# Adaptive Event-Triggered Finite-Time Tracking of Output-Constrained High-Order Nonlinear Systems with Time-Varying Powers

Authors: Fan Liu; You Wu
Journal: Complexity
 (2022)

---

## Abstract

This paper studies the adaptive event-triggered finite-time tracking of output-constrained high-order nonlinear systems with time-varying powers. Due to the presence of multiple unknown powers and the consideration of event-triggered control, all the existing control methods of output-constrained nonlinear systems are inapplicable. By introducing nonlinear mappings, finite-time performance functions, and low-power and high-power terms into adding a power integrator technique and the relative threshold strategy, an adaptive state-feedback controller is designed to eliminate the effects caused by the output constraint and time-varying powers. It is proved that all the closed-loop signals are bounded, the asymmetric time-varying output constraint is not violated, and the tracking error converges to a prescribed arbitrarily small region around zero in a preassigned finite time. Furthermore, the Zeno phenomenon can be avoided. Two simulation examples demonstrate the effectiveness of this control scheme.


---

## Body

## 1. Introduction

Due to hardware limitations, performance requirements, or safety specifications, state/output constraints are always involved in many nonlinear systems. For example, the speed and acceleration of motor vehicles should be restricted to prevent possible accidents. In the process of operation, the violation of state/output constraints may decrease system performance or even make the system unstable. Therefore, the research of constrained control for nonlinear systems becomes extremely important and urgent. In the past decade, barrier Lyapunov function (BLF) and nonlinear mapping (NM), which were firstly put forward in <u>[1, 2]</u>, respectively, have become two valid tools for handling state/output constraints of nonlinear systems. The value of BLF/NM will tend to infinity when the state/output closes to some constraints. As long as BLF/NM is bounded, the state/output does not exceed these constraints. Compared with the BLF-based constrained control method, NM-based counterpart can directly deal with the original state/output constraints, and thus, the undesirable feasibility conditions in [3] can be removed. Based on these two methods, fruitful results were obtained; see [4–19] and other papers.Compared with traditional feedback linearized nonlinear systems, due to the presence of higher powers, the Jacobian linearization of more general high-order nonlinear systems (also known asp-normal nonlinear systems) may be neither feedback linearized nor controllable. Such inherent obstacles make the control design more challenging and difficult. Fortunately, with the development of a power integrator technique [20], these obstacles can be delicately overcome. By combining such a technique with different types of BLF/NM, some interesting results on the stabilization and practical tracking of constrained high-order nonlinear systems have been obtained in [21–28]. However, the powers of considered systems are constants and precisely known.Just like the boiler-turbine unit in [29] and the underactuated, weakly coupled, and unstable mechanical system in [30], due to various operating conditions and the potential aging of the hardening spring, the powers of these two systems are unknown and variable. In view of these applications, the study of nonlinear systems with unknown powers is of vital importance. Recently, the authors in [31] studied nonlinear systems with unknown constant powers, and the stabilization of nonlinear systems with a single unknown time-varying power was discussed in [32]. In the presence of multiple unknown time-varying powers, Cui, Xie, and Lie <u>[33, 34]</u> deeply investigated finite-time stabilization and adaptive stabilization. Subsequently, [35–40] proposed the constrained control methods for nonlinear systems with multiple unknown powers.Different from the traditional continuous sampling control in all the aforementioned works, the event-triggered control mechanism is a useful data scheduling method in networked control systems and only requires control signals to be discontinuously sent to the actuator rather than periodically sampled. Such a feature can effectively diminish the communication overload and computational cost. In view of these benefits, a number of efforts have been made for constructing the event-triggered controllers. Particularly, Xing et al. [41] proposed three original adaptive event-triggered control strategies for strict-feedback nonlinear systems without state/output constraints. Further in-depth studies on the event-triggered control were discussed in some latest results [42–44]. Specially, Zhang et al. and Liu and Li <u>[43, 44]</u> constructed different kinds of event-triggered controllers for two state-/output-constrained nonlinear systems. However, they are unavailable for high-order nonlinear systems with unknown powers. Besides, in view of faster convergence rates, higher accuracies, and better disturbance rejection properties [45], the finite-time stability needs to be further studied.Based on these discussions, an interesting problem arises: is it possible to design an adaptive event-triggered finite-time tracking controller for output-constrained high-order nonlinear systems with time-varying powers?In this paper, we will substantially solve this problem. Main contributions and difficulties are emphasized as follows:(1)
This is the first paper to study the event-triggered control of output-constrained nonlinear systems with time-varying powers. Due to the presence of multiple unknown powers and the consideration of event-triggered control, more complex nonlinear terms will inevitably produce in control design, and the constrained controllers in the existing results are all inapplicable. To overcome these essential difficulties, some NMs are first adopted to convert the original output-constrained system into a new one. Then, by introducing sign function, finite-time performance functions, and low-power and high-power terms into adding a power integrator technique and combining the relative threshold strategy, an innovative adaptive event-triggered state-feedback controller is designed to guarantee the performances of the closed-loop system. Furthermore, the Zeno phenomenon does not occur.(2)
Compared with <u>[1–19, 22–28, 35–40, 43, 44]</u> on nonlinear constrained control, the considered system is more general since it possesses parametric uncertainties, multiple unknown time-varying powers, and asymmetric time-varying output constraint simultaneously.(3)
Compared with <u>[20–28, 31–40]</u> on the continuous sampling control of nonlinear systems, more attractive adaptive event-triggered control is firstly adopted to reduce redundant data transmissions and consume less communication resources in the constrained control framework; see Example 2 for the detailed discussion.The rest of this paper is organized as follows. Section2 gives a motivation example and preliminaries. Section 3 presents the main result of this paper, following two simulation examples in Section 4. Section 5 concludes this paper.Notations:ℤ+, ℝ+, ℝ, and ℝn denote the set of all positive integers, the set of all nonnegative real numbers, the set of real numbers, and the real n-dimensional space, respectively. C1 is the set of all functions with continuous partial derivatives. For a real vector x=x1,…,xn⊤∈ℝn, the norm ‖x‖ is defined by x=∑i=1nxi21/2. For any s∈ℝ, sgn(s) denotes its sign function, which satisfies sgn(s) = 1 if s > 0, sgn(s) = 0 if s = 0, and sgn(s) = −1 if s < 0, and ⌈s⌉p ≜ sgn(s)|s|p for positive p. For a C1 function Vx:ℝn⟶ℝ+, it is positive definite if V(x) ≥ 0, and V(x) = 0 if and only if x = 0. The arguments of functions are sometimes simplified, for example, a function f(x(t)) can be written as f(x), f(⋅), or f.


## 2. Motivation Example and Preliminaries

### 2.1. Motivation Example

Consider the underactuated, weakly coupled, and unstable system in Figure1 [30]. This mechanical system contains a mass m1 on a horizontal smooth surface and an inverted pendulum m2 supported by a massless rod. The mass m1 is interconnected to the wall by a linear spring and to the inverted pendulum by a nonlinear spring. Let x be the displacement of mass m1 and θ be the angle of the pendulum from the vertical such that at x = 0 and θ = 0. The springs are unstretched. A control force u acts on mass m1. The equation of motion for this system is described as(1)θ..=glsinθ+ksm2lx−lsinθptcosθ,x¨=−km1x−ksm1x−lsinθpt+um1,where l is the length of the rod, g is the acceleration of gravity, k and ks are spring coefficients, and y = x − l sin(θ) in Figure 1. Assume that m1, m2, l, and ks are unknown constant parameters which belong to a known interval c¯,c¯ with c¯≥c¯>0. Suppose that θ is small; equation (1) becomes(2)θ..=glθ+ksm2lx−lθpt,x¨=−km1x−ksm1x−lθpt+um1.Figure 1
The underactuated, weakly coupled, and unstable system.Then, by using the change of coordinates(3)x1=θ,x2=θ.,x3=x−lθ,x4=x˙3,system (2) becomes the following system:(4)x˙1=x2,x˙2=ksm2lx3pt+glx1,x˙3=x4,x˙4=um1−km1lx1+x3+ksm1+ksm2x3pt−gx1.The unknown time-varying powerp(t) reflects the potential aging of the hardening spring, and the output y = x1 of system (4) needs to be constrained in (−π/2, π/2) to fit actual demands, which clearly illustrates the research motivation of output-constrained nonlinear systems with unknown powers.The corresponding networked control system including the actuator, plant (4), sensor, controller, and two communication networks is depicted in Figure 2.Figure 2
Sketch of the networked control system for plant (4).


### 2.2. Preliminaries

Lemmas1–6 are used to enlarge inequalities in the following state-feedback control design and analysis.Lemma 1 (see [46]).
Letr1(t), r2(t), and α(x, y) be some positive continuous real-valued functions. For any ,(5)xr1tyr2t≤r1tr1t+r2tαx,yxr1t+r2t+r2tr1t+r2tα−r1t/r2tx,yyr1t+r2t.Lemma 2 (see [46]).
Letr(t) ≥ 1 be a continuous real-valued function. For any xi∈ℝ, i = 1, …, n,(6)∑i=1nxirt≤∑i=1nxirt≤nrt−1∑i=1nxirt.Lemma 3 (see [46]).
Ifr(t) is a continuous real-valued function and satisfies r¯t≤rt≤r¯t, where r¯t and r¯t are positive real-valued functions, for any x∈ℝ,(7)xrt≤xr¯t+xr¯t.Lemma 4 (see [47]).
Letr(t) ≥ 1 be a continuous real-valued function. For any ,(8)xrt−yrt≤rt1+2rt−2x−yrt+x−y∥yrt−1,where |y|r(t)−1 ≜ 0 if y = 0 and r(t) = 1.Lemma 5 (see [48]).
For given continuous functionf(x, y) with x∈ℝm and y∈ℝn, there exist smooth functions a(x) ≥ 0, b(y) ≥ 0, c(x) ≥ 1, and d(y) ≥ 1 such that(9)fx,y≤ax+by,fx,y≤cxdy.Lemma 6 (see [49]).
For anyɛ > 0 and ν∈ℝ, there hold(10)0≤ν−νtanhνε≤0.2875ε,−νtanhνε≤0.


## 2.1. Motivation Example

Consider the underactuated, weakly coupled, and unstable system in Figure1 [30]. This mechanical system contains a mass m1 on a horizontal smooth surface and an inverted pendulum m2 supported by a massless rod. The mass m1 is interconnected to the wall by a linear spring and to the inverted pendulum by a nonlinear spring. Let x be the displacement of mass m1 and θ be the angle of the pendulum from the vertical such that at x = 0 and θ = 0. The springs are unstretched. A control force u acts on mass m1. The equation of motion for this system is described as(1)θ..=glsinθ+ksm2lx−lsinθptcosθ,x¨=−km1x−ksm1x−lsinθpt+um1,where l is the length of the rod, g is the acceleration of gravity, k and ks are spring coefficients, and y = x − l sin(θ) in Figure 1. Assume that m1, m2, l, and ks are unknown constant parameters which belong to a known interval c¯,c¯ with c¯≥c¯>0. Suppose that θ is small; equation (1) becomes(2)θ..=glθ+ksm2lx−lθpt,x¨=−km1x−ksm1x−lθpt+um1.Figure 1
The underactuated, weakly coupled, and unstable system.Then, by using the change of coordinates(3)x1=θ,x2=θ.,x3=x−lθ,x4=x˙3,system (2) becomes the following system:(4)x˙1=x2,x˙2=ksm2lx3pt+glx1,x˙3=x4,x˙4=um1−km1lx1+x3+ksm1+ksm2x3pt−gx1.The unknown time-varying powerp(t) reflects the potential aging of the hardening spring, and the output y = x1 of system (4) needs to be constrained in (−π/2, π/2) to fit actual demands, which clearly illustrates the research motivation of output-constrained nonlinear systems with unknown powers.The corresponding networked control system including the actuator, plant (4), sensor, controller, and two communication networks is depicted in Figure 2.Figure 2
Sketch of the networked control system for plant (4).


## 2.2. Preliminaries

Lemmas1–6 are used to enlarge inequalities in the following state-feedback control design and analysis.Lemma 1 (see [46]).
Letr1(t), r2(t), and α(x, y) be some positive continuous real-valued functions. For any ,(5)xr1tyr2t≤r1tr1t+r2tαx,yxr1t+r2t+r2tr1t+r2tα−r1t/r2tx,yyr1t+r2t.Lemma 2 (see [46]).
Letr(t) ≥ 1 be a continuous real-valued function. For any xi∈ℝ, i = 1, …, n,(6)∑i=1nxirt≤∑i=1nxirt≤nrt−1∑i=1nxirt.Lemma 3 (see [46]).
Ifr(t) is a continuous real-valued function and satisfies r¯t≤rt≤r¯t, where r¯t and r¯t are positive real-valued functions, for any x∈ℝ,(7)xrt≤xr¯t+xr¯t.Lemma 4 (see [47]).
Letr(t) ≥ 1 be a continuous real-valued function. For any ,(8)xrt−yrt≤rt1+2rt−2x−yrt+x−y∥yrt−1,where |y|r(t)−1 ≜ 0 if y = 0 and r(t) = 1.Lemma 5 (see [48]).
For given continuous functionf(x, y) with x∈ℝm and y∈ℝn, there exist smooth functions a(x) ≥ 0, b(y) ≥ 0, c(x) ≥ 1, and d(y) ≥ 1 such that(9)fx,y≤ax+by,fx,y≤cxdy.Lemma 6 (see [49]).
For anyɛ > 0 and ν∈ℝ, there hold(10)0≤ν−νtanhνε≤0.2875ε,−νtanhνε≤0.


## 3. Main Result

### 3.1. Problem Formulation and Assumptions

Inspired by (4), we consider more general nonlinear systems(11)x˙i=gix¯ixi+1pit+fix¯i,d,i=1,…,n−1,x˙n=gnxupnt+fnx,u,d,y=x1,with the asymmetric time-varying output constraint(12)ydt−F1t<yt<ydt+F2t,∀t≥0,where x¯i=x1,…,xi⊤∈ℝi, i = 1, …, n, x = xn, u∈ℝ, and y∈ℝ are measurable states, control input, and output, respectively, d∈ℝr is an unknown constant vector, system powers pit:ℝ+⟶ℝ+, i = 1, …, n, are unknown time-varying functions with pn(t) = 1, fi:ℝi×ℝr⟶ℝ, i = 1, …, n−1, fn:ℝn×ℝ×ℝr⟶ℝ are unknown locally Lipschitz continuous functions, gi:ℝi⟶ℝ, i = 1, …, n, are known C1 function, and ydt:ℝ+⟶ℝ denotes the desired trajectory; finite-time performance functions F1t:ℝ+⟶ℝ+ and F2t:ℝ+⟶ℝ+ are defined as(13)Fjt=Fj0−ttse1−ts/ts−t+Fjs,t∈0,ts,Fjs,t∈ts,∞,j=1,2,with Fj0 > 5/4 and Fjs > 0 being design parameters. System (11) is called as the high-order nonlinear system if there exists at least one pi(t) > 1. From Remark 2 in [50], it is easy to prove that Fj(t) > 0 is decreasing and smooth, and there exist known positive constants ρj1, ρj2, and ρj3 such that |Fj(t)| ≤ ρj1, F˙jt≤ρj2, and F¨jt≤ρj3, ∀t ≥ 0. It is worth mentioning that (13) plays a crucial role on the achievement of finite-time tracking, and the preassigned finite time ts > 0 can be chosen prior to control implementation, which is independent of initial conditions and design parameters.The control objective is to design aC1 adaptive event-triggered state-feedback controller(14)u=ux,yd,Θ∧,F,F˙,Θ∧.i=ζx¯i,yd,Θ∧¯i,F,F˙,where Θ∧¯i=Θ∧1,…,Θ∧i⊤, Θ∧i∈ℝ, i = 1, …, n, are auxiliary variables used to handle parametric uncertainties, Θ∧=Θ∧¯n, F=F1,F2⊤, and ζ(⋅) is a continuous function such that the following properties hold:(O1)
All the closed-loop signals are bounded on [0,∞)(O2)
Asymmetric time-varying output constraint (12) is not violated(O3)
The practical finite-time tracking can be fulfilled, i.e., for any small constantsF1s > 0 and F2s > 0, there exists a preassigned finite time ts > 0 such that the tracking error e(t) = y(t)−yd(t) satisfies et∈Ωe≜et∈ℝ:−F1s≤et≤F2s, ∀t ≥ tsRemark 1.
Although asymptotic tracking has better steady-state performance, practical tracking has a larger applicable scope than asymptotic one since it requires less restrictions on the desired trajectoryyd(t) and the considered system. Actually, no continuous controller exists to achieve the global or even local asymptotic tracking for the considered system (5). This can be verified by a two-dimensional high-order nonlinear system in [51]. Hence, this paper focuses on the practical tracking problem rather than the asymptotic case.
To achieve (O1)–(O3), the following assumptions are needed.Assumption 1.
Unknown time-varying powerspi(t), i = 1, …, n, satisfy p ≥ pi(t) ≥ 1, where p is a known constant.Assumption 2.
Fori = 1, …, n, there exist unknown constants ϑi > 0 and known C1 nonnegative functions f¯ix¯i such that(15)fi≤ϑif¯ix¯i,i=1,…,n.Assumption 3.
There exists an unknown constantM > 0 such that ydt+y˙dt≤M, ∀t ≥ 0.Remark 2.
Assumption1 means that p1(t), ⋯, pn(t) have a common upper bound p. In addition, the restrictive relationship p1(t) ≥ p2(t) ≥ ⋯ ≥ pn(t) is no longer needed in this paper.
Inequality (15) in Assumption 2 indicates the restriction on system nonlinearities f1, …, fn, which allow the existence of parametric uncertainties ϑ1, …, ϑn and do not necessarily vanish in the origin. Hence, f1, …, fn require less prior knowledge to implement the output tracking.
Assumption3 describes the boundedness of yd(t) and y˙dt. Compared with the existing works on state-/output-constrained tracking control, no any bounded restriction is imposed on high-order derivatives of yd(t) in this paper.


### 3.2. System Transformation

To begin with, we infer from (12) that the constraint on y = x1 is equivalent to −F1(t) < e(t) < F2(t), ∀t ≥ 0. To ensure output constraint (12) and then achieve finite-time tracking, inspired by [2] and (13), we introduce NMs(16)ξ1=T1e=eh1e,ξi=Tixi=xihi,i=2,…,n,where h1(e) = (F1 + e) (F2−e), hi = 1, i = 2, …, n. It is easy to show that T1(e) is strictly increasing, smooth, and diffeomorphic with respect to e. Besides, ξ1⟶∞ when e⟶−F1 and e⟶F2. In other words, (6) is not violated when the proper initial condition and the boundedness of the transformed state ξ1 can be guaranteed in the closed-loop system. From (5) and (10), we obtain a new transformed system:(17)ξ˙1=H1x1,yd,Fg1ξ2p1t+f1−y˙d+G1x1,yd,F,F˙,ξ˙i=giξi+1pit+fi,i=2,…,n−1,ξ˙n=gnu+fn,where H1=F1F2+e2/h12 and G1=−F˙1F2+F1F˙2+F˙2−F˙1ee/h12.


### 3.3. State-Feedback Control Design

In view of, we letX0 = β0 = α0 = 0 and specify the coordinate transformations(18)zi=ξi−αi−1Xi−1,αiXi=−βiXizi+zip,i=1,…,n,where X1=x1,yd,Θ∧1,F,F˙, Xi=x¯i,yd,Θ∧¯i,F,F˙, i = 2, …, n, and the estimations Θ∧i≥0 of are generated by the adaptive laws:(19)Θ∧˙i=τiXi−σiΘ∧i,Θ∧i0≥0,i=1,…,n,with σi > 0 being design parameters, and nonnegative C1 functions βi(⋅) and τi(⋅) will be determined later.Remark 3.
Compared with [34–37, 39, 40] on the adaptive control of nonlinear systems with time-varying powers, we relax the requirement Θ∧i≥1 to Θ∧i≥0 in (13). Moreover, the simultaneous introduction of low-power term zi and high-power term zip in (12) is the key to deal with unknown time-varying powers p1(t), …, pn(t).
Based on (11)–(13), we provide the detailed procedure of control design in a recursive manner.Step 1.
Take the Lyapunov functionV1=z12/2+Θ˜12/2 with Θ˜1=Θ1−Θ∧1. By (11)–(13), we have(20)V˙1=z1H1g1ξ2p1t+f1−y˙d+z1G1−Θ˜1Θ∧˙1=z1H1g1α1p1t+z1H1g1ξ2p1t−α1p1t+z1H1f1−z1H1y˙d+z1G1−Θ˜1τ1+σ1Θ˜1Θ∧1.
For the sake of consistency, we specifyH2 = ⋯ = Hn = 1. According to Assumptions 1–3, Lemmas 1–3, and Θ1=max1,ϑ11+p,M1+p, one gets(21)z1H1f1≤z1H1ϑ1f¯1≤Θ1ϕ11X1z11+p+M0,(22)−z1H1y˙d≤z1H1M≤Θ1ϕ12X1z11+p+M0,(23)z1G1≤z1∥G1≤Θ1ϕ13X1z11+p+M0,(24)HigiHigi−1+Higi−1/ppit≥HigiHigi−pit+Higi−pit/p≥1,(25)zizi+zippit≥zizipit+zippit=zi1+pit+zi1+ppit≥zi1+p,(26)σiΘ˜iΘ∧i=σiΘ˜iΘi−Θ˜i≤−σiΘ˜i22+σiΘi22,for each i = 1, …, n, where M0 = p(1/(1 + p))1/p/(1 + p) > 0 is a constant and ϕ1j ≥ 0, j = 1, 2, and 3, are known C1 functions independent of p1(t), …, pn(t). Since p1(t) ≥ 1 and Θ∧1t≥0, one has 1+Θ∧1ϕ1p1t≥1+Θ∧1ϕ1 with ϕ1 = ϕ11 + ϕ12 + ϕ13. From this, by choosing β1=H1g1−1+H1g1−1/p1+Θ∧1ϕ1 and noting α1=−β1z1+z1p, (18), and (19) when i = 1, it yields(27)z1H1g1α1p1t=z1H1g1sgnα1α1p1t=−z1H1g1β1p1tz1+z1pp1t≤−1+Θ∧1ϕ1z11+p.
Substituting (21)–(23), (27), and τ1 ≜ ϕ1|z1|1 + p into (20) and using (26) when i = 1, it yields(28)V˙1≤−1+Θ∧1ϕ1z11+p+z1H1g1ξ2p1t−α1p1t+Θ1ϕ1z11+p+3M0−Θ˜1ϕ1z11+p−σ1Θ˜122+σ1Θ122−z11+p−σ1Θ˜122+z1H1g1ξ2p1t−α1p1t+M1,where M1=3M0+σ1Θ12/2>0.
Stepi (i = 2, …, n): for the sake of consistency, we specify Hj = 1, j = 2, …, i. Suppose that there exists a positive definite and C1 function Vi−1 such that(29)V˙i−1≤−∑j=1i−1zj1+p−∑j=1i−1σj2Θ˜j2+zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t+Mi−1,where Mi−1 > 0 is an unknown constant. In the following, we prove that (29) still holds at this step. Take the Lyapunov function Vi=Vi−1+zi2/2+Θ˜i2/2 with Θ˜i=Θi−Θ∧i. By (11)–(13) and (29), we have(30)V˙i≤∑j=1i−1zj1+p−∑j=1i−1σj2Θ˜j2+zigiαipit+zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t+ziHigiξi+1pit−αipit+zifi−∑j=1i−1∂αi−1∂ξjfj−zi∂αi−1∂ydy˙d−zi∑j=1i−1∂αi−1∂ξjgjξj+1pjt−zi∑j=1i−1∂αi−1∂Θ∧jΘ∧˙j−zi∑l=01∂αi−1∂FlFl+1−Θ˜iτi+σiΘ˜iΘ∧i+Mi−1.
According to Lemmas1–3, Assumptions1–3, (13) and (19), Θi=max1,ϑ11+p,…,ϑi1+p,M1+p, and Θ∧j≥0, one gets(31)zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t≤pi−1t1+2pi−1t−2zi−1Hi−1gi−1zipi−1t−1+αi−1pi−1t−1zi≤Θiϕi1Xizi1+p+M0,(32)zifi−∑j=1i−1∂αi−1∂ξjfj≤ziϑif¯i+∑j=1i−1∂αi−1∂ξjϑjf¯j≤Θiϕi2Xizi1+p+iM0,(33)−zi∑j=1i−1∂αi−1∂ξjgjξj+1pjt≤zi∑j=1i−1∂αi−1∂ξjgi∥ξj+1|pjt≤Θiϕi3Xizi1+p+i−1M0,(34)−zi∂αi−1∂ydy˙d≤zi∂αi−1∂ydM≤Θiϕi4Xizi1+p+M0,(35)−zi∂αi−1∂Θ∧jΘ∧˙j≤zi∂αi−1∂Θ∧jτj+σjΘ∧j≤Θiϕi5Xizi1+p+i−1M0,(36)−zi∑l=01∂αi−1∂FlFl+1≤zi∑j=12∑l=01∂αi−1∂Flρj,l+2≤Θiϕi6Xizi1+p+4M0,where ϕij(⋅) ≥ 0, j = 1, …, 6, are known C1 functions independent of p1(t), …, pn(t). Substituting (25)–(30) and τi ≜ ϕi|zi|1+p into (30) and using (26), it yields(37)V˙i≤−∑j=1i−1zj1+P−∑j=1j=1σj2Θ˜j2+zigiαipit+ziHigiξi+1pit−αipit+Θiϕizi1+p+4+3iM0−Θ˜iϕizi1+p−σiΘ˜i22+σiΘi22+Mi−1≤−∑j=1i−1zj1+p−∑j=1iσj2Θ˜j2+zigiαipit+ziHigiξi+1pit−αipit+Θ∧iϕizi1+p+Mi,where Mi=Mi−1+4+3iM0+σiΘi2/2>0. Similar to (27), by choosing βi=gi−1+gi−1/p1+Θ∧iϕi and employing (18) and (19), Hi = 1, and 1+Θ∧iϕipit≥1+Θ∧iϕi with ϕi=∑j=16ϕij, the virtual controller αi=−βizi+zip leads to(38)zigiαipit≤−1+Θ∧iϕizi1+p.
By virtue of (37), (38) becomes(39)V˙i≤−∑j=1izj1+p−∑j=1iσj2Θ˜j2+ziHigiξi+1pit−αipit+Mi.
At Stepn, inspired by the relative threshold strategy in [41], we adopt the following event-triggered controller:(40)ut=ωtk,∀t∈tk,tk+1,tk+1=inft>tk∥e1t>δut+m,ωt=−1+δαntanhgnznαnε+m¯tanhgnznm¯ε,where e1(t) = ω(t)−u(t) denotes the measurement error, 0 < δ < 1, m > 0, ɛ > 0, and m¯>m/1−δ are all design parameters, and tk, k∈ℤ+, is the triggered time. It is shown that the control signal u(t) will not change during t ∈ [tk, tk + 1) and be updated from ω(tk) to ω(tk + 1) at t = tk + 1. For t ∈ [tk, tk + 1), it follows from [34] that |ω(t) − u(t)| ≤ δ|u(t)| + m, and thus, there exist two continuous functions λ1(t) and λ2(t) satisfying |λ1(t)| ≤ 1 and |λ2(t)| ≤ 1 such that ω(t) = (1 + λ1δ)u(t) + λ2m, i.e.,(41)ut=ωt−λ2tm1+λ1tδ.
Since for anya∈ℝ and ɛ > 0, −a tanh(a/ɛ) ≤ 0 holds from Lemma 6, by [34], we get zngnω≤0. As λ1(t) ∈ [−1, 1] and λ2(t) ∈ [−1, 1], we have zngnω/1+λ1tδ ≤ zn<i>gnω</i>/(1 + δ) and λ2tm/1+λ1tδ≤m/1−δ<m¯. Therefore, by considering (37) when i = n, ξn+1 ≜ u, pn = 1, zngnαn≤−1+Θ∧nϕnzn1+p from (32), (34), and (35), and Lemma 6, one obtains(42)V˙n≤−∑j=1n−1zj1+p−∑j=1nσj2Θ˜j2+zngnωt−λ2tm1+λ1tδ+Θ∧nϕnzn1+p+Mn≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+zngn−αntanhzngnαnε−αn+αn−m¯tanhzngnm¯ε−m¯+m¯+Θ∧nϕnzn1+p+Mn≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+zngnm¯+zngnm1−δ+Mn+0.557ε≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+Mn+0.557ε.


### 3.4. Stability and Constraint Analysis

We state the main result of this paper.Theorem 1.
Consider the closed-loop system consisting of (5), (13), and (34) under Assumptions 1–3. For i = 1, …, n, if the initial condition satisfies(43)−F10<e0<F20,Θ∧i0≥0,then properties (O1)–(O3) in Section 3.1 hold, and the Zeno phenomenon does not occur, i.e., there exists an interexecution time t∗ > 0 such that {tk + 1−tk} ≥ t∗, ∀k∈ℤ+.Proof.
The closed-loop system is rewritten asς˙t=hςt,ut, where ςt=xt,Θ∧t. It is clear that h(⋅) is locally Lipschitz in ς(t). Hence, ς(t) is well defined on the maximal interval [0, tf) with 0 < tf < ∞.
The following proof is divided into four parts.Part 1.
By Lemma3 and Assumption 1, we have zj2≤1+zj1+p, i.e., −zj1+p≤−zj2+1, j = 1, …, n. Hence, (36) becomes(44)V˙nt≤−λ1Vnt+λ2,∀t∈0,tf,where λ1 = min1≤j ≤ n{2, σj} and λ2 = Mn + 0.557ɛ + n. By [37], we deduce that(45)Vnt≤e−λ1tVn0+∫0te−λ1t−τλ2dτ≤λ2λ1+Vn0−λ2λ1e−λ1t≤λ2λ1+Vn0.
Therefore, the boundedness ofVn(t) is ensured on [0, tf), so are zi(t) and Θ∧it, i = 1, …, n. In view of the boundedness of ξ1(t) = z1(t) on [0, tf) and (10), for −F1(0) < e(0) < F2(0), yd(t) −F1(t) < x1(t) < yd(t) + F2(t) is obtained for all t ∈ [0, tf). The continuous virtual controller α1(X1(t)) is bounded because of the boundedness of for all t ∈ [0, tf), and x2(t) = ξ2(t) = z2(t) + α1(t) is also bounded on t ∈ [0, tf). Similarly, we can recursively prove that αi−1(Xi−1(t)) and xi(t) = ξi(t) = zi(t) + αi−1(t), i = 3, …, n, are all bounded on [0, tf). Hence, we conclude that the actual controller u(Xn(t)) is also bounded on [0, tf).Part 2.
Next, we prove thattf = ∞. If tf < ∞, at least one signal in the closed-loop system will tend to ∞ when t = tf, which is a contradiction to the boundedness of xt,Θ∧t,ut on [0, tf). Therefore, tf = ∞ is proved, and ξ1 is bounded on [0, ∞), which means that there exist positive constants υ1, υ2 and positive functions F⌣1t, F⌣2t satisfying Fjt−F⌣jt≥υj>0, j = 1, 2, such that(46)−F1t<−F⌣1t≤et≤F⌣2t<F2t,holds for all t ≥ 0. By choosing the proper initial condition −F1(0) < e(0) < F2(0) and Θ∧i0≥0 in (43) and repeating the previous control design and analysis process, −F⌣1t≤et≤F⌣2t and yd(t) − F1(t) < y(t) < yd(t) + F2(t), ∀t ≥ 0, can be ensured. Hence, all the closed-loop signals are bounded on [0, ∞), and asymmetric time-varying output constraint (6) is not violated. It is worth mentioning that inequality (37) prevents all the terms with h1(e(t)) = (F1(t) + e(t)) (F2(t) − e(t)) in the denominator from becoming unbounded, and thus, the zero division does not occur.Part 3.
From (40), −F1(t) < e(t) < F2(t) holds for all t ≥ 0, which, together with (7), implies that e(t) ∈ Ωe, ∀t ≥ ts, with Ωe being defined in (O3). Hence, the practical finite-time tracking can be fulfilled.Part 4.
Finally, we prove that the Zeno phenomenon does not occur. Sincee(t) = ω(t) − u(t), ∀t ∈ [tk, tk + 1), and u(t) remains u(k) in [tk, tk + 1), we have de1/dt=de1×e11/2/dt=sgne1e˙1≤ω˙. From (34), we obtain(47)ω˙t=−1+δα˙ntanhzngnαnε+αnzng˙nαn+zngnα˙n+z˙ngnαncosh2zngnαn/εε+m¯zng˙nm¯+gnz˙nm¯cosh2zngnm¯/εε.
Sincez˙nt, α˙nt, and gnt are continuous, we deduce that ω˙t is a continuous function. In view of the boundedness of all the closed-loop signals, there exists a constant ω¯>0 such that ω˙t≤ω¯, ∀t ≥ 0. Due to e1(tk) = 0 and limt⟶tk+1e1t=δut+m, the lower bound of the interexecution time t∗ satisfies t∗≥δ|ut|+m/ω¯≥0, and the Zeno phenomenon can be avoided.


## 3.1. Problem Formulation and Assumptions

Inspired by (4), we consider more general nonlinear systems(11)x˙i=gix¯ixi+1pit+fix¯i,d,i=1,…,n−1,x˙n=gnxupnt+fnx,u,d,y=x1,with the asymmetric time-varying output constraint(12)ydt−F1t<yt<ydt+F2t,∀t≥0,where x¯i=x1,…,xi⊤∈ℝi, i = 1, …, n, x = xn, u∈ℝ, and y∈ℝ are measurable states, control input, and output, respectively, d∈ℝr is an unknown constant vector, system powers pit:ℝ+⟶ℝ+, i = 1, …, n, are unknown time-varying functions with pn(t) = 1, fi:ℝi×ℝr⟶ℝ, i = 1, …, n−1, fn:ℝn×ℝ×ℝr⟶ℝ are unknown locally Lipschitz continuous functions, gi:ℝi⟶ℝ, i = 1, …, n, are known C1 function, and ydt:ℝ+⟶ℝ denotes the desired trajectory; finite-time performance functions F1t:ℝ+⟶ℝ+ and F2t:ℝ+⟶ℝ+ are defined as(13)Fjt=Fj0−ttse1−ts/ts−t+Fjs,t∈0,ts,Fjs,t∈ts,∞,j=1,2,with Fj0 > 5/4 and Fjs > 0 being design parameters. System (11) is called as the high-order nonlinear system if there exists at least one pi(t) > 1. From Remark 2 in [50], it is easy to prove that Fj(t) > 0 is decreasing and smooth, and there exist known positive constants ρj1, ρj2, and ρj3 such that |Fj(t)| ≤ ρj1, F˙jt≤ρj2, and F¨jt≤ρj3, ∀t ≥ 0. It is worth mentioning that (13) plays a crucial role on the achievement of finite-time tracking, and the preassigned finite time ts > 0 can be chosen prior to control implementation, which is independent of initial conditions and design parameters.The control objective is to design aC1 adaptive event-triggered state-feedback controller(14)u=ux,yd,Θ∧,F,F˙,Θ∧.i=ζx¯i,yd,Θ∧¯i,F,F˙,where Θ∧¯i=Θ∧1,…,Θ∧i⊤, Θ∧i∈ℝ, i = 1, …, n, are auxiliary variables used to handle parametric uncertainties, Θ∧=Θ∧¯n, F=F1,F2⊤, and ζ(⋅) is a continuous function such that the following properties hold:(O1)
All the closed-loop signals are bounded on [0,∞)(O2)
Asymmetric time-varying output constraint (12) is not violated(O3)
The practical finite-time tracking can be fulfilled, i.e., for any small constantsF1s > 0 and F2s > 0, there exists a preassigned finite time ts > 0 such that the tracking error e(t) = y(t)−yd(t) satisfies et∈Ωe≜et∈ℝ:−F1s≤et≤F2s, ∀t ≥ tsRemark 1.
Although asymptotic tracking has better steady-state performance, practical tracking has a larger applicable scope than asymptotic one since it requires less restrictions on the desired trajectoryyd(t) and the considered system. Actually, no continuous controller exists to achieve the global or even local asymptotic tracking for the considered system (5). This can be verified by a two-dimensional high-order nonlinear system in [51]. Hence, this paper focuses on the practical tracking problem rather than the asymptotic case.
To achieve (O1)–(O3), the following assumptions are needed.Assumption 1.
Unknown time-varying powerspi(t), i = 1, …, n, satisfy p ≥ pi(t) ≥ 1, where p is a known constant.Assumption 2.
Fori = 1, …, n, there exist unknown constants ϑi > 0 and known C1 nonnegative functions f¯ix¯i such that(15)fi≤ϑif¯ix¯i,i=1,…,n.Assumption 3.
There exists an unknown constantM > 0 such that ydt+y˙dt≤M, ∀t ≥ 0.Remark 2.
Assumption1 means that p1(t), ⋯, pn(t) have a common upper bound p. In addition, the restrictive relationship p1(t) ≥ p2(t) ≥ ⋯ ≥ pn(t) is no longer needed in this paper.
Inequality (15) in Assumption 2 indicates the restriction on system nonlinearities f1, …, fn, which allow the existence of parametric uncertainties ϑ1, …, ϑn and do not necessarily vanish in the origin. Hence, f1, …, fn require less prior knowledge to implement the output tracking.
Assumption3 describes the boundedness of yd(t) and y˙dt. Compared with the existing works on state-/output-constrained tracking control, no any bounded restriction is imposed on high-order derivatives of yd(t) in this paper.


## 3.2. System Transformation

To begin with, we infer from (12) that the constraint on y = x1 is equivalent to −F1(t) < e(t) < F2(t), ∀t ≥ 0. To ensure output constraint (12) and then achieve finite-time tracking, inspired by [2] and (13), we introduce NMs(16)ξ1=T1e=eh1e,ξi=Tixi=xihi,i=2,…,n,where h1(e) = (F1 + e) (F2−e), hi = 1, i = 2, …, n. It is easy to show that T1(e) is strictly increasing, smooth, and diffeomorphic with respect to e. Besides, ξ1⟶∞ when e⟶−F1 and e⟶F2. In other words, (6) is not violated when the proper initial condition and the boundedness of the transformed state ξ1 can be guaranteed in the closed-loop system. From (5) and (10), we obtain a new transformed system:(17)ξ˙1=H1x1,yd,Fg1ξ2p1t+f1−y˙d+G1x1,yd,F,F˙,ξ˙i=giξi+1pit+fi,i=2,…,n−1,ξ˙n=gnu+fn,where H1=F1F2+e2/h12 and G1=−F˙1F2+F1F˙2+F˙2−F˙1ee/h12.


## 3.3. State-Feedback Control Design

In view of, we letX0 = β0 = α0 = 0 and specify the coordinate transformations(18)zi=ξi−αi−1Xi−1,αiXi=−βiXizi+zip,i=1,…,n,where X1=x1,yd,Θ∧1,F,F˙, Xi=x¯i,yd,Θ∧¯i,F,F˙, i = 2, …, n, and the estimations Θ∧i≥0 of are generated by the adaptive laws:(19)Θ∧˙i=τiXi−σiΘ∧i,Θ∧i0≥0,i=1,…,n,with σi > 0 being design parameters, and nonnegative C1 functions βi(⋅) and τi(⋅) will be determined later.Remark 3.
Compared with [34–37, 39, 40] on the adaptive control of nonlinear systems with time-varying powers, we relax the requirement Θ∧i≥1 to Θ∧i≥0 in (13). Moreover, the simultaneous introduction of low-power term zi and high-power term zip in (12) is the key to deal with unknown time-varying powers p1(t), …, pn(t).
Based on (11)–(13), we provide the detailed procedure of control design in a recursive manner.Step 1.
Take the Lyapunov functionV1=z12/2+Θ˜12/2 with Θ˜1=Θ1−Θ∧1. By (11)–(13), we have(20)V˙1=z1H1g1ξ2p1t+f1−y˙d+z1G1−Θ˜1Θ∧˙1=z1H1g1α1p1t+z1H1g1ξ2p1t−α1p1t+z1H1f1−z1H1y˙d+z1G1−Θ˜1τ1+σ1Θ˜1Θ∧1.
For the sake of consistency, we specifyH2 = ⋯ = Hn = 1. According to Assumptions 1–3, Lemmas 1–3, and Θ1=max1,ϑ11+p,M1+p, one gets(21)z1H1f1≤z1H1ϑ1f¯1≤Θ1ϕ11X1z11+p+M0,(22)−z1H1y˙d≤z1H1M≤Θ1ϕ12X1z11+p+M0,(23)z1G1≤z1∥G1≤Θ1ϕ13X1z11+p+M0,(24)HigiHigi−1+Higi−1/ppit≥HigiHigi−pit+Higi−pit/p≥1,(25)zizi+zippit≥zizipit+zippit=zi1+pit+zi1+ppit≥zi1+p,(26)σiΘ˜iΘ∧i=σiΘ˜iΘi−Θ˜i≤−σiΘ˜i22+σiΘi22,for each i = 1, …, n, where M0 = p(1/(1 + p))1/p/(1 + p) > 0 is a constant and ϕ1j ≥ 0, j = 1, 2, and 3, are known C1 functions independent of p1(t), …, pn(t). Since p1(t) ≥ 1 and Θ∧1t≥0, one has 1+Θ∧1ϕ1p1t≥1+Θ∧1ϕ1 with ϕ1 = ϕ11 + ϕ12 + ϕ13. From this, by choosing β1=H1g1−1+H1g1−1/p1+Θ∧1ϕ1 and noting α1=−β1z1+z1p, (18), and (19) when i = 1, it yields(27)z1H1g1α1p1t=z1H1g1sgnα1α1p1t=−z1H1g1β1p1tz1+z1pp1t≤−1+Θ∧1ϕ1z11+p.
Substituting (21)–(23), (27), and τ1 ≜ ϕ1|z1|1 + p into (20) and using (26) when i = 1, it yields(28)V˙1≤−1+Θ∧1ϕ1z11+p+z1H1g1ξ2p1t−α1p1t+Θ1ϕ1z11+p+3M0−Θ˜1ϕ1z11+p−σ1Θ˜122+σ1Θ122−z11+p−σ1Θ˜122+z1H1g1ξ2p1t−α1p1t+M1,where M1=3M0+σ1Θ12/2>0.
Stepi (i = 2, …, n): for the sake of consistency, we specify Hj = 1, j = 2, …, i. Suppose that there exists a positive definite and C1 function Vi−1 such that(29)V˙i−1≤−∑j=1i−1zj1+p−∑j=1i−1σj2Θ˜j2+zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t+Mi−1,where Mi−1 > 0 is an unknown constant. In the following, we prove that (29) still holds at this step. Take the Lyapunov function Vi=Vi−1+zi2/2+Θ˜i2/2 with Θ˜i=Θi−Θ∧i. By (11)–(13) and (29), we have(30)V˙i≤∑j=1i−1zj1+p−∑j=1i−1σj2Θ˜j2+zigiαipit+zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t+ziHigiξi+1pit−αipit+zifi−∑j=1i−1∂αi−1∂ξjfj−zi∂αi−1∂ydy˙d−zi∑j=1i−1∂αi−1∂ξjgjξj+1pjt−zi∑j=1i−1∂αi−1∂Θ∧jΘ∧˙j−zi∑l=01∂αi−1∂FlFl+1−Θ˜iτi+σiΘ˜iΘ∧i+Mi−1.
According to Lemmas1–3, Assumptions1–3, (13) and (19), Θi=max1,ϑ11+p,…,ϑi1+p,M1+p, and Θ∧j≥0, one gets(31)zi−1Hi−1gi−1ξipi−1t−αi−1pi−1t≤pi−1t1+2pi−1t−2zi−1Hi−1gi−1zipi−1t−1+αi−1pi−1t−1zi≤Θiϕi1Xizi1+p+M0,(32)zifi−∑j=1i−1∂αi−1∂ξjfj≤ziϑif¯i+∑j=1i−1∂αi−1∂ξjϑjf¯j≤Θiϕi2Xizi1+p+iM0,(33)−zi∑j=1i−1∂αi−1∂ξjgjξj+1pjt≤zi∑j=1i−1∂αi−1∂ξjgi∥ξj+1|pjt≤Θiϕi3Xizi1+p+i−1M0,(34)−zi∂αi−1∂ydy˙d≤zi∂αi−1∂ydM≤Θiϕi4Xizi1+p+M0,(35)−zi∂αi−1∂Θ∧jΘ∧˙j≤zi∂αi−1∂Θ∧jτj+σjΘ∧j≤Θiϕi5Xizi1+p+i−1M0,(36)−zi∑l=01∂αi−1∂FlFl+1≤zi∑j=12∑l=01∂αi−1∂Flρj,l+2≤Θiϕi6Xizi1+p+4M0,where ϕij(⋅) ≥ 0, j = 1, …, 6, are known C1 functions independent of p1(t), …, pn(t). Substituting (25)–(30) and τi ≜ ϕi|zi|1+p into (30) and using (26), it yields(37)V˙i≤−∑j=1i−1zj1+P−∑j=1j=1σj2Θ˜j2+zigiαipit+ziHigiξi+1pit−αipit+Θiϕizi1+p+4+3iM0−Θ˜iϕizi1+p−σiΘ˜i22+σiΘi22+Mi−1≤−∑j=1i−1zj1+p−∑j=1iσj2Θ˜j2+zigiαipit+ziHigiξi+1pit−αipit+Θ∧iϕizi1+p+Mi,where Mi=Mi−1+4+3iM0+σiΘi2/2>0. Similar to (27), by choosing βi=gi−1+gi−1/p1+Θ∧iϕi and employing (18) and (19), Hi = 1, and 1+Θ∧iϕipit≥1+Θ∧iϕi with ϕi=∑j=16ϕij, the virtual controller αi=−βizi+zip leads to(38)zigiαipit≤−1+Θ∧iϕizi1+p.
By virtue of (37), (38) becomes(39)V˙i≤−∑j=1izj1+p−∑j=1iσj2Θ˜j2+ziHigiξi+1pit−αipit+Mi.
At Stepn, inspired by the relative threshold strategy in [41], we adopt the following event-triggered controller:(40)ut=ωtk,∀t∈tk,tk+1,tk+1=inft>tk∥e1t>δut+m,ωt=−1+δαntanhgnznαnε+m¯tanhgnznm¯ε,where e1(t) = ω(t)−u(t) denotes the measurement error, 0 < δ < 1, m > 0, ɛ > 0, and m¯>m/1−δ are all design parameters, and tk, k∈ℤ+, is the triggered time. It is shown that the control signal u(t) will not change during t ∈ [tk, tk + 1) and be updated from ω(tk) to ω(tk + 1) at t = tk + 1. For t ∈ [tk, tk + 1), it follows from [34] that |ω(t) − u(t)| ≤ δ|u(t)| + m, and thus, there exist two continuous functions λ1(t) and λ2(t) satisfying |λ1(t)| ≤ 1 and |λ2(t)| ≤ 1 such that ω(t) = (1 + λ1δ)u(t) + λ2m, i.e.,(41)ut=ωt−λ2tm1+λ1tδ.
Since for anya∈ℝ and ɛ > 0, −a tanh(a/ɛ) ≤ 0 holds from Lemma 6, by [34], we get zngnω≤0. As λ1(t) ∈ [−1, 1] and λ2(t) ∈ [−1, 1], we have zngnω/1+λ1tδ ≤ zn<i>gnω</i>/(1 + δ) and λ2tm/1+λ1tδ≤m/1−δ<m¯. Therefore, by considering (37) when i = n, ξn+1 ≜ u, pn = 1, zngnαn≤−1+Θ∧nϕnzn1+p from (32), (34), and (35), and Lemma 6, one obtains(42)V˙n≤−∑j=1n−1zj1+p−∑j=1nσj2Θ˜j2+zngnωt−λ2tm1+λ1tδ+Θ∧nϕnzn1+p+Mn≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+zngn−αntanhzngnαnε−αn+αn−m¯tanhzngnm¯ε−m¯+m¯+Θ∧nϕnzn1+p+Mn≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+zngnm¯+zngnm1−δ+Mn+0.557ε≤−∑j=1nzj1+p−∑j=1nσj2Θ˜j2+Mn+0.557ε.


## 3.4. Stability and Constraint Analysis

We state the main result of this paper.Theorem 1.
Consider the closed-loop system consisting of (5), (13), and (34) under Assumptions 1–3. For i = 1, …, n, if the initial condition satisfies(43)−F10<e0<F20,Θ∧i0≥0,then properties (O1)–(O3) in Section 3.1 hold, and the Zeno phenomenon does not occur, i.e., there exists an interexecution time t∗ > 0 such that {tk + 1−tk} ≥ t∗, ∀k∈ℤ+.Proof.
The closed-loop system is rewritten asς˙t=hςt,ut, where ςt=xt,Θ∧t. It is clear that h(⋅) is locally Lipschitz in ς(t). Hence, ς(t) is well defined on the maximal interval [0, tf) with 0 < tf < ∞.
The following proof is divided into four parts.Part 1.
By Lemma3 and Assumption 1, we have zj2≤1+zj1+p, i.e., −zj1+p≤−zj2+1, j = 1, …, n. Hence, (36) becomes(44)V˙nt≤−λ1Vnt+λ2,∀t∈0,tf,where λ1 = min1≤j ≤ n{2, σj} and λ2 = Mn + 0.557ɛ + n. By [37], we deduce that(45)Vnt≤e−λ1tVn0+∫0te−λ1t−τλ2dτ≤λ2λ1+Vn0−λ2λ1e−λ1t≤λ2λ1+Vn0.
Therefore, the boundedness ofVn(t) is ensured on [0, tf), so are zi(t) and Θ∧it, i = 1, …, n. In view of the boundedness of ξ1(t) = z1(t) on [0, tf) and (10), for −F1(0) < e(0) < F2(0), yd(t) −F1(t) < x1(t) < yd(t) + F2(t) is obtained for all t ∈ [0, tf). The continuous virtual controller α1(X1(t)) is bounded because of the boundedness of for all t ∈ [0, tf), and x2(t) = ξ2(t) = z2(t) + α1(t) is also bounded on t ∈ [0, tf). Similarly, we can recursively prove that αi−1(Xi−1(t)) and xi(t) = ξi(t) = zi(t) + αi−1(t), i = 3, …, n, are all bounded on [0, tf). Hence, we conclude that the actual controller u(Xn(t)) is also bounded on [0, tf).Part 2.
Next, we prove thattf = ∞. If tf < ∞, at least one signal in the closed-loop system will tend to ∞ when t = tf, which is a contradiction to the boundedness of xt,Θ∧t,ut on [0, tf). Therefore, tf = ∞ is proved, and ξ1 is bounded on [0, ∞), which means that there exist positive constants υ1, υ2 and positive functions F⌣1t, F⌣2t satisfying Fjt−F⌣jt≥υj>0, j = 1, 2, such that(46)−F1t<−F⌣1t≤et≤F⌣2t<F2t,holds for all t ≥ 0. By choosing the proper initial condition −F1(0) < e(0) < F2(0) and Θ∧i0≥0 in (43) and repeating the previous control design and analysis process, −F⌣1t≤et≤F⌣2t and yd(t) − F1(t) < y(t) < yd(t) + F2(t), ∀t ≥ 0, can be ensured. Hence, all the closed-loop signals are bounded on [0, ∞), and asymmetric time-varying output constraint (6) is not violated. It is worth mentioning that inequality (37) prevents all the terms with h1(e(t)) = (F1(t) + e(t)) (F2(t) − e(t)) in the denominator from becoming unbounded, and thus, the zero division does not occur.Part 3.
From (40), −F1(t) < e(t) < F2(t) holds for all t ≥ 0, which, together with (7), implies that e(t) ∈ Ωe, ∀t ≥ ts, with Ωe being defined in (O3). Hence, the practical finite-time tracking can be fulfilled.Part 4.
Finally, we prove that the Zeno phenomenon does not occur. Sincee(t) = ω(t) − u(t), ∀t ∈ [tk, tk + 1), and u(t) remains u(k) in [tk, tk + 1), we have de1/dt=de1×e11/2/dt=sgne1e˙1≤ω˙. From (34), we obtain(47)ω˙t=−1+δα˙ntanhzngnαnε+αnzng˙nαn+zngnα˙n+z˙ngnαncosh2zngnαn/εε+m¯zng˙nm¯+gnz˙nm¯cosh2zngnm¯/εε.
Sincez˙nt, α˙nt, and gnt are continuous, we deduce that ω˙t is a continuous function. In view of the boundedness of all the closed-loop signals, there exists a constant ω¯>0 such that ω˙t≤ω¯, ∀t ≥ 0. Due to e1(tk) = 0 and limt⟶tk+1e1t=δut+m, the lower bound of the interexecution time t∗ satisfies t∗≥δ|ut|+m/ω¯≥0, and the Zeno phenomenon can be avoided.


## 4. Two Simulation Examples

Example 1.
Consider a reduced-order model of the boiler-turbine unit:(48)x˙1=x2p1t,x2=1+x12up2t+ϑ2x121+x22,y=x1,where x1, x2 are the drum and reheater pressures, u is the position of the control valve, 1 ≤ p1(t) ≤ 11/9, p2(t) = 1 are time-varying powers, and ϑ2 > 0 is an unknown constant. The desired trajectory is chosen as yd(t) = 0.1 sin(t). Clearly, system (48) is a special case of system (13), and Assumptions1–3 are satisfied with p = 11/9, g1=1, g2=1+x12, f¯1=0, and f¯2=x12. In order to achieve the practical finite-time tracking and satisfy the actual demands, the time-varying output constraint yd(t)−F1(t) < yd(t) < yd(t) + F2(t), ∀t ≥ 0, needs to be guaranteed, where F1(t) and F2(t) are defined in (13) with ts = 20s, F10 = F20 = 1.2, and F1s = F2s = 0.06.
According to control design in Section3.3, by setting e = −yd, h1 = (F1 + e) (F2−e), ξ1 = e/h1, ξ2 = x2, H1=F1F2+e2/h12, G1=−F˙1F2+F1F˙2+F˙2−F˙1ee/h12, z1 = ξ1, and z2 = ξ2−α1, the event-triggered controller and adaptive laws are obtained as(49)ut=ωtk,∀t∈tk,tk+1,tk+1=inft>tk∥e1t>δut+m,ωt=−1+δα2tanhz2α2ε+m¯tanhz2m¯ε,Θ∧˙i=ϕizi20/9−σiΘ∧i,i=1,2,wheree1=ω−u,αi=−βizi+zi11/9,i=1,2β1=H1−1+H1−9/111+Θ^1ϕ1,β2=g2−1+g2−9/111+Θ^2ϕ2,ϕ1=H120/9+G120/9,ϕ2=∑j=16ϕ2j,ϕ21=ϱ1H120/9z120/92+β1sz1+z111/9+sz2,ϕ22=f¯220/9,ϕ23=dx120/91+x2220/81,ϕ24=dx120/9,ϕ25=dx220/9τ1+σ1Θ^120/9,ϕ26=2520dx3+1500dx420/9,ϱ1=9×22/910112011/9119×1+2−7/920/9,dx1=H1−2+911H1−20/11∂H1∂x11+Θ^1ϕ1sz1+z111/9+H1−1+H1−9/11Θ^1209H111/9∂H1∂x1+G111/9∂G1∂x1sz1+z111/9+β1e2+F12h121+z12/9,dx2=H1−1+H1−9/11ϕ1sz1+z111/9,dx3=H1−2+911H1−20/11∂H1∂F11+Θ^1ϕ1sz1+z111/9+H1−1+H1−9/11Θ^1209H111/9∂H1∂F1+G111/9∂G1∂F1sz1+z111/9+β121+e21/2h121+z12/9,dx4=H1−1+H1−9/11Θ^1∂G1∂F˙1sz1+z111/9.
Figures3 and 4 provide the responses of the closed-loop system and the time interval of each triggered event. From Figure 3, we know that the closed-loop signals x1, x2, Θ∧1, Θ∧2, and u are bounded, the asymmetric time-varying output constraint is not violated, and then the practical finite-time tracking can be fulfilled.Figure 3
Responses of the closed-loop system (42) and (43).Figure 4
Time interval of each triggered event.Example 2.
Consider a numerical example(50)x˙1=x2p1t+ϑ1x1,x˙2=up2t+ϑ2x12+sinx2,y=x1,where 1 ≤ p1(t) ≤ 21/19, p2(t) = 1 are time-varying powers and ϑ1 and ϑ2 are unknown positive constants. The desired trajectory is chosen as yd(t) = 0.4 sin(t). Clearly, system (48) is a special case of system (11), and Assumptions1–3 are satisfied with p = 21/19, g1=g2=1, f¯1=1+x121/2, and f¯2=1+x12. In order to achieve the practical finite-time tracking and satisfy the actual demands, the time-varying output constraint yd(t)−F1(t) < y(t) < yd(t) + F2(t), ∀t ≥ 0, needs to be guaranteed, where F1(t) and F2(t) are defined in (13) with ts = 40s, F10 = F20 = 2.4, and F1s = F2s = 0.05.
Similar to Example1, the event-triggered controller and adaptive laws are obtained as(51)ut=ωtk,∀t∈tk,tk+1,tk+1=inft>tk∥e1t>δut+m,ωt=−1+δα2tanhz2α2ε+m¯tanhz2m¯ε,Θ∧˙i=ϕizi40/19−σiΘ∧i,i=1,2,wheree1=ω−u,αi=−βizi+zi12/19⊕,i=1,2,β1=H1−1+H1−19/211+Θ^1ϕ1,β2=21+ϕ2Θ^2,ϕ1=H1f¯140/19+H140/19+G140/19,ϕ2=∑j=16ϕ2j,ϕ21=ϱ1H140/19z140/192+β1sz1+z121/19+sz2,ϕ22=f¯240/19+dx1f¯140/19,ϕ23=dx140/191+x2840/361,ϕ24=dx140/19,ϕ25=dx240/19τ1+σ1Θ^140/19,ϕ26=174040/19dx340/19+1400040/19dx440/19,ϱ1=221/191940214021/19211+2−17/191940/19,szi=1+zi21/2,i=1,2,dx1=β1H14019+21sz119+4019Θ^1sz1+z140/19H1−1+H1−19/21∂H1∂x11+f¯140/19+H140/19f¯121/19∂f¯1∂x1+G140/19∂G1∂x1+1+Θ^1ϕ1sz1+z140/19H1−2+H1−40/21∂H1∂x1,dx2=ϕ1H1−1+H1−19/21sz1+z121/19,dx3=β14019+21sz1192F˙11+e21/2+4019Θ^1sz1+z140/19H121/19f¯140/19∂H1∂F1+H121/19∂H1∂F1+G121/19∂G1∂F1+1+Θ^1ϕ1sz1+z140/19H1−2+H1−40/21∂H1∂F1,dx4=4021Θ^1H1−1+H1−19/21G121/192F11+e21/2h12sz1+z121/19.
Figures5 and 6 provide the responses of the closed-loop system (44) and (45) and the time interval of each triggered event. By calculation, the number of triggered control signal transmissions is 207, and the amount of continuous time control sampling is 2072. Since the control signal can be discontinuously sent to the actuator rather than periodically sampled in the traditional continuous sampling control, the designed event-triggered controller reduces redundant data transmissions and consumes less communication resources in the constrained control framework.Figure 5
Responses of the closed-loop system (44) and (45).Figure 6
Time interval of each triggered event.


## 5. Conclusions

For more general output-constrained high-order nonlinear systems with unknown time-varying powers, this paper investigated the adaptive event-triggered finite-time tracking problem.Some challenging problems are still unsolved:(1)
In this paper,pi(t) ≥ 1 in Assumption 1 is assumed. When pi(t) < 1, i = 1, …, n, system (11) is called as a low-order nonlinear system. Recently, Cui and Xie [33] constructed a state-feedback controller for low-order nonlinear systems with unknown time-varying powers, but it does not consider the output constraint and the event-triggered control. Hence, a more challenging problem is can we design an adaptive event-triggered tracking controller for output-constrained low-order nonlinear systems with unknown powers?(2)
Recently, some important results on systems with stochastic noise, incomplete measurements, coding-decoding mechanisms, and protocol scheduling have been achieved; see [52–67] and other papers. However, they do not consider the effects of the output constraint. Hence, our subsequent works are to consider stochastic systems with incomplete measurements and coding-decoding mechanisms and solve the output-constrained tracking problem with protocol scheduling.



---

*Source: 7466780-2022-07-07.xml*