# Event-Triggered Adaptive Backstepping Control for Strict-Feedback Nonlinear Systems with Zero Dynamics

Authors: Bo Xu; Xiaoping Liu; Huanqing Wang; Yucheng Zhou
Journal: Complexity
 (2019)

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## Abstract

This paper focuses on the problem of event-triggered control for a class of uncertain nonlinear strict-feedback systems with zero dynamics via backstepping technique. In the design procedure, the adaptive controller and the triggering event are designed at the same time to remove the assumption of the input-to-state stability with respect to the measurement errors. Besides, we propose an assumption to deal with the problem of zero dynamics. Three different event-triggered control strategies are designed, which guarantees that all the closed-loop signals are globally bounded. The effectiveness of the proposed methods is illustrated and compared using simulation examples.


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## Body

## 1. Introduction

Nowadays, the control problem of uncertain systems draws more and more attention because of the extensive existence in the practical engineering. Due to the fact that vehicles, robots, or UAVs (Unmanned Aerial Vehicles) are required to work more precisely, there have been plenty of researches towards the control problem. For example, Zhou et al. in [1] proposed two adaptive controllers for uncertain nonlinear systems in the presence of input quantization. An event-triggered output-feedback controller is discussed in [2], for output feedback in switched linear systems. A constrained optimization issue for a class of strict-feedback nonlinear systems is proposed in [3].Among these researches, adaptive control strategy [4] has been proved to be an effective and efficient method for handling uncertainties and nonlinearities. It introduces a parameter estimator to eliminate the influence caused by unknown parameters. Therefore, over the past few years, there have been many adaptive control strategies for uncertain nonlinear systems. For instance, an adaptive controller is proposed for switched nonlinear systems with coupled input nonlinearities and state constraints in [5]. Sun et al. study a class of state-constrained uncertain nonlinear systems which have steady-state behavior and prescribed transient and design a robust adaptive strategy in [6]. In [7], the authors discuss the problems of adaptive output-feedback tracking control for a class of uncertain nonlinear systems with output constraints, unmodeled dynamics, and quantized input. An extremum seeking controller is proposed based on a fractional-order sliding mode in [8], which has a faster convergence speed. Further, Li et al. propose two adaptive control approaches to synchronize the neural networks in finite time in [9]. A fractional-order sliding mode controller is proposed in [10] for robust stabilization of uncertain fractional-order nonlinear systems. For more applications, see [11–18].In the practical systems, input energy is obviously limited. Therefore, energy-saving should also be considered in order to extend the systems’ lifespan. In the past, the time-triggered scheme is widely used, which means the actuation is executed at every periodic instant even the systems do not need any actuations, and it requires abundant times of experiments to find the most suitable parameters. Therefore, it is impractical for many resource-limited systems. The event-triggered control scheme, which is first proposed in [19], can greatly reduce the control execution times and save the computation costs. In event-triggered systems, actuations are not executed periodically, but only when triggering conditions are satisfied. Furthermore, once using the event-triggered control scheme, the Zeno behavior must be taken into consideration. It is also of great importance to find a way to avoid the Zeno behavior to ensure that the system will not make any mistakes while running. Recently, some event-triggered schemes are proposed to handle different problems. For example, an event-triggered control method is proposed in [20] for parametric strict-feedback nonlinear systems involving backstepping and Lyapunov theory. Xing et al. consider the problems of event-triggered based adaptive control for uncertain nonlinear systems in [21] and propose three event-triggered strategies, while Su et al. design an event-triggered controller by using fuzzy logic method in [22]. In [23], by using input-to-state stable (ISS) condition, a state-feedback event-triggered scheme was proposed and the closed-loop system is guaranteed to be asymptotically stable. Wang et al. design a decentralized event-triggered scheme in [24] for both linear and nonlinear subsystems considering transmission delays. While in networked-control systems, an event-triggered scheme for robust set stabilization is proposed in [25]. What is more, event-triggered schemes also show advantages in consensus and synchronization of multiagent systems in [26–29]. Besides, some results on event-triggered control have also been applied on real systems (see in [30–37]). They extend the range of applications of the event-triggered control scheme.The zero dynamics, which are well understood and extended to nonlinear finite-dimension systems in [38], are the dynamics of the system when choosing a certain input u that makes output y be identically 0. It is an expansion of the zero-point concept in linear systems and it is an intrinsic character of nonlinear systems which describes how internal states act when the output is 0. The stability of zero dynamics in sampled-data nonlinear systems is widely discussed in [39, 40], while a funnel controller is proposed in [41] for linear systems considering zero dynamics. For stochastic systems, a controller is designed in [42, 43] using event-triggered scheme and backstepping method based on Lyapunov theory. In continuous nonlinear systems, since the zero dynamics are related to the systems dynamics and the minimum interexecution time which is also influenced by the systems dynamics, has to be considered, it increases the complexity to design a controller for such systems. However, to the best of our knowledge, all of the existing event-triggered control strategies were designed without considering zero dynamics in nonstochastic systems. The existence of unstable zero dynamics may restrict the control performance, such as sensitivity, robustness, and stabilization. And the stabilization of zero dynamics in continuous systems is harder to keep than in discrete-time systems [44]. Therefore, it is of great significance to study the stability of zero dynamics for analyzing the internal states of nonstochastic and consensus nonlinear systems.Inspired by above discussions, an event-triggered based adaptive control is proposed for strict-feedback nonlinear systems with unknown parameters and zero dynamics. Compared with the existing results, the main contributions are given as follows:( 1 ) The problem of event-triggered adaptive control is first considered for a class of continuous nonlinear systems with zero dynamics. The nonlinearity part in the system does not have to be globally Lipschitz.( 2 ) Three different triggering conditions are designed, and it is proved that the proposed controllers can make the system asymptotically stable. Then, we prove that every actuation is triggered in finite time, which means the Zeno behavior is successfully avoided.The rest of the paper is organized as follows. Section2 proposes the problem formulation and gives the objective of this note. Then, three different design schemes and main theoretical results are described in Section 3. Section 4 gives the simulation results. The conclusion is summarized in Section 5.


## 2. Problem Formulation

Consider a class of strict-feedback nonlinear systems with unknown parameter and zero dynamics:(1)η˙=f0η+g0ηx1x˙i=xi+1+fix-i+φiTx-iθ,i=1,…,n-1x˙n=ut+fnx-n+φnTx-nθy=x1where η denotes the zero dynamics of the system. f0(η) and g0(η) are known nonlinear functions with respect to η. And, x=[x1,…,xn]T∈Rnx¯i=[x1,…,xi]T∈Ri are the states of the system, while u(t):R1→R1 is the system input. θ∈Rr is a vector which contains all the unknown parameters in the system. fi(·):Ri→R1 and φi(·):Rn→Rr,(i=1,…,n) are known nonlinear functions with γ-th order (γ⩾n+1-i) smooth with respect to x1,…,xi. For convenience, we define fi(·) and φi(·) as fi and φi, respectively.Assumption 1 (see [45]).
For the zero dynamicsη˙=f0(η)+g0(η)x1 in (1), there exists a smooth and definite positive function Vη and a smooth state-feedback control law x1=ϕ0(η), with ϕ0(0)=0, such that(2)∂Vη∂ηf0η+g0ηϕ0η≤-cηVηSince system (1) contains the zero dynamics and unknown parameters, the existing control approaches are not suitable. Therefore, the control objective of this work is to design an adaptive law θ^˙ and control signal u(t) for the system (1), which make the output y asymptotically stable, and all the states in the systems are required to be globally bounded.


## 3. Event-Triggered Control Scheme

Similar to [21], three different adaptive controller design strategies are proposed based on different event-triggered conditions, that is, fixed threshold strategy, relative threshold strategy, and switching threshold strategy. In this research, the considered control system is affected by the zero dynamics which exists widely in many practical applications. For each strategy, we start with system (1) using the backstepping technique which is carried out with the help of coordinates transformation:(3)zi=xi-αi-1where αi-1(i=1,2,3,…,n) denote the virtual control laws and zi(i=1,2,…,n) are the dynamics of the new coordinates. The design process is started as follows.Step 1.
From the coordinate transformationz1=x1-α0, the time derivative of z1 is(4)z˙1=x2+f1+φ1Tθ-α˙0=z2+α1+f1+φ1Tθ-α˙0,where z2=x2-α1.Then, choose a Lyapunov function as(5)V1=Vη+12z12+12θ~TΓ-1θ~where θ~=θ-θ^ is the estimation error, θ^ denotes the estimation of the uncertain constant vector θ, and Γ denotes a positive definite matrix. Therefore, the derivative of V1 is(6)V˙1≤∂Vη∂ηf0η+g0ηx1+z1z2+α1+f1+φ1Tθ-α˙0-θ~TΓ-1θ^˙≤∂Vη∂ηf0η+g0ηϕ0η+∂Vη∂ηg0x1-ϕ0η+z1z2+α1+f1+φ1Tθ-α˙0-θ~TΓ-1θ^˙≤∂Vη∂ηf0η+g0ηϕ0η+z1z2+∂Vη∂ηg0+α1+f1+φ1Tθ-α˙0-θ~TΓ-1θ^˙where α0=ϕ0(η). Next, choose a virtual law as(7)α1=-c1z1-∂Vη∂ηg0-f1-φ1Tθ^+α˙0where c1 is a positive constant. Further, substituting (7) into (6) results in(8)V˙1≤-cηVη-c1z12+z1z2+θ~Tτ1-Γ-1θ^˙where τ1=φ1Tz1 is a tuning function.Step 2.
The derivative ofz2 is given as(9)z˙2=x˙2-α˙1=z3+α2+f2+φ2Tθ-∂α1∂ηη˙-∂α1∂x1x2+f1+φ1Tθ-∂α1∂θ^θ^˙where z3=x3-α2.Then, the Lyapunov function is chosen asV2=V1+1/2z22. The derivative of V2 is(10)V˙2=V˙1+z2z˙2≤V˙1+z2x3+f2+φ2Tθ-α˙1≤-cηVη-c1z12+z1z2+θ~Tτ1-Γ-1θ^˙+z2z3+α2+f2+φ2Tθ-∂α1∂ηη˙-∂α1∂x1x2+f1+φ1Tθ-∂α1∂θ^θ^˙≤-cηVη-c1z12+θ~Tτ1-Γ-1θ^˙+z2z3+z1+α2+f2-∂α1∂ηη˙-∂α1∂x1x2+f1+φ2T-φ1T∂α1∂x1θ-∂α1∂θ^θ^˙≤-cηVη-c1z12+θ~Tτ1-Γ-1θ^˙+z2φ2-φ1∂α1∂x1+z2z3+z1+α2+f2-∂α1∂ηη˙-∂α1∂x1x2+f1+φ2T-φ1T∂α1∂x1θ^-∂α1∂θ^θ^˙Similar to Step1, we choose the virtual law as(11)α2=-z1-c2z2-f2+∂α1∂ηη˙+∂α1∂x1x2+f1-φ2T-φ1T∂α1∂x1θ^+∂α1∂θ^Γτ2-ςθ^where c2 is a positive constant and ς is a positive constant. Further, substituting (11) into (10) results in(12)V˙2≤-cηVη-c1z12-c2z22+z2z3+z2∂α1∂θ^Γτ2-θ^˙-Γςθ^+θ~Tτ2-Γ-1θ^˙where τ2=τ1+(φ2T-∂α1/∂x1φ1T)z2 is a tuning function.Step i   ( i = 3 , … , n ). In this step, we repeat the procedure in a recursive way. The derivative of zi is given as(13)z˙i=x˙i-α˙i-1=xi+1+fi+φiTθ-∂αi-1∂ηη˙-∑j=1i-1∂αi-1∂xjxj+1+fj+φjTθ-∂αi-1∂θ^θ^˙Then, the Lyapunov function is chosen asVi=Vi-1+1/2zi2, and the derivative of Vi is(14)V˙i=V˙i-1+ziz˙i≤V˙i-1+zixi+1+fi+φiTθ-∂αi-1∂ηη˙-∑j=1i-1∂αi-1∂xjxj+1+fj+φjTθ-∂αi-1∂θ^θ^˙≤-cηVη-∑j=1i-1cjzj2+zi-1zi+θ~Tτi-1-Γ-1θ^˙+∑j=2i-1zj∂αj-2∂θ^Γτi-1-θ^˙-ςΓθ^+zixi+1+fi+φiTθ-∂αi-1∂ηη˙-∑j=1i-1∂αi-1∂xjxj+1+fj+φjTθ-∂αi-1∂θ^θ^˙≤-cηVη-∑j=1i-1cjzj2+θ~Tτi-1-Γ-1θ^˙+∑j=2i-1zj∂αj-2∂θ^Γτi-1-θ^˙-ςΓθ^+zixi+1+zi-1+fi-∂αi-1∂ηη˙-∑j=1i-1∂αi-1∂xjxj+1+fj+φiT-∑j=1i-1∂αi-1∂xjφjTθ-∂αi-1∂θ^θ^˙≤-cηVη-∑j=1i-1cjzj2+θ~Tτi-1-Γ-1θ^˙+φi-∑j=1i-1∂αi-1∂xjφjzi+zizi+1+αi+zi-1+fi-∂αi-1∂ηη˙-∑j=1i-1∂αi-1∂xjxj+1+fj+φiT-∑j=1i-1∂αi-1∂xjφjTθ^-∂αi-1∂θ^θ^˙+∑j=2i-1zj∂αj-1∂θ^Γτi-1-θ^˙-ςΓθ^The virtual law is chosen as(15)αi=-zi-1-cizi-fi+∂αi-1∂ηη˙+∑j=1i-1∂αi-1∂xjxj+1+fj+∂αi-1∂θ^Γτi-ςθ^-φiT-∑j=1i-1∂αi-1∂xjφjTθ^+∑j=2i-1zj∂αj-1∂θ^Γφi-∑j=1i-1∂αi-1∂xjφjwhere ci is a positive constant. Next, substituting (15) into (14) results in(16)V˙i≤-cηVη-∑j=1icjzj2+zizi+1+∑j=2izj∂αj-1∂θ^Γτi-θ^˙-ςΓθ^+θ~Tτi-Γ-1θ^˙where τi=τi-1+φiT-∑j=1i-1∂αi-1/∂xjφjTzi is a tuning function.In the last stepn, αn=u, which means the actual control input u appears and it is at our disposal. The objective is now to design an appropriate controller and the triggering event.


### 3.1. Fixed Threshold Strategy

Under this strategy, the adaptive controller is proposed as(17)wt=αn-d¯tanh⁡zndε(18)θ^˙=Γτn-Γςθ^and the triggering event is given as(19)ut=wtk,∀t∈tk,tk+1(20)tk+1=inf⁡t∈R∣et≥d,t1=0where e(t)=w(t)-u(t) denotes the error between the current control signal and last control signal. d¯,d, and ε are all positive constants with d¯≥d.tk, k∈Z+ is the update time of the controller, which means when (20) is triggered, the time will jump to tk+1, and the control value u(tk+1) will be applied to the system. In the period between tk and tk+1, the control value will keep being w(tk) and will not be changed.Now we analyze the stability of the event-triggered controller in system (1), and the design process to avoid the Zeno behavior is also showed. The theorem and proof are proposed as follows.Theorem 2.
Consider a certain of nonlinear closed-loop system (1) and adaptive controller based on the event-triggered scheme (17)-(20) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
The Lyapunov function is chosen as(21)V=Vη+12∑i=1nzi2+12θ~TΓ-1θ~where θ~=θ-θ^. During the execution interval [tk,tk+1), we have wt-ut≤d. Assuming that there exists a continuous time-varying parameter σ(t), which satisfies σ(tk)=0, σ(tk+1)=±1, and σt≤1,∀t∈tk,tk+1, such that w(t)=u(t)+σ(t)d. For convenience, σ(t) and w(t) are defined as σ and w. With this in mind, we have(22)V˙=znz˙n+V˙n-1≤znut+fn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVηBy recalling w(t)=u(t)+σ(t)d, substituting (17) into (22) gives(23)V˙≤znαn-d¯tanh⁡zndε-σd+fn+φnTθ-α˙n-1+θ~Tτn-1-Γ-1θ^˙+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-cηVη-∑i=1n-1cizi2+znzn-1≤znθ~TφnT-∑i=1n-1∂αn-1∂xiφiT+∑i=2n-1zi∂αi-1∂θ^ΓφnT-∑i=1n-1∂αn-1∂xiφiT+zn-d¯tanh⁡zndε-σd-cηVη+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1ncizi2+θ~Tτn-1-Γ-1θ^˙+zn∂αn-1∂θ^Γτn-θ^˙-Γςθ^Notice that τn-τn-1=φnT-∑j=1n-1∂αn-1/∂xjφjTzn. Then, V˙ satisfies(24)V˙≤zn-d¯tanh⁡zndε-σd+θ~Tτn-τn-1+∑i=2n-1zi∂αi-1∂θ^Γτn-τn-1+θ~Tτn-1-Γ-1θ^˙+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1ncizi2-cηVη+zn∂αn-1∂θ^Γτn-θ^˙-Γςθ^≤zn-d¯tanh⁡zndε-σd+θ~Tτn-Γ-1θ^˙+∑i=2nzi∂αi-1∂θ^Γτn-θ^˙-Γςθ^-∑i=1ncizi2-cηVη
Based on the property [46], the hyperbolic tangent function tanh⁡(·) has(25)0≤μ-μtanh⁡με≤0.2785εwhere ε>0 and μ∈R. As above, we design θ^˙ as θ^˙=Γτn-Γςθ^. Based on the Assumption 1, its derivative is given by(26)V˙≤-∑i=1ncizi2-cηVη+ςθ~Tθ^+0.2785εwhere ς is selected based on [47]. Let ξ=min⁡2c1,…,2cn,cη,ς/λmax(Γ-1), where λmax(Γ-1) is the maximum eigenvalue of Γ-1. And with the help of Young’s inequalities in [37], θ~Tθ^≤-1/2θ~Tθ~+1/2θTθ. Finally, let Δ1=0.2785ε+ς/2θTθ. Then, we get(27)V˙≤-ξV+Δ1Based on the LaSalle’s invariance principle in [48] and the result in [46], z1,…,zn, and θ^ are proved to be bounded. Therefore, it can be concluded that the control signal u(t) is also bounded. That is, all the closed-loop signals are globally bounded. Then, assuming that there exists a t∗>0 such that ∀k∈Z+,tk+1-tk≥t∗, and noticing e(t)=w(t)-u(t), the following can be verified(28)ddte=ddte∗e1/2=sign⁡ee˙≤w˙From (17), we get that w˙=α˙n-d¯z˙n/cosh2⁡(znd¯/ε). Since fi(·) and φi(·) are at least (n+1-i)th order smooth functions, w˙ is continuous. Next, since w˙ is a function of x and θ, and all the closed-loop signals are globally bounded, there must be a constant ζ>0 such that w˙≤ζ. Noticing e(tk)=0 and limt→tk+1e(t)=d, there exists a lower bound of t∗ which satisfies t∗≥d/ζ; that is, the Zeno behavior is successfully avoided.


### 3.2. Relative Threshold Strategy

In practical systems, when considering a stabilization problem, the value of the control signal always requires to be considered. In the fixed threshold strategy, no matter how big the control magnitude is, the errore(t) is always bounded by a given constant, which may not be applicable to all practical systems. Based on this consideration, a varying threshold comes to mind. Therefore, the following relative threshold control strategy is proposed:(29)wt=-1+δαntanh⁡znαnε+d1¯tanh⁡znd1¯ε(30)θ^˙=Γτn-Γςθ^and the triggering event is given as(31)ut=wtk,∀t∈tk,tk+1(32)tk+1=inf⁡t∈R∣et≥ρut+d1,t1=0where tk,k∈Z+,ε,ρ(0<ρ<1),δ(0<δ<1),d1 and d¯1>d1/1-ρ are all positive parameters. d1 is added specially to guarantee the Zeno behavior will not happen, which will be elaborated in the following proof. Using the control strategy, it reaches the following theorem.Theorem 3.
Consider a certain of nonlinear closed-loop system (1) and adaptive controller based on the event-triggered scheme (29)–(32) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
From (32), we can get that w(t)=(1+σ1(t)ρ)u(t)+σ2(t)d1 during the interval [tk,tk+1). Similar to the fixed threshold strategy, σ1(t) and σ2(t) are time-varying parameters which satisfy σ1t≤1 and σ2t≤1. Therefore, it reaches that u(t)=w(t)/1+σ1(t)ρ-σ2(t)d1/1+σ1(t)ρ. For convenience, σ1(t), σ2(t), and w(t) are defined as σ1, σ2, and w. Then, we get(33)V˙=znz˙n+V˙n-1≤znw-σ2d11+σ1ρ+fn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVη
Since∀m∈R and ε>0, -mtanh⁡(m/ε)≤0, it can be obtained that znw≤0. Notice that znw/1+σ1ρ≤znw/1+ρ and σ2d1/1+σ1ρ≤d1/1-ρ. Therefore, similar to (24), substituting (29) into (33) gives(34)V˙≤znw1+ρ+znd11-ρ+znfn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVη≤znαn-znαntanh⁡znαnε+znd¯1-znd¯1tanh⁡znd¯1ε+znd11-ρ+znfn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-znαn-znd¯1-cηVη≤znαn+fn+φnTθ-α˙n-1-znd¯1+znd11-ρ+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙+0.557ε-cηVηThen, substituting (15) with i=n into (34) gives(35)V˙≤-∑i=1ncizi2-znd¯1+znd11-ρ+0.557ε+θ~Tτn-Γ-1θ^˙+∑i=2nzi∂αi-1∂θ^Γτn-θ^˙-Γςθ^-cηVη≤-∑i=1ncizi2-cηVη+ςθ~Tθ^+0.557εSimilar to (27), it reaches(36)V˙≤-ξV+Δ2where Δ2=0.557ε+ς/2θTθ. As the proof in Theorem 2, all the closed-loop signals are globally bounded, and the interexecution time t∗≥ρut+d1/k, which means the Zeno behavior is successfully avoided.


### 3.3. Switching Threshold Strategy

In the above, the relative threshold strategy shows its advantages. For example, when the value of control signalu is large, the triggering threshold is large too, which ensures that the system can get longer update intervals, while when the u is close to zero, the system can get precise control. Nevertheless, once the value of control signal u is excessively big, the system will get excessively large errors of control signal. Therefore, the control signal will jump suddenly when an event is triggered, which will give the system a large impulse and affect the performance of the system. Hence, with these considerations, we give the switching threshold strategy as follows:(37)ut=wtk,∀t∈tk,tk+1(38)tk+1=inf⁡t∈R∣et≥ρut+d1,ut<Dinf⁡t∈R∣et≥d,ut≥D,where D is a positive constant parameter designed by users and ρ,d, and d1 are same parameters which are designed before. This strategy is a combination of first two strategies. From (38), it can be seen that when the value of control signal ut is less than D, the relative strategy is applied; therefore the system will obtain a more accurate control. While when the value of ut is high, the control method switches to the fixed threshold strategy so that the error e(t) will ensure the system performance. Apparently, the advantages of the fixed threshold strategy and the relative strategy are brought to this strategy.In this strategy, it can be obtained that(39)e¯=sup⁡et≤max⁡ρD+d1,d∀t∈tk,tk+1Combining the switching threshold strategy above and (39), the following results are stated.Theorem 4.
Consider a certain of nonlinear closed-loop system (1) under Assumption 1 and a switching adaptive controller between the fixed threshold and relative threshold with a user-designed parameter D with d¯≥e¯. The system is globally stable; that is, all the signals are bounded. And the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
Since the same control law of the first two strategies are adopted by the switching threshold strategy, the system is also globally stable, which means all the signals are bounded. And by lettingw˙≤, the interexecution t∗ satisfies t∗≥max⁡d1,d/k. That completes the proof.


## 3.1. Fixed Threshold Strategy

Under this strategy, the adaptive controller is proposed as(17)wt=αn-d¯tanh⁡zndε(18)θ^˙=Γτn-Γςθ^and the triggering event is given as(19)ut=wtk,∀t∈tk,tk+1(20)tk+1=inf⁡t∈R∣et≥d,t1=0where e(t)=w(t)-u(t) denotes the error between the current control signal and last control signal. d¯,d, and ε are all positive constants with d¯≥d.tk, k∈Z+ is the update time of the controller, which means when (20) is triggered, the time will jump to tk+1, and the control value u(tk+1) will be applied to the system. In the period between tk and tk+1, the control value will keep being w(tk) and will not be changed.Now we analyze the stability of the event-triggered controller in system (1), and the design process to avoid the Zeno behavior is also showed. The theorem and proof are proposed as follows.Theorem 2.
Consider a certain of nonlinear closed-loop system (1) and adaptive controller based on the event-triggered scheme (17)-(20) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
The Lyapunov function is chosen as(21)V=Vη+12∑i=1nzi2+12θ~TΓ-1θ~where θ~=θ-θ^. During the execution interval [tk,tk+1), we have wt-ut≤d. Assuming that there exists a continuous time-varying parameter σ(t), which satisfies σ(tk)=0, σ(tk+1)=±1, and σt≤1,∀t∈tk,tk+1, such that w(t)=u(t)+σ(t)d. For convenience, σ(t) and w(t) are defined as σ and w. With this in mind, we have(22)V˙=znz˙n+V˙n-1≤znut+fn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVηBy recalling w(t)=u(t)+σ(t)d, substituting (17) into (22) gives(23)V˙≤znαn-d¯tanh⁡zndε-σd+fn+φnTθ-α˙n-1+θ~Tτn-1-Γ-1θ^˙+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-cηVη-∑i=1n-1cizi2+znzn-1≤znθ~TφnT-∑i=1n-1∂αn-1∂xiφiT+∑i=2n-1zi∂αi-1∂θ^ΓφnT-∑i=1n-1∂αn-1∂xiφiT+zn-d¯tanh⁡zndε-σd-cηVη+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1ncizi2+θ~Tτn-1-Γ-1θ^˙+zn∂αn-1∂θ^Γτn-θ^˙-Γςθ^Notice that τn-τn-1=φnT-∑j=1n-1∂αn-1/∂xjφjTzn. Then, V˙ satisfies(24)V˙≤zn-d¯tanh⁡zndε-σd+θ~Tτn-τn-1+∑i=2n-1zi∂αi-1∂θ^Γτn-τn-1+θ~Tτn-1-Γ-1θ^˙+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1ncizi2-cηVη+zn∂αn-1∂θ^Γτn-θ^˙-Γςθ^≤zn-d¯tanh⁡zndε-σd+θ~Tτn-Γ-1θ^˙+∑i=2nzi∂αi-1∂θ^Γτn-θ^˙-Γςθ^-∑i=1ncizi2-cηVη
Based on the property [46], the hyperbolic tangent function tanh⁡(·) has(25)0≤μ-μtanh⁡με≤0.2785εwhere ε>0 and μ∈R. As above, we design θ^˙ as θ^˙=Γτn-Γςθ^. Based on the Assumption 1, its derivative is given by(26)V˙≤-∑i=1ncizi2-cηVη+ςθ~Tθ^+0.2785εwhere ς is selected based on [47]. Let ξ=min⁡2c1,…,2cn,cη,ς/λmax(Γ-1), where λmax(Γ-1) is the maximum eigenvalue of Γ-1. And with the help of Young’s inequalities in [37], θ~Tθ^≤-1/2θ~Tθ~+1/2θTθ. Finally, let Δ1=0.2785ε+ς/2θTθ. Then, we get(27)V˙≤-ξV+Δ1Based on the LaSalle’s invariance principle in [48] and the result in [46], z1,…,zn, and θ^ are proved to be bounded. Therefore, it can be concluded that the control signal u(t) is also bounded. That is, all the closed-loop signals are globally bounded. Then, assuming that there exists a t∗>0 such that ∀k∈Z+,tk+1-tk≥t∗, and noticing e(t)=w(t)-u(t), the following can be verified(28)ddte=ddte∗e1/2=sign⁡ee˙≤w˙From (17), we get that w˙=α˙n-d¯z˙n/cosh2⁡(znd¯/ε). Since fi(·) and φi(·) are at least (n+1-i)th order smooth functions, w˙ is continuous. Next, since w˙ is a function of x and θ, and all the closed-loop signals are globally bounded, there must be a constant ζ>0 such that w˙≤ζ. Noticing e(tk)=0 and limt→tk+1e(t)=d, there exists a lower bound of t∗ which satisfies t∗≥d/ζ; that is, the Zeno behavior is successfully avoided.


## 3.2. Relative Threshold Strategy

In practical systems, when considering a stabilization problem, the value of the control signal always requires to be considered. In the fixed threshold strategy, no matter how big the control magnitude is, the errore(t) is always bounded by a given constant, which may not be applicable to all practical systems. Based on this consideration, a varying threshold comes to mind. Therefore, the following relative threshold control strategy is proposed:(29)wt=-1+δαntanh⁡znαnε+d1¯tanh⁡znd1¯ε(30)θ^˙=Γτn-Γςθ^and the triggering event is given as(31)ut=wtk,∀t∈tk,tk+1(32)tk+1=inf⁡t∈R∣et≥ρut+d1,t1=0where tk,k∈Z+,ε,ρ(0<ρ<1),δ(0<δ<1),d1 and d¯1>d1/1-ρ are all positive parameters. d1 is added specially to guarantee the Zeno behavior will not happen, which will be elaborated in the following proof. Using the control strategy, it reaches the following theorem.Theorem 3.
Consider a certain of nonlinear closed-loop system (1) and adaptive controller based on the event-triggered scheme (29)–(32) under Assumption 1. The system is globally stable; that is, all the signals are bounded, and the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
From (32), we can get that w(t)=(1+σ1(t)ρ)u(t)+σ2(t)d1 during the interval [tk,tk+1). Similar to the fixed threshold strategy, σ1(t) and σ2(t) are time-varying parameters which satisfy σ1t≤1 and σ2t≤1. Therefore, it reaches that u(t)=w(t)/1+σ1(t)ρ-σ2(t)d1/1+σ1(t)ρ. For convenience, σ1(t), σ2(t), and w(t) are defined as σ1, σ2, and w. Then, we get(33)V˙=znz˙n+V˙n-1≤znw-σ2d11+σ1ρ+fn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVη
Since∀m∈R and ε>0, -mtanh⁡(m/ε)≤0, it can be obtained that znw≤0. Notice that znw/1+σ1ρ≤znw/1+ρ and σ2d1/1+σ1ρ≤d1/1-ρ. Therefore, similar to (24), substituting (29) into (33) gives(34)V˙≤znw1+ρ+znd11-ρ+znfn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-cηVη≤znαn-znαntanh⁡znαnε+znd¯1-znd¯1tanh⁡znd¯1ε+znd11-ρ+znfn+φnTθ-α˙n-1+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙-znαn-znd¯1-cηVη≤znαn+fn+φnTθ-α˙n-1-znd¯1+znd11-ρ+∑i=2n-1zi∂αi-1∂θ^Γτn-1-θ^˙-Γςθ^-∑i=1n-1cizi2+znzn-1+θ~Tτn-1-Γ-1θ^˙+0.557ε-cηVηThen, substituting (15) with i=n into (34) gives(35)V˙≤-∑i=1ncizi2-znd¯1+znd11-ρ+0.557ε+θ~Tτn-Γ-1θ^˙+∑i=2nzi∂αi-1∂θ^Γτn-θ^˙-Γςθ^-cηVη≤-∑i=1ncizi2-cηVη+ςθ~Tθ^+0.557εSimilar to (27), it reaches(36)V˙≤-ξV+Δ2where Δ2=0.557ε+ς/2θTθ. As the proof in Theorem 2, all the closed-loop signals are globally bounded, and the interexecution time t∗≥ρut+d1/k, which means the Zeno behavior is successfully avoided.


## 3.3. Switching Threshold Strategy

In the above, the relative threshold strategy shows its advantages. For example, when the value of control signalu is large, the triggering threshold is large too, which ensures that the system can get longer update intervals, while when the u is close to zero, the system can get precise control. Nevertheless, once the value of control signal u is excessively big, the system will get excessively large errors of control signal. Therefore, the control signal will jump suddenly when an event is triggered, which will give the system a large impulse and affect the performance of the system. Hence, with these considerations, we give the switching threshold strategy as follows:(37)ut=wtk,∀t∈tk,tk+1(38)tk+1=inf⁡t∈R∣et≥ρut+d1,ut<Dinf⁡t∈R∣et≥d,ut≥D,where D is a positive constant parameter designed by users and ρ,d, and d1 are same parameters which are designed before. This strategy is a combination of first two strategies. From (38), it can be seen that when the value of control signal ut is less than D, the relative strategy is applied; therefore the system will obtain a more accurate control. While when the value of ut is high, the control method switches to the fixed threshold strategy so that the error e(t) will ensure the system performance. Apparently, the advantages of the fixed threshold strategy and the relative strategy are brought to this strategy.In this strategy, it can be obtained that(39)e¯=sup⁡et≤max⁡ρD+d1,d∀t∈tk,tk+1Combining the switching threshold strategy above and (39), the following results are stated.Theorem 4.
Consider a certain of nonlinear closed-loop system (1) under Assumption 1 and a switching adaptive controller between the fixed threshold and relative threshold with a user-designed parameter D with d¯≥e¯. The system is globally stable; that is, all the signals are bounded. And the system can be adjusted by selecting proper parameters. In addition, there must be a time t∗>0 such that t∗ is greater than or equal to the execution period tk+1-tk, k∈Z+, which means the Zeno behavior is successfully avoided.Proof.
Since the same control law of the first two strategies are adopted by the switching threshold strategy, the system is also globally stable, which means all the signals are bounded. And by lettingw˙≤, the interexecution t∗ satisfies t∗≥max⁡d1,d/k. That completes the proof.


## 4. Simulation Results

Example 1.
In this section, based on the system given in [49], we combine zero dynamics with it. A stabilization case is illustrated with three proposed event-triggered strategies and time-triggered strategy, and each strategy’s performance is compared. Consider the following systems:(40)η˙=f0η+g0ηx1x˙1=x2+θx12x˙2=uy=x1where θ denotes an unknown parameter. Our control goal is now to make the system output y=x1(t) asymptotically back to zero. We use the above four strategies to balance the performance of the system and the times of triggering events. The design constants are chosen as follows: (1) Fixed threshold strategy: d=3,d¯=4. (2) Relative threshold strategy: ρ=0.15,d1=0.3,d¯1=2. (3) Switching threshold strategy: D=30. (4) Time-triggered strategy: interexecution time: 0.05s. The other parameters are chosen as x1(0)=0,x2(0)=0,θ^(0)=0,θ=5,c1=7,c2=10,Γ=1,ε=10. And since f0(η) and g0(η) are known, we define that f0(η)=-η,g0(η)=1. The Lyapunov function is chosen as Vη=1/2η2; therefore we get(41)V˙η=ηη˙≤η-η+x1≤η-η+z1+α0Letα0=-c0η. Therefore(42)V˙η=-1+c0η2+ηz1and c0 is chosen as c0=50.Figure1 shows the stabilization performance of the output signal y(t) in three event-triggered strategies and time-triggered strategy, and the control signals u(t) and w(t) are shown in Figure 2. The times of triggering events is shown in Table 1, while state signal x2 and the parameter estimate θ^ are, respectively, shown in Figures 3 and 4. It can be seen that the fixed threshold strategy has the best stabilization performance due to its relatively small threshold, but it has the most triggering events. On the contrary, the relative threshold strategy’s performance is not relatively good, while it has the fewest triggering events. And the switching threshold strategy gives a flexibility between the performance of system and times of triggering events. For time-triggered strategy, though it has the best performance at last, it also has the most triggering times, which may result in unnecessary high workloads in practical systems. Besides, the system does not perform well at the beginning.Table 1
Times of triggering events.
Fixed strategy Relative strategy Switching strategy Time-triggering strategy 81 46 70 200Figure 1
Outputy(t).Figure 2
Control signal.Figure 3
State signalx2(t).Figure 4
Parameter estimateθ^.Example 2.
Consider a higher order networked-control system based on [45] with zero dynamics(43)η˙=f0η+g0ηx1x˙1=x2+x12x˙2=θx3x˙3=x1+x2+uy=x1where θ denotes an unknown parameter. The above three event-triggered strategies are applied to the system. The design constants are chosen as follows: (1) Fixed threshold strategy: d=10,d¯=11. (2) Relative threshold strategy: ρ=0.15,d1=5,d¯1=8. (3) Switching threshold strategy: D=60. The other parameters are chosen as x1(0)=0,x2(0)=-1,x3(0)=5,θ^(0)=0,θ=1,c0=50,c1=10,c2=20,Γ=1,ε=1. And f0(η) and g0(η) are defined as f0(η)=η2,g0(η)=η.Figure5 shows the stabilization performance of the output signal y(t) in three event-triggered strategies, and the control signals u(t) and w(t) are shown in Figure 6. The time of triggering events is shown in Table 2, while state signals x2 and x3 and the parameter estimate θ^ are, respectively, shown in Figures 7, 8, and 9. In this simulation example, the switching threshold strategy provides the smallest triggering times, but its performance is not better than the other strategies, while the fixed threshold strategy provides the best performance with the most triggering events. And the relative threshold strategy balances the performance and times of triggering events. It can be seen that the parameters can be designed by users to find the most suitable strategy for different practical systems under different situations.Table 2
Times of triggering events.
Fixed strategy Relative strategy Switching strategy 133 128 82Figure 5
Outputy(t).Figure 6
Control signal.Figure 7
State signalx2(t).Figure 8
State signalx3(t).Figure 9
Parameter estimateθ^.


## 5. Conclusion

This paper has considered an event-triggered adaptive backstepping control for strict-feedback nonlinear systems with zero dynamics. An adaptive control method is utilized to overcome the unknown system parameters, by which all the closed-loop signals are guaranteed to be uniformly bounded. Besides, to overcome the need of ISS assumption, an adaptive controller and event-triggered technique are designed at the same time. And three different event-triggered control strategies are also given. It is shown from the simulation results that all the strategies guarantee that the stability error approaches to a small neighborhood of the origin.



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*Source: 7890968-2019-10-10.xml*