Dynamic event-triggered adaptive fuzzy asymptotic tracking of constrained nonlinear systems with applications

Abstract

This article investigates the fuzzy event-triggered asymptotic tracking control of nonlinear systems with full state constraints. First, to approximate unknown nonlinearity, the fuzzy logic system is employed. Second, the “explosion of complexity” issue is tackled by developing an improved dynamic surface technique. Additionally, a new Barrier Lyapunov function is introduced to make sure state constraints are not violated. Particularly, an efficient dynamic event-triggered control scheme is presented by introducing an auxiliary dynamic variable, which can not only greatly save communication resources but also achieve asymptotic tracking. Ultimately, validations are illustrated through two simulation examples.

Keywords

uncertain nonlinear system asymptotic tracking dynamic surface control barrier Lyapunov function dynamic event-triggered control

1. Introduction

With the prevalence of nonlinear control systems in practical engineering, the stabilization and tracking control of such systems has become a prominent field of research. For nonlinear systems (NSs) with uncertain parameters, Zhang et al. (2015) proposed an adaptive control algorithm by using the robust technique. For uncertain NSs Zhou et al. (2006), the challenge of unknown dead-zone nonlinearities was addressed by using smooth inverse functions. For the case of unknown actuator nonlinearities and bounded disturbances, the tracking problem was studied for NSs in Zhang et al. (2014). Noted that the above results become invalid when the nonlinear functions cannot be linearly parameterized or are completely unknown. To overcome the above difficulties, Wang and Mendel (1992) first proposed the notion of fuzzy basis function and applied it to approximate unknown nonlinearities of systems. Recently, numerous meaningful results were proposed by combining backstepping technology with fuzzy logic system (FLS) (Li et al. (2019a); Xu et al. (2023); Wang et al. (2021); Tong et al. (2020); Deng et al. (2024); Yao et al. (2023); Sun and Li (2023)). However, it should be emphasized that as the system order grows, the derivation of controller becomes complicated. In traditional backstepping-based design, the implementations often frequently encounter the issue of “explosion of complexity,” which arises from the repetitive differentiation of virtual controllers.

Subsequently, the adaptive dynamic surface control (DSC) was first introduced in Swaroop et al. (1997), which was recognized as an efficient tool to overcome the above difficulties. In Ma and Ma (2020), the controller was designed for NSs based on adaptive DSC. To alleviate the effects of filter performance on closed-loop stability, a modified neural DSC method was introduced in Sun et al. (2015). Furthermore, the DSC technique by using FLS for NSs were presented in Liu et al. (2023); Hu and Liu (2023); Wang and Wang (2014). Although some important works have been done with the DSC technique, the above literature is limited to bounded tracking. In order to overcome the aforementioned deficiency, a novel state-feedback controller was designed by utilizing DSC scheme in Liu (2018), where the asymptotic tracking was realized. Nevertheless, the method in Liu (2018) is available when the control coefficients of the system are known. Thus, further research is needed to build a DSC strategy for NSs with unknown virtual control coefficients (UVCC) that can accomplish asymptotic tracking.

In real-world applications, the controller always needs to maintain certain performance indicators, which requires that system states can only change in a specific boundary. Otherwise, violating state constraints may degrade system performance. To solve this problem, Tee et al. (2009) first introduced the Barrier Lyapunov function (BLF) to resolve output constraints for the first time. Since then, as a tool that can effectively avoid violating state constraints, BLF has attracted extensive research attention. Based on BLF, Sun et al. (2018) investigated full state constraints for stochastic switched system. Inspired by the early works Sun et al. (2018) and Tee et al. (2009), the tangent barrier Lyapunov function (BLF-Tans) was adopted to ensure each state were not exceed the boundary for switched NSs in Liu et al. (2022). Abundant results on state constraints control for NS have been made in Song et al. (2023); Li et al. (2019b, 2022); Wu et al. (2023); Liu et al. (2021). Note that the controllers in the above control frameworks adopted time-sampling.

In practical engineering, it is essential to consider two significant factors: control energy consumption and communication load. Event-triggered control (ETC) strategy as an alternative way has appeared, which mitigates unnecessary transmission. Motivated by this fact, for nonlinear continuous-time systems, adaptive ETC strategy was studied in Li and Yang (2017). A novel saturated ETC scheme was presented avoid continuous communication in Wang et al. (2023) for constrained NSs. Although significant progress has been made, it is worth noting that numerous ETC results on uncertain strict-feedback NSs adopt static event conditions. Recently, there has been a growing interest in dynamic ETC methods across different fields (Xing and Wen (2023); Cao et al. (2024, 2022); Ge et al. (2020); Xia et al. (2023)). Compared to static ETC control, dynamic ETC approaches tend to yield larger average inter-event intervals. However, there are relatively few dynamic ETC results for strict-feedback uncertain NSs, and most of current researches mainly concentrates on bounded-error tracking. Motivated by the above perspectives, a dynamic ETC-based asymptotic tracking algorithm for nonlinear constrained systems with UVCC is presented. The contributions are summarized as

(i) Unlike Liu (2018), where the states were unconstrained and only bounded tracking errors were attained. In this paper, a novel BLF is devised by incorporating the lower bound of the UVCC to satisfy full state constraints. Meanwhile, the developed controller developed can also realize asymptotic tracking by utilizing backstepping approach and DSC technique.

(ii) In contrast to the triggering function with a constant term discussed in Wang et al. (2023), which only leads to a bounded tracking error, this work introduces an auxiliary dynamic variable that ensures a zero tracking error. Furthermore, unlike the time-triggered control results in Li et al. (2019a); Jin and Li (2021) where the data transmitted periodically with the fixed sampling period, the presented dynamic ETC scheme updates the states and control laws by using only the triggered samples, thereby significantly reducing the communication resources required.

(iii) Different from Liu (2018), we design a new first-order filter for nonlinear constrained system with UVCC in this paper, which can not only tackle the “explosion of complexity” issue, but also offset the influence of boundary layer error.

2. Problem statement and preliminaries

2.1. Problem formulation

Consider nonlinear strict-feedback systems as follows

\begin{aligned} {\dot{x}}_{i} & = f_{i} ({\bar{x}}_{i}) + g_{i} ({\bar{x}}_{i}) x_{i + 1}, \\ {\dot{x}}_{n} & = f_{n} (x_{n}) + g_{n} (x_{n}) u, \\ y & = x_{1}, \end{aligned}

(1)

where

{\bar{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in R^{i}

, i = 1, …, n is state, u ∈ R and y ∈ R are control input and output,

f_{i} ({\bar{x}}_{i}) \in R

is unknown nonlinear function,

g_{i} ({\bar{x}}_{i}) \in R

represent unknown control coefficients. x_i are required to satisfy

| x_{i} | < k_{r_{i}}

k_{r_{i}} > 0

is a constant.

2.1.1. Control objective

Construct a dynamic ETC tracking control method for (1) to ensure the boundedness of each signal and the state constraints are not broken, while asymptotic tracking can be achieved.

Assumption 1

The desired trajectory y_d(t) and its derivatives ${\dot{y}}_{d} (t)$ , $y_{d}^{(j)} (t)$ are continuous and bounded, that is, $| y_{d} (t) | \leq D_{0} < k_{r_{1}}$ , and $| y_{d}^{(j)} (t) | \leq D_{j}$ , j = 1, 2, …, n, where D₀, D₁, …, D_n are positive constants.

Assumption 2

The sign of g_i is known. There exists unknown constant ${\underset{̲}{g}}_{i} > 0$ satisfying ${\underset{̲}{g}}_{i} \leq | g_{i} ({\bar{x}}_{i}) |$ . Without losing generality, assume $g_{i} ({\bar{x}}_{i}) > 0$ .

Lemma 1

Zhang et al. (2024): For any variable $a \in R$ and any scalar function $σ (t) : R_{\geq 0} \to R^{+}$ , it holds

0 \leq | a | - \frac{a^{2}}{\sqrt{a^{2} + σ^{2} (t)}} < σ (t) .

(2)

Lemma 2

Polycarpou and Weaver (1995): There are bounded and positive continuous function ɛ and κ ∈ R, which satisfy

0 \leq | κ | - κ \tanh (\frac{κ}{ε}) \leq ϱ ε,

(3)

where ϱ = 0.2785.

Lemma 3

Xu et al. (2023): There exists continuous function f(X) and positive constant ϱ, under the ideal constant weights, for all X ∈ Ω_X satisfy

\sup | f (X) - W^{T} ψ (X) | = | ζ (X) | \leq ϱ .

(4)

3. Controller design and stability analysis

3.1. Controller design

Step 1: Introduce the state changes as z₁ = x₁ − y_d, the derivative can be described as

{\dot{z}}_{1} = g_{1} (x_{2} - α_{1}) + g_{1} α_{1} + f_{1} - {\dot{y}}_{d} .

(5)

We choose the symmetric BLF candidate V₁ as

V_{1} = \frac{1}{2 {\underset{̲}{g}}_{1}} \log \frac{k_{a_{1}}^{2}}{k_{a_{1}}^{2} - z_{1}^{2}} + \frac{1}{2 γ_{1}} {\tilde{θ}}_{1}^{2},

(6)

where γ₁ > 0 is a known parameter,

{\hat{θ}}_{1}

is the estimation of θ₁, and

{\tilde{θ}}_{1}

θ_{1} - {\hat{θ}}_{1}

is the estimation error. In what follows, we define a compact set

Ω_{z_{1}} ≔ {| z_{1} | < k_{a_{1}}}

k_{a_{1}} = k_{r_{1}} - D_{0}

is a constant with

k_{a_{1}} > 0

The derivative of V₁ is

\dot{V_{1}} = \frac{h_{z_{1}}}{{\underset{̲}{g}}_{1}} [g_{1} z_{2} + g_{1} α_{1} + {\underset{̲}{g}}_{1} e_{1} + H_{1}] - \frac{1}{γ_{1}} {\tilde{θ}}_{1} {\dot{\hat{θ}}}_{1},

(7)

where

h_{z_{1}} = \frac{z_{1}}{k_{a_{1}}^{2} - z_{1}^{2}}

, H₁ is defined as

H_{1} = g_{1} e_{1} - {\underset{̲}{g}}_{1} e_{1} + f_{1} - {\dot{y}}_{d}

To avoid “explosion of complexity,” a novel filter virtual controller $s_{i}^{*}$ is designed, which is obtained by passing α_i through the first-order filter

\begin{aligned} τ_{i} {\dot{s}}_{i}^{*} & = - e_{i} - τ_{i} {\hat{Ξ}}_{i} \tanh (\frac{e_{i}}{ϵ}) - τ_{i} h_{z_{i}}, \\ s_{i}^{*} (0) & = α_{i} (0), \end{aligned}

(8)

where

{\hat{Ξ}}_{i}

indicates an estimation of Ξ_i, τ_i is a time constant,

e_{i} ≔ s_{i}^{*} - α_{i}

indicates the ith boundary layer error, ϵ denotes a differentiable function satisfies

ϵ > 0, \int_{0}^{t} ϵ (s) d s \leq \bar{ϵ}, t \geq 0,

(9)

where

\bar{ϵ}

is a constant with

\bar{ϵ} > 0

Remark 1

Unlike the traditional adaptive backstepping technique, the advantage of DSC scheme is that the filtered signal ${\dot{s}}^{*}$ in each step can overcome the obstacles of repeated derivation, which greatly simplifies the calculation steps and solves complexity explosion issue.

Since H₁ includes unknown functions, the FLS is employed to approximate H₁

H_{1} = ξ_{1}^{T} ϕ_{1} (X_{1}),

(10)

where

ξ_{1} = {[W_{1}^{T}, ζ_{1} (X_{1})]}^{T}

ϕ_{1} (X_{1}) = {[ψ_{1} (X_{1}), 1]}^{T}

and

X_{1} = {[x_{1}, y_{d}, {\dot{y}}_{d}]}^{T}

Then, we define $θ_{1} = ‖ ξ_{1} ‖ / {\underset{̲}{g}}_{1}$ , based on Lemma 1, $1 / {\underset{̲}{g}}_{1} h_{z_{1}} ξ_{1}^{T} ϕ_{1} (X_{1})$ can be rewritten as

\begin{aligned} \frac{1}{{\underset{̲}{g}}_{1}} h_{z_{1}} ξ_{1}^{T} ϕ_{1} (X_{1}) & \leq θ_{1} | h_{z_{1}} | ‖ ϕ_{1} ({\bar{X}}_{1}) ‖, \\ \leq \frac{θ_{1} h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1}}{\sqrt{h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1} + σ_{1}^{2}}} + σ_{1} θ_{1}, \end{aligned}

(11)

where σ₁ satisfy

\lim_{t \to \infty} \int_{t 0}^{t} σ_{1} (s) d s \leq {\bar{σ}}_{1} < \infty

with

{\bar{σ}}_{1} > 0

Based on (10) and (11), (7) becomes

\begin{aligned} {\dot{V}}_{1} \leq & - k_{1} h_{z_{1}}^{2} + \frac{g_{1}}{{\underset{̲}{g}}_{1}} h_{z_{1}} z_{2} + \frac{g_{1}}{{\underset{̲}{g}}_{1}} h_{z_{1}} α_{1} + h_{z_{1}} e_{1} + h_{z_{1}} υ_{1} \\ + σ_{1} θ_{1} + \frac{1}{γ_{1}} {\tilde{θ}}_{1} (\frac{γ_{1} h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1}}{\sqrt{h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1} + σ_{1}^{2}}} - {\dot{\hat{θ}}}_{1}), \end{aligned}

(12)

where

υ_{1} = k_{1} h_{z_{1}} + {\hat{θ}}_{1} h_{z_{1}} ϕ_{1}^{T} ϕ_{1} / \sqrt{h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1} + σ_{1}^{2}}

, k₁ is a positive constant.

It is apparent that

\begin{aligned} \frac{g_{1}}{{\underset{̲}{g}}_{1}} h_{z_{1}} α_{1} + h_{z_{1}} υ_{1} = & - \frac{g_{1}}{{\underset{̲}{g}}_{1}} \frac{h_{z_{1}}^{2} υ_{1}^{2}}{\sqrt{h_{z_{1}}^{2} υ_{1}^{2} + σ_{1}^{2}}} + h_{z_{1}} υ_{1}, \\ \leq - \frac{h_{z_{1}}^{2} υ_{1}^{2}}{| h_{z_{1}} | | υ_{1} | + σ_{1}} + h_{z_{1}} υ_{1}, \\ \leq σ_{1} . \end{aligned}

(13)

Construct the virtual controller and update law as

α_{1} = - \frac{h_{z_{1}} υ_{1}^{2}}{\sqrt{h_{z_{1}}^{2} υ_{1}^{2} + σ_{1}^{2}}},

(14)

{\dot{\hat{θ}}}_{1} = \frac{γ_{1} h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1}}{\sqrt{h_{z_{1}}^{2} ϕ_{1}^{T} ϕ_{1} + σ_{1}^{2}}} - γ_{1} σ_{1} {\hat{θ}}_{1} .

(15)

By substituting (13)–(15) into (12), one can get

{\dot{V}}_{1} \leq - k_{1} h_{z_{1}}^{2} + h_{z_{1}} e_{1} + (1 + θ_{1}) σ_{1} + σ_{1} {\tilde{θ}}_{1} {\hat{θ}}_{1} + \frac{g_{1}}{{\underset{̲}{g}}_{1}} h_{z_{1}} z_{2} .

(16)

Step i (2 ≤ i ≤ n − 1): The ith surface error is $z_{i} = x_{i} - s_{i - 1}^{*}$ , and the derivative of z_i is

\begin{aligned} {\dot{z}}_{i} = & g_{i} (x_{i + 1} - α_{i}) + g_{i} α_{i} + f_{i} + \frac{e_{i - 1}}{τ_{i - 1}} \\ + {\hat{Ξ}}_{i - 1} \tanh (\frac{e_{i - 1}}{ϵ}) + h_{z_{i - 1}} . \end{aligned}

(17)

Constructing the symmetric BLF candidate V_i as

V_{i} = V_{i - 1} + \frac{1}{2 {\underset{̲}{g}}_{i}} \log \frac{k_{a_{i}}^{2}}{k_{a_{i}}^{2} - z_{i}^{2}} + \frac{1}{2 γ_{i}} {\tilde{θ}}_{i}^{2},

(18)

where γ_i > 0 is known parameters. We define a compact set

Ω_{z_{i}} ≔ {| z_{i} | < k_{a_{i}}}

k_{a_{i}} = k_{r_{i}} - D_{i - 1} + D_{e}

is a constant with

k_{a_{i}} > 0

, D_e > 0 is a constant.

Then, we arrive at

\begin{align} \dot{V_{i}} & \leq \sum_{j = 1}^{i - 1} [- k_{j} h_{z_{j}}^{2} + h_{z_{j}} e_{j} + (1 + θ_{j}) σ_{j} + σ_{j} {\tilde{θ}}_{j} {\hat{θ}}_{j}] \\ + \frac{h_{z_{i}}}{{\underset{̲}{g}}_{i}} [g_{i} z_{i + 1} + {\underset{̲}{g}}_{i} e_{i} + g_{i} α_{i} + H_{i}] - \frac{1}{γ_{i}} {\tilde{θ}}_{i} {\dot{\hat{θ}}}_{i} \end{align},

(19)

where

h_{z_{i}} = \frac{z_{i}}{k_{a_{i}}^{2} - z_{i}^{2}}

, H_i is defined as

H_{i} = - {\underset{̲}{g}}_{i} e_{i} + g_{i} e_{i} + f_{i} + h_{z_{i - 1}} + \frac{e_{i - 1}}{τ_{i - 1}} + {\hat{Ξ}}_{i - 1} \tanh (e_{i - 1} / ϵ) + \frac{g_{i - 1}}{{\underset{̲}{g}}_{i - 1}} {\underset{̲}{g}}_{i} h_{z_{i - 1}}

Similar to (10) and (11), one has

\begin{align} \dot{V_{i}} & \leq \sum_{j = 1}^{i} (- k_{j} h_{z_{j}}^{2} + h_{z_{j}} e_{j}) + \sum_{j = 1}^{i - 1} (1 + θ_{j}) σ_{j} + h_{z_{i}} υ_{i} \\ + \frac{g_{i}}{{\underset{̲}{g}}_{i}} h_{z_{i}} z_{i + 1} + \frac{g_{i}}{{\underset{̲}{g}}_{i}} h_{z_{i}} α_{i} + θ_{i} σ_{i} + \sum_{j = 1}^{i - 1} σ_{j} {\tilde{θ}}_{j} {\hat{θ}}_{j} \\ + \frac{1}{γ_{i}} {\tilde{θ}}_{i} (\frac{γ_{i} h_{z_{i}}^{2} ϕ_{i}^{T} ϕ_{i}}{\sqrt{h_{z_{i}}^{2} ϕ_{i}^{T} ϕ_{i} + σ_{i}^{2}}} - {\dot{\hat{θ}}}_{i}) \end{align},

(20)

where

υ_{i} = k_{i} h_{z_{i}} + \frac{{\hat{θ}}_{i} h_{z_{i}} ϕ_{i}^{T} ϕ_{i}}{\sqrt{h_{z_{i}}^{2} ϕ_{i}^{T} ϕ_{i} + σ_{i}^{2}}}

, k_i > 0 is a constant.

Design the virtual controller and update law as

α_{i} = - \frac{h_{z_{i}} υ_{i}^{2}}{\sqrt{h_{z_{i}}^{2} υ_{i}^{2} + σ_{i}^{2}}},

(21)

{\dot{\hat{θ}}}_{i} = \frac{γ_{i} h_{z_{i}}^{2} ϕ_{i}^{T} ϕ_{i}}{\sqrt{h_{z_{i}}^{2} ϕ_{i}^{T} ϕ_{i} + σ_{i}^{2}}} - γ_{i} σ_{i} {\hat{θ}}_{i} .

(22)

Refer to step 1, it is clear that $\frac{g_{i}}{{\underset{̲}{g}}_{i}} h_{z_{i}} α_{i} + h_{z_{i}} υ_{i} \leq σ_{i}$ , substituting (21)–(22) into (20), it gives

\begin{aligned} {\dot{V}}_{i} \leq & \sum_{j = 1}^{i} [- k_{j} h_{z_{j}}^{2} + h_{z_{j}} e_{j} + (1 + θ_{j}) σ_{j} + σ_{j} {\tilde{θ}}_{j} {\hat{θ}}_{j}] \\ + \frac{g_{i}}{{\underset{̲}{g}}_{i}} h_{z_{i}} z_{i + 1} . \end{aligned}

(23)

Step n : The nth error can be defined as z_n = x_n − s_n−1, ${\dot{z}}_{n}$ is calculated as

{\dot{z}}_{n} = g_{n} u + f_{n} + \frac{e_{n - 1}}{τ_{n - 1}} + {\hat{Ξ}}_{n - 1} \tanh (\frac{e_{n - 1}}{ϵ}) + h_{z_{n - 1}} .

(24)

Design the symmetric BLF V_n as

V_{n} = V_{n - 1} + \frac{1}{2 {\underset{̲}{g}}_{n}} \log \frac{k_{a_{n}}^{2}}{k_{a_{n}}^{2} - z_{n}^{2}} + \frac{1}{2 γ_{n}} {\tilde{θ}}_{n}^{2},

(25)

where γ_n > 0 is design parameter. Define a compact set

Ω_{z_{n}} ≔ {| z_{n} | < k_{a_{n}}}

k_{a_{n}} = k_{r_{n}} - D_{n - 1} + D_{e}

is a constant with

k_{a_{n}} > 0

Similar to step i, repeat the above steps, it yields

\begin{align} \dot{V_{n}} & \leq \sum_{j = 1}^{n - 1} [- k_{j} h_{z_{j}}^{2} + (1 + θ_{j}) σ_{j} + σ_{j} {\tilde{θ}}_{j} {\hat{θ}}_{j} + h_{z_{j}} e_{j}] \\ + \frac{g_{n}}{{\underset{̲}{g}}_{n}} h_{z_{n}} u + \frac{θ_{n} h_{z_{n}}^{2} ϕ_{n}^{T} ϕ_{n}}{\sqrt{h_{z_{n}}^{2} ϕ_{n}^{T} ϕ_{n} + σ_{n}^{2}}} \\ + θ_{n} σ_{n} - \frac{1}{γ_{n}} {\tilde{θ}}_{n} {\dot{\hat{θ}}}_{n} \end{align} .

(26)

Introduce the dynamic variable η in the following

\dot{η} = - β η + ξ (σ \sum_{i = 1}^{n} k_{i} h_{z i}^{2} - | o (t) |),

(27)

where β > 0, η denotes a differentiable function satisfies

η > 0, \int_{0}^{t} η (s) d s \leq \bar{η}, t \geq 0

\bar{η}

is a constant with

\bar{ϵ} > 0

, η (0) > 0, ξ ∈ [0, 1] is constants, and o(t) = ω(t) − u(t). Based on (27), design the ETC mechanism as

\begin{align} u (t) & = ω (t_{k}), \forall t \in [t_{k}, t_{k + 1}), \end{align}

(28)

\begin{align} t_{k + 1} & = \sup {t > t_{k} | ρ (| o (t) | - σ \sum_{i = 1}^{n} k_{i} h_{z i}^{2}) \leq η} \end{align},

(29)

where t_k (k ∈ z⁺) is the controller update time, ρ > 0 is a constant. When (29) is satisfied, update the control signal to u (t) = ω (t_k+1).

The virtual controller and update law are chosen as

\begin{align} ω (t) & = α_{n}, \end{align}

(30)

\begin{align} α_{n} & = - \frac{h_{z_{n}} υ_{n}^{2}}{\sqrt{h_{z_{n}}^{2} υ_{n}^{2} + σ_{n}^{2}}}, \end{align}

(31)

\begin{align} {\dot{\hat{θ}}}_{n} & = \frac{γ_{n} h_{z_{n}}^{2} ϕ_{n}^{T} ϕ_{n}}{\sqrt{h_{z_{n}}^{2} ϕ_{n}^{T} ϕ_{n} + σ_{n}^{2}}} - γ_{n} σ_{n} {\hat{θ}}_{n} . \end{align}

(32)

Substituting (30)–(32) into (26), refer to step 1, it is clear that $g_{n} / {\underset{̲}{g}}_{n} h_{z_{n}} α_{n} + h_{z_{n}} υ_{n} \leq σ_{n}$ , thus, it gives

\begin{aligned} {\dot{V}}_{n} \leq & \sum_{j = 1}^{n} [- k_{j} h_{z_{j}}^{2} + (1 + θ_{j}) σ_{j}] + \sum_{j = 1}^{n - 1} h_{z_{j}} e_{j} \\ + \sum_{j = 1}^{n} σ_{j} \tilde{θ_{j}} {\hat{θ}}_{j} - \frac{g_{n}}{{\underset{̲}{g}}_{n}} o (t) . \end{aligned}

(33)

Remark 2

It should be emphasized that the nonlinear filter in Liu (2018) contains control coefficients, therefore, it will be invalid for the nonlinear system with UVCC. Thus, a new nonlinear filter is designed in this paper, which can overcome the above limitation. Moreover, the term $- τ_{i} {\hat{Ξ}}_{i} \tanh (e_{i} / ϵ)$ is used to offset the influence of boundary layer error e_i, which is important to achieve zero error tracking.

3.2. Stability analysis

Theorem 1

Given the system (1), under Assumption 1–2, for any initial condition V (0) ≤ p, p > 0 is a constant, and if ρ satisfies ρ > 1 − ξ/β, then the proposed control strategy can realize asymptotic tracking. In addition, all states are constrained and the Zeno behavior will be avoided.

Proof

Describe compact sets

\begin{aligned} Ω_{1} & = {{[y_{d}, {\dot{y}}_{d}, {\ddot{y}}_{d}]}^{T} : y_{d}^{2} + {\dot{y}}_{d}^{2} + {\ddot{y}}_{d}^{2} \leq B_{0}} \\ Ω_{2} & = {V (t) \leq p} \end{aligned}

(34)

where the constant B₀ > 0, Ω₁ × Ω₂ ∈ R⁴ⁿ⁺¹, and functions ɛ (t),

\dot{ε} (t)

are bounded. Thus, there exists constants Ξ_i > 0 satisfying |B_i (⋅)| ≤ Ξ_i on Ω₁ × Ω₂,

| {\hat{Ξ}}_{i} |

indicates an estimation of |Ξ_i|, and |Ξ_i| are unknown.

The derivative of $e_{i} = s_{i}^{*} - α_{i}$ is

{\dot{e}}_{i} = - \frac{e_{i}}{τ_{i}} - {\hat{Ξ}}_{i} \tanh (\frac{e_{i}}{ϵ}) - h_{z_{i}} + B_{i} (\cdot),

(35)

From (27) and (29), we can obtain $\dot{η} \geq - β η - \frac{ξ}{ρ} η = - (β + \frac{ξ}{ρ}) η$ . Thus, it produces

η \geq η (0) e^{- (β + \frac{ξ}{ρ}) t} > 0 .

(36)

The symmetric BLF candidate V can be constructed as

V = V_{n} + \sum_{i = 1}^{n - 1} (\frac{1}{2} e_{i}^{2} + \frac{1}{2 β_{i}} {\tilde{Ξ}}_{i}^{2}) + η .

(37)

where β_i > 0 are design parameters.

Differentiating V yields

\begin{aligned} \dot{V} \leq & \sum_{i = 1}^{n} [- k_{i} h_{z_{i}}^{2} + σ_{i} (1 + θ_{i}) + σ_{i} \tilde{θ_{i}} {\hat{θ}}_{i}] - \frac{g_{n}}{{\underset{̲}{g}}_{n}} o (t) \\ - \sum_{i = 1}^{n - 1} \frac{{\tilde{Ξ}}_{i}}{β_{i}} [{\dot{\hat{Ξ}}}_{i} + \frac{β_{i} e_{i} {\hat{Ξ}}_{i} \tanh (\frac{e_{i}}{ϵ})}{{\tilde{Ξ}}_{i}}] \\ - \sum_{i = 1}^{n - 1} (\frac{e_{i}^{2}}{τ_{i}} - Ξ_{i} | e_{i} |) + \dot{η} . \end{aligned}

(38)

Construct the update law as

{\dot{\hat{Ξ}}}_{i} = β_{i} e_{i} \tanh (\frac{e_{i}}{ϵ}) .

(39)

From Lemma 2, one can deduced that

- e_{i} Ξ_{i} \tanh (\frac{e_{i}}{ϵ}) + e_{i} Ξ_{i} \leq Ξ_{i} ϱ ϵ .

(40)

In accordance to $\tilde{θ} = θ - \hat{θ}$ , we have

{\tilde{θ}}_{i} {\hat{θ}}_{i} = - {\tilde{θ}}_{i}^{2} + {\tilde{θ}}_{i} θ_{i} \leq \frac{θ_{i}^{2}}{4} .

(41)

Putting (39)–(41) into (38), and by (29), there holds $| o (t) | \leq \frac{η}{ρ} + σ \sum_{i = 1}^{n} k_{i} h_{z_{i}}^{2}$ . In addition, from ρ > 1 − ξ/β, it has

\begin{align} \dot{V} & \leq \sum_{i = 1}^{n} [- (1 - σ) k_{i} h_{z_{i}}^{2} + σ_{i} {(1 + \frac{θ_{i}}{2})}^{2}] - \sum_{i = 1}^{n - 1} \frac{e_{i}^{2}}{τ_{i}} \\ + \sum_{i = 1}^{n - 1} Ξ_{i} ϱ ϵ + (\frac{1 - ξ}{ρ} - β) η . \end{align}

(42)

Integrating (42) over [0, t] obtains

\begin{align} V (t) & \leq V (0) - \int_{0}^{t} [(1 - σ) \sum_{i = 1}^{n} k_{i} h_{z_{i}}^{2} (s) + \sum_{i = 1}^{n - 1} \frac{e_{i}^{2} (s)}{τ_{i}}] d (s) \\ + \sum_{i = 1}^{n} {(1 + \frac{θ_{i}}{2})}^{2} \int_{0}^{t} σ_{i} (s) d s + \sum_{i = 1}^{n - 1} Ξ_{i} ϱ \int_{0}^{t} ϵ (s) d s \\ + (\frac{1 - ξ}{ρ} - β) \int_{0}^{t} η (s) d s . \end{align}

(43)

From (43), we can obtain the boundness of V_n (t) on [0, + ∞), which means $z_{i}, s_{i}, {\tilde{θ}}_{i}, η$ are also bounded. As $\tilde{θ} = θ - \hat{θ}$ , we can obtain the boundness of θ. Considering the equation z₁ = x₁ − y_d, and y_d ≤ D₀, there holds $x_{1} \leq | z_{1} | + | y_{d} | < k_{a_{1}} + D_{0}$ . Then, we define $k_{a_{1}} = k_{r_{1}} - D_{0}$ , and we have $x_{1} \leq k_{r_{1}}$ , From the equation $z_{2} = x_{2} - s_{1}^{*}$ , and we define $D_{e} \geq \max {{\bar{e}}_{1}, \dots, {\bar{e}}_{i}}, i = 1, \dots, n$ , and α₁ ≤ D₂, it yields that $x_{2} \leq | z_{2} | + | s_{1}^{*} | < k_{a_{2}} + D_{2} + D_{e}$ , we can get that $x_{2} \leq k_{r_{2}}$ . Similarly, we can infer that $x_{i} \leq k_{r_{i}}$ , which means that state constraints are satisfied and each signal is bounded. Next, the proof that system (1) can realize asymptotic tracking is given below

\begin{aligned} \lim_{t \to \infty} \sum_{i = 1}^{n} (1 - σ) k_{i} \int_{t 0}^{t} h_{z_{i}}^{2} d s \leq V (0) + \sum_{i = 1}^{n - 1} Ξ_{i} ϱ \bar{ϵ} \\ + \sum_{i = 1}^{n} {(1 + \frac{θ_{i}}{2})}^{2} \bar{σ_{i}} + (\frac{1 - ξ}{ρ} - β) \bar{η} . \end{aligned}

(44)

From the boundedness of $k_{i} h_{z_{i}}^{2}$ , by utilizing the Barbalat lemma to (44), the conclusion is given as

\lim_{t \to \infty} z_{i} = 0, i = 1, \dots, n .

(45)

Next, the exclusion of Zeno behavior will be provided. Based on o (t) = ω (t) − u (t), it can derive that

\frac{d}{d t} | o | = \frac{d}{d t} \sqrt{o * o} = s i g n (o) \dot{o} \leq | \dot{ω} | .

(46)

where

\dot{ω}

is a bounded function satisfying

\dot{ω} \leq φ

, φ > 0 is a constant.

As $\lim_{t \to \infty} \sum_{i = 1}^{n} k_{i} h_{z_{i}}^{2} \geq 0$ , according to (36), to make (29) hold, one sufficient condition is got

| o | \leq \frac{η (0)}{ρ} e^{- (β + \frac{ξ}{ρ}) t} .

(47)

Let $t_{k} < \bar{t} < t_{k + 1}$ , we have $e (\bar{t}) = 0$ . Thus, t_k+1 should meet

t_{k + 1} - t_{k} \geq t_{k + 1} - \bar{t} \geq \frac{η (0)}{ρ} e^{- (β + \frac{ξ}{ρ}) t} / φ .

(48)

As t ∈ [t_k, t_k+1) ∈ [0, T], one has

t_{k + 1} - t_{k} \geq \frac{η (0)}{ρ} e^{- (β + \frac{ξ}{ρ}) T} / φ = μ > 0 .

(49)

Thus, the Zeno behavior is removed.

Remark 3

In this article, we design an adaptive dynamic ETC algorithm for nonlinear constrained system. We have mainly achieved the following control objectives: (1) the asymptotic tracking is realized; (2) all states are constrained within a compact set; (3) communication resources can be greatly conserved and the Zeno behavior does not exist; (4) the “explosion of complexity” problem is solved.

4. Simulation example

Example 1

(Numerical example): Consider a second-order nonlinear system as

\begin{aligned} {\dot{x}}_{1} & = x_{1} e^{- 0.75 x_{1}} + (1 + x_{1}^{2}) x_{2} \\ {\dot{x}}_{2} & = 2 x_{2}^{2} x_{1} + (3.5 + \cos (x_{1} x_{2})) u \end{aligned}

(50)

where

g_{1} ({\bar{x}}_{1}) = 1 + x_{1}^{2}

g_{2} ({\bar{x}}_{2}) = 3.5 + \cos (x_{1} x_{2})

are unknown. Therefore, some existing adaptive control methods (Liu (2018); Wang et al. (2019); Ma et al. (2019)) are not applicable to system (1). Select the target trajectory as y_d = 0.5 sin (t).

Select the values of parameters as k₁ = 40, k₂ = 40, γ₁ = 15, γ₂ = 5, β₁ = 1, τ₁ = 0.3, $k_{r_{1}} = 1$ , $k_{r_{2}} = 1.5$ , ρ = σ = β = 0.1, ξ = 1, σ₁ = σ₂ = 0.1e^−0.01t, ϵ = 0.001e^−0.27t, and the initial condition are η (0) = 0.1, x (0) = [0.1,0.1]^T, ${\hat{θ}}_{1} (0) = {\hat{θ}}_{2} (0) = {\hat{Ξ}}_{1} (0) = 0.1$ .

Figures 1 –6 reveal the simulation results. From Figures 1 and 2, the controller can achieve asymptotic tracking. The curves of x₂ is plotted in Figure 3. We can see from Figures 1 and 3 that full state constraints are not broken. In Figure 4, the input u is presented. Figure 5 presents that the adaptive parameters ${\hat{θ}}_{1}$ , ${\hat{θ}}_{2}$ , ${\hat{Ξ}}_{1}$ can converge to positive constants. The time intervals are presented in Figure 6. To make an impartial comparison, the identical design parameters are select in the simulation. In Figure 2, the tracking errors under the control method in Liu (2018) and our presented method are depicted, we can see the devised control scheme outperforms Liu (2018) in terms of tracking performance.

Figure 1.

Tracking trajectory: y (t) and y_d (t).

Figure 2.

Tracking errors: y − y_d.

Figure 3.

State trajectories:x₂.

Figure 4.

Control input: u.

Figure 5.

Adaptive parameters.

Figure 6.

Time interval of triggering events.

Example 2

(Application example): Consider a networked based one-link robotic manipulator in Wang et al. (2019). The form of dynamic model is described as follows

N \ddot{q} + E \dot{q} + m g L \sin q = δ + δ_{d},

(51)

where q,

\dot{q}

, and

\ddot{q}

are the angle, velocity, and acceleration of the angular, respectively. u is the system input. N is the mechanical inertia, E is the friction coefficient, m denotes the load mass, L represents length of the link, g is the gravitational acceleration, δ is the computed-torque control law, and δ_d represents the disturbance. Choose the values of above parameters as N = 1, E = 1, g = 10, m = 1, L = 1.

The system dynamic is expressed as shown below

\begin{aligned} {\dot{x}}_{1} = x_{2}, \\ {\dot{x}}_{2} = - x_{2} - 10 s i n (x_{1}) + u + δ_{d}, \end{aligned}

(52)

where

δ_{d} = x_{1} e^{x_{2}}

is the disturbance.

In this simulation, select the initial values as k₁ = 80, k₂ = 80, γ₁ = 15, γ₂ = 9, β₁ = 0.5, τ₁ = 0.5, $k_{a_{1}} = 1,5$ , $k_{a_{2}} = 2$ , ρ = 0.5, σ = 0.8, ξ = 1, β = 0.1, σ₁ = 0.03e^−0.01t, σ₂ = 0.04e^−0.01t, ϵ(t) = 0.02e^−0.25t. Set the initial values as ${\hat{Ξ}}_{1} (0) = 1$ , η(0) = 0.1, $\hat{θ} (0) = {[1,1]}^{T}$ . In addition, x (0) = [0,0.1]^T.

We can see from Figures 7 and 8 that the designed control scheme can achieve asymptotic tracking. The curve of state variable x₂ is shown in Figure 9. Figure 10 exhibits that the adaptive parameters ${\hat{θ}}_{1}$ , ${\hat{θ}}_{2}$ , ${\hat{Ξ}}_{1}$ . In Figure 11, the curve of input u is presented. As observed in Figures 7 and 9, the full state constraints are achieved. The total event numbers in this paper and Jin and Li (2021) are exhibited in Figures 12 and 13. The proposed dynamic ETC scheme can achieve satisfactory system performance by using 263 events, while the total event number of the static ETC method in Jin and Li (2021) is 840. This proves that compared with the static ETC method, our control method can greatly save communication resources.

Figure 7.

Tracking trajectory: y (t) and y_d (t).

Figure 8.

Tracking errors: y − y_d.

Figure 9.

State trajectories:x₂.

Figure 10.

Adaptive parameters.

Figure 11.

Control input: u.

Figure 12.

Time interval of triggering events.

Figure 13.

Time interval of triggering events in.

5. Conclusions

This paper puts forward a dynamic event-based asymptotic tracking control algorithm for UNSs. Specifically, the fuzzy adaptive DSC method can tackle the calculating explosion problem. By incorporating the lower bound of the UVCC, the new BLF is important to obtain asymptotic tracking and guarantees full state constraints are not violated. Moreover, an efficient dynamic ETC scheme is presented by introducing an auxiliary dynamic variable, and the Zeno phenomenon is perfectly avoided. The feasibility of the developed method is proven in simulation results.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Funds of National Science of China (Grant 62373176 and Grant No. 61973146), in part by the Applied Basic Research Program in Liaoning Province (No. 2022JH2/101300276); and in part by the Key Project of the Educational Department of Liaoning Province (Grant No. JYTZD2023084).

ORCID iD

Yuan-Xin Li

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