Fuzzy Adaptive Fixed-Time Bipartite Consensus Self-Triggered Control for Multi-QUAVs with Deferred Full-State Constraints

Highlights

What are the main findings?

• Development of a universal nonlinear transformation function (UNTF) that eliminates the restrictions of feasibility conditions.
• Design of a zero-free self-triggered mechanism (STM) that removes continuous monitoring requirements and allows on-demand control signal updates.

What is the implication of the main finding?

• Establishment of a unified framework with broad applicability to diverse scenarios, including unconstrained, symmetric/asymmetric constraints, and constant/time-varying constraints.
• Effective balancing between control costs and consensus performance to address practical engineering applications.

Abstract

This paper investigates the interval type-2 (IT2) fuzzy adaptive fixed-time bipartite consensus self-triggered control for multiple quadrotor unmanned aerial vehicles with deferred full-state constraints and input saturation under cooperative-antagonistic interactions. First, a uniform nonlinear transformation function, incorporating a shifting function, is constructed to achieve the deferred asymmetric constraints on the vehicle states and eliminate the restrictions imposed by feasibility criteria. Notably, the proposed framework provides a unified solution for unconstrained, constant/time-varying, and symmetric/asymmetric constraints without necessitating controller reconfiguration. By employing interval type-2 fuzzy logic systems and an improved self-triggered mechanism, an IT2 fuzzy adaptive fixed-time self-triggered controller is designed to allow the control signals to perform on-demand self-updating without the need for additional hardware monitors, effectively mitigating bandwidth over-consumption. Stability analysis indicates that all states in the closed-loop attitude system are fixed-time bounded while strictly adhering to deferred time-varying constraints. Finally, illustrative examples are presented to validate the effectiveness of the proposed control scheme.

1. Introduction

The cooperative advantages of multiple quadrotor unmanned aerial vehicles (MQUAVs) have become increasingly prominent compared with the single-vehicle operations, driven by advancements in intelligent control technologies, with applications spanning both military and civil domains [1]. Note that the airframe attitude stability, a core element of safety control for MQUAVs, determines the stability of the position subsystem and serves as the fundamental guarantee for mission execution reliability [2]. However, the attitude consensus control of MQUAVs encounters several fundamental challenges: inherent strong coupling and parametric uncertainties in vehicle dynamics, and persistent unknown external disturbances [3,4]. Existing investigations indicate that adaptive control technologies incorporating fuzzy logic systems and neural networks have provided viable solutions to address these challenges [5,6,7,8]. Notably, current significant research focuses on consensus control under cooperative interactions; however, limited discussions are conducted on cooperative–antagonistic interactions prevalent in practical network scenarios [9]. Consequently, the bipartite consensus control that employs signed graphs to characterize cooperative–antagonistic interactions has gradually stepped into the limelight, with substantial theoretical advancements reported [10,11,12]. A critical observation is that the above outcomes only guarantee the asymptotic stability of the closed-loop system, that is, the expected performance will be realized as time approaches infinity, while the indeterminate settling time poses potential challenges for real-world applications [13].

Building upon these considerations, several notable adaptive bipartite consensus finite-time control (FTC) schemes were developed, with empirical validations confirming that these strategies can effectively improve the convergence speed and disturbance rejection capabilities [14,15,16]. As demonstrated in [16], a neural bipartite consensus FTC scheme was designed for heterogeneous nonlinear multi-agent systems. Nevertheless, it is well-established that the settling time estimation functions provided by the FTC strategy depend on the initial conditions of the system [17], which is particularly problematic for MQUAV bipartite consensus control, where differential initial states or information unavailability can complicate settling time prediction, thereby impeding the evaluation of consensus performance. To address this obstacle, motivated by pioneering work [18], some excellent fixed-time control results were proposed [19,20,21]. To just name a few, a composite learning fixed-time attitude consensus strategy was presented for MQUAVs in [19], but the potential control singularity problem remains unresolved. In [20], a singularity-avoidance fixed-time control method incorporating a piecewise function was devised to mitigate the control singularity problem arising from the derivation of virtual control laws. However, the aforementioned fixed-time results have not adequately accounted for the state constraints that are prevalent in practical implementations.

Note that the vehicle states inevitably encounter input/output constraints arising from flight safety specifications, uncontrolled environments, and mechanical limitations. Without comprehensive consideration of state constraints, controlled vehicles may experience system instability or even catastrophic failure [22]. This key concern has driven extensive research in full-state constrained control, resulting in significant theoretical advancements [23,24,25,26,27,28,29]. In seminal work [23], an adaptive backstepping control strategy based on the barrier Lyapunov function (BLF) was developed for output-constrained nonlinear systems to force the states to remain within the established boundaries. Building on the groundbreaking result, several adaptive control schemes based on tan-type [24], integral [25], asymmetric integral fractional [26], log-type [27], symmetric/asymmetric time-varying [28,29] BLFs were presented to address the output and full-state constraint problems of nonlinear systems. The necessity of prior knowledge regarding virtual controller boundaries is crucial for the aforementioned BLF-based constructive results, as this implicit feasibility condition complicates both the control synthesis and the stability analysis. To obtain a promising control solution without feasibility conditions, a nonlinear transformation function (NTF)-based control strategy was developed for constrained MIMO nonlinear systems [30]. In [31,32], a log-type nonlinear state-dependent transformation and nonlinear state-dependent mapping strategies were proposed to eliminate the limitations of feasibility criteria. In [33], an NTF-based fuzzy distributed formation control method was reported for constrained MQUAVs. It is vital to emphasize that the aforesaid feasibility-free results generally assume that the imposed constraints are inherently obeyed at the start of system operation [34]. However, practical implementations encounter three critical scenarios: (1) constraint enforcement with incomplete a priori knowledge of the system state, (2) initial state constraints that are not satisfied and require adjustment by the control strategy, and (3) constraints that arise during operation due to external conditions or safety factors. Note that the above constraint strategies cannot directly deal with the bipartite consensus control with deferred full-state constraints, which remains an interesting problem worth exploring.

The prevailing view suggests that hardware-induced on-board bandwidth limitations are widespread, particularly due to the frequent interaction demands of multi-vehicle coordination. However, the conventional time-driven method adopts a periodic communication, resulting in redundant data transmission and limited bandwidth contention among system nodes [35]. To this end, the event-triggered control (ETC) strategy has emerged and wildly investigated, enabling intermittent updates of control instructions while optimizing bandwidth utilization [36,37,38,39,40,41]. Notably, a dynamic ETC strategy was proposed for QUAV attitude control in [38]. In [41], a neural adaptive fixed-time dynamic ETC scheme with improved funnel function was developed for nonlinear systems. It is manifest that implementing the ETC scheme requires continuous supervision of measurement errors, necessitating the introduction of a monitor into the system hardware. Self-triggered control (STC) has been deliberated, which produces the next control execution interval by calculating the current information of the system without monitoring the triggering protocol [42,43]. In [42], a fuzzy neural wavelet adaptive STC scheme was designed for a quadrotor. It is noteworthy that studies on bipartite consensus STC for MQUAVs have not yet been fully developed, particularly regarding the trade-off between consensus performance and bandwidth resources. Therefore, extending existing conclusions to address bipartite consensus control of MQUAVs with input saturation, bandwidth limitations, and deferred full-state constraints represents a meaningful and challenging endeavor.

Motivated by the above discussions, an interval type-2 (IT2) fuzzy adaptive fixed-time bipartite consensus control scheme with a self-triggered mechanism is designed for MQUAVs with deferred full-state constraints and input saturation, which counteracts two key challenges: (1) feasibility constraint limitations and unmanageable deferred constraints existing in BLF-based results and (2) persistent monitoring required in ETC schemes. The major features of this work are summarized below.

• This paper presents an IT2 fuzzy adaptive fixed-time bipartite consensus self-triggered control scheme for MQUAVs with deferred full-state constraints and input saturation by constructing interval type-2 fuzzy logic systems (IT2FLSs). Moreover, both compensation signals are designed to address the impacts of unknown disturbances, approximation errors, and saturation bias.
• Different from previous outcomes [33,34], a uniform NTF (UNTF) is designed by embedding a shifting function to drive the vehicle states to constraint regions within a specified time and eliminates feasibility condition restrictions in traditional BLF-based schemes [25,27,28,29]. Notably, the suggested control strategy can accommodate bipartite consensus control of MQUAVs with unconstrained, constant/time-varying, symmetric/asymmetric cases without modifying the design framework.
• Compared with the existing asymptotic/finite-time control results [11,17], an IT2 fuzzy adaptive fixed-time bipartite consensus self-triggered controller is devised to ensure the fixed-time stability of the closed-loop attitude system and provides a pared-down settling time estimation solution without relying on initial conditions. The implementation of the designed self-triggered mechanism (STM) not only facilitates adjustable on-demand updates of the control signals but also eliminates the continuous monitoring effort required for the ETC results [27,40,44].

2. Problem Formulation

2.1. Graph Theory

Consider a directed graph $\mathbb{G}=(\mathcal{N},\mathcal{E})$ representing the interactions among $\mathcal{Q}$ followers, where $\mathcal{N}=\{1,2,\dots ,\mathcal{Q}\}$ and $\mathcal{E}\subseteq \mathcal{N}\times \mathcal{N}$ are the sets of nodes and edges. Let ${\mathcal{S}}_i=\{s\in \mathcal{N}:(s,i)\in \mathcal{E},s\ne i\}$ denote the neighboring set of node i, where $(s,i)\in \mathcal{E}$ signifies the information flow from node s to node i. $\mathbb{A}=\left[a_{is}\right]\in {\mathbb{R}}^{\mathcal{Q}\times \mathcal{Q}}$ is defined such that $a_{is}\ne 0$ if $(s,i)\in \mathcal{E}$, and $a_{is}=0$ otherwise. Note that $a_{is}>0$ indicates a cooperative relationship, while $a_{is}<0$ represents an antagonistic relationship. The Laplacian matrix associated with $\mathbb{G}$ is $\mathbb{L}=\mathbb{D}-\mathbb{A}\in {\mathbb{R}}^{\mathcal{Q}\times \mathcal{Q}}$, where $\mathbb{D}=\mathrm{diag}\{{\nu }_1,{\nu }_2,\dots ,{\nu }_{\mathcal{Q}}\}$ with ${\nu }_i={\sum }_{s\in {\mathcal{S}}_i}|a_{is}|$. The graph $\mathbb{G}$ is structurally balanced if there is a bipartition $\{{\mathcal{N}}_1,{\mathcal{N}}_2\}$ of $\mathcal{N}$ (i.e., ${\mathcal{N}}_1\cup {\mathcal{N}}_2=\mathcal{N}$, ${\mathcal{N}}_1\cap {\mathcal{N}}_2=\varnothing$) such that $a_{is}\ge 0$ for $\forall i,s\in {\mathcal{N}}_{\eta }(\eta \in \{1,2\})$; $a_{is}\le 0$ for $\forall i\in {\mathcal{N}}_{\eta },s\in {\mathcal{N}}_{\overline{\eta }},\eta \ne \overline{\eta }(\eta ,\overline{\eta }\in \{1,2\})$. The diagonal matrix $\tau =\mathrm{diag}\{{\tau }_1,{\tau }_2,\dots ,{\tau }_{\mathcal{Q}}\}$ denotes the group membership of nodes, where ${\tau }_i=1$ if $i\in {\mathcal{N}}_1$ and ${\tau }_i=-1$ if $i\in {\mathcal{N}}_2$. The augmented graph is $\overline{\mathbb{G}}=(\overline{\mathcal{N}},\overline{\mathcal{E}})$, where $\overline{\mathcal{N}}=\mathcal{N}\cup \left\{0\right\}$ and $\overline{\mathcal{E}}\subseteq \overline{\mathcal{N}}\times \overline{\mathcal{N}}$. $\mathbb{B}=\mathrm{diag}\{|{\mathsf{ı}}_1|,|{\mathsf{ı}}_2|,\dots ,|{\mathsf{ı}}_{\mathcal{Q}}|\}$ denotes the interactions between the leader (labeled as 0) and the followers; ${\mathsf{ı}}_i\ne 0$ signifies that follower i can receive information from the leader, otherwise ${\mathsf{ı}}_i=0$. Here ${\mathsf{ı}}_i>0$ indicate cooperation, while ${\mathsf{ı}}_i<0$ signifies competition. The presence of a spanning tree in a directed graph $\mathbb{G}$ means that at least one node has directed paths to all other nodes.

2.2. Model Description

Consider the MQUAVs consisting of $\mathcal{Q}$ followers, and the attitude dynamics is formulated as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\ddot{\varphi }}_i=& {\displaystyle \frac{{\mathcal{H}}_{i\varphi }}{{\mathcal{O}}_{ix}}}+{\displaystyle \frac{{\dot{\theta }}_i{\dot{\psi }}_i({\mathcal{O}}_{iy}-{\mathcal{O}}_{iz})-{\dot{\varphi }}_i{\mathbb{Q}}_{i\varphi }}{{\mathcal{O}}_{ix}}}+{\mathcal{W}}_{i\varphi }\hfill \\ \hfill {\ddot{\theta }}_i=& {\displaystyle \frac{{\mathcal{H}}_{i\theta }}{{\mathcal{O}}_{iy}}}+{\displaystyle \frac{{\dot{\varphi }}_i{\dot{\psi }}_i({\mathcal{O}}_{iz}-{\mathcal{O}}_{ix})-{\dot{\theta }}_i{\mathbb{Q}}_{i\theta }}{{♭}_{i,y}}}+{\mathcal{W}}_{i\theta }\hfill \\ \hfill {\ddot{\psi }}_i=& {\displaystyle \frac{{\mathcal{H}}_{i\psi }}{{\mathcal{O}}_{iz}}}+{\displaystyle \frac{{\dot{\varphi }}_i{\dot{\theta }}_i({\mathcal{O}}_{ix}-{\mathcal{O}}_{iy})-{\dot{\psi }}_i{\mathbb{Q}}_{i\psi }}{{\mathcal{O}}_{iz}}}+{\mathcal{W}}_{i\psi }\hfill \end{array}\right.\end{array}$$

(1)

where $i=1,2,\dots ,\mathcal{Q}$, $({\varphi }_i,{\theta }_i,{\psi }_i)$ denotes the Euler angles of the ith QUAV. The symbols ${\mathcal{O}}_{ix}$, ${\mathcal{O}}_{iy}$, and ${\mathcal{O}}_{iz}$ denote the inertia coefficients. For $♭\in \{\varphi ,\theta ,\psi \}$, ${\mathcal{H}}_{i♭}$, ${\mathbb{Q}}_{i♭}$, and ${\mathcal{W}}_{i♭}$ represent the control torque, aerodynamic damping coefficient, and the time-varying bounded disturbances, respectively.

For subsequent investigation, defining ${\mathcal{L}}_{i1}={[{\mathcal{L}}_{i11},{\mathcal{L}}_{i21},{\mathcal{L}}_{i31}]}^{\top }={[{\varphi }_i,{\theta }_i,{\psi }_i]}^{\top }$ and ${\mathcal{L}}_{i2}={[{\mathcal{L}}_{i12},{\mathcal{L}}_{i22},{\mathcal{L}}_{i32}]}^{\top }={[{\dot{\varphi }}_i,{\dot{\theta }}_i,{\dot{\psi }}_i]}^{\top }$, the dynamics (1) are reformulated as

$$\left\{\begin{array}{c}{\dot{\mathcal{L}}}_{i1}={\mathcal{L}}_{i2}\hfill \\ {\dot{\mathcal{L}}}_{i2}={\mathcal{F}}_i+{\mathcal{J}}_i+{\mathcal{W}}_i\hfill \\ {\mathcal{Y}}_i={\mathcal{L}}_{i1}\hfill \end{array}\right.$$

(2)

where ${\mathcal{F}}_i={[{\mathcal{F}}_{i1},{\mathcal{F}}_{i2},{\mathcal{F}}_{i3}]}^{\top }=[({\dot{\theta }}_i{\dot{\psi }}_i\left({\mathcal{O}}_{iy}-{\mathcal{O}}_{iz}\right)-{\dot{\varphi }}_i{\mathbb{Q}}_{i\varphi })/{\mathcal{O}}_{ix},({\dot{\varphi }}_i{\dot{\psi }}_i({\mathcal{O}}_{iz}-{\mathcal{O}}_{ix})-{\dot{\theta }}_i{\mathbb{Q}}_{i\theta })$$/{\mathcal{O}}_{iy},({\dot{\varphi }}_i{\dot{\theta }}_i({\mathcal{O}}_{ix}-{\mathcal{O}}_{iy})-{\dot{\psi }}_i{\mathbb{Q}}_{i\psi })/{\mathcal{O}}_{iz}{]}^{\top }, {\mathcal{J}}_i={[{\mathcal{J}}_{i1},{\mathcal{J}}_{i2},{\mathcal{J}}_{i3}]}^{\top }={[{\displaystyle \frac{{\mathcal{H}}_{i\varphi }}{{\mathcal{O}}_{ix}}},{\displaystyle \frac{{\mathcal{H}}_{i\theta }}{{\mathcal{O}}_{iy}}},{\displaystyle \frac{{\mathcal{H}}_{i\psi }}{{\mathcal{O}}_{iz}}}]}^{\top }$, ${\mathcal{W}}_i={[{\mathcal{W}}_{i\varphi },{\mathcal{W}}_{i\theta },{\mathcal{W}}_{i\psi }]}^{\top }={[{\mathcal{W}}_{i1},{\mathcal{W}}_{i2},{\mathcal{W}}_{i3}]}^{\top }$${\mathcal{Y}}_i={[{\mathcal{Y}}_{i1},{\mathcal{Y}}_{i2},{\mathcal{Y}}_{i3}]}^{\top }$ is the attitude output of the ith QUAV. In addition, starting from $t\ge T_{i\alpha \mathsf{ȷ}}$, all states of the controlled vehicle are to be kept in the region ${\mathrm{\Phi }}_{i\alpha \mathsf{ȷ}}:=\{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\in \mathbb{R}|-{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)<{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\le {\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)\}$, $\alpha =1,2,3$, $\mathsf{ȷ}=1,2$, where ${\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)>0$ and ${\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)>0$ denote time-varying functions.

Suppose that the ith vehicle suffers from the following saturation nonlinear constraints:

$${\mathcal{J}}_{i\alpha }=\mathrm{sat}\left({ϵ}_{i\alpha }\right)=\left\{\begin{array}{cc}{ϵ}_{i\alpha },\hfill & |{ϵ}_{i\alpha }|\le {\overline{ϵ}}_{i\alpha }\hfill \\ {\overline{ϵ}}_{i\alpha }\mathrm{sgn}\left({ϵ}_{i\alpha }\right),\hfill & |{ϵ}_{i\alpha }|>{\overline{ϵ}}_{i\alpha }\hfill \end{array}\right.$$

(3)

where ${\overline{ϵ}}_{i\alpha }$ is the known saturation boundary. The following approximation is considered to solve the sharp corner:

$$\begin{array}{c}\hfill {\mathcal{J}}_{i\alpha }={\kappa }_{i\alpha }\left({ϵ}_{i\alpha }\right)+{\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right),\end{array}$$

(4)

where ${\kappa }_{i\alpha }\left({ϵ}_{i\alpha }\right)={\overline{ϵ}}_{i\alpha }tanh({ϵ}_{i\alpha }/{\overline{ϵ}}_{i\alpha })$, ${\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right)={\mathcal{J}}_{i\alpha }-{\kappa }_{i\alpha }\left({ϵ}_{i\alpha }\right)$ satisfying $|{\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right)|\le {\overline{ϵ}}_{i\alpha }(1-tanh\left(1\right))\triangleq {\overline{\mathrm{\Delta }}}_{i\alpha }$.

2.3. IT2FLSs

In this paper, IT2FLSs are deployed to identify the nonlinear terms of controlled vehicles. Therein, the IT2 fuzzy rule base is made up of ℜ rules, with the mth rule being defined as

$${\mathcal{R}}^m:\mathrm{IF}{c}_1\mathrm{is}{\tilde{\diamond }}_1^m,c_2\mathrm{is}{\tilde{\diamond }}_2^{m}3,\dots ,c_p\mathrm{is}{\tilde{\diamond }}_p^m,$$

$$\mathrm{THEN}y\mathrm{is}{\tilde{\mathfrak{Y}}}^m,$$

where $1\le m\le \Re$, $c={[c_1,c_2,\dots ,c_p]}^{\top }$, and $c_n(n=1,2,\dots ,p)$ and y denote the input and output of the IT2FLSs. ${\tilde{\diamond }}_n^m$ and ${\tilde{\mathfrak{Y}}}^m=[{\mathfrak{Y}}_l^m,{\mathfrak{Y}}_r^m]$ represent the antecedent and consequent sets. By used the fuzzy inference engine, the total trigger interval of the rule ${\mathcal{R}}^m$ is ${\mathcal{H}}^m\left(c\right)=[{\underline{h}}^m\left(c\right),{\overline{h}}^m\left(c\right)]$ with ${\underline{h}}^m\left(c\right)={\prod }_{n=1}^p{\underline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)$ and ${\overline{h}}^m\left(c\right)={\prod }_{n=1}^p{\overline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)$. The membership functions ${\underline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)$ and ${\overline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)$ are defined as

$$\begin{array}{c}\hfill {\underline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)={\underline{\mathfrak{V}}}_n^{m}exp\left[-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{c_n-{\mathfrak{K}}_n^m}{{\underline{\mathfrak{U}}}_n^m}}\right)}^2\right],\end{array}$$

(5)

$$\begin{array}{c}\hfill {\overline{\pounds }}_{{\tilde{\diamond }}_n^m}\left(c_n\right)={\overline{\mathfrak{V}}}_n^{m}exp\left[-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{c_n-{\mathfrak{K}}_n^m}{{\overline{\mathfrak{U}}}_n^m}}\right)}^2\right],\end{array}$$

(6)

where ${\underline{\mathfrak{V}}}_n^m$, ${\overline{\mathfrak{V}}}_n^m$, ${\underline{\mathfrak{U}}}_n^m$, ${\overline{\mathfrak{U}}}_n^m$, and ${\mathfrak{K}}_n^m$ are design parameters satisfying $0<{\underline{\mathfrak{V}}}_n^m\le {\overline{\mathfrak{V}}}_n^m<1$ and ${\underline{\mathfrak{U}}}_n^m<{\overline{\mathfrak{U}}}_n^m$.

By virtue of the unnormalized type-reduction strategy in [45], the output of the IT2FLSs is formulated as

$$y={\displaystyle \frac{1}{2}}\sum _{m=1}^{\Re }\left({\underline{h}}^m{\mathfrak{Y}}_l^m+{\overline{h}}^m{\mathfrak{Y}}_r^m\right)=\left[\begin{array}{cc}{\lambda }_l^{\top }& {\lambda }_r^{\top }\end{array}\right]\left[\begin{array}{c}{\displaystyle \frac{{\mu }_l}{2}}\\ {\displaystyle \frac{{\mu }_r}{2}}\end{array}\right]={\lambda }^{\top }\mu ,$$

(7)

where ${\lambda }_l={[{\mathfrak{Y}}_l^1,{\mathfrak{Y}}_l^2,\dots ,{\mathfrak{Y}}_l^{\Re }]}^{\top }$, ${\lambda }_r={[{\mathfrak{Y}}_r^1,{\mathfrak{Y}}_r^2,\dots ,{\mathfrak{Y}}_r^{\Re }]}^{\top }$, ${\mu }_l={[{\underline{h}}^1,{\underline{h}}^2,\dots ,{\underline{h}}^{\Re }]}^{\top }$, and ${\mu }_r={[{\overline{h}}^1,{\overline{h}}^2,\dots ,{\overline{h}}^{\Re }]}^{\top }$.

The control objective is to present an adaptive fixed-time bipartite consensus control scheme under self-triggered communication for MQUAVs subject to deferred full-state constraints and input saturation, ensuring that all states in the closed-loop attitude system are fixed-time bounded and vehicle states are driven within the specified regions ${\mathrm{\Phi }}_{i\alpha \mathsf{ȷ}}:=\{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\in \mathbb{R}|-{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)<{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\le -{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)\}$ after $t>T_{i\alpha \mathsf{ȷ}}$, $\alpha =1,2,3$, $\mathsf{ȷ}=1,2$.

Assumption A1

([33]). The graph $\mathbb{G}$ has a spanning tree with the leader assigned as the root node, and $\mathbb{L}+\mathbb{B}$ is invertible.

Assumption A2

([30]). For $♭=\varphi ,\theta ,\psi$, the leader signals $y_{0♭}$, ${\dot{y}}_{0♭}$, and ${\ddot{y}}_{0♭}$ are continuous and available with the known set ${\mathrm{\Omega }}_{y_{0♭}}=\{y_{0♭}\in \mathbb{R},{\dot{y}}_{0♭}\in \mathbb{R},{\ddot{y}}_{0♭}\in \mathbb{R}|y_{0♭}^2+{\dot{y}}_{0♭}^2+{\ddot{y}}_{0♭}^2\le {\overline{y}}_{0♭}\}$. There are continuous positive functions ${\underline{F}}_{0♭}\left(t\right)$ and ${\overline{F}}_{0♭}\left(t\right)$ satisfying ${\underline{F}}_{0♭}\left(t\right)\le {\underline{\mathcal{C}}}_{i\alpha 1}\left(t\right)$ and ${\overline{F}}_{0♭}\left(t\right)\le {\overline{\mathcal{C}}}_{i\alpha 1}\left(t\right)$ such that $y_{0♭}\in {\mathrm{\Phi }}_{0♭}:=\{y_{0♭}\in \mathbb{R}|-{\underline{F}}_{0♭}\left(t\right)<y_{0♭}\le {\overline{F}}_{0♭}\left(t\right)\}$.

Lemma 1

([18]). For the system $\dot{\Im }\left(t\right)=\mathcal{F}(\Im )$ with $\Im \left(0\right)={\Im }_0$ and the scales ${\mathcal{Q}}_1>0$, ${\mathcal{Q}}_2>0$, $0<c<1$, $\overline{c}>1$, and $P>0$, if the positive-definite function V satisfies $\dot{V}\le -{\mathcal{Q}}_1{V}^c-{\mathcal{Q}}_2{V}^{\overline{c}}+P$, then the system $\dot{\Im }\left(t\right)=\mathcal{F}(\Im )$ is practically fixed-time stable, and the residual set is stated as $\mathrm{\Omega }\in \{V(\Im )\le min\{P/\left({\left((1-q){\mathcal{Q}}_1\right)}^{1/c}\right),P/\left({\left((1-q){\mathcal{Q}}_2\right)}^{1/\overline{c}}\right)\}\}$ with $0<q<1$, and the settling time is derived as $t\le t_{max}=1/\left({\mathcal{Q}}_{1}q(1-c)\right)+1/\left({\mathcal{Q}}_{2}q(\overline{c}-1)\right)$.

Lemma 2

([46]). For any variable $\mathcal{T}$ and any positive uniform continuous bounded function $\mathsf{Ϝ}\left(t\right)$, one has

$$\begin{array}{c}\hfill 0\le |\mathcal{T}|-{\displaystyle \frac{{\mathcal{T}}^2}{\sqrt{{\mathcal{T}}^2+{\mathsf{Ϝ}}^2\left(t\right)}}}<\mathsf{Ϝ}\left(t\right),\end{array}$$

(8)

where $\mathsf{Ϝ}\left(t\right)$ satisfies ${\int }_{t_0}^t\mathsf{Ϝ}\left(\tau \right)d\tau \le \overline{\mathsf{Ϝ}}<\infty$ with $\overline{\mathsf{Ϝ}}$ being any positive parameter.

Lemma 3

([47]). For $\mathcal{X}\ge \overline{\mathcal{X}}$ and $l>1$, one obtains

$$\begin{array}{c}\hfill \overline{\mathcal{X}}{\left(\mathcal{X}-\overline{\mathcal{X}}\right)}^l\le {\displaystyle \frac{l}{1+l}}\left({\mathcal{X}}^{1+l}-{\overline{\mathcal{X}}}^{1+l}\right).\end{array}$$

(9)

Lemma 4

([48]). For $0<\mathcal{A}\le 1$, $\mathcal{B}>1$, and ${♭}_j\in \mathbb{R}$, $j=1,\dots ,†$, one has

$$\begin{array}{c}\hfill {\left(\sum _{j=1}^{†}|{♭}_i|\right)}^{\mathcal{A}}\le \sum _{j=1}^{†}|{♭}_i{|}^{\mathcal{A}},{†}^{1-\mathcal{B}}{\left(\sum _{j=1}^{†}|{♭}_i|\right)}^{\mathcal{B}}\le \sum _{j=1}^{†}{|{♭}_i|}^{\mathcal{B}}.\end{array}$$

(10)

3. Main Results

3.1. Nonlinear Transformation Function

To enforce the deferred full-state constraints on the vehicle states, the following shifting function is introduced:

$${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(t\right)=\left\{\begin{array}{cc}1-(1-\mathsf{ı}){\left({\displaystyle \frac{T_{i\alpha \mathsf{ȷ}}^{*}-t}{T_{i\alpha \mathsf{ȷ}}^{*}}}\right)}^{n+2},\hfill & t\in [0,T_{i\alpha \mathsf{ȷ}}^{*})\hfill \\ 1,\hfill & t\in [T_{i\alpha \mathsf{ȷ}}^{*},\infty )\hfill \end{array}\right.$$

(11)

where $0<T_{i\alpha \mathsf{ȷ}}^{*}\le T_{i\alpha \mathsf{ȷ}}$, and n corresponds to the system order, ı is a small design parameter satisfying

$$\begin{array}{c}\hfill 0<\mathsf{ı}<min\left\{1,{\displaystyle \frac{{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right)}{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\left(0\right)}},{\displaystyle \frac{{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right)}{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\left(0\right)}}\right\}.\end{array}$$

(12)

Remark 1.

Note that the introduced shifting function has the following properties:

1.

${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(t\right)$ is a monotonically increasing function on the interval $[0,T_{i\alpha \mathsf{ȷ}}^{*})$, and ${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(0\right)=\mathsf{ı}$.

2.

${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(t\right)$ reaches its maximum value 1 at $t=T_{i\alpha \mathsf{ȷ}}^{*}$, and remains constant for $t\ge T_{i\alpha \mathsf{ȷ}}^{*}$.

3.

For $t\in [0,\infty )$, ${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}^{\left(a\right)}\left(t\right)$ are ${\mathcal{C}}^{n+1-a}$ and bounded, $a=0,1,\dots ,n+1$.

Construct the UNTF with the shifting function as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill s_{i\alpha \mathsf{ȷ}}=& {\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\hfill \\ \hfill {\mathcal{S}}_{i\alpha \mathsf{ȷ}}=& {\displaystyle \frac{{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right){\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)s_{i\alpha \mathsf{ȷ}}}{({\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)-s_{i\alpha \mathsf{ȷ}})({\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)+s_{i\alpha \mathsf{ȷ}})}}\hfill \end{array}\right.\end{array}$$

(13)

where $\alpha =1,2,3$, $\mathsf{ȷ}=1,2$.

Taking the derivative of ${\mathcal{S}}_{i\alpha \mathsf{ȷ}}$ yields

$$\begin{array}{c}\hfill {\dot{\mathcal{S}}}_{i\alpha \mathsf{ȷ}}={\gamma }_{i\alpha \mathsf{ȷ}}+{\overline{\gamma }}_{i\alpha \mathsf{ȷ}}({\dot{\mathsf{\aleph }}}_{i\alpha \mathsf{ȷ}}{\mathcal{L}}_{i\alpha \mathsf{ȷ}}+{\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}{\dot{\mathcal{L}}}_{i\alpha \mathsf{ȷ}})\end{array}$$

(14)

with ${\gamma }_{i\alpha \mathsf{ȷ}}={\displaystyle \frac{{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}{\mathcal{L}}_{i\alpha \mathsf{ȷ}}^2{\dot{\underline{\mathcal{C}}}}_{i\alpha \mathsf{ȷ}}}{{({\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}+{\mathcal{L}}_{i\alpha \mathsf{ȷ}})}^2({\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}-{\mathcal{L}}_{i\alpha \mathsf{ȷ}})}}-{\displaystyle \frac{{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}{\mathcal{L}}_{i\alpha \mathsf{ȷ}}^2{\dot{\overline{\mathcal{C}}}}_{i\alpha \mathsf{ȷ}}}{({\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}+{\mathcal{L}}_{i\alpha \mathsf{ȷ}}){({\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}-{\mathcal{L}}_{i\alpha \mathsf{ȷ}})}^2}}$ and ${\overline{\gamma }}_{i\alpha \mathsf{ȷ}}={\displaystyle \frac{{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}({\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}+{\mathcal{L}}_{i\alpha \mathsf{ȷ}}^2)}{{({\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}+{\mathcal{L}}_{i\alpha \mathsf{ȷ}})}^2{({\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}-{\mathcal{L}}_{i\alpha \mathsf{ȷ}})}^2}}$.

Then, the system (2) can be rewritten as

$$\left\{\begin{array}{c}{\dot{\mathcal{S}}}_{i1}={\mathrm{\Gamma }}_{i1}+{\mathcal{S}}_{i2}\hfill \\ {\dot{\mathcal{S}}}_{i2}={\mathrm{\Gamma }}_{i2}+{\overline{\gamma }}_{i2}{\mathcal{J}}_i+{\mathcal{W}}_i^{*}\hfill \\ {\mathcal{Y}}_i^{*}={\mathcal{S}}_{i1}\hfill \end{array}\right.$$

(15)

where ${\mathcal{S}}_{i1}={[{\mathcal{S}}_{i11},{\mathcal{S}}_{i21},{\mathcal{S}}_{i31}]}^{\top }$, ${\mathcal{S}}_{i2}={[{\mathcal{S}}_{i12},{\mathcal{S}}_{i22},{\mathcal{S}}_{i32}]}^{\top }$, ${\mathrm{\Gamma }}_{i1}={[{\mathrm{\Gamma }}_{i11},{\mathrm{\Gamma }}_{i21},{\mathrm{\Gamma }}_{i31}]}^{\top }$ with ${\mathrm{\Gamma }}_{i\alpha 1}={\gamma }_{i\alpha 1}+{\overline{\gamma }}_{i\alpha 1}{\dot{\mathsf{\aleph }}}_{i\alpha 1}{\mathcal{L}}_{i\alpha 1}+{\overline{\gamma }}_{i\alpha 1}{\dot{\mathsf{\aleph }}}_{i\alpha 1}{\mathcal{L}}_{i\alpha 2}-{\mathcal{S}}_{i\alpha 2}$, ${\mathrm{\Gamma }}_{i2}={[{\mathrm{\Gamma }}_{i12},{\mathrm{\Gamma }}_{i22},{\mathrm{\Gamma }}_{i32}]}^{\top }$ with ${\mathrm{\Gamma }}_{i\alpha 2}={\gamma }_{i\alpha 2}+{\overline{\gamma }}_{i\alpha 2}{\dot{\mathsf{\aleph }}}_{i\alpha 2}{\mathcal{L}}_{i\alpha 2}+{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}{\mathcal{F}}_{i\alpha }$, ${\overline{\gamma }}_{i2}=\mathrm{diag}\{{\overline{\gamma }}_{i12}{\mathsf{\aleph }}_{i12},{\overline{\gamma }}_{i22}{\mathsf{\aleph }}_{i22},{\overline{\gamma }}_{i32}{\mathsf{\aleph }}_{i32}\}$, ${\mathcal{W}}_i^{*}={\overline{\gamma }}_{i2}{\mathcal{W}}_i={[{\mathcal{W}}_{i1}^{*},{\mathcal{W}}_{i2}^{*},{\mathcal{W}}_{i3}^{*}]}^{\top }$, and ${\mathcal{Y}}_i^{*}={[{\mathcal{Y}}_{i1}^{*},{\mathcal{Y}}_{i2}^{*},{\mathcal{Y}}_{i3}^{*}]}^{\top }$.

3.2. Controller Design

At first, the coordinates transformation is formulated as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\beta }_{i\alpha 1}=& \sum _{s\in {\mathcal{S}}_i}|a_{is}|({\mathcal{Y}}_{i\alpha }^{*}-\mathrm{sign}\left(a_{is}\right){\mathcal{Y}}_{s\alpha }^{*})+|{\mathsf{ı}}_i|({\mathcal{Y}}_{i\alpha }^{*}-\mathrm{sign}\left({\mathsf{ı}}_i\right){\mathcal{Y}}_{0\alpha }^{*})\hfill \\ \hfill {\beta }_{i\alpha 2}=& {\mathcal{S}}_{i\alpha 2}-{\hslash }_{i\alpha 1}-{\breve{\hslash }}_{i\alpha }\hfill \end{array}\right.\end{array}$$

(16)

where $i=1,2,\dots ,\mathcal{Q}$, $\alpha =1,2,3$. ${\mathcal{Y}}_{0\alpha }^{*}={\displaystyle \frac{{\overline{\mathcal{C}}}_{i\alpha 1}\left(t\right){\underline{\mathcal{C}}}_{i\alpha 1}\left(t\right)y_{0\alpha }}{({\overline{\mathcal{C}}}_{i\alpha 1}\left(t\right)-y_{0\alpha })({\underline{\mathcal{C}}}_{i\alpha 1}\left(t\right)+y_{0\alpha })}}$. ${\breve{\hslash }}_{i\alpha }$ is an auxiliary signal, which will be given later.

Step 1. Recalling (2), (13), and (16), we can obtain

$$\begin{array}{cc}\hfill {\dot{\beta }}_{i\alpha 1}=& k_i({\mathrm{\Gamma }}_{i\alpha 1}+{\beta }_{i\alpha 2}+{\hslash }_{i\alpha 1})-\sum _{s\in {\mathcal{S}}_i}{a}_{is}{\mathrm{\Gamma }}_{s\alpha 1}\hfill \\ \hfill & -\sum _{s\in {\mathcal{S}}_i}{a}_{is}{\mathcal{S}}_{s\alpha 2}-{\mathsf{ı}}_i{\dot{\mathcal{Y}}}_{0\alpha }^{*}\hfill \\ \hfill =& k_i({\beta }_{i\alpha 2}+{\hslash }_{i\alpha 1})+f_{i\alpha }-\sum _{s\in {\mathcal{S}}_i}{a}_{is}{\mathcal{S}}_{s\alpha 2}-{\mathsf{ı}}_i{\dot{\mathcal{Y}}}_{0\alpha }^{*},\hfill \end{array}$$

(17)

where $k_i={\nu }_i+|{\mathsf{ı}}_i|$, $f_{i\alpha }=k_i{\mathrm{\Gamma }}_{i\alpha 1}-{\sum }_{s\in {\mathcal{S}}_i}{a}_{is}{\mathrm{\Gamma }}_{s\alpha 1}$. For the unknown item $f_{i\alpha }$, an IT2FLS ${\lambda }_{i\alpha 1}^{\top }{\mu }_{i\alpha 1}\left(c_{i\alpha 1}\right)$ is deployed to approximate $f_{i\alpha }$ such that

$$\begin{array}{c}\hfill f_{i\alpha }={\lambda }_{i\alpha 1}^{\top }{\mu }_{i\alpha 1}\left(c_{i\alpha 1}\right)+o_{i\alpha 1}\left(c_{i\alpha 1}\right),\end{array}$$

(18)

where the estimation error $o_{i\alpha 1}\left(c_{i\alpha 1}\right)$ satisfies $|o_{i\alpha 1}\left(c_{i\alpha 1}\right)|\le {\overline{o}}_{i\alpha 1}$.

Select the Lyapunov function $V_{i\alpha 1}$ as

$$\begin{array}{c}\hfill V_{i\alpha 1}={\displaystyle \frac{1}{2}}{\beta }_{i\alpha 1}^2+{\displaystyle \frac{1}{2}}{\tilde{\lambda }}_{i\alpha 1}^{\top }{\mathrm{\Lambda }}_{i\alpha 1}^{-1}{\tilde{\lambda }}_{i\alpha 1},\end{array}$$

(19)

where ${\tilde{\lambda }}_{i\alpha 1}={\lambda }_{i\alpha 1}-{\widehat{\lambda }}_{i\alpha 1}$, ${\widehat{\lambda }}_{i\alpha 1}$ is the estimation of ${\lambda }_{i\alpha 1}$, and ${\mathrm{\Lambda }}_{i\alpha 1}$ is a positive-definite diagonal matrix.

Calculating the derivative of $V_{i\alpha 1}$ gives

$$\begin{array}{cc}\hfill {\dot{V}}_{i\alpha 1}=& {\beta }_{i\alpha 1}[k_i({\beta }_{i\alpha 2}+{\hslash }_{i\alpha 1})-\sum _{s\in {\mathcal{S}}_i}{a}_{is}{\mathcal{S}}_{s\alpha 2}-{\mathsf{ı}}_i{\dot{\mathcal{Y}}}_{0\alpha }^{*}\hfill \\ \hfill & +{\lambda }_{i\alpha 1}^{\top }{\mu }_{i\alpha 1}\left(c_{i\alpha 1}\right)+o_{i\alpha 1}\left(c_{i\alpha 1}\right)]-{\tilde{\lambda }}_{i\alpha 1}^{\top }{\mathrm{\Lambda }}_{i\alpha 1}^{-1}{\dot{\widehat{\lambda }}}_{i\alpha 1}.\hfill \end{array}$$

(20)

From Young’s inequality, it follows that

$$\begin{array}{c}\hfill {\beta }_{i\alpha 1}[k_i{\beta }_{i\alpha 2}+o_{i\alpha 1}\left(c_{i\alpha 1}\right)]\le {\displaystyle \frac{1}{2}}{\beta }_{i\alpha 2}^2+{\displaystyle \frac{1}{2b_{i\alpha 1}^0}}{\overline{o}}_{i\alpha 1}^2+{\displaystyle \frac{k_i^2+b_{i\alpha 1}^0}{2}}{\beta }_{i\alpha 1}^2.\end{array}$$

(21)

The parameter updating law ${\dot{\widehat{\lambda }}}_{i\alpha 1}$ is devised as

$$\begin{array}{cc}\hfill {\dot{\widehat{\lambda }}}_{i\alpha 1}=& {\mathrm{\Lambda }}_{i\alpha 1}({\beta }_{i\alpha 1}{\mu }_{i\alpha 1}\left(c_{i\alpha 1}\right)-d_{i\alpha 1}{\widehat{\lambda }}_{i\alpha 1}),\hfill \end{array}$$

(22)

where $d_{i\alpha 1}>0$ is a design parameter.

Design the virtual control law as

$$\begin{array}{cc}\hfill {\hslash }_{i\alpha 1}=& -{\displaystyle \frac{1}{k_i}}[{\displaystyle \frac{k_i^2+b_{i\alpha 1}^0}{2}}{\beta }_{i\alpha 1}+b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c-1}tanh\left({\displaystyle \frac{b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}}{h_{i\alpha 1}}}\right)+b_{i\alpha 1}^2{\beta }_{i\alpha 1}^{2\overline{c}-1}\hfill \\ \hfill & +{\widehat{\lambda }}_{i\alpha 1}^{\top }{\mu }_{i\alpha 1}\left(c_{i\alpha 1}\right)-\sum _{s\in {\mathcal{S}}_i}{a}_{is}{\mathcal{S}}_{s\alpha 2}-{\mathsf{ı}}_i{\dot{\mathcal{Y}}}_{0\alpha }^{*}],\hfill \end{array}$$

(23)

where $b_{i\alpha 1}^0$, $b_{i\alpha 1}^1$, $b_{i\alpha 1}^2$, and $h_{i\alpha 1}$ are positive design parameters.

By Lemma 1 in Ref. [49], one has

$$\begin{array}{cc}\hfill -b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}tanh\left({\displaystyle \frac{b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}}{h_{i\alpha 1}}}\right)\le & -|b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}|+0.2785h_{i\alpha 1}\hfill \\ \hfill \le & -b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}+0.2785h_{i\alpha 1}.\hfill \end{array}$$

(24)

Substituting (21)–(24) into (20) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{i\alpha 1}\le & -b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}-b_{i\alpha 1}^2{\beta }_{i\alpha 1}^{2\overline{c}}+{\displaystyle \frac{1}{2}}{\beta }_{i\alpha 2}^2+{\displaystyle \frac{1}{2b_{i\alpha 1}^0}}{\overline{o}}_{i\alpha 1}^2\hfill \\ \hfill & +d_{i\alpha 1}{\tilde{\lambda }}_{i\alpha 1}^{\top }{\widehat{\lambda }}_{i\alpha 1}+0.2785h_{i\alpha 1}.\hfill \end{array}$$

(25)

Step 2. The following auxiliary signal ${\breve{\hslash }}_{i\alpha }$ is embedded to moderate the influence of saturation nonlinearity:

$$\begin{array}{c}\hfill {\dot{\breve{\hslash }}}_{i\alpha }=-{\breve{\hslash }}_{i\alpha }+{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}({\kappa }_{i\alpha }\left({ϵ}_{i\alpha }\right)-{ϵ}_{i\alpha }).\end{array}$$

(26)

From (15) and (16), the derivative of ${\beta }_{i\alpha 2}$ can be expressed as

$$\begin{array}{c}\hfill {\dot{\beta }}_{i\alpha 2}={\overline{f}}_{i\alpha }+{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}({ϵ}_{i\alpha }+{\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right))+{\mathcal{W}}_{i\alpha }^{*}++{\breve{\hslash }}_{i\alpha }.\end{array}$$

(27)

where ${\overline{f}}_{i\alpha }={\mathrm{\Gamma }}_{i\alpha 2}-{\dot{\hslash }}_{i\alpha 1}$. Similar to Step 1, for ${\overline{o}}_{i\alpha 2}>0$, an IT2FLS ${\lambda }_{i\alpha 2}^{\top }{\mu }_{i\alpha 2}\left(c_{i\alpha 2}\right)$ is adopted to identify the nonlinear function ${\mathrm{\Gamma }}_{i\alpha 2}$, i.e.,

$$\begin{array}{c}\hfill {\overline{f}}_{i\alpha }={\lambda }_{i\alpha 2}^{\top }{\mu }_{i\alpha 2}\left(c_{i\alpha 2}\right)+o_{i\alpha 2}\left(c_{i\alpha 2}\right),\end{array}$$

(28)

where $o_{i\alpha 2}\left(c_{i\alpha 2}\right)$ is the approximation error satisfying $|o_{i\alpha 2}\left(c_{i\alpha 2}\right)|\le {\overline{o}}_{i\alpha 2}$.

Choose the Lyapunov function $V_{i\alpha 2}$ as

$$\begin{array}{cc}\hfill V_{i\alpha 2}=& {\displaystyle \frac{1}{2}}{\beta }_{i\alpha 2}^2+{\displaystyle \frac{1}{2}}{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}^2+{\displaystyle \frac{1}{2}}{\tilde{\lambda }}_{i\alpha 2}^{\top }{\mathrm{\Lambda }}_{i\alpha 2}^{-1}{\tilde{\lambda }}_{i\alpha 2},\hfill \end{array}$$

(29)

where ${\mathrm{\Lambda }}_{i\alpha 2}$ is positive-definite diagonal matrix. ${\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}={\mathsf{{\rm Y}}}_{i\alpha 2}-{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}$, ${\tilde{\lambda }}_{i\alpha 2}={\lambda }_{i\alpha 2}-{\widehat{\lambda }}_{i\alpha 2}$, ${\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}$ and ${\widehat{\lambda }}_{i\alpha 2}$ are the estimations of ${\mathsf{{\rm Y}}}_{i\alpha 2}$ and ${\lambda }_{i\alpha 2}$, respectively.

Differentiating $V_{i\alpha 2}$ gives

$$\begin{array}{cc}\hfill {\dot{V}}_{i\alpha 2}=& {\beta }_{i\alpha 2}[{\lambda }_{i\alpha 2}^{\top }{\mu }_{i\alpha 2}\left(c_{i\alpha 2}\right)+{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}{ϵ}_{i\alpha }+o_{i\alpha 2}\left(c_{i\alpha 2}\right)\hfill \\ \hfill & +{\mathcal{W}}_{i\alpha }^{*}+{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}{\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right)+{\breve{\hslash }}_{i\alpha }].\hfill \end{array}$$

(30)

Utilizing Young’s inequality yields

$$\begin{array}{c}\hfill {\beta }_{i\alpha 2}{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}{\mathrm{\Delta }}_{i\alpha }\left({ϵ}_{i\alpha }\right)\le {\displaystyle \frac{{\overline{\gamma }}_{i\alpha 2}^2{\mathsf{\aleph }}_{i\alpha 2}^2}{2}}{\beta }_{i\alpha 2}^2+{\displaystyle \frac{1}{2}}{\overline{\mathrm{\Delta }}}_{i\alpha }^2.\end{array}$$

(31)

Founded on Lemma 2, we can deduce that

$$\begin{array}{cc}\hfill {\beta }_{i\alpha 2}(o_{i\alpha 2}\left(c_{i\alpha 2}\right)+{\mathcal{W}}_{i\alpha }^{*})\le & {\mathsf{{\rm Y}}}_{i\alpha 2}{\varrho }_{i\alpha 2}\left(t\right)+{\displaystyle \frac{{\mathsf{{\rm Y}}}_{i\alpha 2}{\beta }_{i\alpha 2}^2}{\sqrt{{\beta }_{i\alpha 2}^2+{\varrho }_{i\alpha 2}^2\left(t\right)}}},\hfill \end{array}$$

(32)

where ${\mathsf{{\rm Y}}}_{i\alpha 2}={\overline{o}}_{i\alpha 2}+{\overline{\mathcal{W}}}_{i\alpha }^{*}$.

The parameter updating laws are designed as

$$\begin{array}{cc}\hfill {\dot{\widehat{\lambda }}}_{i\alpha 2}=& {\mathrm{\Lambda }}_{i\alpha 2}({\beta }_{i\alpha 2}{\mu }_{i\alpha 2}\left(c_{i\alpha 2}\right)-d_{i\alpha 2}{\widehat{\lambda }}_{i\alpha 2}),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\dot{\widehat{\mathsf{{\rm Y}}}}}_{i\alpha 2}=& {\displaystyle \frac{{\beta }_{i\alpha 2}^2}{\sqrt{{\beta }_{i\alpha 2}^2+{\varrho }_{i\alpha 2}^2\left(t\right)}}}-d_{i\alpha 2}^1{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}-d_{i\alpha 2}^2{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}^{2\overline{c}-1},\hfill \end{array}$$

(34)

To alleviate the burden on on-board bandwidth and improve resource utilization, the following self-triggered mechanism is designed:

$$\begin{array}{cc}\hfill & {ϵ}_{i\alpha }\left(t\right)={ϵ}_{i\alpha }^{*}\left(t_m^i\right),\forall t\in [t_{\alpha ,m}^i,t_{\alpha ,m+1}^i)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & t_{\alpha ,m+1}^i=t_{\alpha ,m}^i+{\displaystyle \frac{g_{i\alpha }tanh\left({ϵ}_{i\alpha }\left(t\right)\right)+{\overline{g}}_{i\alpha }}{max\{{\mathsf{Ϝ}}_{i\alpha },|{\dot{ϵ}}_{i\alpha }\left(t\right)|\}}},\hfill \end{array}$$

(36)

where $t_{\alpha ,m}^i,t_{\alpha ,m+1}^i\in {\mathbb{Z}}^{+}$, $g_{i\alpha }tanh\left({ϵ}_{i\alpha }\left(t\right)\right)+{\overline{g}}_{i\alpha }$ denotes the execution interval between two consecutive trigger signals. $max\{{\mathsf{Ϝ}}_{i\alpha },|{\dot{ϵ}}_{i\alpha }\left(t\right)|\}$ is the change rate of the control interval. $0<g_{i\alpha }<1$, ${\overline{g}}_{i\alpha }>0$, and ${\mathsf{Ϝ}}_{i\alpha }>0$ are design parameters.

It follows from (36) that $|{ϵ}_{i\alpha }\left(t_{\alpha ,m+1}^i\right)-{ϵ}_{i\alpha }\left(t\right)|\le g_{i\alpha }tanh\left({ϵ}_{i\alpha }\left(t\right)\right)+{\overline{g}}_{i\alpha }$, so we have

$$\begin{array}{c}\hfill |{ϵ}_{i\alpha }^{*}\left(t\right)-{ϵ}_{i\alpha }\left(t\right)|\le g_{i\alpha }+{\overline{g}}_{i\alpha }.\end{array}$$

(37)

For the interval $[t_{\alpha ,m}^i,t_{\alpha ,m+1}^i)$, one has

$$\begin{array}{c}\hfill {ϵ}_{i\alpha }\left(t\right)={ϵ}_{i\alpha }^{*}\left(t\right)-{\mathrm{\Delta }}_{i\alpha }\left(t\right)(g_{i\alpha }+{\overline{g}}_{i\alpha }),\end{array}$$

(38)

where ${\mathrm{\Delta }}_{i\alpha }\left(t\right)={\displaystyle \frac{{ϵ}_{i\alpha }^{*}\left(t\right)-{ϵ}_{i\alpha }\left(t\right)}{g_{i\alpha }+{\overline{g}}_{i\alpha }}}$ satisfying $|{\mathrm{\Delta }}_{i\alpha }\left(t\right)|<1$.

The intermediate control signal ${ϵ}_{i\alpha }^{*}$ is designed as

$$\begin{array}{c}\hfill {ϵ}_{i\alpha }^{*}=-{\displaystyle \frac{1}{{\overline{\gamma }}_{i\alpha 2}{\mathsf{\aleph }}_{i\alpha 2}}}\left[{\sigma }_{i\alpha 2}tanh\left({\displaystyle \frac{{\beta }_{i\alpha 2}{\sigma }_{i\alpha 2}}{h_{i\alpha 2}}}\right)+G_{i\alpha }tanh\left({\displaystyle \frac{{\beta }_{i\alpha 2}{G}_{i\alpha }}{h_{i\alpha 2}}}\right)\right]\end{array}$$

(39)

with

$$\begin{array}{cc}\hfill {\sigma }_{i\alpha 2}=& -j_{i\alpha 2}^0{\beta }_{i\alpha 2}-j_{i\alpha 2}^1{\beta }_{i\alpha 2}^{2c-1}-j_{i\alpha 2}^2{\beta }_{i\alpha 2}^{2\overline{c}-1}-{\breve{\hslash }}_{i\alpha 2}\hfill \\ \hfill & -{\displaystyle \frac{{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}{\beta }_{i\alpha 2}}{\sqrt{{\beta }_{i\alpha 2}^2+{\varrho }_{i\alpha 2}^2\left(t\right)}}}-{\widehat{\lambda }}_{i\alpha 2}^{\top }{\mu }_{i\alpha 2}\left(c_{i\alpha 2}\right),\hfill \end{array}$$

(40)

where $j_{i\alpha 2}^0={\displaystyle \frac{{\overline{\gamma }}_{i\alpha 2}^2{\mathsf{\aleph }}_{i\alpha 2}^2+1}{2}}$, $G_{i\alpha }\ge g_{i\alpha }+{\overline{g}}_{i\alpha }$, and $h_{i\alpha 2}$, $j_{i\alpha 2}^1$, and $j_{i\alpha 2}^2$ are positive design parameters, and ${\varrho }_{i\alpha 2}\left(t\right)$ is a time-varying bounded function.

Inserting (33), (34), (38), and (39) into (30) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{i\alpha 2}\le & -{\displaystyle \frac{1}{2}}{\beta }_{i\alpha 2}^2-j_{i\alpha 2}^1{\beta }_{i\alpha 2}^{2c}-j_{i\alpha 2}^2{\beta }_{i\alpha 2}^{2\overline{c}}+d_{i\alpha 2}{\tilde{\lambda }}_{i\alpha 2}^{\top }{\widehat{\lambda }}_{i\alpha 2}+d_{i\alpha 2}^1{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}\hfill \\ \hfill & +d_{i\alpha 2}^2{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}^{2\overline{c}-1}+{\displaystyle \frac{1}{2}}{\overline{\mathrm{\Delta }}}_{i\alpha }^2+{\mathsf{{\rm Y}}}_{i\alpha 2}{\varrho }_{i\alpha 2}\left(t\right)+0.557h_{i\alpha 2}.\hfill \end{array}$$

(41)

Until now, the UNTF-based fuzzy adaptive fixed-time bipartite consensus control scheme is presented for MQUAVs suffering from deferred full-state constraints, and the key results obtained can be summarized as the following theorem.

3.3. Stability Analysis

Theorem 1.

Regarding the MQUAVs with deferred full-state constraints and input saturation, under Assumptions 1 and 2, the suggested UNTF-based adaptive fuzzy self-triggered fixed-time bipartite consensus control strategy, encompassing the virtual control law (23), parameter updating laws (22), (33) and (34), auxiliary signal (26), and actual controller (35), has the following properties: (1) the closed-loop attitude control system is PFTB, (2) all states of the controlled vehicles strictly adhere to the deferred time-varying constraint boundaries after the specified time $T_{i\alpha \mathsf{ȷ}}$, and (3) the devised STM is Zeno-free.

Proof. 

Construct $V={\sum }_{i=1}^{\mathcal{Q}}{\sum }_{\alpha =1}^3{V}_{i\alpha 1}+V_{i\alpha 2}$ as the overall Lyapunov function. Upon review of (25) and (41), differentiating V yields

$$\begin{array}{cc}\hfill \dot{V}\le & \sum _{i=1}^{\mathcal{Q}}\sum _{\alpha =1}^3-b_{i\alpha 1}^1{\beta }_{i\alpha 1}^{2c}-b_{i\alpha 1}^2{\beta }_{i\alpha 1}^{2\overline{c}}-j_{i\alpha 2}^1{\beta }_{i\alpha 2}^{2c}-j_{i\alpha 2}^2{\beta }_{i\alpha 2}^{2\overline{c}}+d_{i\alpha 1}{\tilde{\lambda }}_{i\alpha 1}^{\top }{\widehat{\lambda }}_{i\alpha 1}\hfill \\ \hfill & +d_{i\alpha 2}{\tilde{\lambda }}_{i\alpha 2}^{\top }{\widehat{\lambda }}_{i\alpha 2}+d_{i\alpha 2}^1{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}+d_{i\alpha 2}^2{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}^{2\overline{c}-1}+{\overline{P}}_{i\alpha },\hfill \end{array}$$

(42)

where ${\overline{P}}_{i\alpha }={\displaystyle \frac{1}{2b_{i\alpha 1}^0}}{\overline{o}}_{i\alpha 1}^2+0.2785h_{i\alpha 1}+{\displaystyle \frac{1}{2}}{\overline{\mathrm{\Delta }}}_{i\alpha }^2+{\mathsf{{\rm Y}}}_{i\alpha 2}{\varrho }_{i\alpha 2}\left(t\right)+0.557h_{i\alpha 2}$.

Consider the following facts:

$$\begin{array}{cc}\hfill & {\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\widehat{\lambda }}_{i\alpha \mathsf{ȷ}}\le -{\displaystyle \frac{1}{2}}{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}+{\displaystyle \frac{1}{2}}{\lambda }_{i\alpha \mathsf{ȷ}}^{\top }{\lambda }_{i\alpha \mathsf{ȷ}},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & {\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}\le -{\displaystyle \frac{1}{2}}{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}^2+{\displaystyle \frac{1}{2}}{\mathsf{{\rm Y}}}_{i\alpha 2}^2.\hfill \end{array}$$

(44)

By Lemma 6 in Ref. [50], for $\mathsf{ȷ}=1,2$, one has

$$\begin{array}{cc}\hfill -\sum _{i=1}^{\mathcal{Q}}\sum _{\alpha =1}^3{\displaystyle \frac{{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}}{2}}\le & r_{i\alpha \mathsf{ȷ}}-{\displaystyle \frac{\overline{\vartheta }}{2}}{\left(\sum _{i=1}^{\mathcal{Q}}\sum _{\alpha =1}^3{\displaystyle \frac{1}{2{\vartheta }_{i\alpha }}}{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}\right)}^c\hfill \\ \hfill & -{\displaystyle \frac{\overline{\vartheta }}{2{\left(3\mathcal{Q}\right)}^{\overline{c}-1}}}{\left(\sum _{i=1}^{\mathcal{Q}}\sum _{\alpha =1}^3{\displaystyle \frac{1}{2{\vartheta }_{i\alpha }}}{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}\right)}^{\overline{c}},\hfill \end{array}$$

(45)

where $\overline{\vartheta }=min\left\{{\vartheta }_{i\alpha }\right\}$ with ${\vartheta }_{i\alpha }>1$, $r_{i\alpha \mathsf{ȷ}}={\displaystyle \frac{\overline{\vartheta }}{2}}(1-c)c^{{\displaystyle \frac{c}{1-c}}}+{\displaystyle \frac{\overline{\vartheta }}{2}}{\left({\displaystyle \frac{{\overline{\lambda }}_{i\alpha \mathsf{ȷ}}}{2}}\right)}^{\overline{c}}$ with $||{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}^{\top }{\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}||\le {\overline{\lambda }}_{i\alpha \mathsf{ȷ}}$.

In view of Lemma 2 in Ref. [33], we have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{1}{2}}{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}^2\right)}^c& \le {\displaystyle \frac{1}{2}}{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}^2+(1-c)c^{{\displaystyle \frac{c}{1-c}}}.\hfill \end{array}$$

(46)

Drawing from Lemma 3, one yields

$$\begin{array}{cc}\hfill {\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}^{2\overline{c}-1}& \le {\displaystyle \frac{2\overline{c}-1}{2\overline{c}}}({\mathsf{{\rm Y}}}_{i\alpha 2}^{2\overline{c}}-{\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}^{2\overline{c}}).\hfill \end{array}$$

(47)

Substituting (43)–(47) into (42) and applying Lemma 4 gives

$$\begin{array}{c}\hfill \dot{V}\le -{\mathcal{Q}}_1{V}^c-{\mathcal{Q}}_2{V}^{\overline{c}}+P,\end{array}$$

(48)

where ${\mathcal{Q}}_1=min\{2^c{b}_{i\alpha 1}^1,2^c{j}_{i\alpha 2}^1,d_{i\alpha \mathsf{ȷ}}/2{\lambda }_{max}\left({\mathrm{\Lambda }}_{i\alpha \mathsf{ȷ}}^{-1}\right),d_{i\alpha 2}^1\}$, ${\mathcal{Q}}_2={\left(15\mathcal{Q}\right)}^{1-\overline{c}}min\{2^{\overline{c}}{b}_{i\alpha 1}^2,2^{\overline{c}}{j}_{i\alpha 2}^2,d_{i\alpha \mathsf{ȷ}}/2{\lambda }_{max}\left({\mathrm{\Lambda }}_{i\alpha \mathsf{ȷ}}^{-1}\right),2^{\overline{c}}{d}_{i\alpha 2}^2{\displaystyle \frac{2\overline{c}-1}{2\overline{c}}}\}$, and $P={\overline{P}}_{i\alpha }+{\sum }_{i=1}^{\mathcal{Q}}{\sum }_{\alpha =1}^3({\displaystyle \frac{1}{2}}{d}_{i\alpha 1}{\lambda }_{i\alpha 1}^{\top }{\lambda }_{i\alpha 1}+{\displaystyle \frac{1}{2}}{d}_{i\alpha 2}{\lambda }_{i\alpha 2}^{\top }{\lambda }_{i\alpha 2}+d_{i\alpha 1}(r_{i\alpha 1}+r_{i\alpha 2})+d_{i\alpha 1}^1(1-c)c^{{\displaystyle \frac{c}{1-c}}}+{\displaystyle \frac{2\overline{c}-1}{2\overline{c}}}{d}_{i\alpha 1}^2{\mathsf{{\rm Y}}}_{i\alpha 2}^{2\overline{c}})$.

It can be extrapolated from Lemma 1 that the closed-loop attitude system is practical fixed-time stable, and the state ${\beta }_{i\alpha \mathsf{ȷ}}$, ${\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}$, and ${\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}$ will evolve to the following resign:

$$\begin{array}{c}\hfill \mathrm{\Omega }=\left\{V|V\le min\left\{{\left({\displaystyle \frac{P}{(1-q){\mathcal{Q}}_1}}\right)}^{{\displaystyle \frac{1}{c}}},{\left({\displaystyle \frac{P}{(1-q){\mathcal{Q}}_2}}\right)}^{{\displaystyle \frac{1}{\overline{c}}}}\right\}\right\}\end{array}$$

within a fixed time $t<t_{max}={\displaystyle \frac{1}{q{\mathcal{Q}}_1(1-c)}}+{\displaystyle \frac{1}{q{\mathcal{Q}}_2(\overline{c}-1)}}$. Given that ${\tilde{\lambda }}_{i\alpha \mathsf{ȷ}}={\lambda }_{i\alpha \mathsf{ȷ}}-{\widehat{\lambda }}_{i\alpha \mathsf{ȷ}}$ and ${\widetilde{\mathsf{{\rm Y}}}}_{i\alpha 2}={\mathsf{{\rm Y}}}_{i\alpha 2}-{\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}$, the boundedness of the ${\widehat{\lambda }}_{i\alpha \mathsf{ȷ}}$ and ${\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}$ is guaranteed. Consequently, ${\hslash }_{i\alpha 1}$, ${\breve{\hslash }}_{i\alpha }$, and ${ϵ}_{i\alpha }$ are also bounded. Therefore, all variables in the closed-loop system are fixed-time bounded. Then, one can deduce that ${\beta }_{i\alpha \mathsf{ȷ}}$ will converge into the region

$$\begin{array}{c}\hfill |{\beta }_{i\alpha \mathsf{ȷ}}|\le \left\{\sqrt{2{\left({\displaystyle \frac{P}{(1-q){\mathcal{Q}}_1}}\right)}^{{\displaystyle \frac{1}{c}}}},\sqrt{2{\left({\displaystyle \frac{P}{(1-q){\mathcal{Q}}_2}}\right)}^{{\displaystyle \frac{1}{\overline{c}}}}}\right\}.\end{array}$$

Recalling the definition of the shifting function ${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(t\right)$ and the UNTF ${\mathcal{S}}_{i\alpha \mathsf{ȷ}}$, it can be observed that for $t>T_{i\alpha \mathsf{ȷ}}^{*}$, $s_{i\alpha \mathsf{ȷ}}={\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(t\right){\mathcal{L}}_{i\alpha \mathsf{ȷ}}={\mathcal{L}}_{i\alpha \mathsf{ȷ}}$. Additionally, for any initial state $s_{i\alpha \mathsf{ȷ}}\left(0\right)={\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(0\right){\mathcal{L}}_{i\alpha \mathsf{ȷ}}\left(0\right)\in (-{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right),{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right))$, ${\mathcal{S}}_{i\alpha \mathsf{ȷ}}\to \infty$ if and only if $s_{i\alpha \mathsf{ȷ}}\to -{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)$ or $s_{i\alpha \mathsf{ȷ}}\to {\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)$, provided that ${\mathsf{\aleph }}_{i\alpha \mathsf{ȷ}}\left(0\right)<min\left\{1,{\displaystyle \frac{{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right)}{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\left(0\right)}},{\displaystyle \frac{{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(0\right)}{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\left(0\right)}}\right\}$. Therefore, based on the aforesaid analysis, it is concluded that under initial conditions $s_{i\alpha \mathsf{ȷ}}\left(0\right)\in {\mathrm{\Phi }}_{i\alpha \mathsf{ȷ}}$, the vehicle dynamics ${\mathcal{L}}_{i\alpha \mathsf{ȷ}}$ will evolve into the specified regions ${\mathrm{\Phi }}_{i\alpha \mathsf{ȷ}}:=\{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\in \mathbb{R}|-{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)<{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\le -{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)\}$ for $t\ge T_{i\alpha \mathsf{ȷ}}^{*}$, provided that ${\widehat{\lambda }}_{i\alpha \mathsf{ȷ}}$, ${\widehat{\mathsf{{\rm Y}}}}_{i\alpha 2}$${\hslash }_{i\alpha 1}$, ${\breve{\hslash }}_{i\alpha }$, and ${ϵ}_{i\alpha }$ are bounded. This indicates that the deferred full-state constraints on the vehicle dynamics ${\mathcal{L}}_{i\alpha \mathsf{ȷ}}$ can be achieved.

Define ${\mathcal{Y}}_0^{*}=[{\mathcal{Y}}_{01}^{*},{\mathcal{Y}}_{02}^{*},{\mathcal{Y}}_{03}^{*}]$ and $\tau ={[{\tau }_1,{\tau }_2,\dots ,{\tau }_{\mathcal{Q}}]}^{\top }$, the synchronization error can be expressed as $\beta =(\mathbb{L}+\mathbb{B})({\mathcal{Y}}^{*}-\tau \otimes {\mathcal{Y}}_0^{*})$, where ⊗ represents the Kronecker product,

$$\beta =\left[\begin{array}{ccc}{\beta }_{111}& {\beta }_{121}& {\beta }_{131}\\ ⋮& ⋮& ⋮\\ {\beta }_{\mathcal{Q}11}& {\beta }_{\mathcal{Q}21}& {\beta }_{\mathcal{Q}31}\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}{\mathcal{Y}}^{*}=\left[\begin{array}{ccc}{\mathcal{Y}}_{11}^{*}& {\mathcal{Y}}_{12}^{*}& {\mathcal{Y}}_{13}^{*}\\ ⋮& ⋮& ⋮\\ {\mathcal{Y}}_{\mathcal{Q}1}^{*}& {\mathcal{Y}}_{\mathcal{Q}2}^{*}& {\mathcal{Y}}_{\mathcal{Q}3}^{*}\end{array}\right].$$

Since $\mathbb{L}+\mathbb{B}$ is nonsingular, one can intuitively obtain

$$\begin{array}{c}\hfill ||{\mathcal{Y}}^{*}-\tau \otimes {\mathcal{Y}}_0^{*}||\le {\displaystyle \frac{1}{{\lambda }_{min}(\mathbb{L}+\mathbb{B})}}||\beta ||,\end{array}$$

(49)

where ${\lambda }_{min}(\mathbb{L}+\mathbb{B})$ denotes the minimum singular value of the matrix $\mathbb{L}+\mathbb{B}$. It is evident that the suggested control scheme can ensure that the consensus errors converge to a small neighborhood near the origin within a fixed time.

According to (36), the boundedness of ${\displaystyle \frac{g_{i\alpha }tanh\left({ϵ}_{i\alpha }\left(t\right)\right)+{\overline{g}}_{i\alpha }}{max\{{\mathsf{Ϝ}}_{i\alpha },|{\dot{ϵ}}_{i\alpha }\left(t\right)|\}}}$ is guaranteed since ${ϵ}_{i\alpha }\left(t\right)$ is bounded. Therefore, one can easily extrapolate that the trigger interval $t_{i\alpha }^{*}$ between two consecutive executions satisfies $t_{i\alpha }^{*}=t_{\alpha ,m+1}^i-t_{\alpha ,m}^i>0$, which indicates that the design STM is Zeno-free. This completes the proof. □

Remark 2.

It can be derived from the stability analysis that the ideal dynamic response is closely related to the user-defined control parameters. Specifically, by increasing the values of $b_{i\alpha 1}^1$, $b_{i\alpha 1}^2$, $j_{i\alpha 2}^1$, and $j_{i\alpha 2}^2$ while decreasing $b_{i\alpha 1}^0$, $d_{i\alpha 1}$, $d_{i\alpha 2}$, $d_{i\alpha 2}^1$, $d_{i\alpha 2}^2$, and ${\varrho }_{i\alpha 2}\left(t\right)$ within appropriate ranges, the convergence speed can be improved, and the synchronization errors converge to a smaller region around the origin. Critically, an overemphasis on performance metrics elevates control effort and induces chattering behavior, whereas prudent parameter selection is more conducive to practical implementation.

4. Simulation Results

To verify the feasibility of the proposed fixed-time bipartite consensus control scheme, comprehensive simulation results are presented in this section.

Simulation studies are conducted on the MATLAB R2020a/SIMULINK platform, with the fixed step size set at 0.001 s. Consider MQUAVs consisting of a leader (marked L0) and four followers (marked as F1–F4), and the controlled vehicles are classified into two groups, ${\mathcal{N}}_1=\{\mathrm{F}1,\mathrm{F}3\}$ and ${\mathcal{N}}_2=\{\mathrm{F}2,\mathrm{F}4\}$, based on the cooperative–antagonist relationship, and the topology is shown in Figure 1.

The model parameters and external disturbances are selected as ${\mathcal{O}}_{iz}=0.149\mathrm{kg}·{\mathrm{m}}^2$, ${\mathcal{O}}_{ix}={\mathcal{O}}_{iy}=0.082\mathrm{kg}·{\mathrm{m}}^2$, ${\mathcal{W}}_{i\varphi }={\mathcal{W}}_{i\psi }=0.3cos(\pi t/9)$, ${\mathcal{W}}_{i\theta }=0.2sin(\pi t/10)$, ${\mathbb{Q}}_{i\varphi }={\mathbb{Q}}_{i\theta }={\mathbb{Q}}_{i\psi }=0.6\mathrm{kg}/\mathrm{rad}$. The initial conditions of the vehicles are set to $[{\varphi }_1\left(0\right),{\theta }_1\left(0\right),{\psi }_1\left(0\right)]=[-1.8,2.2,-2.1]$, $[{\varphi }_2\left(0\right),{\theta }_2\left(0\right),{\psi }_2\left(0\right)]=[3,-2.2,1.6]$, $[{\varphi }_3\left(0\right),{\theta }_3\left(0\right),{\psi }_3\left(0\right)]=[1.1,-0.5,2.4]$, and $[{\varphi }_4\left(0\right),{\theta }_4\left(0\right),{\psi }_4\left(0\right)]=[-1.5,-2,2.2]$. The leader trajectory is selected as $[{\mathcal{Y}}_{01},{\mathcal{Y}}_{02},{\mathcal{Y}}_{03}]=[0.4sin\left(2t\right),0.6cos\left(2t\right),0.4cos\left(1.5t\right)]$. The time-varying constraint boundaries are chosen as $[{\underline{\mathcal{C}}}_{i11}\left(t\right),{\underline{\mathcal{C}}}_{i21}\left(t\right),{\underline{\mathcal{C}}}_{i31}\left(t\right)]=[1.7-0.3cos(\pi t/3),2.1-0.3cos(\pi t/3),2-0.2sin(\pi t/2)]$, $[{\overline{\mathcal{C}}}_{i11}\left(t\right),{\overline{\mathcal{C}}}_{i21}\left(t\right),{\overline{\mathcal{C}}}_{i31}\left(t\right)]=[2.3+0.5sin\left(\pi t\right),2.1+0.2sin\left(\pi t\right),2+0.2sin\left(\pi t\right)]$, $[{\underline{\mathcal{C}}}_{i12}\left(t\right),{\underline{\mathcal{C}}}_{i22}\left(t\right),{\underline{\mathcal{C}}}_{i32}\left(t\right)]=[5-1.2cos\left(\pi t\right),6-0.9sin\left(\pi t\right),7-1.2cos(\pi t/2)]$, and $[{\overline{\mathcal{C}}}_{i12}\left(t\right),{\overline{\mathcal{C}}}_{i22}\left(t\right),{\overline{\mathcal{C}}}_{i32}\left(t\right)]=[5+0.8sin(\pi t/3),6+cos(\pi t/3),7+0.8sin(\pi t/2)]$. The saturation threshold is set as ${\overline{ϵ}}_{i\alpha }=50$. The control parameters are chosen as $T_{i\alpha \mathsf{ȷ}}=3$, $c=997/1001$, $\overline{c}=1057/1001$, $d_{i\alpha \mathsf{ȷ}}=0.2$, $b_{i\alpha 1}^0=j_{i\alpha 2}^0=10$, $b_{i\alpha 1}^1=j_{i\alpha 2}^1=9$, $b_{i\alpha 1}^2=j_{i\alpha 2}^2=9$, $h_{i\alpha 1}=0.02$, $h_{i\alpha 2}=3$, ${\varrho }_{i\alpha 2}\left(t\right)=exp(-10t)$, $d_{i\alpha 2}^1=d_{i\alpha 2}^1=0.02$, $G_{i\alpha }=3$, $g_{i\alpha }=0.1$, ${\overline{g}}_{i\alpha }=0.5$, and ${\mathsf{Ϝ}}_{i\alpha }=10$. The fuzzy membership functions are chosen as ${\underline{\pounds }}_{{\tilde{\diamond }}_{i\alpha \mathsf{ȷ}}^m}\left(c_{i\alpha \mathsf{ȷ}}\right)=0.5exp[-{\displaystyle \frac{1}{2}}{(\left(c_{i\alpha \mathsf{ȷ}}-{\mathfrak{K}}_{i\alpha \mathsf{ȷ}}^m\right)/2)}^2]$, ${\overline{\pounds }}_{{\tilde{\diamond }}_{i\alpha \mathsf{ȷ}}^m}\left(c_{i\alpha \mathsf{ȷ}}\right)=0.8exp[-{\displaystyle \frac{1}{2}}{(\left(c_{i\alpha \mathsf{ȷ}}-{\mathfrak{K}}_{i\alpha \mathsf{ȷ}}^m\right)/4)}^2]$, where $m=1,2,\dots ,5$, ${\mathfrak{K}}_{i\alpha \mathsf{ȷ}}^m=10(3-n)$, $c_{i\alpha 1}={[{\mathcal{L}}_{i\alpha 1},{\mathcal{L}}_{i\alpha 2},{\mathcal{L}}_{s\alpha 1},{\mathcal{L}}_{s\alpha 2}]}^{\top }$, $c_{i\alpha 2}={[{\mathcal{L}}_{i12},{\mathcal{L}}_{i22},{\mathcal{L}}_{i32}]}^{\top }$.

The simulation results are presented in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. Specifically, Figure 2 and Figure 3 illustrate the response curves of the Euler angles and the angular velocities for F1–F4, respectively. From Figure 2, it can be observed that the output trajectories of the controlled vehicles realize bipartite consensus control based on the cooperative–antagonistic interactions defined by the topology network. By observing the subplots of Figure 2 and Figure 3, one can identify that the initial attitude angles and angular velocities of the vehicles do not all lie within the given time-varying constraint boundaries upon the system starts operating. By implementing the proposed control scheme, all the states are driven into the region ${\mathrm{\Phi }}_{i\alpha \mathsf{ȷ}}:=\{{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\in \mathbb{R}|-{\underline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)<{\mathcal{L}}_{i\alpha \mathsf{ȷ}}\le -{\overline{\mathcal{C}}}_{i\alpha \mathsf{ȷ}}\left(t\right)\}$ after $T_{i\alpha \mathsf{ȷ}}=3\mathrm{s}$ and adhere to the asymmetric state constraints.

To demonstrate the advantages of the proposed control scheme, a comparative investigation is conducted against the existing BLF-based fuzzy adaptive event-triggered control (FAETC) scheme [44]. Identical initial conditions, external disturbances, and asymmetric time-varying constraint boundaries are maintained for all controlled vehicles to ensure rigorous comparison. Meanwhile, three performance indicators, the integral absolute error (IAE): ${\int }_0^T{\sum }_{\alpha =1}^3|{\beta }_{i\alpha 1}|dt/3$, integral time absolute error (ITAE): ${\int }_0^{T}t{\sum }_{\alpha =1}^3|{\beta }_{i\alpha 1}|dt/3$, and root mean square error (RMSE): $\sqrt{{\int }_0^{T}t{\sum }_{\alpha =1}^3{\beta }_{i\alpha 1}^{2}dt/3T}$, are adopted to quantitatively evaluate the control performance of both schemes. Figure 4 shows trajectories of the synchronization error for F1–F4. Compared to the FAETC scheme, the proposed control strategy exhibits better steady-state performance. The quantitative results summarized in

Table 1. Performance comparison under different control methods.

Control Method	IAE				ITAE				RMSE
	F1	F2	F3	F4	F1	F2	F3	F4	F1	F2	F3	F4
FAETC in [44]	1.816	1.552	2.07	2.105	12.98	12.89	12.88	12.96	0.247	0.2409	0.2587	0.2601
Proposed	0.805	0.7638	0.9405	1.793	2.327	4.111	1.56	8.326	0.0846	0.1233	0.1104	0.2075

and visualized in Figure 5 confirm that the proposed scheme consistently outperforms the FAETC scheme across all performance indicators.

Figure 6, Figure 7, Figure 8 and Figure 9 plot the control input curves of the controlled vehicles, respectively, from which it can be inferred that the control signals achieve non-periodic updating based on the suggested STM, while strictly adhering to the specified saturation constraints. The trigger intervals for the controlled vehicles are presented in Figure 10, Figure 11, Figure 12 and Figure 13. Furthermore, the data transmission rate (DTR) is introduced to quantitatively analyze the advantages of the proposed STM, where $\mathrm{DTR}={\mathcal{J}}_{i\alpha }/N\times 100\%$ with N representing the data transmission count under the time-triggered mechanism (TTM). Additionally, the trigger times under three communication protocols, TTM, the relative threshold event-triggered mechanism (ETM) reported in [44], and the proposed STM, are presented in

Table 2. Triggers times for controlled vehicles.

Items	F1			F2			F3			F4
	${\mathcal{J}}_{\mathbf{11}}$	${\mathcal{J}}_{\mathbf{12}}$	${\mathcal{J}}_{\mathbf{13}}$	${\mathcal{J}}_{\mathbf{21}}$	${\mathcal{J}}_{\mathbf{22}}$	${\mathcal{J}}_{\mathbf{23}}$	${\mathcal{J}}_{\mathbf{31}}$	${\mathcal{J}}_{\mathbf{32}}$	${\mathcal{J}}_{\mathbf{33}}$	${\mathcal{J}}_{\mathbf{41}}$	${\mathcal{J}}_{\mathbf{42}}$	${\mathcal{J}}_{\mathbf{43}}$
TTM	20,001	20,001	20,001	20,001	20,001	20,001	20,001	20,001	20,001	20,001	20,001	20,001
ETM in [44]	1502	1487	978	1455	1600	1075	1512	1006	1006	1564	1619	1114
Proposed	604	1036	550	900	971	690	914	1029	954	986	1141	919
DTR	3.02%	5.18%	2.75%	4.5%	4.85%	3.45%	4.57%	5.14%	4.77%	4.93%	5.7%	4.59%

. It follows from Table 2 that the suggested STC protocol effectively reduces the amount of data transmission compared to the TTM and ETM.

Simulation results indicate that the presented UNTF-based IT2 fuzzy adaptive fixed-time self-triggered control scheme realizes the bipartite consensus for MQUAVs under deferred full-state constraints. This control strategy effectively addresses the challenges posed by asymmetric time-varying constraints without requiring feasibility conditions, while also significantly improving the bandwidth utilization.

5. Conclusions

This paper presents a UNTF-based fuzzy adaptive fixed-time bipartite consensus control scheme with self-triggered communication for MQUAVs subject to deferred full-state constraints and input saturation. By deploying the proposed control strategy, the feasibility criterion present in traditional BLF-based results can be completely removed, and its applicability is extended to bipartite consensus control of unconstrained, constant/time-varying, and symmetric/asymmetric constrained MQUAVs without structural redesign. Stability analysis and illustrative results verify that the proposed fixed-time bipartite consensus self-triggered control protocol can ensure that the vehicle states are driven to the time-varying constraint region within a specified time, and the Zeno-free STM effectively reduces the actuator update frequency and the DTR. Given that potential conflicts between predefined flight paths and environmental constraints may impose an unreachable region compromising vehicle flight safety, future efforts will focus on high-order control barrier function-based adaptive optimal safety-critical formation control for MQUAVs.