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

Liu, Fan; Wu, You

doi:https://doi.org/10.1155/2022/7466780

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 7466780 | https://doi.org/10.1155/2022/7466780

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

Fan Liu¹and You Wu¹

Academic Editor: Zidong Wang

Received02 Nov 2021

Revised21 Dec 2021

Accepted30 Dec 2021

Published07 Jul 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.

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 [1, 2], 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 as p-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 [33, 34] 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 [43, 44] 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 [1–19, 22–28, 35–40, 43, 44] 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 [20–28, 31–40] 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. Section 2 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: , , , and 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. is the set of all functions with continuous partial derivatives. For a real vector , the norm ‖x‖ is defined by . For any , 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 function , 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 Figure 1 [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 aswhere l is the length of the rod, is the acceleration of gravity, k and k_s are spring coefficients, and y = x − l sin(θ) in Figure 1. Assume that m₁, m₂, l, and k_s are unknown constant parameters which belong to a known interval with . Suppose that θ is small; equation (1) becomes

Then, by using the change of coordinatessystem (2) becomes the following system:

The unknown time-varying power p(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.

2.2. Preliminaries

Lemmas 1–6 are used to enlarge inequalities in the following state-feedback control design and analysis.

Lemma 1 (see [46]). Let r₁(t), r₂(t), and α(x, y) be some positive continuous real-valued functions. For any ,

Lemma 2 (see [46]). Let r(t) ≥ 1 be a continuous real-valued function. For any , i = 1, …, n,

Lemma 3 (see [46]). If r(t) is a continuous real-valued function and satisfies , where and are positive real-valued functions, for any ,

Lemma 4 (see [47]). Let r(t) ≥ 1 be a continuous real-valued function. For any ,where |y|^r(t)−1 ≜ 0 if y = 0 and r(t) = 1.

Lemma 5 (see [48]). For given continuous function f(x, y) with and , there exist smooth functions a(x) ≥ 0, b(y) ≥ 0, c(x) ≥ 1, and d(y) ≥ 1 such that

Lemma 6 (see [49]). For any ɛ > 0 and , there hold

3. Main Result

3.1. Problem Formulation and Assumptions

Inspired by (4), we consider more general nonlinear systemswith the asymmetric time-varying output constraintwhere , i = 1, …, n, x = x_n, , and are measurable states, control input, and output, respectively, is an unknown constant vector, system powers , i = 1, …, n, are unknown time-varying functions with p_n(t) = 1, , i = 1, …, n − 1, are unknown locally Lipschitz continuous functions, , i = 1, …, n, are known function, and denotes the desired trajectory; finite-time performance functions and are defined aswith F_j0 > 5/4 and F_js > 0 being design parameters. System (11) is called as the high-order nonlinear system if there exists at least one p_i(t) > 1. From Remark 2 in [50], it is easy to prove that F_j(t) > 0 is decreasing and smooth, and there exist known positive constants ρ_j1, ρ_j2, and ρ_j3 such that |F_j(t)| ≤ ρ_j1, , and , ∀t ≥ 0. It is worth mentioning that (13) plays a crucial role on the achievement of finite-time tracking, and the preassigned finite time t_s > 0 can be chosen prior to control implementation, which is independent of initial conditions and design parameters.

The control objective is to design a adaptive event-triggered state-feedback controllerwhere , , i = 1, …, n, are auxiliary variables used to handle parametric uncertainties, , , 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 constants F_1s > 0 and F_2s > 0, there exists a preassigned finite time t_s > 0 such that the tracking error e(t) = y(t)−y_d(t) satisfies , ∀t ≥ t_s

Remark 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 trajectory y_d(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 powers p_i(t), i = 1, …, n, satisfy p ≥ p_i(t) ≥ 1, where p is a known constant.

Assumption 2. For i = 1, …, n, there exist unknown constants ϑ_i > 0 and known nonnegative functions such that

Assumption 3. There exists an unknown constant M > 0 such that , ∀t ≥ 0.

Remark 2. Assumption 1 means that p₁(t), ⋯, p_n(t) have a common upper bound p. In addition, the restrictive relationship p₁(t) ≥ p₂(t) ≥ ⋯ ≥ p_n(t) is no longer needed in this paper.
Inequality (15) in Assumption 2 indicates the restriction on system nonlinearities f₁, …, f_n, which allow the existence of parametric uncertainties ϑ₁, …, ϑ_n and do not necessarily vanish in the origin. Hence, f₁, …, f_n require less prior knowledge to implement the output tracking.
Assumption 3 describes the boundedness of y_d(t) and . Compared with the existing works on state-/output-constrained tracking control, no any bounded restriction is imposed on high-order derivatives of y_d(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 NMswhere h₁(e) = (F₁ + e) (F₂−e), h_i = 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:where and .

3.3. State-Feedback Control Design

In view of, we let X₀ = β₀ = α₀ = 0 and specify the coordinate transformationswhere , , i = 2, …, n, and the estimations of are generated by the adaptive laws:with σ_i > 0 being design parameters, and nonnegative functions β_i(⋅) and τ_i(⋅) 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 to in (13). Moreover, the simultaneous introduction of low-power term z_i and high-power term in (12) is the key to deal with unknown time-varying powers p₁(t), …, p_n(t).
Based on (11)–(13), we provide the detailed procedure of control design in a recursive manner.

Step 1. Take the Lyapunov function with . By (11)–(13), we haveFor the sake of consistency, we specify H₂ = ⋯ = H_n = 1. According to Assumptions 1–3, Lemmas 1–3, and , one getsfor each i = 1, …, n, where M₀ = p(1/(1 + p))^1/p/(1 + p) > 0 is a constant and ϕ_1j ≥ 0, j = 1, 2, and 3, are known functions independent of p₁(t), …, p_n(t). Since p₁(t) ≥ 1 and , one has with ϕ₁ = ϕ₁₁ + ϕ₁₂ + ϕ₁₃. From this, by choosing and noting , (18), and (19) when i = 1, it yieldsSubstituting (21)–(23), (27), and τ₁ ≜ ϕ₁|z₁|^1 + ^p into (20) and using (26) when i = 1, it yieldswhere .
Stepi (i = 2, …, n): for the sake of consistency, we specify H_j = 1, j = 2, …, i. Suppose that there exists a positive definite and function V_i−1 such thatwhere M_i−1 > 0 is an unknown constant. In the following, we prove that (29) still holds at this step. Take the Lyapunov function with . By (11)–(13) and (29), we haveAccording to Lemmas 1–3, Assumptions1–3, (13) and (19), , and , one getswhere ϕ_ij(⋅) ≥ 0, j = 1, …, 6, are known functions independent of p₁(t), …, p_n(t). Substituting (25)–(30) and τ_i ≜ ϕ_i|z_i|^1+p into (30) and using (26), it yieldswhere . Similar to (27), by choosing and employing (18) and (19), H_i = 1, and with , the virtual controller leads toBy virtue of (37), (38) becomesAt Step n, inspired by the relative threshold strategy in [41], we adopt the following event-triggered controller:where e₁(t) = ω(t)−u(t) denotes the measurement error, 0 < δ < 1, m > 0, ɛ > 0, and are all design parameters, and t_k, , is the triggered time. It is shown that the control signal u(t) will not change during t ∈ [t_k, t_k ₊ ₁) and be updated from ω(t_k) to ω(t_k ₊ ₁) at t = t_k ₊ ₁. For t ∈ [t_k, t_k ₊ 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.,Since for any and ɛ > 0, −a tanh(a/ɛ) ≤ 0 holds from Lemma 6, by [34], we get . As λ₁(t) ∈ [−1, 1] and λ₂(t) ∈ [−1, 1], we have ≤ z_n__nω_/(1 + δ) and . Therefore, by considering (37) when i = n, ξ_n+1 ≜ u, p_n = 1, from (32), (34), and (35), and Lemma 6, one obtains

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 satisfiesthen properties (O1)–(O3) in Section 3.1 hold, and the Zeno phenomenon does not occur, i.e., there exists an interexecution time > 0 such that {t_k ₊ ₁−t_k} ≥ , .

Proof. The closed-loop system is rewritten as , where . It is clear that h(⋅) is locally Lipschitz in ς(t). Hence, ς(t) is well defined on the maximal interval [0, t_f) with 0 < t_f < ∞.
The following proof is divided into four parts.

Part 1. By Lemma 3 and Assumption 1, we have , i.e., , j = 1, …, n. Hence, (36) becomeswhere λ₁ = min_1≤j _≤ _n{2, σ_j} and λ₂ = M_n + 0.557ɛ + n. By [37], we deduce thatTherefore, the boundedness of V_n(t) is ensured on [0, t_f), so are z_i(t) and , i = 1, …, n. In view of the boundedness of ξ₁(t) = z₁(t) on [0, t_f) and (10), for −F₁(0) < e(0) < F₂(0), y_d(t) −F₁(t) < x₁(t) < y_d(t) + F₂(t) is obtained for all t ∈ [0, t_f). The continuous virtual controller α₁(X₁(t)) is bounded because of the boundedness of for all t ∈ [0, t_f), and x₂(t) = ξ₂(t) = z₂(t) + α₁(t) is also bounded on t ∈ [0, t_f). Similarly, we can recursively prove that α_i−1(X_i−1(t)) and x_i(t) = ξ_i(t) = z_i(t) + α_i−1(t), i = 3, …, n, are all bounded on [0, t_f). Hence, we conclude that the actual controller u(X_n(t)) is also bounded on [0, t_f).

Part 2. Next, we prove that t_f = ∞. If t_f < ∞, at least one signal in the closed-loop system will tend to ∞ when t = t_f, which is a contradiction to the boundedness of on [0, t_f). Therefore, t_f = ∞ is proved, and ξ₁ is bounded on [0, ∞), which means that there exist positive constants υ₁, υ₂ and positive functions , satisfying , j = 1, 2, such thatholds for all t ≥ 0. By choosing the proper initial condition −F₁(0) < e(0) < F₂(0) and in (43) and repeating the previous control design and analysis process, and y_d(t) − F₁(t) < y(t) < y_d(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) ∈ Ω_e, ∀t ≥ t_s, with Ω_e 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. Since e(t) = ω(t) − u(t), ∀t ∈ [t_k, t_k ₊ ₁), and u(t) remains u(k) in [t_k, t_k ₊ ₁), we have . From (34), we obtainSince , , and are continuous, we deduce that is a continuous function. In view of the boundedness of all the closed-loop signals, there exists a constant such that , ∀t ≥ 0. Due to e₁(t_k) = 0 and , the lower bound of the interexecution time satisfies , and the Zeno phenomenon can be avoided.

4. Two Simulation Examples

Example 1. Consider a reduced-order model of the boiler-turbine unit: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 y_d(t) = 0.1 sin(t). Clearly, system (48) is a special case of system (13), and Assumptions1–3 are satisfied with p = 11/9, , , , and . In order to achieve the practical finite-time tracking and satisfy the actual demands, the time-varying output constraint y_d(t)−F₁(t) < y_d(t) < y_d(t) + F₂(t), ∀t ≥ 0, needs to be guaranteed, where F₁(t) and F₂(t) are defined in (13) with t_s = 20s, F₁₀ = F₂₀ = 1.2, and F_1s = F_2s = 0.06.
According to control design in Section 3.3, by setting e = −y_d, h₁ = (F₁ + e) (F₂−e), ξ₁ = e/h₁, ξ₂ = x₂, , , z₁ = ξ₁, and z₂ = ξ₂−α₁, the event-triggered controller and adaptive laws are obtained aswhereFigures 3 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, , , and u are bounded, the asymmetric time-varying output constraint is not violated, and then the practical finite-time tracking can be fulfilled.

Example 2. Consider a numerical examplewhere 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 y_d(t) = 0.4 sin(t). Clearly, system (48) is a special case of system (11), and Assumptions1–3 are satisfied with p = 21/19, , , and . In order to achieve the practical finite-time tracking and satisfy the actual demands, the time-varying output constraint y_d(t)−F₁(t) < y(t) < y_d(t) + F₂(t), ∀t ≥ 0, needs to be guaranteed, where F₁(t) and F₂(t) are defined in (13) with t_s = 40s, F₁₀ = F₂₀ = 2.4, and F_1s = F_2s = 0.05.
Similar to Example 1, the event-triggered controller and adaptive laws are obtained aswhereFigures 5 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.

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, p_i(t) ≥ 1 in Assumption 1 is assumed. When p_i(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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Taishan Scholar Project of Shandong Province of China under grant ts201712040, the National Natural Science Foundation of China under grant 62073186, the National Key R&D Program of China under grant 2018YFC2001700, and the Natural Science Foundation of Shandong Province under grant 501100007129.

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Copyright © 2022 Fan Liu and You Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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