where φi(tj) =∫ tj

tj−1ϕi(t) dt and φn(tj) =

∫ tj

tj−1ϕn(t) dt.

It can be inferred from (7) that the recorded data are

only transmitted to the controller at the triggering in-stant through the integrators in Fig. 1. By substituting(5) to (6), it further renders

si(tj) =

∫ tj

tj−1

WTi ϕi(x) + σi dt

=WTi φi(tj) +Di(tj), i = 1, · · · , n

(8)

whereDi(tj) =∫ tj

tj−1σi dt. According to (8), the update

laws of Wi are designed as

W+i (tj) = lwiWi(tj) + γwisi(tj)φi(tj), t = tj

Wi(t) = W+i (tj), t ∈ (tj , tj+1]

(9)

where γwi > 0 denotes the learning rate, lwiWi(tj) with

lwi > 0 is used to prevent the parameter drift. The se-

lection principle of γwi and lwi is discussed in Appendix

A.

According to (1), the prediction error is designed as

pi(tj) =

∫ tj

tj−1

xi − WTi ϕi(x)− xi+1 − σi dt,

i = 1, · · · , n− 1

pn(tj) =

∫ tj

tj−1

xn − WTn ϕn(x)− u− σn dt

(10)

6 Yingjie Deng et al.

Similar with (7), it can be transformed to

pi(tj) =xi(tj)− xi(tj−1)− W+i (tj−1)

Tφi(tj)

−∫ tj

tj−1

xi+1 dt− σ+i (tj−1)(tj − tj−1),

i = 1, · · · , n− 1

pn(tj) =xn(tj)− xn(tj−1)− W+n (tj−1)

Tφn(tj)

−∫ tj

tj−1

u dt− σ+n (tj−1)(tj − tj−1)

(11)

(11) illustrates that all the data are achievable for piand pn at the triggering instant. By substituting (1) to(10), it renders

pi(tj) =

∫ tj

tj−1

WTi ϕi(x) + σi dt

=WTi φi(tj) +Di(tj)− σi(tj − tj−1), i = 1, · · · , n

(12)

According to (12), the update laws of σi are designed

as

σ+i (tj) = (lσi + γσi(tj − tj−1))σi(ti) + γσipi(tj), t = tj

σi(t) = σ+i (tj), t ∈ (tj , tj+1]

(13)

where γσi > 0 and lσi > 0 have the same definitions

with those in (9). The proposed update laws of (9) and

(12) can guarantee the boundedness of ‖Wi‖ and |σi|all along the control process. The proof is provided in

Appendix A.

3.3 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 z1 =

x1 − yd and zi = xi − βi−1 with i = 1, · · · , n, where βi

is the filtered variable of the virtual control law αi andqi = βi − αi. The control design is divided into three

parts.

Step 1 : According to (4), the backstepping methodis employed to eliminate the tracking error of z1, and

the first virtual control law in t ∈ (tj , tj+1] is designedas

α1 = −ka1z1 − WT1 ϕ1(x(tj))− σ1 + yd (14)

where ka1 is the positive tuning parameter to be dis-

cussed later. Differentiating z1 along (4) and (14), it

renders

z1 = −ka1z1 + z2 + q1 + k1e1 + k1x1 (15)

Step i : Similarly to eliminate zi with i = 2, · · · , n−1, the virtual control law in the ith step is designed as

αi = −kaizi − zi−1 − WTi ϕi(x(tj))− σi + βi−1 (16)

where ka2 is the positive tuning parameter to be dis-

cussed later. βi and βi with i = 1, · · · , n− 1 are gener-

ated by the following second-order command filter [54].

βi =ηi

ηi =− 2ζωηi − ω2(βi − αi)(17)

where ζ > 0 is the damping coefficient and ω > 0 is the

natural frequency, and it has βi = ηi. According to the

definition of qi, (16) and (17), ∆qi at tj is expressed as

∆qi =β+i (tj)− α+

i (tj)− βi(tj) + αi(tj)

=αi(tj)− α+i (tj)

=W+i (tj)

Tϕi(tj)− WTi (tj)ϕi(tj−1) + σ+

i (tj)

− σi(tj)

(18)

By invoking (3), the following relationship holds

WTi ϕi =

√

WTi ϕiϕT

i Wi ≤√

λmax(ϕiϕTi )‖Wi‖2

≤‖Wi‖ ≤ ‖Wi‖+ ‖Wi‖(19)

By invoking Assumption 1 and Lemma 1, (18) can be

further expressed as

∆qi ≤2‖Wi‖+ 2‖Wi‖+ 2|σi|+ 2|σi|≤2‖Wi‖+ 2‖Wi‖+ 2|σi|+ 2bεi + 2bdi

(20)

As is proved in Appendix A that ‖Wi‖ and |σi| are

bounded all along, it is reasonable that to assume a

positive constant bqi such that |∆qi| ≤ bqi always holds.

Then, it is proved in Appendix B that (17) has the

following property.

Lemma 2 There exists a positive constant bqi such that

|qi| ≤ bqi always holds. By increasing ω in (17), it has

bqi → bqi .

Remark 2 Because xi is continuous in (4) and αi is

filtered by (17), the components of −kaizi and −zi−1

in (14) and (16) are continuous at tj . Different with the

first-order DSC filter in [52], the second-order command

filter of (17) can also guarantee the continuity of βi−1

in (16), such that the sophisticated analysis of ∆βi isavoided.

Differentiating zi along (4) and (16), it renders

zi = −kaizi + zi+1 + qi − zi−1 + kiei + kixi (21)

Title Suppressed Due to Excessive Length 7

Step n: Similarly to eliminate zn, the real control

law of u in the final step is designed as

u = −kanzn − zn−1 − WTn ϕn(x(tj))− σn + βn−1 (22)

By differentiating zn along (4) and (16), it renders

zn = −kanzn − zn−1 + knen + knxn (23)

Remark 3 Following from (1), the essence of “algebra-ic loop” problem implies that the xi can not converge

to the αi−1(x) in the backstepping method which is

synchronously changed. Whereas in the event-triggered

adaptive model (4), the convergence of xi to βi−1 willnot cause the synchronous change of αi−1(x, xi−1). Here,

xi can only affect the derivative of x, which is same asxi can only affect the derivative of xi−1 in the strict-

feedback system. Moreover in the proposed scheme, x is

not continuously sampled in αi−1. Thus, the “algebraic

loop” problem will never occur.

4 Triggering Condition and Stability Analysis

4.1 Design of Triggering Condition

Select the Lyapunov function as V1 =∑n

i=1 x2i /2. Dif-

ferentiating V1 along (5), it renders

V1 ≤− (k1 −5

2)x2

1 −n−1∑

i=2

(ki − 3)x2i − (kn − 5

2)x2

n

+

n∑

i=1

(1

2WT

i (ϕi − ϕi(tj))(ϕTi − ϕT

i (tj))Wi +1

2k2i e

2i )

+

n∑

i=1

(1

2‖Wi‖2 +

1

2σ2i )

(24)

Select the Lyapunov function as V2 =∑n

i=1 z2i /2.

Differentiating V2 along (15), (21) and (23), it renders

V2 =−n∑

i=1

kaiz2i +

n−1∑

i=1

qizi +

n∑

i=1

(kieizi + kixizi)

≤−n∑

i=1

(kai − 1− k2i2)z2i +

n−1∑

i=1

q2i2

+

n∑

i=1

(k2i e

2i

2+

x2i

2)

(25)

Define V = V1+V2. By adding up (24) and (25) and

invoking Lemma 2, the differential of V is obtained as

V ≤− (k1 − 3)x21 −

n−1∑

i=2

(ki −7

2)x2

i − (kn − 3)x2n

−n∑

i=1

(kai − 1− k2i2)z2i +

n−1∑

i=1

b2qi2

+

n∑

i=1

bli

+

n∑

i=1

(1

2WT

i (ϕi − ϕi(tj))(ϕTi − ϕT

i (tj))Wi + k2i e2i )

(26)

where bli = sup‖Wi‖2/2 + σ2i /2. Because ‖Wi‖ and

|σi| are bounded all along according to the proof inAppendix A, bli is an unknown constant.

According to (26), the triggering condition is de-

signed as

n∑

i=1

(1

2WT

i (ϕi − ϕi(tj))(ϕTi − ϕT

i (tj))Wi + k2i e2i ) < µ

(27)

where µ is the positive triggering threshold. The nex-

t triggering instant is activated while (27) is violated.

The proof of the existence of minimum inter-event time

δt (namely to avoid the “Zeno” behavior) adopts thesame thought in [49–51], which is briefly introduced as

follows.

Proof of δt: It is observed that the left side of (27)

equates to 0 at the triggering instant. Because all the

states are defined in a compact set, the left side of (27)

must have a limited growth rate. As µ is a positive

constant, it takes at least the time of δt to increasefrom 0 to µ.

4.2 Stability Analysis

The main result of this paper is concluded as the fol-

lowing theorem.

Theorem 1 Assume all the states of (1) are defined

in a compact set, namely Ωx = x|xTx ≤ bΩx where

bΩsis a positive constant. If the update laws of (9) and

(13), the real control law of (22), and the triggering con-dition of (27) are used, all the tracking errors are provedto satisfy SGUUB. Moreover, by tuning the parameters

appropriately, the output y can track the reference yd in

an arbitrarily small precision.

Proof : By substituting (27) to (26), and selecting a =

2mink1 − 3, ki − 7/2, kn − 3, kai − 1 − k2i /2 and b =∑n−1

i=1 b2qi/2 +∑n

i=1 bli + µ, (26) can be simplified as

V ≤ −aV + b (28)

8 Yingjie Deng et al.

According to the comparison theorem, it can be inferred

from (28) that for i = 1, · · · , n

|zi| ≤√2V ≤

√

2V (t0)e−a(t−t0) +2b

a(1− e−a(t−t0))

(29)

Thus, the SGUUB of zi is proved. Because y − yd =

z1+ x1 and |xi| ≤√2V too, it is inferred that |y−yd| ≤

2√2V . If a → +∞ in (29), it has 2

√2V → 0 with

t → +∞. The convergence of y to yd is proved.

The parameter selection follows two steps: 1) se-lect k1 > 3, ki > 7/2, kn > 3 and µ > 0 to ensure

the acceptable estimation performance of (4); 2) select

kai > 1 + k2i /2 to ensure the satisfactory tracking per-

formance.

Remark 4 According to the definition of z1, it is in-ferred that z1 → 0 ⇒ x1 → yd if xi → xi. Because xi is

affected by the approximation errors and the measure-

ment errors according to (5), the tracking performance

is positively associated with the learning performance

and decreasing the triggering threshold µ in (27). Nev-ertheless, decreasing µ will increase the communication

load as the inter-event time is shorten.

5 Numerical Experiment

In this section, the proposed ETC compound learning

scheme is applied to the ball and beam system in [55],

and its setup is shown in Fig. 2.

M

DC motor

Gear 1

Gear 2

Beam 1

Beam 2

Stand

Base

Ball

+

_

u1

o

2o

3o

d

1x

3x

Fig. 2 Diagram of ball and beam system

In Fig. 2, o1, o2 and o3 are three fixed pivots. By

regulating the voltage of the DC motor (namely u),

the position of the ball in beam 2 (namely x1) can be

regulated as needed. This system possesses the mathe-

Table 1 MODEL PARAMETERS

Symbols Definitions Values

mB Mass of ball 0.0217kgmb Mass of beam 0.334kgR Radius of ball 0.00873mlb Beam 2 length 0.4md Radius of large gear 0.04mJB Ball inertia 6.803× 10−7kgm2

Jb Beam inertia 0.017813kgm2

Kb Back-EMF constant 0.1491Nm/ARa Armature resistance 18.91Ωg Acceleration of gravity 9.8m/sng Gear ratio 4.2

matical model of

x1 =x2

x2 =Ax1x24 −Agx3 + d2

x3 =x4

x4 =B cos(x3)(C cos(lbx3

d)u−Dx4 cos(

lbx3

d)

− E −Hx1)−BGx1x2x4 + d4

(30)

where x2 is the translational speed of the ball alongbeam 2, x3 is the angle of beam 2, x4 is the angular

speed, A = 1/(1 + JB/(mBR2)), B = 1/(JB + Jb +

mBx21), C = ngKblb/(Rad), D = (ngKblb)

2/(Rad2),

E = 0.5lbmbg, H = mBg, and G = 2mB . The def-

initions and the values of the model parameters areprovided in Table I. To correspond with the structure

of (1), let f2(x) = Ax1x24 − Agx3 − x3 and f4(x) =

−B cos(x3)(Dx4 cos(lbx3/d) + E + Hx1) − BGx1x2x4

express the unknown dynamics in (30), and the un-

known dynamics and the disturbances in the hierar-

chies of x1 and x3 are omitted. To eliminate the un-

known input gain, reconstruct the control input as U =

BC cos(x3) cos(lbx3/d)u.

The time-varying disturbances are described by thefollowing second-order Markov process.

d2 = −d2 − d2 + 0.1ω2

d4 = −d4 − d4 + ω4

(31)

where ω2 and ω4 are two independent Gauss white noisewith the variance of 1. The initial values of d2, d2, d4and d4 are assigned as 0. The FLS is set with 7 fuzzy

rules, and the membership function for each fuzzy set

is described as

µF li= exp

(

− (xi − l + 4)2

4

)

, l = 1, · · · , 7 (32)

It should be noted that f2(x) and f4(x) in (30) employ

the same set of fuzzy membership functions in this sim-

ulation. The reference signal is selected as a sinusoidal

Title Suppressed Due to Excessive Length 9

function of time, namely yd = 0.2 + 0.05 sin(0.1257(t−t0)). The initial values of states are assigned as x1(t0) =0.3m, x2(t0) = 0m/s, x3(t0) = 0 and x4 = 0/s. The

initial values of the estimation states in (4) and the

learning parameters in (9) and (13) are assigned as ze-

ros.

Not to deteriorate the algorithm effectiveness, the

simulation adopts the constant numerical calculation

period of 0.01s. To demonstrate the differences, two

comparative algorithms are used in this simulation. The

first one is the time-triggered compound learning, marked

as “TT-CL”. In this scheme, µ in (27) is set as 0. Ac-cording to [30], the continuous update law of Wi and

the disturbance observer of σi are presented as

˙Wi = γwisi(t)φi(t)− lwiWi, i = 1, · · · , n (33)

σi = lσi(xi − δi),

δi = WTi ϕi(x) + σi + xi+1, i = 1, · · · , n− 1

σn = lσn(xn − δn), δn = WTn ϕn(x) + σn + u

(34)

where si(t) in (33) is the prediction error using the datacollected in the latest 0.1s. The second algorithm adopts

the predictor-based learning in [23], which is marked as“ET-PL”. In this scheme, the compound disturbance σi

is not addressed, and the update law of Wi is designed

as

W+i (tj) = lwiWi(tj) + γwixi(tj)ϕi(tj−1), t = tj

Wi(t) = W+i (tj), t ∈ (tj , tj+1]

(35)

where the event-triggered adaptive model (4) functions

as the predictor and xi can be deemed as the prediction

error.

The parameters are uniformly set as k1 = k3 = 4,

k2 = k4 = 5, µ = 0.2, ka1 = ka2 = ka3 = ka4 = 15,lw2 = lw4 = lσ2 = lσ4 = 0.1 and γw2 = γw4 = γσ2 =

γσ4 = 0.1 in the schemes. Define the total uncertainty

as Υi = fi(x)+di and its estimate as Υi = WTi ϕi(x)+σi,

and let Υ = [Υ2, Υ4]T. To quantify the performance, four

indexes are defined as

Jz(IAE) =

∫ t

t0

|y(τ)− yd(τ)| dτ , Jc(IAE) =

∫ t

t0

‖Υ‖ dτ,

Jz(ITAE) =

∫ t

t0

τ |y − yd| dτ , Jc(ITAE) =

∫ t

t0

τ‖Υ‖ dτ

(36)

The values of these indexes are provided in Table II.

The simulation results are exhibited as follows.

Fig. 3 shows the evolution of disturbances. The mag-

nitudes of them correspond with the magnitudes of

0 10 20 30 40 50-0.2

-0.1

0

0.1

0.2

st

2d

4d

id

Fig. 3 Disturbance di

0 10 20 30 40 500.1

0.15

0.2

0.25

0.3

0.35

TT-CL

ET-PL

ET-CL

st

dy

y

Fig. 4 Output y

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0 10 20 30 40 50-0.3

-0.2

-0.1

0

0.1

st

st

1x

1x

2x

2x

Am

pli

tud

eA

mp

litu

de

(a) x1 and x2

0 10 20 30 40 50-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50-1

-0.5

0

0.5

1

st

st

3x

3x

4x

4x

Am

pli

tud

eA

mp

litu

de

(b) x3 and x4

Fig. 5 States of event-triggered adaptive model in ET-CL

0 5 10 15 20 25 30 35 40 45 50-1

0

1

2

3

4

5

6

7

TT-CL

ET-PL

ET-CL

20 22 24 261.5

2

2.5

3

st

u

Fig. 6 Control input u

10 Yingjie Deng et al.

Table 2 PERFORMANCE INDEXES

Indexes TT-CL ET-PL ET-CLJz(IAE) 0.215 0.710 0.254Jz(ITAE) 2.155 13.699 3.154Jc(IAE) 17.379 364.606 192.462Jc(ITAE) 65.365 8325.970 4698.126

Total sampling times 5001 593 1005

0 10 20 30 40 50-1

-0.5

0

0.5

0 10 20 30 40 50-100

-50

0

50

st

st

Am

pli

tud

eA

mp

litu

de

2

2

4

4

Fig. 7 Learning performance of TT-CL

0 10 20 30 40 50-0.5

0

0.5

0 10 20 30 40 50-80

-60

-40

-20

0

st

st

Am

pli

tud

eA

mp

litu

de

2

2

4

4

Fig. 8 Learning performance of ET-PL

total uncertainties. Fig. 4 demonstrates the output ofthree schemes. It is clear that TT-CL and ET-CL havebetter tracking performance than ET-PL due to the

better learning performance of them. This point is ver-

ified in Jz(IAE) and Jz(ITAE) of Table II. Because the

ETC is employed, the outputs of ET-CL and ET-PL

inevitably suffer from fluctuation. Fig. 5 shows the evo-lution of xi in (4) of ET-CL, in which a fast and accu-

0 10 20 30 40 50-1

-0.5

0

0.5

0 10 20 30 40 50-100

-50

0

50

st

st

Am

pli

tud

eA

mp

litu

de

2

2

4

4

Fig. 9 Learning performance of ET-CL

0 10 20 30 40 500

0.2

0.4

0.6

0.8

TT-CL

ET-PL

ET-CL

0 10 20 30 40 500

5

10

15

st

st

IAE

zJ

ITA

Ez

J

(a) Jz(IAE) and Jz(ITAE)

0 10 20 30 40 500

100

200

300

400

TT-CL

ET-PL

ET-CL

0 10 20 30 40 500

2000

4000

6000

8000

10000

st

st

IAE

cJ

ITA

Ec

J

(b) Jc(IAE) and Jc(ITAE)

Fig. 10 Evolution of performance indexes

0 10 20 30 40 50-0.02

0

0.02

0.04

0.06

0 10 20 30 40 50-60

-40

-20

0

st

st

2W

4W

(a) Fuzzy weights Wi

0 10 20 30 40 50-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50-30

-20

-10

0

10

st

st

2

4

(b) Compound disturbanceσi

Fig. 11 Update laws of ET-CL

rate convergence to xi is observed. Fig. 6 demonstrates

the control inputs of three schemes. The discontinuityof ET-CL and ET-PL is observed in the local enlarged

view.

Fig. 7-9 compare the leaning performance of three

schemes. Among them, TT-CL has the best learning

performance, and ET-CL has much better learning per-formance than ET-PL. This point is verified in Jc(IAE)

Title Suppressed Due to Excessive Length 11

0 10 20 30 40 500

1000

2000

3000

4000

5000

6000

TT-CL

ET-PL

ET-CL

st

Tim

es

Fig. 12 Accumulation of sampling times

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

st

Am

pli

tud

e

Fig. 13 Inter-event time of ET-CL

and Jc(ITAE) of Table II. Fig. 10 shows the evolution of

performance indexes. It is clear that the indexes of ET-

PL grow faster than the other two. Fig. 11 demonstrates

the variation of update laws in ET-CL, in which the

event-triggered renewal is observed. Fig. 12 compares

the accumulation of sampling times in three schemes.

A considerable communication saving is observed in theevent-triggered schemes. The statistics of total sam-pling times are provided in Table II. Fig. 13 shows theevolution of inter-event time in ET-CL. The minimum

inter-event time is observed at the initial stage with

the constant calculation period of 0.01s. The maximum

inter-event time is recorded as 0.26s.

6 CONCLUSION

This paper proposes an ETC scheme of the nonstrict-

feedback nonlinear system with the compound learning.

An event-triggered adaptive fuzzy model is fabricated

to direct the backstepping design of control laws, such

that the “algebraic loop” problem is solved. Two pre-

diction errors are derived from the adaptive model and

the original system, which can collect the data during

the previous inter-event time. By resorting to the pre-

diction errors, the estimates of the fuzzy weights and

the compounding disturbances are updated in an event-

triggered manner. In the control laws, the second-order

command filters are employed to substitute for the vir-tual control laws and their derivatives, such that theproblem of “jumps of virtual control laws” is solved. An

adaptive triggering condition is constructed to guaran-

tee the closed-loop stability. Finally, the effectiveness of

the proposed scheme is 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 the nautical practice.

A Proof of bounded estimation errors

Select the Lyapunov function as Vwi = WTi Wi. By invoking

(8), (9) and the Young’s inequality, ∆Vwi is derived as

∆Vwi =− WTi Wi + (Wi − lwiWi − γwiφiφ

Ti Wi − γwiφiDi)

T

× (Wi − lwiWi − γwiφiφTi Wi − γwiφiDi)

=(lwiWTi + (1− lwi)W

Ti − γwiW

Ti φiφ

Ti − γwiDiφ

Ti )

× (lwiWi + (1− lwi)WTi − γwiφiφ

Ti Wi − γwiφiDi)

− WTi Wi

≤4(1− lwi)2WT

i Wi + 3l2wiWTi Wi + 4γ2

wiD2i φ

Ti φi

+ 3γ2wiW

Ti φiφ

Ti φiφ

Ti Wi − WT

i Wi

≤− (1− 3l2wi − 3γ2win

2φi)WT

i Wi + 4γ2winφi

bDi

+ 4(1− lwi)2WT

i Wi

(A-1)

where ‖φi‖2 ≤ nφi, D2

i ≤ bDiand λmax(φiφT

i φiφTi ) ≤ n2

φi.

Because the maximum inter-event time Tmax is finite and‖ϕi‖ and σi are bounded, it is tenable that the positive con-stants of nφi

and bDiexist.

Select the Lyapunov function as Vσi = σ2i . By invoking

(12), (13) and the Young’s inequality, ∆Vσi is derived as

∆Vσi=(σi − lσiσi − γσiW

Ti φi − γσiDi)

2 − σ2i

=(lσiσi + (1− lσi)σi − γσiWTi φi − γσiDi)

2 − σ2i

≤4(1− lσi)2σ2

i + 4l2σiσ2i + 4γ2

σiWTi φiφ

Ti Wi − σ2

i

+ 4γ2σiD

2i

≤− (1− 4l2σi)σ2i + 4(1− lσi)

2bσi+ 4γ2

σibDi

+ 4γ2σinφi

WTi Wi

(A-2)

where bσi= (bdi

+ bεi)2. Define Vi = Vwi + Vσi. By adding

up (A-1) and (A-2), it renders

∆Vi ≤− (1− 3l2wi − 3γ2win

2φi

− 4γ2σinφi

)WTi Wi + 4γ2

σibDi

− (1− 4l2σi)σ2i + 4γ2

winφibDi

+ 4(1− lwi)2WT

i Wi

+ 4(1− lσi)2bσi

(A-3)

By selecting lσi < 1/2, 3γ2win

2φi

+ 4γ2σinφi

< 1 and lwi <

(1−3γ2win

2φi

−4γ2σinφi

)/√3, and letting ai = min1−3l2wi−

3γ2win

2φi

− 4γ2σinφi

, 1 − 4l2σi and bi = 4γ2winφi

bDi+ 4(1 −

lwi)2WTi Wi+4(1−lσi)2bσi

+4γ2σibDi

, (A-3) can be simplified

12 Yingjie Deng et al.

as ∆Vi ≤ −aiVi+bi. Because σi is changing during the inter-event time, one can not infer the boundedness of |σi| from∆Vi ≤ −aiVi + bi. A further processing is needed.

Because ˙Wi = 0 in t ∈ (tj , tj+1], it is obvious that

ΘVi = ∆Vi + σ2i (tj+1) − σ+

i (tj)2. Assume |σi| grows withthe fastest speed during the inter-vent time. According to As-sumption 1 and (13), it is tenable to assume | ˙σi| = |σi| ≤ bσi

in t ∈ (tj , tj+1], where bσiis a unknown constant. Thus, the

following inequality holds.

σ2i (tj+1)− σ+

i (tj)2 ≤ 2bσi

Tmax|σ+i (tj)|+ b2σi

T 2max

≤ 2bσTmax

√

Vi +∆Vi + b2σiT 2max

(A-4)

Thus, a new interconnected system is obtained as

∆Vi ≤ −aiVi + bi

ΘVi ≤ −aiVi + 2bσiTmax

√

Vi +∆Vi + b2σiT 2max + bi

(A-5)

If Vi(tj) > bi/ai, it has ∆Vi < 0 for the first inequali-ty in (A-5). In this case, the second inequality in (A-5) istransformed to ΘVi ≤ −aiVi + 2bσi

Tmax

√Vi + b2σi

T 2max +

bi. By solving this inequality, it has ΘVi < 0 for Vi(tj) >

(bσiTmax +

√

b2σiT 2max(1 + ai) + aibi)2/a2

i . Thus, it is con-

cluded that ΘVi < 0 for Vi(tj) > maxbi/ai, (bσiTmax +

√

b2σiT 2max(1 + ai) + aibi)2/a2

i . Because V +i (tj) < Vi(t) <

Vi(tj+1) in t ∈ (tj , tj+1], it is inferred that Vi(t) is boundedall along and is ultimately bounded by maxbi/ai, (bσi

Tmax+√

b2σiT 2max(1 + ai) + aibi)2/a2

i . The proof is completed.

It is obvious that the larger Tmax (namely by increasingµ) will lead to the smaller ai and the larger ultimate boundof Vi, which suggests the slower learning rate and the worseidentification accuracy.

B Proof of Lemma 2

For convenience, set ζ = 1 in (17). By subtracting [αi, 0]T

from (17), it renders

[

qiηi

]

=

[

0 1−ω2 −2ω

] [

qiηi

]

−[

αi

0

]

(B-1)

Define A = [0, 1;−ω2,−2ω]. It is calculated that A has tworepeated eigenvalues of λ(A) = −ω. Through the Hamilton-Cayley theorem, it can be derived that

a1(t)λ(A) + a0(t) = eλ(A)t

a1(t) = teλ(A)t(B-2)

Thus, a0(t) = (1 + ωt)e−ωt and a1(t) = te−ωt are obtained.Ulteriorly, the state transformation matrix of (B-1) in t ∈(tj , tj+1] is obtained as Φ(t) = a1(t− tj)A+a0(t− tj). Then,the analytical solution of (B-1) is derived as

[

qi(t)ηi(t)

]

= Φ(t)

[

q+i (tj)ηi(tj)

]

+

∫

t

tj

Φ(τ)

[

αi(t− tj − τ)0

]

dτ

(B-3)

Because all the states are defined in a compact set, it is ten-able to assume |αi(t)| ≤ bαi

, where bαiis an unknown con-

stant. Then, the following inequality can be derived from (B-3).

|qi(t)| ≤|q+i (tj)|(1 + ω(t− tj))e−ω(t−tj)

+

(

2

ω− 2

ωe−ω(t−tj) − (t− tj)e

−ω(t−tj)

)

bαi

+ |ηi(tj)|(t− tj)e−ω(t−tj)

|ηi(t)| ≤|q+i (tj)|ω2(t− tj)e−ω(t−tj)

+ |(1− ω(t− tj))e−ω(t−tj)| × |ηi(tj)|

+ (1− e−ω(t−tj) − w(t− tj)e−ω(t−tj))bαi

(B-4)

For (B-4), an assumption is made at first that |q+i (tj)| ≤µqi

+ bqiand |ηi(tj)| ≤ µηi

+ bαi, where µqi

and µηican two

arbitrarily small positive constants. For the first inequality in(B-4), it is inferred that |qi(t)| → 0 with ω → +∞. Thus,there must exist ω = ωqi

such that |qi(tj + δt)| ≤ µqiholds,

where δt is the minimum inter-event time proved in SectionIV. Invoking |∆qi| ≤ bqi

, |q+i (tj)| ≤ µqi+ bqi

holds. For thesecond inequality in (B-4), it is inferred that |ηi(t)| ≤ bαi

withω → +∞. Thus, there must exist ω = ωηi

such that |ηi(tj +δt)| ≤ µηi

+ bαiholds. By selecting ω = maxωqi

, ωηi, this

assumption holds.

From the first inequality in (B-4), it can be found that|(1 + ω(t − tj))e−ω(t−tj)| < 1, |2/ω − 2/ωe−ω(t−tj) − (t −tj)e−ω(t−tj)| < 2/ω and |(t − tj)e−ω(t−tj)| < 1/(ωe). Byinvoking the above assumption, the first inequality in (B-4)can be transformed to

|qi(t)| ≤ µqi+ bqi

+µηi

+ bαi

ωe+

2bαi

ω(B-5)

Let bqi= µqi

+ bqi+ (µηi

+ bαi)/(ωe) + 2bαi

/ω. It is obviousthat bqi

→ bqiwith ω → +∞. The proof is completed.

Acknowledgements This work is supported by the Foun-dation for Innovative Research Groups of the Natural ScienceFoundation of Hebei Province (No.E2020203174) and HebeiProvince Natural Science Fund (No.E2019203431) titled with“Research on Multi-Field Coupling Dynamics Modeling forRescue Helicopter Simulator”. We’d like to thank ProfessorChen Bing in Institute of Complexity Science, Qingdao Uni-versity for his suggestions on the nonstrict-nonlinear system.

Compliance with ethical standards

Conflict of interest There is no potential conflict of inter-est in this paper.

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Figures

Figure 1

Framework of proposed ETC

Figure 2

Diagram of ball and beam system

Figure 3

Disturbance di

Figure 4

Output y

Figure 5

States of event-triggered adaptive model in ET-CL

Figure 6

Control input u

Figure 7

Learning performance of TT-CL

Figure 8

Learning performance of ET-PL

Figure 9

Learning performance of ET-CL

Figure 10

Evolution of performance indexes

Figure 11

Update laws of ET-CL

Figure 12

Accumulation of sampling times

Figure 13

Inter-event time of ET-CL