event-triggered neuroadaptive output-feedback control for

Noname manuscript No.(will be inserted by the editor)

Event-triggered neuroadaptive output-feedback control fornonstrict-feedback nonlinear systems with given performancespecifications

Di Yang · Weijun Liu · Chen Guo

Received: date / Accepted: date

Abstract This paper focuses on the event-triggered neu-roadaptive output-feedback tracking control issue for nonstrict-feedback nonlinear systems with given performance speci-fications. By constructing a neural observer to estimate un-measurable states, a novel event-triggered controller is pre-sented together with a piecewise threshold rule. The pre-sented event-triggered mechanism has two thresholds to re-duce communication resources between the controller andactuator. An improved speed transformation function is in-troduced to make the output tracking error converge to apreassigned small region at predesigned converging modewithin preset finite time. Combining the variable separationmethod based on the structural property of radial basis func-tion (RBF) and backstepping technology, the algebraic loopproblem caused by the nonstrict-feedback structure is over-come. The command filtered technology with filtering er-ror compensating signal is applied to address the “explo-sion of complexity” problem. Furthermore, Lyapunov sta-bility analysis demonstrates that under the presented event-triggered controller, all signals in the closed-loop system aresemiglobally bounded, and the Zeno-behaviour is ruled out

D. Yang ·W. LiuSchool of Mechanical Engineering, Shenyang University of Technol-ogy, Shenyang 110870, Liaoning, People’s Republic of Chinae-mail: [email protected]

W. Liu ()e-mail: [email protected]

D. YangSchool of Chemical Process Automation, Shenyang University ofTechnology, Liaoyang 111003, Liaoning, People’s Republic of China

C. GuoSchool of Marine Electrical Engineering, Dalian Maritime University,Dalian 116026, Liaoning, People’s Republic of Chinae-mail: [email protected]

strictly. Numerical simulations are finally provided to illus-trate the presented control scheme.

Keywords Nonstrict-feedback nonlinear systems · Adap-tive neural control · Given performance specifications ·Speed transformation function · Event-triggered control ·Command filtered backstepping

1 Introduction

Adaptive backstepping technology has become a resultfulinstrument to design the control scheme for uncertain strict-feedback systems, and a lot of significant research resultshave been achieved (see [1]–[7]). Whereas many practicalengineering systems, such as the helicopter model [8], theball and beam system [9], and uncertain robot systems [10],are in nonstrict-feedback form. The algebraic loop problem,caused by the nonstrict-feedback structure, will render thetraditional adaptive backstepping control method discussedin strict-feedback systems invalid. Thus, to solve this prob-lem, the Butterworth low-pass filter was adopted in [11]–[12]for pure-feedback nonlinear systems. During the control lawdesign process, it is known that the introduction of addi-tional low-pass filter will inevitably lead to the increase inalgorithm complexity. Without introducing the additional fil-ter, some adaptive neural networks/fuzzy control strategieswere developed in [13]–[16] for nonstrict-feedback nonlin-ear systems via utilizing the variable separation technology.The restrictive condition, requiring the unknown functionsto conform to the monotonically increasing property [13]–[16],was overcome in [17]–[18] according to the structual prop-erty of fuzzy-logic systems or neural networks. Such resultswere further generalized to the cases of prescribed perfor-mance control and finite-time control. To ensure that thetracking error enters into a prescribed range, an adaptivefuzzy controller was constructed in [19] for nontriangular

2 Di Yang et al.

nonlinear systems, in which the tracking error with con-straint was converted into a new variable. Via applying afuzzy observer to estimate unmeasurable states, authors in[20] investigated the finite-time adaptive control issue fornonlinear systems in nontriangular form, where the dynamicsurface control scheme was adopted to overcome the “ex-plosion of complexity” problem resulted from the iterativedifferentiations for virtual control functions.

Although the prescribed performance control can achievethat the output error enters into a predefined compact set atthe prescribed converging mode [21]–[23], it cannot guar-antee that the convergence of output error is realized in afinite time. In order for the tracking error to converge withina finite time, the finite-time controller in [24]–[26] attemptsto give a new idea, but the settling time, depending on theinitial condition of system states, cannot be given arbitrarilyand chosen in advance. Fortunately, the work [27] applied aspeed function to transform the original output error into anaccelerated one, and then the convergence performance oforiginal output error can be adjusted via stabilizing the ac-celerated dynamic system. This idea was further extended in[28] to design prescribed performance controller for MIMOstrict-feedback systems, where the tracking error can con-verge to a preassigned compact set in a pre-given time. Morerecently, via blending finite-time performance function andintermediate transformation, the finite-time controller withprescribed performance was presented for nonlinear strict-feedback systems in [29]–[31] and for nonlinear nonstrict-feedback systems in [32], respectively. However, the resultspresented in [27]–[32] are dependent on the full-state infor-mation, and do not take into account the event-triggered con-trol issue.

Due to the increasing popularity of the network control,event-triggered scheme has been widely acknowledged as aneffective alternative to traditional periodic sampling scheme,which can greatly reduce transmission burden in the com-munication network. Thus, some significant results aboutevent-triggered control scheme can be found in [33]–[35].Among them, the authors in [33] developed an output-feedbackcontroller with prescribed performance for pure-feedbacksystems using the fixed threshold event-triggered mecha-nism. It should be noted that the threshold value in [33] doesnot change during the control process. For further decreas-ing the data transmission and increasing the efficiency in re-source utilization, an event-triggered adaptive controller wasdesigned in [34] for strict-feedback systems with actuatorfailures by utilizing the neural state observer, where a vary-ing threshold method, in the light of the value of control sig-nal, was proposed to update the control signal. Based on thelinear state observer, an event-triggered adaptive neural net-work controller was proposed in [35] for nonstrict-feedbacksystems with unknown control directions via adopting rela-tive threshold strategy. However, the complexity of the event-

triggered control algorithm based on relative threshold willincrease in order to guarantee the stability of whole systembecause the triggering condition contains the control signal.

According to the discussions above, it is very neces-sary to develop a new event-triggered neuroadaptive output-feedback controller with given performance specificationsfor nonstrict-feedback uncertain nonlinear systems, whichdrives the current work. Specifically, there are three compli-cated problems to be solved in the article:

1) How to use a speed transformation function to designan event-triggered output-feedback controller with given per-formance specifications for nonstrict-feedback uncertain non-linear systems?

2) How to effectively reduce data transmission and im-prove resource utilization while avoiding the inclusion ofcontrol signal in the triggering condition?

3) How to introduce command filtering technology withfiltering error compensating signal to deal with the “explo-sion of complexity” problem during the control strategy de-sign process?

In this work, a new event-triggered adaptive control schemecombined with neural state observer and command filter-ing technology will be investigated for nonstrict-feedbackuncertain nonlinear systems with given performance speci-fications. Furthermore, the variable separation method andbackstepping technology will be adopted to deal with the al-gebraic loop problem arising from nonstrict-feedback struc-ture. The main features and contributions of the proposedcontroller lie in the following aspects:

1) In comparison to the observer-based finite-time adap-tive tracking control in [4] and [20], observer-based pre-scribed performance tracking control in [36], the proposedoutput-feedback control scheme can ensure that not only thetracking error enters into the prescribed bounded set withina preset time, but also decay rate during the preset time in-terval is predesigned and controllable explicitly. Further, thepreset time is independent of the initial condition of systemstates.

2) In comparison to the speed transformation function-based tracking control strategy with given performance spec-ifications in [27] and [28], finite-time tracking control strat-egy with prescribed performance in [31] for strict-feedbacknonlinear systems, this paper applies the output informationand universal approximation ability of neural networks toconstruct state observer, and a novel event-triggered trackingcontroller is presented for more general nonstrict-feedbacknonlinear systems.

3) Compared with the event-triggered prescribed perfor-mance controller using the fixed threshold strategy in [33]for pure-feedback nonlinear systems, the designed event-triggered mechanism has two thresholds, which can effec-tively reduce the data transmission and avoid including thecontrol signal in the trigger condition. Furthermore, the com-

Event-triggered neuroadaptive output-feedback control for nonstrict-feedback nonlinear systems with given performance specifications 3

mand filtering technology is employed to deal with the “ex-plosion of complexity” and “filtering error compensation”problem by utilizing filtering error compensating signal.

2 Problem formulation

2.1 System description

Consider the following uncertain nonstrict-feedback system:

xk = xk+1 +hk (X) , k = 1,2, ...,n−1xn = u+hn (X)

y = x1

(1)

where X = [x1, ...,xn]T ∈ Rn and u ∈ R represent plant state

vector and control input, respectively. y ∈ R is the output ofplant. hk (X) , k = 1,2, ...,n represent unknown smooth non-linear functions. It is assumed that nonstrict-feedback sys-tem (1) is completely observable and controllable and onlysystem output y can be directly measured.

Remark 1 Because the nonlinear function hk (X) in each sub-system contains the whole plant state vector X = [x1, ...,xn]

T,the plant (1) is a nonstrict-feedback system. It is worth not-ing that the control methods developed in [5] and [31] areeffective for strict-feedback systems with full-state infor-mation. If the backstepping technology in [5] and [31] isadopted to develop control law for nonstrict-feedback sys-tem (1), the algebraic loop problem will arise, making itvery difficult to design virtual controller. In practice, manypractical plants are with nonstrict-feedback structure and un-measured states. Therefore, the system (1) considered in thearticle is very necessary.

The purpose of this paper is to present an event-triggeredneuroadaptive output-feedback control scheme with givenperformance specifications for (1), such that:

1) All signals in the closed-loop system are semigloballybounded.

2) The tracking error χ1 = y−yd , where yd is the desiredreference signal, converges to a prescribed compact set atpredesigned decay rate within a pre-given finite time.

3) A novel event-triggered mechanism is designed toeffectively reduce data transmission between the controllerand actuator and the Zeno behaviour is strictly ruled out.

To achieve the control objective, we require the follow-ing assumption and lemmas.

Assumption 1 The desired reference signal yd and its timederivative yd are smooth, bounded, and available.

Remark 2 Compared with traditional backstepping method,the control scheme in this article does not have strict require-ments on the desired reference signal, because only the in-formation of yd and yd is used in the design process.

Lemma 1 [37] For hyperbolic tangent function tanh(·), thefollowing property holds

0≤−h tanh(

hϖ

)+ |h| ≤ 0.2785ϖ (2)

where ϖ > 0, h ∈ R, −h tanh(h/ϖ)≤ 0.

Lemma 2 [38] Consider a continuous nonlinear functionh(X) defined on a compact set Ω . Then for ∀ε > 0, thereexists a RBF neural network satisfying

supx∈Ω

∣∣h(X)−θTφ (X)

∣∣≤ ε (3)

where θ = [θ1, ...,θM]T represents the ideal weight vector,φ (X) = [p1 (X) , ..., pM (X)]T denotes the basis function vec-tor, M > 1 is node number of neural network, and pk (X) is

constructed as pk (X) = exp[−(X− vk)

T (X− vk)/γ2k

], k =

1, 2, ...,M, vk = [vk1, ...,vkn]T is the center vector and γk is

the width of Gaussian function.

Remark 3 According to the definition of pk (X) and φ (X), itcan be demonstrated that 0< pk (X)≤ 1 and 0< φ(X)T

φ (X)

≤M, the characteristic of which will be utilized to separatethe whole state variable in the system function hk (X).

2.2 Speed function

Inspired by [27] and [28], an improved speed function isintroduced as follows:

µ (t) =

t3r

(1−µtr)(tr−t)3µ0+µ0µtrt3

r, t ∈ [0, tr)

1µ0µtr

, t ∈ [tr, +∞)(4)

where µ0 and µtr < 1 are positive parameters, tr > 0 andµ0µtr denote finite settling time and ultimate bound of track-ing error, respectively.

Lemma 3 [28] The unique properties of speed function in(4) can be described as follows.

1) For t ∈ [0, tr), µ (t) is strictly increasing with µ (0) =1/µ0 and µ ∈ [1/µ0, 1/µ0µtr] for t ∈ [0, +∞).

2) At t = tr, µ (t) reaches its maximum value 1/µ0µtrand remains to be 1/µ0µtr for t ∈ [tr, +∞).

3) For t ∈ [0, +∞), µ(i) (t) (i = 0, 1, 2) are C2−i andbounded, and µ−1 (t) µ (t) is bounded.

Remark 4 Although some prescribed performance controllers,using exponential performance function ρ (t)= (ρ0−ρ∞)e−ωt

+ρ∞ (see [21]–[23]), have been developed to deal with thetracking error, they cannot guarantee that the tracking er-ror enters into a preassigned compact set within a knowntime. However, the speed function considered in this articlecan make the convergence of tracking error have the givenperformance specifications, which is more challenging anddifficult.

4 Di Yang et al.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

Time[s]

1/µ

tr = 2, µtr = 0.02, µ0 = 2.4tr = 2, µtr = 0.1, µ0 = 2.4tr = 5, µtr = 0.1, µ0 = 2.4

Fig. 1 Trajectories of 1/µ in different cases.

To achieve the pregiven performance specifications, thefollowing state transformation is defined:

s1 = tan(

π

2µχ1

), χ1 (0)< µ0. (5)

From the definition of (5), we have

χ1=2

πµactans1 (6)

and

χ1 =−2µ

πµ2 arctans1 +2

πµ

s1

1+ s21

= x2 +h1 (X)− yd

(7)

which yields

s1 = Ξx2 +Ξh1 (X)−Ξ yd +Ξ2µ

πµ2 arctans1 (8)

where Ξ = πµ(1+ s2

1)/2 > 0.

Remark 5 In contrast to the speed function in [27] and [28],a new parameter µ0 is introduced to obtain µ (0) = µ0 ratherthan µ (0) = 1, in which the restrictive condition of the ini-tial tracking error χ1 (0) is relaxed largely. In addition, dif-ferent from the existing state transformation in [39] and [12],a distinguishing feature, via using (5), is to ensure that thevalue of function Ξ is always positive, which is very neces-sary to design a stable filtering error compensation system.

Remark 6 The advantage of using state transformation is toconvert the constrained tracking error χ1 into an equivalentunconstrained signal. It should be noted from (5) that if s1 isbounded, then −1/µ < χ1 < 1/µ where the curve of 1/µ isshown in Fig.1. Therefore, by designing the control methodto ensure the boundedness of s1, the pregiven performancespecifications of χ1 can be realized indirectly.

2.3 Event-triggered mechanism

Inspired by [33], a novel event-triggered mechanism withtwo thresholds is designed as follows.

u(t) = τ (tq) , tq ≤ t < tq+1 (9)

tq+1 =

inf

t > tq∣∣ |ς (t)| ≥ ρax

, if t < T †

inf

t > tq∣∣ |ς (t)| ≥ ρin

, if t ≥ T † (10)

where ς (t) = τ (t)−u(t) denotes the measurement error be-tween the intermediate control τ (t) and control input u(t),ρax, ρin, T † are positive design parameters and ρax > ρin, tr >T †, tq, q ∈ z+, denotes controller updating moment, i.e.,once (10) is triggered, the actuator will be updated by τ

(tq+1

).

In the period t ∈[tq, tq+1

), the actuator holds as a constant

τ (tq). During the initial stage t ∈[0, T †

), a relatively large

threshold is chosen to deal with large control signal causedby initial tracking error to obtain longer event-triggered in-tervals. In another condition when tracking error graduallycomes to stability, the control signal should be close to zero.Thus, to achieve a better system stabilization performance,a smaller measurement error is designed to obtain a moreprecise control signal.

Remark 7 Compare with fixed threshold event-triggered strat-egy in [33], the relative threshold method [35] can effec-tively reduce the number of triggering events because theoutput of controller is a time-varying signal during systemoperation. However, the relative threshold method, in whichthe control signal is introduced into triggering condition,will complicate the expression of the control algorithm. Inorder to effectively reduce the number of triggering eventsand avoid introducing the control signal into triggering con-dition, a novel event-triggered mechanism with two thresh-olds is designed in this work, whose superiority will be ver-ified in the following comparative simulations.

3 Control design and stability analysis

3.1 Neural state observer design

Due to unmeasurable state variables and unknown nonlinearfunctions, the RBF neural networks are utilized to be uncer-tainty approximators embedded in the observer, and therebycontributing to a neural state observer. Before constructingthe neural state observer, the nonstrict-feedback system (1)can be rewritten as X = AX +Ly+

n∑

k=1Bkhk (X)+Bu

y =CX(11)

where


A =

−l1... In−1−ln 0 · · · 0

,

Bk = [0, 0, ...,0, 1︸︷︷︸k

, 0, ..., 0]T, B = [0, ...,0, 1]T, L = [l1, l2

, ..., ln]T, and C = [1, 0, ...,0]. Via selecting proper L, a strict

Hurwitz matrix can be obtained. Therefore, given QT = Q >

0, there is a positive definite matrix PT = P > 0 satisfying

ATP+PA =−2Q. (12)

Construct a neural observer as follows:

˙X = AX +Ly+n

∑k=1

Bkhk(X |θk

)+Bu (13)

where X = [x1, ..., xn]T are defined to estimate X = [x1, ...,xn]

T,and under Lemma 2, the nonlinear functions in (1) can be es-timated as hk (X |θk )= θ T

k φk (X) and hk(X |θk

)= θ T

k φk(X).

Similar to [40]–[42], the variable errors εk and δk aredefined as

εk = hk (X)− hk(X∣∣θ ∗k )

δk = hk (X)− hk(X |θk

) (14)

where θ ∗k denotes the optimal parameter vector, and we as-sume that there exist positive constants εk and δk, k= 1, 2, ...,n,such that |εk| ≤ εk and |δk| ≤ δk.

Denote X = X − X = [x1, ..., xn]T as the observer error.

Then from (11) and (13), we can obtain the error dynamics:

˙X = AX +n

∑k=1

Bk(hk (X)− hk

(X |θk

))= AX +δ

(15)

where δ = [δ1, ...,δn]T.

Remark 8 Since only the output y can be used to design thecontrol law, a neural state observer is constructed to estimateunmeasured state vector. Compared with the linear state ob-server in [43] and [10], the observer (13) uses neural net-works to approximate useful information of uncertain func-tion hk (X) for obtaining better estimation performance

3.2 Event-triggered control design

To avoid the repeated differentiations virtual controllers inthe backstepping design, the following tracking errors andcommand filtered system are defined:

z1 = s1

zk = xk−λk, k = 2, ...,n(16)

where λk denotes the output of the following command filterwith virtual controller αk−1

ωk−1λk +λk = αk−1, λk (0) = αk−1 (0) , k = 2, ...,n (17)

where ωk−1 is a positive design parameter.The following virtual controllers and actual controller

are given for t ∈[tq, tq+1

)α1 =−

c1z1

Ξ− a13

2η1Ξ −a11η1Ξ −a12η1Ξ

−θT1 φ1 (x1)−

2µ

πµ2 arctans1 + yd

α2 =−c2z2−a21η2−a23

2η2−

a22

2η2

−θT2 φ2

(X2)− l2x1 + λ2−Ξz1

αk =−ckzk−ak1ηk−ak3

2ηk−

ak2

2ηk−θ

Tk φk

(Xk)

− lkx1 + λk− zk−1, k = 3, ...,n−1

αn =−cnzn−an3

2ηn−

an2

2ηn−θ

Tn φn

(X)− lnx1

+ λn− zn−1

τ (t) = αn− ρ tanh(

ηnρ

ϖ

)u(t) = τ (tq)

(18)

where ρ > ρax > ρin, ϖ > 0, ck > 0, ak2 > 0, ak3 > 0 anda j1 > 0, k = 1, ...,n, j = 1, ...,n− 1 are design parameters.ηk denotes the compensated tracking error given as

ηk = zk−ξk, k = 1, 2, ...,n. (19)

where ξk is error compensation signal designed in (21). Theupdating process of parameters can be designed as

θ1 = π1Ξη1φ1 (x1)−σ1θ1

θk = πkηkφk(Xk)−σkθk, k = 2, ...,n

(20)

where Xk = [x1, x2, ..., xk]T, k = 2, ...,n, Xn = X , πk > 0 and

σk > 0 are design parameters.To eliminate the adverse effect of filtering errors λk −

αk−1 (k = 2, ...,n), we design the error compensation mech-anism as

ξ1 =−c1ξ1 +Ξξ2 +Ξ (λ2−α1)−Ξι1 sign(ξ1)

ξ2 =−c2ξ2−Ξξ1 +λ3−α2 +ξ3− ι2 sign(ξ2)

ξk =−ckξk−ξk−1 +λk+1−αk +ξk+1− ιk sign(ξk)

(k = 3, ...,n−1)

ξn =−cnξn−ξn−1− ιn sign(ξn)

(21)

with ξk (0) = 0(k = 1, ...,n), ιk is a positive design parame-ter.

Remark 9 Although the dynamic surface technique was em-ployed in [20] to solve the “explosion of complexity” prob-lem for nonstrict-feedback nonlinear systems, the adverseeffect of filtering error λk−αk−1 was not addressed, whichmight degrade control quality. In order to cope with the ad-verse effect, the error compensation mechanism in (21) isdeveloped at each step of the control strategy.

6 Di Yang et al.

3.3 Stability analysis

In this section, the specifics of developed event-triggeredcontrol strategy are presented according to the backsteppingtechnique, where the n recursive steps are involved.

Step 1: Based on the compensated tracking errors (19),the derivative of η1 is given as

η1 = z1− ξ1

= Ξ x2 +Ξ (z2 +λ2)+Ξ

(θ∗1

Tφ1(X)

+ε1

−yd +2µ

πµ2 arctans1

)− ξ1

= Ξ x2 +Ξ (z2 +λ2)+Ξε1− ξ1

+Ξ

(θ∗1

Tφ1(X)−θ

∗1

Tφ1 (x1)

)+Ξθ

T1 φ1 (x1)

+ΞθT1 φ1 (x1)+Ξ

(2µ

πµ2 arctans1− yd

)(22)

where θ1 = θ ∗1 −θ1. Constructing the Lyapunov function

V1 =Vx +12

η21 +

12π1

θ21 (23)

where Vx = XTPX /2. Its time derivative is obtained as

V1 = Vx +η1η1−1π1

θT1 θ1

=−XTQX + XTPδ +η1 (Ξ x2 +Ξη2 +Ξα1

+Ξ

(θ∗1

Tφ1(X)−θ

∗1

Tφ1 (x1)

)+Ξθ

T1 φ1 (x1)

+ΞθT1 φ1 (x1)+Ξε1 +Ξ

(2µ

πµ2 arctans1− yd

)+c1ξ1 +Ξι1 sign(ξ1))−

1π1

θT1 θ1

(24)

By Young’s inequality, and applying the property of RBF,that is 0 < φ1(·)T

φ1 (·)≤M1, one has

η1Ξ

(θ∗1

Tφ1(X)−θ

∗1

Tφ1 (x1)

)≤ a11η

21 Ξ

2 +

∥∥θ ∗1∥∥2M1

a11(25)

η1Ξε1 +η1Ξι1 sign(ξ1)≤ a12η21 Ξ

2 +ε2

12a12

+ι21

2a12(26)

η1Ξ x2 ≤a13

2η

21 Ξ

2 +

∥∥X∥∥2

2a13(27)

XTPδ ≤ a14

2‖Pδ‖2 +

∥∥X∥∥2

2a14(28)

Substituting (25)-(28) into (24), yields

V1 ≤−XT(

λmin (Q)− 12a13

− 12a14

)X +Ξη1η2

+Ξη1θT1 φ1 (x1)−

1π1

θT1 θ1 +η1Ξ (α1

+a13

2η1Ξ +a11η1Ξ +a12η1Ξ +θ

T1 φ1 (x1)

+2µ

πµ2 arctans1− yd +c1ξ1

Ξ

)+

a14

2‖Pδ‖2

+

∥∥θ ∗1∥∥2M1

a11+

ε21

2a12+

ι21

2a12

(29)

Then, substituting α1 and θ1 into (29), one has

V1 ≤−µ1XTX− c1η21 +Ξη1η2 +

σ1

π1θ

T1 θ1 +Λ1 (30)

where Λ1 = a14‖Pδ‖2/2+∥∥θ ∗1

∥∥2M1/a11 + ε21/(2a12)

+ι21/(2a12) and µ1 = λmin (Q)−1/(2a13)−1/(2a14)> 0.Step 2: The derivative of η2 is

η2 = z2− ξ2

= x3 + h2(X |θ2

)+ l2x1− λ2− ξ2

= z3 +λ3 +θ∗2

Tφ2(X)− θ

T2 φ2

(X)+ l2x1− λ2

− ξ2−θ∗2

Tφ2(X2)+θ

T2 φ2

(X2)+ θ

T2 φ2

(X2)

(31)

where θ2 = θ ∗2 −θ2. The Lyapunov function is constructedas

V2 =V1 +12

η22 +

12π2

θ22 (32)

Then we have

V2 = V1 +η2η2−1π2

θT2 θ2

= V1−1π2

θT2 θ2 +η2

(η3 +α2 +θ

∗2

Tφ2(X)

− θT2 φ2

(X)−θ

∗2

Tφ2(X2)+θ

T2 φ2

(X2)

+ θT2 φ2

(X2)+ l2x1− λ2 + c2ξ2 +Ξξ1

+ι2 sign(ξ2)) .

(33)

Similar to step 1, we can obtain

η2

(θ∗2

Tφ2(X)−θ

∗2

Tφ2(X2))≤ a21η

22 +‖θ ∗2 ‖2M2

a21(34)

η2ι2 sign(ξ2)≤a22

2η

22 +

ι22

2a22(35)

−η2θT2 φ2

(X)≤ a23

2η

22 +

θ T2 θ2M2

2a23(36)


Substituting (34)–(36) into (33), yields

V2 ≤−µ1XTX− c1η21 +

σ1

π1θ

T1 θ1 +Λ1 +η2η3

+η2

(α2 +a21η2 +

a23

2η2 +

a22

2η2

+θT2 φ2

(X2)+ l2x1− λ2 + c2ξ2 +Ξz1

)+η2θ

T2 φ2

(X2)− 1

π2θ

T2 θ2 +

θ T2 θ2M2

2a23

+‖θ ∗2 ‖2M2

a21+

ι22

2a22

(37)

Then, substituting α2 and θ2 into (37), one has

V2 ≤−µ1XTX− c1η21 − c2η

22 +η2η3

+σ1

π1θ

T1 θ1 +

σ2

π2θ

T2 θ2 +

θ T2 θ2M2

2a23+Λ2

(38)

where Λ2 = Λ1 +‖θ ∗2 ‖2M2/a21 + ι22/(2a22).

Step k (k = 3, ...,n−1): The derivative of ηk is

ηk = zk− ξk

= xk+1 + hk(X |θk

)+ lkx1− λk− ξk

= zk+1 +λk+1 +θ∗k

Tφk(X)− θ

Tk φk

(X)+ lkx1

− λk− ξk−θ∗k

Tφk(Xk)+θ

Tk φk

(Xk)

+ θTk φk

(Xk)

(39)

where θk = θ ∗k − θk. The Lyapunov function is constructedas

Vk =Vk−1 +12

η2k +

12πk

θ2k (40)

Then we have

Vk = Vk−1 +ηkηk−1πk

θTk θk

= Vk−1−1πk

θTk θk

+ηk

(ηk+1 +αk +θ

∗k

Tφk(X)− θ

Tk φk

(X)

−θ∗k

Tφk(Xk)+θ

Tk φk

(Xk)+ θ

Tk φk

(Xk)

+lkx1− λk + ckξk +ξk−1 + ιk sign(ξk)).

(41)

Similar to step 2, we can obtain

ηk

(θ∗k

Tφk(X)−θ

∗k

Tφk(Xk))≤ ak1η

2k +

∥∥θ ∗k∥∥2Mk

ak1(42)

ηkιk sign(ξk)≤ak2

2η

2k +

ι2k

2ak2(43)

−ηkθTk φk

(X)≤ ak3

2η

2k +

θ Tk θkMk

2ak3(44)


Vk ≤ Vk−1 +ηkηk+1 +ηk

(αk +ak1ηk +

ak3

2ηk

+ak2

2ηk +θ

Tk φk

(Xk)+ lkx1− λk + ckξk

+ξk−1)+ηkθTk φk

(Xk)− 1

πkθ

Tk θk

+θ T

k θkMk

2ak3+

∥∥θ ∗k∥∥2Mk

ak1+

ι2k

2ak2

(45)

Then, substituting αk and θk into (45), one has

Vk ≤−µ1XTX−k

∑i=1

ciη2i +ηkηk+1 +

k

∑i=1

σi

πiθ

Ti θi

+k

∑i=2

θ Ti θiMi

2ai3+Λk

(46)

where Λk = Λk−1 +∥∥θ ∗k

∥∥2Mk/ak1 + ι2k /(2ak2).

Remark 10 It can be noticed that the function hk(X |θk

)in

the neural observer (13) contains the whole state estimateX = [x1, ..., xn]

T. To ensure that only partial state estimateXk = [x1, ..., xk]

T is included in the virtual controller αk, weemploy Young’s inequality and the structural property ofRBF to deal with the whole variable X , which solves theproblem of algebraic loop. Compared with [15]–[16] and[12], the method in this article neither needs the restrictivecondition that the unknown function satisfies monotonicallyincreasing property, nor needs to introduce additional low-pass filter.

Step n: The actual controller will be given in this finalstep, the derivative of ηn is

ηn = zn− ξn

= u+ hn(X |θn

)+ lnx1− λn− ξn

= u+θTn φn

(X)− θ

Tn φn

(X)+ θ

Tn φn

(X)

+ lnx1− λn− ξn.

(47)

Choose the Lyapunov function

Vn =Vn−1 +12

η2n +

12πn

θ2n . (48)

Then we have

Vn = Vn−1 +ηnηn−1πn

θTn θn

= Vn−1−1πn

θTn θn +ηnθ

Tn φn

(X)

+ηn(u+θ

Tn φn

(X)− θ

Tn φn

(X)+ lnx1

−λn + cnξn +ξn−1 + ιn sign(ξn)).

(49)

8 Di Yang et al.

Applying Young’s inequality yields

ηnιn sign(ξn)≤an2

2η

2n +

ι2n

2an2(50)

−ηnθTn φn

(X)≤ an3

2η

2n +

θ Tn θnMn

2an3. (51)


Vn ≤ Vn−1 +ηn

(u+

an3

2ηn +

an2

2ηn +θ

Tn φn

(X)

+lnx1− λn + cnξn +ξn−1

)+ηnθ

Tn φn

(X)

− 1πn

θTn θn +

θ Tn θnMn

2an3+

ι2n

2an2.

(52)

In the interval t ∈[tq, tq+1

), from (10) ,we know that

|τ (t)−u(t)|< ρax. (53)

So, there is a time-varying parameter |κ (t)| ≤ 1 satisfiy-ing

u(t) = τ (t)−κ (t)ρax. (54)

By substituting τ (t) and (54) into (52), we have

Vn ≤ Vn−1 +ηn

(αn− ρ tanh

(ηnρ

ϖ

)−κ (t)ρax

+an3

2ηn +

an2

2ηn +θ

Tn φn

(X)+ lnx1

−λn + cnξn +ξn−1

)+ηnθ

Tn φn

(X)

− 1πn

θTn θn +

θ Tn θnMn

2an3+

ι2n

2an2.

(55)

Substitute αn and θn into (55), yields

Vn ≤−µ1XTX−n

∑k=1

ckη2k +

n

∑k=1

σk

πkθ

Tk θk

+n

∑k=2

θ Tk θkMk

2ak3+Λn +ηn

(−ρ tanh

(ηnρ

ϖ

)−κ (t)ρax)

(56)

where Λn = Λn−1 + ι2n/(2an2). From θk = θ ∗k −θk, one has

θTk θk ≤

12

θ∗k

Tθ∗k −

12

θTk θk (57)

then

n

∑k=1

σk

πkθ

Tk θk ≤−

n

∑k=1

σk

2πkθ

Tk θk +

n

∑k=1

σk

2πkθ∗k

Tθ∗k (58)

Based on Lemma 1, ρ > ρax and |κ (t)| ≤ 1, from (56)and (58), we have

Vn ≤−µ1XTX−n

∑k=1

ckη2k +

n

∑k=1

σk

πkθ

Tk θk

+n

∑k=2

θ Tk θkMk

2ak3+Λn−ηnρ tanh

(ηnρ

ϖ

)+ |ηnρ|

≤ −µ1XTX−n

∑k=1

ckη2k −(

σ1

2π1θ

T1 θ1+

n

∑k=2

(σk

2πk− Mk

2ak3

)θ

Tk θk

)+Λ

′n

(59)

where Λ ′n = Λn +n∑

k=1σkθ ∗k

Tθ ∗k /2πk +0.2785ϖ .

According to the above analysis, the main result of thisarticle is stated as follows.

Theorem 1 For the nonstrict-feedback uncertain nonlinearsystems (1) with Assumption 1, design the neural state ob-server (13), the command filtered system (17), error com-pensation system (21), the virtual controllers and actual con-troller (18), the adaptive updating laws (20), then the pro-posed event-triggered control strategy can guarantee thatthe tracking error converges to a prescribed small regionnear zero within preset finite time tr at predesigned con-verging mode during the period t ∈ [0, tr) and all the sig-nals xk, xk, zk, ηk, ξk, αk, k = 1, ..., n, τ (t) and u(t)are bounded. In addition, ∀q ∈ z+, there is a positve instanttℵ > 0 satisfying

tq+1− tq

≥ tℵ.

Proof :To prove the stability of the error compensationsystem (21), we choose the Lyapunov function as Vξ =

n∑

k=1ξ 2

k /2. Then, the derivative of Vξ is obtained as

Vξ =−c1ξ21 +Ξξ1ξ2 +Ξ (λ2−α1)ξ1−Ξι1 |ξ1|

− c2ξ22 −Ξξ1ξ2 +(λ3−α2)ξ2 +ξ2ξ3

− ι2 |ξ2|+ · · ·− ckξ2k −ξk−1ξk

+(λk+1−αk)ξk +ξkξk+1− ιk |ξk|+ · · ·− cnξ

2n −ξn−1ξn− ιn |ξn|

=−n

∑k=1

ckξ2k −Ξ (ι1 |ξ1|− (λ2−α1)ξ1)

−n−1

∑k=2

(ιk |ξk|− (λk+1−αk)ξk)− ιn |ξn|

(60)

Similar to [44] and [45], we can get that |(λk+1−αk)| ≤βk, k = 1, ..., n−1, can be achieved with the positive con-stant βk. Since Ξ > 0, via choosing suitable parameter ιk >


βk, one has

Vξ ≤−n

∑k=1

ckξ2k (61)

To analysis the stability of the closed-loop system, weconstruct the whole Lyapunov function as

V =Vn +Vξ . (62)

The derivative of V is obtained as

V ≤− 2µ1

λmax (P)XTPX

2−2

n

∑k=1

ckη2

k2

−(

σ1θ T

1 θ1

2π1+

n

∑k=2

(σk−

2πkMk

2ak3

)θ T

k θk

2πk

)

−2n

∑k=1

ckξ 2

k2

+Λ′n

≤−ΘV +Λ′n

(63)

where Θ =min2µ1/λmax (P) , 2ck, σ1, σk−2πkMk/(2ak3).Furthermore, select the design parameters σk, πk, ak3 so thatσk−2πkMk/(2ak3)> 0. Consequently, from (63), we have

V (t)≤V (0)e−Θ t +Λ ′nΘ

. (64)

Then it can be obtained that ηk, θk, ξk and θk, k =

1, ..., n are bounded. From zk = ηk + ξk., zk is bounded.Next, x1 and x1 are bounded as yd is bounded, which im-plies that α1 and λ2 are bounded. Similarly, we can provexk, xk, αk, τ (t) and u(t) are also bounded. Therefore, all thesignals inside the closed-loop system are bounded.

Furthermore, since z1 = s1 is bounded, we obtain

|χ1|<∣∣∣∣ 1µ

∣∣∣∣=

∣∣∣ (1−µtr )(tr−t)3µ0+µ0µtr t3

rt3r

∣∣∣ , t ∈ [0, tr)

µ0µtr , t ∈ [tr, +∞)

(65)

which implies that the error |χ1| reduces to µ0µtr within apregiven time tr at predesigned converging mode governedby (1−µtr)(tr− t)3

µ0/t3r .

Next, we prove that there exists tℵ > 0 such that

tq+1−tq≥ tℵ, ∀q∈ z+. Based on ς (t)= τ (t)−u(t), ∀t ∈

[tq, tq+1

),

one has

ddt|ς (t)|= d

dt

√ς (t) ς (t)

= sign(ς (t)) ς (t)

≤ |τ (t)| .

(66)

From (18), we can demonstrate that τ (t) is differentiableand its time derivative τ (t) is a function with all boundedsignals, which implies that there is a positive constant γ > 0

satisfiying |τ (t)| ≤ γ . In addition, ς (tq)= 0 and limt→q+1

ς (t)≥ρin, so the lower bound of event-triggered time intervals tℵ

must satisfy tℵ ≥ ρin/γ > 0. Therefore, the Zeno-behaviouris ruled out.

Remark 11 It can be demonstrated from (65) that in the in-terval t ∈

[tq, tq+1

), if the value of µ0 determined by the

initial tracking error χ1 (0) does not change, then the con-verging rate of |χ1| is affected not only by µtr but also by tr.A faster tracking process can be achieved by decreasing µtror shortening preset time tr, which will be shown in the latersimulation comparison experiment.

Remark 12 The developed event-triggered output feedbackcontrol strategy can obtain given performance specificationsfor the tracking error. Such solution, very useful in practi-cal engineering, has never been proposed for the uncertainnonstrict-feeback systems, as described in (1).

4 Simulation Examples

This section is to demonstrate the effectiveness and superi-ority of the developed control strategy by two examples.

Example 1: Consider the nonstrict-feedback nonlinearsystem as follows

x1 = x2 +h1 (X)

x2 = u+h2 (X)

y = x1

(67)

where h1 (X)=x2e−0.5x1 , h2 (X)=x1 sin(x2

2). The initial states

are given as x1 (0)=2, x2 (0)=0, x1 (0)=1.5 and x1 (0) =−2.The desired reference signal is yd = sin(t).

Construct the following neural state observer

˙x1 = x2 + l1 (y− x1)+θ T1 φ1

(X)

˙x2 = u+ l2 (y− x1)+θ T2 φ2

(X) (68)

where the observer gain vector is designed as L = [l1, l2] =[29.9643, 113.4592]. θ T

1 φ1(X)

and θ T2 φ2

(X)

contain 125nodes and 75 nodes respectively. For each of two neuralnetworks, centers are spaced evenly in [−1, 2]× [−2, 1.5]and widths are equal to 3.85. The parameter update laws aregiven as

θ1 = π1Ξη1φ1 (x1)−σ1θ1

θ2 = π2η2φ2(X)−σ2θ2

(69)

with θ1 (0) = 0125×1 and θ2 (0) = 075×1.

10 Di Yang et al.

Then, we design virtual controllers, actual controller andevent-triggered mechanism as

α1 =−c1z1

Ξ− a13

2η1Ξ −a11η1Ξ −a12η1Ξ

−θT1 φ1 (x1)−

2µ

πµ2 arctans1 + yd

α2 =−c2z2−a23

2η2−

a22

2η2−θ

T2 φ2

(X)

− l2x1 + λ2−Ξz1

τ (t) = α2− ρ tanh(

η2ρ

ϖ

)u(t) = τ (tq)

tq+1 =

inf

t > tq∣∣ |ς (t)| ≥ ρax

, if t < T †

inf

t > tq∣∣ |ς (t)| ≥ ρin

, if t ≥ T †

(70)

where the command filter system is given as

ω1λ2 +λ2 = α1, λ2 (0) = α1 (0) (71)

and the error compensation mechanism is constructed as

ξ1 =−c1ξ1 +Ξξ2 +Ξ (λ2−α1)−Ξι1 sign(ξ1)

ξ2 =−c2ξ2−Ξξ1− ι2 sign(ξ2)(72)

with ξk (0) = 0(k = 1, 2).In the simulation, choosing the design parameters as c1=0.1,

c2=5, ρ=9, ϖ=15, ω1=0.01, ι1=ι2=0.1, tr=5, µtr=0.05, µ0=2.5,T †=1, ρax=8, ρin=0.2, π1=π2=0.01, σ1=σ2=1, a11=a12=0.025,a13=0.1, a23=0.9, a22=0.1.

To show the effectiveness of event-triggered mechanism,comparative simulations between the fixed threshold approach,relative threshold strategy and the presented method are car-ried out. The fixed threshold approach can be obtain from[33] and it is given as follows:

τ (t) = α2− ρ tanh(

η2ρ

ϖ

)u(t) = τ (tq)

tq+1 = inf

t > tq∣∣ |ς (t)| ≥ ρ f ix

(73)

where ρ = 9 and the value of ρ f ix is selected as 0.2, 4.2 and8, respectively, for comparative simulations.

The relative threshold strategy from [34] is given as fol-lows:

τrel (t) =−(1+σrel)

(α2 tanh

(η2α2

ϖrel

)+ρrel tanh

(η2ρrel

ϖrel

))urel (t) = τrel (tq)

tq+1 = inf

t > tq∣∣ |ς (t)| ≥ σrel |urel (t)|+ρrel

(74)

where ρrel = 0.09, σrel = 0.4, ϖrel = 5 and ρrel = 0.05.

0 5 10 15 20

−1

0

1

2

Time[s]

ydx1

Fig. 2 Trajectories of yd and x1 in Example 1.

0 5 10 15 20

−1

0

1

2

Time[s]

x1x1

0 0.5 11

1.5

2

Fig. 3 Trajectories of x1 and x1 in Example 1.

Figs. 2–8 describe simulation results, where Fig. 5 showsthat the tracking error χ1 converges to a small region aroundzero within a preset time tr=5 at predesigned decay rate. Italso points out that the proposed method can obtain supe-rior tracking performance according to faster response andsmaller oscillation. For the proposed method, Fig. 3 and Fig.4 show the system state and estimation of them, the controlsignal is shown in Fig. 6 and the trajectories of the norm ofthe adaptive parameters ‖θ1‖ and ‖θ2‖ are shown in Fig. 7and Fig. 8.

Furthermore, performance comparisons of the presentedmethod, the fixed threshold approach and relative thresh-old strategy are summarized in Table I. The integrated timeabsolute and integrated absolute error in [46], labelled asITAE and IAE, are employed to evaluate dynamic perfor-mance and stable precision of tracking controllers, respec-

tively, where ITAE=20∫0

υ |χ1 (υ)|dυ and IAE=20∫0|χ1 (υ)|dυ .

Table I demonstrates that the presented method is superiorto the other two event-triggered mechanisms since it has thelower ITAE and IAE while requiring less triggering num-bers.

Example 2: To verify the effectiveness of the proposedcontroller in the practical system, the network-based one-link manipulator obtained from [47] is considered as fol-lows:

Jv+Fvv+Gsin(v) = τv (75)

where v and v are the velocity and position of the rigid link,respectively, Fv = 1N ·s/rad denotes the overall damping


0 5 10 15 20−3

−2

−1

0

1

2

Time[s]

x2x2


Fig. 5 Trajectories of χ1 under different methods.

0 5 10 15 20

−60

−40

−20

0

Time[s]

u

0 0.2 0.4 0.6 0.8 1

−60

−40

−20

0

14 14.2 14.4 14.6 14.8 15−1

−0.5

0

Fig. 6 Trajectory of controller u in Example 1.

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

Time[s]

‖θ1‖

Fig. 7 Trajectory of ‖θ1‖ in Example 1.

coefficient, G denotes the gravity, J = 1kg ·m2 denotes therotational inertia of motor.

Define u = τv, x1 = v, and x2 = v, from manipulator sys-tem (75), one has

x1 = x2 +h1 (X)

x2 = u+h2 (X)(76)

0 5 10 15 200

0.005

0.01

0.015

0.02

Time[s]

‖θ2‖


Table 1 Performance comparisons of different methods

Method ITAE IAE Triggering numbers

Proposed method 1.08 1.29 227

Fixed threshold with ρ f ix = 0.2 1.18 1.39 354

Fixed threshold with ρ f ix = 4.2 1.46 1.43 232

Fixed threshold with ρ f ix = 8 1.56 1.31 235

Relative threshold 1.79 1.43 236

where h1 (X)= 0.2sin(x1)cos(x2) and h2 (X)=−x2−10sin(x1).In (76), we intentionally introduce disturbed term h1 (X).The desired reference signal is yd = 0.3sin(t).

In the simulation, choose the design parameters as l1 =20, l2 = 200, c1=0.1, c2=5, ρ=9, ϖ=15, ω1=0.01, ι1=ι2=0.1,tr=2, µtr=0.02, µ0=2.4, T †=1, ρax=8, ρin=0.2, π1=0.01, π2=0.02,σ1=σ2=1, a11=a12=0.025, a13=0.1, a23=0.9, a22=0.1. θ T

1 φ1(X)

and θ T2 φ2

(X)

contain 20 nodes and 30 nodes respectively.For each of two neural networks, centers are spaced evenlyin [−2.5, 0.5]× [−1, 7] and widths are equal to 8.8. Theinitial values are set as x1 (0)=−2, x2 (0)=0, x1 (0)=−1.5,x2 (0) = 1, ξk (0) = 0(k = 1, 2), θ1 (0) = 020×1 and θ2 (0) =030×1.

The simulation results are given in Figs. 9–15, where thetracking performance and tracking error are shown in Fig. 9and Fig. 12. The system state and estimation of them areshown in Fig. 10 and Fig. 11. Fig. 13 describes the con-trol signal. Fig. 14 and Fig. 15 show that the trajectories ofthe norm of the adaptive parameters ‖θ1‖ and ‖θ2‖ remainbounded.

Furthermore, we select different design parameters tr andµtr to demonstrate that these factors can affect the trackingperformance via the proposed controller. The comparativeresults are given in Fig. 12, which is explained as follows.

1) For the design parameter µtr=0.1, a faster error con-vergence rate can be achieved by selecting the smaller tr,and for tr=2, we can also obtain a faster transient processby choosing the smaller µtr, which verifies the conclusionin Remark 11.

2) Decreased the value of µtr will result in the bettersteady-state performance when tr=2.

12 Di Yang et al.

0 2 4 6 8 10

−2

−1.5

−1

−0.5

0

0.5

1

Time[s]

ydx1

Fig. 9 Trajectories of yd and x1 in Example 2.

0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time[s]

x1x1

0 0.5 1

−2

−1

0


0 2 4 6 8 10

0

2

4

6

Time[s]

x2x2


0 2 4 6 8 10

−2

−1

0

1

Time[s]

tr = 2, µtr = 0.02tr = 2, µtr = 0.1tr = 5, µtr = 0.1

0 1 2 3

−2

−1

0

6 7 8 9 10−0.05

0

0.05

Fig. 12 Trajectories of χ1 under different cases.

5 Conclusion

This article has developed an event-triggered adaptive output-feedback controller for nonstrict-feedback nonlinear systemswith given performance specifications, where a novel event-triggered mechanism with two thresholds and an improvedspeed transformation function have been designed. Neuralstate observer and neural networks have been adopted to ap-

0 2 4 6 8 10−50

0

50

100

150

200

Time[s]

u

0 0.2 0.4 0.6 0.8 1

0

100

200

4 4.2 4.4 4.6 4.8

−3

−2.5

−2

−1.5

Fig. 13 Trajectory of controller u in Example 2.

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

Time[s]

‖θ1‖


0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

Time[s]

‖θ2‖


proximate unmeasured states and estimate uncertain func-tions, respectively. The structural property of RBF has beenemployed to deal with the algebraic loop problem causedby the nonstrict-feedback structure. Via flexibly incorporat-ing command filtered technology, error compensation sys-tem with backstepping recursive method, a novel controlstrategy has been developed. All closed-loop signals are guar-anteed to be bounded according to Lyapunov theory, andthere is no Zeno-behavior. The developed control strategycan not only reduce the communication resources betweenthe controller and actuator but also ensure that the outputtracking error converges to a prescribed small region withinpreset time at predesigned converging rate. Finally, two sim-ulation examples demonstrate the effectiveness and superi-ority of the developed control strategy. Further, we will ex-tend the control strategy of this article to nonstrict-feedbackswitched nonlinear systems and nonstrict-feedback stochas-tic nonlinear systems.


Acknowledgment

This work is support by the National Natural Science Foun-dation of China (51879027, 51579024)

Compliance with ethical standards

Conflict of interest The authors declare that they have noconflict of interest.Data availability statement The authors can confirm thatall relevant data are included in the article.

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