A dual-event-triggered adaptive neural network control for resource-constrained UAV

Abstract

This paper proposes a dual-event-triggered adaptive neural-network control strategy for pose regulation of resource-constrained quadrotor unmanned aerial vehicles (UAVs). The proposed method addresses the difficulty of compensating for model uncertainties and external disturbances while optimizing communication and computational resources under limited bandwidth and onboard processing capability. In the developed framework, state information is transmitted from the UAV to the controller only when the prescribed triggering conditions are violated, and control commands are updated only when their deviation from the previously transmitted commands exceeds a prescribed threshold. This dual-event-triggered framework reduces communication and computation from both the state-transmission and control-update channels. In addition, a trigger-interval-weighted neural-network updating law is introduced to improve the applicability of the adaptive mechanism under different operating conditions. Based on Lyapunov stability theory, it is shown that all signals in the closed-loop system are uniformly ultimately bounded and that Zeno behavior is excluded. Simulations under both constant and time-varying reference signals verify the effectiveness of the proposed method in terms of control performance and communication–computation trade-off.

Keywords

event-triggered control adaptive neural network control unmanned aerial vehicles communication-efficient control

Introduction

Quadrotor unmanned aerial vehicles (UAVs), owing to their simple structure and high maneuverability, have been widely employed in moving-target tracking, logistics delivery, power-line inspection, and image-based transmission-line defect detection tasks.^1–6 As mission environments become increasingly complex, control accuracy and resource efficiency have become two key concerns in practical UAV applications. In real-world deployments, however, onboard computation, communication bandwidth, and energy resources are inherently limited. Under these constraints, frequent state transmission and control updating may impose a heavy burden on both communication and computation. Therefore, it is practically important to develop a quadrotor UAV control framework that preserves satisfactory pose-regulation performance while reducing unnecessary system updates.⁷

Event-triggered control provides an effective way to reduce resource consumption by updating control actions only when prescribed triggering conditions are satisfied. Its effectiveness has been demonstrated in uncertain nonlinear systems, mechatronic systems, robotic manipulators, and nonlinear multi-agent systems.^8–12 Recent studies have further extended event-triggered ideas to broader control architectures and application scenarios. For instance, data-driven adaptive event-triggered terminal sliding mode control, two-stage event-triggered model predictive control, and fuzzy event-triggered security control have been developed for nonlinear systems, EV charging stations, and networked systems under cyber-attacks, respectively.^13–15 These results show that event-triggered control is no longer limited to conventional periodic or single-loop update strategies.

A more recent trend is the transition from single-channel event-triggered updating to dual-channel information scheduling. In a networked control loop, the sensor-to-controller channel determines when state or measurement information is transmitted to the controller, whereas the controller-to-actuator channel determines when newly computed control commands are delivered to the plant. Separate scheduling of these two channels is especially relevant when sensing, communication, computation, and actuation are all resource constrained. Along this line, Chen and Dong¹⁶ designed a dual-channel dynamic event-triggered mechanism for discrete-time nonlinear systems. Zhang et al.¹⁷ proposed a global dual-channel dynamic event-triggered control method for uncertain nonlinear strict-feedback systems with prescribed transient performance, where both the sensor-to-controller and controller-to-actuator channels are incorporated into the design. Zhao et al.¹⁸ integrated a dual-triggered mechanism with adaptive neural control for MIMO nonlinear switched systems subject to sensor and actuator attacks.

Dual-channel and event-sampled designs have also been reported in vehicle-control systems, particularly in networked marine vehicles. For maritime autonomous surface ships, event-sampled adaptive fuzzy output-feedback control was developed using intermittent position data, where an event-sampled adaptive fuzzy state observer reconstructs the untransmitted velocity and a dynamic event-triggering mechanism is established in the controller-to-actuator channel.¹⁹ For autonomous underwater vehicles, adaptive dual-channel event-triggered fuzzy control was proposed for formation tracking and obstacle avoidance under limited communication resources, with event-triggering mechanisms designed in both the sensor-to-controller and controller-to-actuator channels.²⁰ For networked unmanned surface vehicles under dual-channel malicious attacks, adaptive neural output-feedback security control was developed, where an event-sampled mechanism in the sensor-control channel handles intermittent transmissions caused by DoS attacks, and a self-triggering mechanism in the control-actuator channel reduces command transmissions.²¹ Beyond vehicle-control applications, dynamic event-triggered adaptive neural practical predefined-time control has also been investigated for uncertain nonlinear systems with unknown control directions, where the control command is transmitted through a dynamically scheduled controller-to-actuator channel.²² Taken together, these studies show that event-triggered designs have evolved from reducing a single control-update process to coordinating sensing-side transmission and actuation-side updating in networked systems, and have also been combined with adaptive learning mechanisms for uncertain nonlinear systems.

In parallel, neural-network-based and adaptive methods have played an important role in compensating unknown nonlinear dynamics and external disturbances. In the UAV domain, adaptive neural-network techniques have been applied to attitude tracking of small unmanned helicopters, quadrotor hovering control, adaptive quadrotor control with experimental validation, fixed-wing UAV tracking under switching disturbances, quadrotor-slung-load fault-tolerant control, UAV aerial-recovery docking, and neural-enhanced trajectory tracking.^23–29 More generally, neural-network-based adaptive control has shown strong uncertainty-compensation capability in nonlinear systems with complex coupling effects.³⁰ Event-triggered guaranteed-performance control based on intermittent information utilization also suggests the potential of reducing update activity under constrained resources.³¹ Nevertheless, most UAV-oriented neural control designs mainly focus on uncertainty compensation, tracking performance, or fault tolerance. Existing dual-channel event-triggered studies, on the other hand, are mostly developed for general nonlinear systems, marine vehicles, security control, or predefined-time stabilization. The coordinated design of asynchronous state-channel triggering, control-channel triggering, and demand-driven neural learning for quadrotor UAV pose regulation under resource constraints has received less attention.

Motivated by the above observations, this paper proposes a dual-event-triggered adaptive neural-network control scheme for resource-constrained quadrotor UAV pose regulation. Specifically, state information is transmitted from the UAV to the controller only when the associated triggering conditions are violated, whereas newly generated control commands are delivered to the UAV only when their deviation from the previously transmitted commands exceeds a prescribed threshold. The proposed framework reduces unnecessary communication and computation through the coordinated design of the state-transmission channel and the control-update channel. In addition, the full pose dynamics of the quadrotor are considered, and the translational virtual inputs are converted into the total thrust and desired attitude commands through the quadrotor kinematic relations. A trigger-interval-weighted neural-network updating law is further introduced to improve the applicability of the adaptive mechanism under different operating conditions while reducing online computational burden.

The main contributions of this paper are summarized as follows:

A dual-channel event-triggered framework is developed for quadrotor UAV pose regulation, in which the state-transmission channel and the control-update channel are triggered asynchronously, thereby avoiding unnecessary updates and reducing the overall system update frequency.

A trigger-interval-weighted neural-network updating law is designed, so that the adaptive mechanism exhibits stronger applicability under different operating conditions while reducing unnecessary online computation.

Under the coexistence of event-triggering and intermittent learning, the boundedness of the closed-loop system is analyzed, and the trade-off between control performance and resource consumption is verified.

Kinematic and dynamic models of quadrotor

In UAV control, accurate kinematic and dynamic modeling is essential for controller design. The quadrotor considered in this paper adopts an “X” configuration, and its motion model is established on the basis of the Newton–Euler equations. Two coordinate frames are used, as shown in Figure 1:

the earth-fixed inertial reference frame $O_{E}$ - $X Y Z$ ;

the body-fixed reference frame $O_{B}$ - $X_{B} Y_{B} Z_{B}$ rigidly attached to the airframe.

Figure 1.

Model and frame of the quadrotor.

Quadrotor motion is generated mainly by the aerodynamic lift $f_{i}$ and the reactive torque $τ_{i}$ produced by the four rotors. These quantities depend on the rotor speeds, with $f_{i} = k_{t} w_{i}^{2}$ and $τ_{i} = k_{d} w_{i}^{2}$ , where $i = 1, \dots, 4$ , $k_{t}$ is the thrust coefficient, $k_{d}$ is the drag-moment coefficient, and $w_{i}$ is the angular speed of the $i$ -th propeller. Here, $U_{1}$ denotes the total thrust along the body $Z$ -axis, whereas $U_{2}$ , $U_{3}$ , and $U_{4}$ denote the roll, pitch, and yaw moments about the body $X$ -, $Y$ -, and $Z$ -axes, respectively.

\begin{aligned} [\begin{matrix} U_{1} \\ U_{2} \\ U_{3} \\ U_{4} \end{matrix}] & = [\begin{matrix} \sum f_{i} \\ l (f_{4} - f_{2}) \\ l (f_{3} - f_{1}) \\ τ_{1} + τ_{3} - τ_{2} - τ_{4} \end{matrix}] \\ = [\begin{matrix} k_{t} \sum w_{i}^{2} \\ k_{t} l (w_{4}^{2} - w_{2}^{2}) \\ k_{t} l (w_{3}^{2} - w_{1}^{2}) \\ k_{d} (w_{1}^{2} + w_{3}^{2} - w_{2}^{2} - w_{4}^{2}) \end{matrix}], \end{aligned}

(1)

where

l

is defined as the distance from the rotor to the center of the fuselage.

The full pose regulation problem of the quadrotor UAV is considered in this work. Accordingly, the six-degree-of-freedom nonlinear dynamics of the quadrotor UAV are described as follows:

{\begin{matrix} \ddot{x} = (\cos ϕ \sin θ \cos ψ + \sin ϕ \sin ψ) \frac{U_{1}}{m} + \frac{d_{x}}{m} \\ \ddot{y} = (\cos ϕ \sin θ \sin ψ - \sin ϕ \cos ψ) \frac{U_{1}}{m} + \frac{d_{y}}{m} \\ \ddot{z} = \cos ϕ \cos θ \frac{U_{1}}{m} - g + \frac{d_{z}}{m} \\ \ddot{ϕ} = \frac{\dot{θ} \dot{ψ} (I_{y} - I_{z}) + U_{2}}{I_{x}} + \frac{d_{ϕ}}{I_{x}} \\ \ddot{θ} = \frac{\dot{ϕ} \dot{ψ} (I_{z} - I_{x}) + U_{3}}{I_{y}} + \frac{d_{θ}}{I_{y}} \\ \ddot{ψ} = \frac{\dot{ϕ} \dot{θ} (I_{x} - I_{y}) + U_{4}}{I_{z}} + \frac{d_{ψ}}{I_{z}}, \end{matrix}

(2)

where

ϕ

θ

, and

ψ

denote the roll, pitch, and yaw angles, respectively;

x

y

, and

z

denote the translational positions of the quadrotor UAV;

m

is the mass;

g

is the gravitational acceleration;

I_{x}

I_{y}

, and

I_{z}

are the moments of inertia;

[d_{x}, d_{y}, d_{z}]^{T}

and

[d_{ϕ}, d_{θ}, d_{ψ}]^{T}

denote the translational and rotational disturbances, respectively.

To facilitate subsequent controller design, the state variables are defined as follows:

\begin{aligned} x_{1} & = x, x_{2} = \dot{x}, x_{3} = y, x_{4} = \dot{y}, x_{5} = z, x_{6} = \dot{z}, \\ x_{7} & = ϕ, x_{8} = \dot{ϕ}, x_{9} = θ, x_{10} = \dot{θ}, x_{11} = ψ, x_{12} = \dot{ψ} . \end{aligned}

(3)

It is evident that the translational dynamics are coupled with the attitude variables. To address the underactuation in the translational motion and facilitate the subsequent controller design, the virtual control inputs $u_{x}$ , $u_{y}$ , and $u_{z}$ are introduced as

\begin{aligned} u_{x} & = U_{1} (\cos ϕ \sin θ \cos ψ + \sin ϕ \sin ψ) \\ u_{y} & = U_{1} (\cos ϕ \sin θ \sin ψ - \sin ϕ \cos ψ) \\ u_{z} & = U_{1} \cos ϕ \cos θ . \end{aligned}

(4)

The quadrotor UAV is an underactuated and strongly coupled system with four control inputs and six outputs. For controller synthesis, the overall dynamics are decomposed into an attitude subsystem and a position subsystem. The outer-loop controller generates the virtual translational inputs $u_{x}$ , $u_{y}$ , and $u_{z}$ , which are subsequently transformed into the total thrust and the desired attitude commands for the inner-loop controller.

The attitude subsystem is as follows:

{\begin{matrix} {\dot{x}}_{7} & = x_{8} \\ {\dot{x}}_{8} & = \frac{U_{2}}{I_{x}} + d_{ϕ} + f_{8} (x_{8}) \\ {\dot{x}}_{9} & = x_{10} \\ {\dot{x}}_{10} & = \frac{U_{3}}{I_{y}} + d_{θ} + f_{10} (x_{10}) \\ {\dot{x}}_{11} & = x_{12} \\ {\dot{x}}_{12} & = \frac{U_{4}}{I_{z}} + d_{ψ} + f_{12} (x_{12}), \end{matrix}

(5)

where

f_{8} (\cdot)

f_{10} (\cdot)

, and

f_{12} (\cdot)

represent unknown nonlinear dynamics in the three attitude channels.

The position subsystem is as follows:

{\begin{matrix} {\dot{x}}_{1} & = x_{2} \\ {\dot{x}}_{2} & = \frac{1}{m} u_{x} + d_{x} + f_{2} (x_{2}) \\ {\dot{x}}_{3} & = x_{4} \\ {\dot{x}}_{4} & = \frac{1}{m} u_{y} + d_{y} + f_{4} (x_{4}) \\ {\dot{x}}_{5} & = x_{6} \\ {\dot{x}}_{6} & = \frac{1}{m} u_{z} - g + d_{z} + f_{6} (x_{6}), \end{matrix}

(6)

where

f_{2} (\cdot)

f_{4} (\cdot)

, and

f_{6} (\cdot)

represent unknown nonlinear dynamics in the three translational channels.

By performing inverse kinematics from the virtual translational inputs $u_{x}$ , $u_{y}$ , and $u_{z}$ to the thrust and desired attitude commands, one obtains

\begin{aligned} U_{1} & = \sqrt{u_{x}^{2} + u_{y}^{2} + u_{z}^{2}} \\ ϕ_{d} & = \arcsin [U_{1}^{- 1} (u_{x} \sin ψ - u_{y} \cos ψ)] \\ θ_{d} & = \arctan [u_{z}^{- 1} (u_{x} \cos ψ + u_{y} \sin ψ)] . \end{aligned}

(7)

For compact representation, the above quadrotor pose dynamics can be further described by six second-order subsystems, corresponding to the translational coordinates $x$ , $y$ , and $z$ and the attitude angles $ϕ$ , $θ$ , and $ψ$ , respectively. The general second-order model of the $i$ -th subsystem is described as

{\begin{matrix} {\dot{ξ}}_{i 1} = ξ_{i 2}, \\ {\dot{ξ}}_{i 2} = b_{i} u_{i} + d_{i} + f_{i} ({\bar{ξ}}_{i}), \\ γ_{i} = ξ_{i 1}, \end{matrix} i = 1, 2, \dots, 6,

(8)

where

{\bar{ξ}}_{i} = [ξ_{i 1}, ξ_{i 2}]^{T} \in R^{2}

denotes the state vector of the

i

-th subsystem,

u_{i} \in R

is the corresponding control input,

γ_{i} \in R

is the subsystem output,

b_{i} \in R

is the input coefficient,

d_{i} \in R

denotes the bounded external disturbance satisfying

| d_{i} | \leq {\bar{d}}_{i}

with unknown constant

{\bar{d}}_{i} > 0

, and

f_{i} (\cdot) : R^{2} \to R

represents the unknown nonlinear dynamics.

Lemma 1.

Ge and Wang³² For each continuous nonlinear function $f_{i} ({\bar{ξ}}_{i})$ corresponding to the six second-order subsystems, defined over the compact set $Ω_{{\bar{ξ}}_{i}} \subset R^{q}$ , there exists an ideal neural network representation $W_{i}^{T} Υ_{i} ({\bar{ξ}}_{i})$ and a constant $ε_{i}$ such that the following relation holds:

f_{i} ({\bar{ξ}}_{i}) = W_{i}^{T} Υ_{i} ({\bar{ξ}}_{i}) + ε_{i} ({\bar{ξ}}_{i}), | ε_{i} ({\bar{ξ}}_{i}) | \leq {\bar{ε}}_{i},

(9)

where

ε_{i} ({\bar{ξ}}_{i})

stands for the approximation error of the

i

-th subsystem. The vector

W_{i} = [w_{i 1}, w_{i 2}, \dots, w_{i p}]^{T} \in R^{p}

represents the ideal weights for approximating

f_{i} (\cdot)

and

p

denotes the number of radial-basis functions. The vector

Υ_{i} ({\bar{ξ}}_{i}) = [υ_{i 1} ({\bar{ξ}}_{i}), υ_{i 2} ({\bar{ξ}}_{i}), \dots, υ_{i p} ({\bar{ξ}}_{i})]^{T}

is the basis function vector for the

i

-th subsystem. Each basis function is explicitly defined as

υ_{i κ} ({\bar{ξ}}_{i}) = \exp [- \frac{({\bar{ξ}}_{i} - C_{i κ})^{T} ({\bar{ξ}}_{i} - C_{i κ})}{σ_{i κ}^{2}}],

(10)

where

κ = 1, \dots, p

C_{i κ} \in R^{q}

denotes the receptive field center, and

σ_{i κ} > 0

denotes the width parameter. Obviously,

Υ_{i}^{T} ({\bar{ξ}}_{i}) Υ_{i} ({\bar{ξ}}_{i}) \leq p

Lemma 2.

Tong et al.³³ The basis function vector $Υ_{i} (\cdot)$ is bounded as $‖ Υ_{i} (\cdot) ‖ \leq {\bar{Υ}}_{i}$ with a constant ${\bar{Υ}}_{i} > 0$ .

Main result

This section starts with some notations. Let $\tilde{*}$ and $\hat{*}$ represent the triggered value and estimation of the variable $*$ , respectively. $*^{+}$ denotes the limit of $*$ as time $t$ grows to $t_{k}$ from its right side. $\land$ , $\lor$ , and $\neg$ stand for the logic operators “AND,” “OR,” and “NOT.”

For $t \in (t_{s, k}, t_{s, k + 1}]$ with $s = 1, 2$ and $k = 0, 1, \dots$ , $s = 1$ corresponds to state triggering and $s = 2$ corresponds to control-output triggering for the $i$ -th subsystem with $i = 1, 2, \dots, 6$ . The triggered values for the $i$ -th subsystem are defined as

\begin{aligned} {\tilde{γ}}_{i}^{+} (t) & = γ_{i} (t_{1, k}), \\ {\tilde{ξ}}_{i 2}^{+} (t) & = ξ_{i 2} (t_{1, k}), \\ {\tilde{γ}}_{i d}^{+} (t) & = γ_{i d} (t_{1, k}), \\ {\tilde{v}}_{i d}^{+} (t) & = v_{i d} (t_{1, k}), \\ u_{i}^{+} (t) & = w_{i} (t_{2, k}), \\ t_{1, k + 1} & = inf {t \in R | E_{i 1} \lor E_{i 2} \lor E_{i 3} \lor E_{i 4} \lor E_{i 5} \lor E_{i 6}}, \\ t_{2, ℓ + 1} & = inf {t \in R | E_{i c}} . \end{aligned}

(11)

The triggering conditions $E_{i 1}$ – $E_{i 6}$ and $E_{i c}$ are defined as follows:

{\begin{matrix} E_{i 1} : | e_{i 1} (t) | > ℏ_{i 1}, e_{i 1} (t) = γ_{i d} (t) - {\tilde{γ}}_{i d} (t) \\ E_{i 2} : | e_{i 2} (t) | > ℏ_{i 2}, e_{i 2} (t) = v_{i d} (t) - {\tilde{v}}_{i d} (t) \\ E_{i 3} : | e_{i 3} (t) | > ℏ_{i 3}, e_{i 3} (t) = ξ_{i 1} (t) - {\tilde{ξ}}_{i 1} (t) \\ E_{i 4} : | e_{i 4} (t) | > ℏ_{i 4}, e_{i 4} (t) = ξ_{i 2} (t) - {\tilde{ξ}}_{i 2} (t) \\ E_{i 5} : | e_{i 5} (t) | > ℏ_{i 5}, e_{i 5} (t) = ξ_{i 1} (t) - γ_{i d} (t) \\ E_{i 6} : | e_{i 6} (t) | > ℏ_{i 6}, e_{i 6} (t) = ξ_{i 2} (t) - v_{i d} (t) \\ E_{i c} : | e_{i c} (t) | > ℏ_{i c} + ϱ_{i c} \cdot | \tanh (e_{i c} (t)) | \\ e_{i c} (t) = u_{i} (t) - w_{i} (t), \end{matrix}

(12)

where

γ_{i d} (t)

denotes the desired reference signal of the

i

-th pose subsystem, and

v_{i d} (t)

denotes the corresponding desired rate reference signal. Specifically,

γ_{i d} (t)

corresponds to

x_{d}

y_{d}

z_{d}

ϕ_{d}

θ_{d}

, and

ψ_{d}

for

i = 1, 2, \dots, 6

, respectively. The variable

w_{i}

represents the ideal control input computed by the controller for the

i

-th subsystem, while

u_{i}

denotes the applied control input updated according to the control-channel triggering condition. The constants

ℏ_{i 1}, ℏ_{i 2}, ℏ_{i 3}, ℏ_{i 4}, ℏ_{i 5}, ℏ_{i 6}, ℏ_{i c} > 0

denote the fixed thresholds associated with each triggering condition, and

ϱ_{i c} > 0

is a design parameter.

The overall structure of the dual-event-triggered networked control system is illustrated in Figure 2. Under the proposed framework, the state channel and the control channel are triggered asynchronously. Specifically, the state-channel triggers $E_{i 1}$ – $E_{i 6}$ determine the transmission instants $t_{1, k}$ , at which the UAV transmits the subsystem information to the controller, including the sampled states $ξ_{i 1} (t_{1, k})$ , $ξ_{i 2} (t_{1, k})$ and the reference signals $γ_{i d} (t_{1, k})$ , $v_{i d} (t_{1, k})$ for $i = 1, 2, \dots, 6$ . Upon reception, the controller updates the corresponding triggered values ${\tilde{ξ}}_{i 1} (t)$ , ${\tilde{ξ}}_{i 2} (t)$ , ${\tilde{γ}}_{i d} (t)$ , and ${\tilde{v}}_{i d} (t)$ via a sample-and-hold mechanism, which are locally maintained and utilized for control computation and adaptation, while the estimation variables such as the neural network weights are kept and updated on the controller side. In contrast, the control-channel trigger $E_{i c}$ specifies the update instants $t_{2, k}$ , at which the controller transmits the newly generated ideal control input $w_{i} (t_{2, k})$ to the UAV, and the applied input is updated as $u_{i} (t_{2, k}^{+}) = w_{i} (t_{2, k})$ and held constant until the next control-triggering instant.

Figure 2.

Structure of the dual-event-triggered UAV networked control system.

Design of dual-event-triggered adaptive neural network control

In the context of the second-order system (8) for the $i$ -th pose subsystem ( $i = 1, 2, \dots, 6$ ), considering that only triggered outputs are available under output feedback, non-differentiable virtual controls must be employed to regulate each subsystem. Consequently, the design of the backstepping-based adaptive neural event-triggered controller should be constructed upon the virtual error $z_{i, j + 1} (t) = {\hat{x}}_{i, j + 1} (t) - {\tilde{α}}_{i, j} (t)$ with $j = 1$ denoting the first step of backstepping rather than the form corresponding to traditional continuous virtual controls. However, since ${\tilde{α}}_{i 1} (t)$ is non-differentiable and thus cannot be directly used in the recursive design, a first-order filter is introduced to address the non-differentiability issue of the virtual control law. Based on the coordinate transformation for the $i$ -th subsystem:

{\begin{matrix} {\tilde{z}}_{i 1} = {\tilde{ξ}}_{i 1} - {\tilde{γ}}_{i d} \\ {\tilde{z}}_{i 2} = {\tilde{ξ}}_{i 2} - α_{i 1, f}, \end{matrix}

(13)

where

α_{i 1, f} (t)

is continuous and derived by the following first-order filter with time constant

τ_{i 1}

for the

i

-th subsystem:

τ_{i 1} {\dot{α}}_{i 1, f} (t) + α_{i 1, f} (t) = {\tilde{α}}_{i 1} (t), α_{i 1, f} (0) = {\tilde{α}}_{i 1} (0) .

(14)

The filter error is defined as $η_{i} (t) = α_{i 1, f} (t) - {\tilde{α}}_{i 1} (t)$ , which will be used for stability analysis.

The practical virtual control law ${\tilde{α}}_{i 1}$ for the $i$ -th subsystem is designed as follows:

{\tilde{α}}_{i 1} = - k_{i 1} {\tilde{z}}_{i 1} + {\tilde{v}}_{i d},

(15)

where

k_{i 1}

is a controller parameter. The ideal control input

w_{i}

for the

i

-th subsystem is designed as follows:

\begin{aligned} w_{i} & = (- k_{i 2} {\tilde{z}}_{i 2} - {\tilde{z}}_{i 1} - {\hat{W}}_{i 2}^{T} Υ_{i 2} ({\tilde{\bar{ξ}}}_{i}) + \frac{{\tilde{α}}_{i 1} - α_{i 1, f}}{τ_{i 1}} \\ - b_{i} μ_{i} \tanh (\frac{b_{i} μ_{i} {\tilde{z}}_{i 2}}{c_{i}})) \frac{1}{b_{i}}, \end{aligned}

(16)

where

k_{i 2}

is a controller parameter,

c_{i} > 0

is a constant, and

μ_{i} = ℏ_{i c} + ϱ_{i c} > 0

is a design parameter. The term

{\hat{W}}_{i 2}

denotes the adaptive neural parameter of the

i

-th subsystem. To improve the applicability of the learning process under different triggering rhythms, the neural parameter is updated in an interval-weighted event-triggered manner. Specifically, the weight correction at each state-triggering instant is scaled by the actual inter-event interval

Δ t_{1, k} = t_{1, k} - t_{1, k - 1}

, rather than by a fixed scenario-dependent coefficient. The updating law is designed as

{\begin{matrix} {\dot{\hat{W}}}_{i 2} = & 0, t \in (t_{1, k}, t_{1, k + 1}), \\ {\hat{W}}_{i 2}^{+} = & {\hat{W}}_{i 2} - Δ t_{1, k} g_{i 2} \frac{{\tilde{z}}_{i 2}}{c_{i 2} + {\tilde{z}}_{i 2}^{2}} Υ_{i 2} ({\tilde{\bar{ξ}}}_{i}) \\ - Δ t_{1, k} σ_{i 2} {\hat{W}}_{i 2}, t = t_{1, k}, \end{matrix}

(17)

where

{\tilde{\bar{ξ}}}_{i} = [{\tilde{ξ}}_{i 1}, {\tilde{ξ}}_{i 2}]^{T}

denotes the triggered state vector of the

i

-th subsystem;

g_{i 2} > 0

σ_{i 2} > 0

and

c_{i 2} > 0

are design parameters for the

i

-th subsystem.

Lemma 3.

Under the interval-weighted event-triggered neural updating law (17), assume that the state-triggering intervals satisfy $0 < {\underline{Δ}}_{i} \leq Δ t_{1, k} \leq {\bar{Δ}}_{i}$ and $0 < σ_{i 2} {\bar{Δ}}_{i} < 1$ . Then the corresponding neural weight estimation error ${\tilde{W}}_{i 2} (t) = W_{i 2} (t) - {\hat{W}}_{i 2} (t)$ is uniformly ultimately bounded for all $t > 0$ . That is, there exists a positive constant $Λ_{i}$ such that $‖ {\tilde{W}}_{i 2} (t) ‖ \leq Λ_{i}$ .

Lemma 4.

Liu et al.³⁴ Under the designed event-triggered control scheme and the first-order filter dynamics approximating the discontinuous virtual control signal ${\tilde{α}}_{i 1} (t)$ , the filter error $η_{i} (t) = α_{i 1, f} (t) - {\tilde{α}}_{i 1} (t)$ is uniformly ultimately bounded. That is, there exists a positive constant ${\bar{η}}_{i} > 0$ such that $| η_{i} (t) | \leq {\bar{η}}_{i}$ for all $t > 0$ .

Theorem 1.

Consider the six second-order subsystems of the quadrotor (8). Under the proposed event-triggered control framework comprising the triggering conditions (12), first-order filters (14), practical virtual control laws (15), and the backstepping-based adaptive neural controller (16) with updated parameters ${\hat{W}}_{i 2}$ (17) for $i = 1, 2, \dots, 6$ , all closed-loop signals remain bounded, and Zeno behavior is avoided.

Stability analysis

1) Proof of Lemma 3: For (17), ${\hat{W}}_{i 2}^{+}$ is given by

{\hat{W}}_{i 2}^{+} = {\hat{W}}_{i 2} - Δ t_{1, k} g_{i 2} ι_{i 2} Υ_{i 2} ({\tilde{\bar{ξ}}}_{i}) - Δ t_{1, k} σ_{i 2} {\hat{W}}_{i 2},

(18)

where

ι_{i 2} = {\tilde{z}}_{i 2} / (c_{i 2} + {\tilde{z}}_{i 2}^{2})

, with

c_{i 2} > 0

, and

Υ_{i 2} ({\tilde{\bar{ξ}}}_{i})

is a bounded regressor. The scalar parameters

g_{i 2} > 0

and

σ_{i 2} > 0

are tuning constants. Here,

Δ t_{1, k} = t_{1, k} - t_{1, k - 1} > 0

denotes the state-triggering interval, and

σ_{i 2}

is selected such that

0 < σ_{i 2} Δ t_{1, k} < 1

at each triggering instant. Throughout this proof,

i = 1, 2, \dots, 6

corresponds to the six pose subsystems.

Define the weight estimation error as ${\tilde{W}}_{i 2} = W_{i 2} - {\hat{W}}_{i 2}$ , where $W_{i 2}$ is the ideal constant weight. The error dynamics form a hybrid system with continuous flows and discrete jumps. During flow periods $t \in (t_{1, k}, t_{1, k + 1})$ , the update law is inactive $({\dot{\hat{W}}}_{i 2} = 0)$ , so ${\tilde{W}}_{i 2} (t)$ remains constant. At each impulse instant $t = t_{1, k}$ , the update induces a jump in the error, yielding the following expression:

\begin{aligned} {\tilde{W}}_{i 2}^{+} = & W_{i 2} - {\hat{W}}_{i 2}^{+} \\ = & W_{i 2} - ({\hat{W}}_{i 2} - Δ t_{1, k} g_{i 2} ι_{i 2} Υ_{i 2} - Δ t_{1, k} σ_{i 2} {\hat{W}}_{i 2}) \\ = & (1 - σ_{i 2} Δ t_{1, k}) {\tilde{W}}_{i 2}^{-} + Δ t_{1, k} g_{i 2} ι_{i 2} Υ_{i 2} \\ + σ_{i 2} Δ t_{1, k} W_{i 2}, \end{aligned}

(19)

where

{\tilde{W}}_{i 2}^{-}

and

{\tilde{W}}_{i 2}^{+}

denote the left and right limits of the error at the triggering instant, respectively.

(i) Flow periods ( $t \neq t_{1, k}, k \in Z^{+}$ ): A Lyapunov function is constructed as $V_{w, i 2} = (1 / 2) {\tilde{W}}_{i 2}^{⊤} {\tilde{W}}_{i 2}$ . The time derivative of $V_{w, i 2}$ is ${\dot{V}}_{w, i 2} = 0$ . Therefore, $V_{w, i 2}$ is constant in the flow periods, and thus ${\tilde{W}}_{i 2}$ is bounded.

(ii) Jumping instants ( $t = t_{1, k}, k \in Z^{+}$ ): At the triggering instants, consider the difference of $V_{w, i 2}$ :

\begin{aligned} Δ V_{w, i 2} & = V_{w, i 2} (t_{1, k}^{+}) - V_{w, i 2} (t_{1, k}^{-}) \\ = \frac{1}{2} [({\tilde{W}}_{i 2}^{+})^{⊤} {\tilde{W}}_{i 2}^{+} - ({\tilde{W}}_{i 2}^{-})^{⊤} {\tilde{W}}_{i 2}^{-}] . \end{aligned}

(20)

Let $s_{i, k} = 1 - σ_{i 2} Δ t_{1, k}$ . Since $0 < σ_{i 2} Δ t_{1, k} < 1$ , it follows that $0 < s_{i, k} < 1$ . Substituting the jump relation of ${\tilde{W}}_{i 2}$ into the Lyapunov difference and using Young’s inequality to bound the cross terms, together with the boundedness of $Υ_{i 2} ({\tilde{\bar{ξ}}}_{i})$ and $W_{i 2}$ , one obtains

Δ V_{w, i 2} \leq - {\bar{σ}}_{i 2, k} ‖ {\tilde{W}}_{i 2}^{-} ‖^{2} + C_{i 2, k},

(21)

where

{\bar{σ}}_{i 2, k} > 0

and

C_{i 2, k} > 0

are bounded constants depending on

Δ t_{1, k}

g_{i 2}

σ_{i 2}

{\bar{Υ}}_{i 2}

{\bar{W}}_{i 2}

, and

c_{i 2}

Since $0 < Δ_{i} \leq Δ t_{1, k} \leq {\bar{Δ}}_{i}$ and $0 < σ_{i 2} {\bar{Δ}}_{i} < 1$ , there exist positive constants ${\bar{σ}}_{i 2}$ and $C_{i 2}$ such that ${\bar{σ}}_{i 2, k} \geq {\bar{σ}}_{i 2}$ and $C_{i 2, k} \leq C_{i 2}$ for all $k$ . Therefore, when $‖ {\tilde{W}}_{i 2}^{-} ‖ > \sqrt{C_{i 2} / {\bar{σ}}_{i 2}}$ , one has $Δ V_{w, i 2} < 0$ . It follows that

‖ {\tilde{W}}_{i 2} (t) ‖ \leq max {‖ {\tilde{W}}_{i 2} (0) ‖, \sqrt{\frac{C_{i 2}}{{\bar{σ}}_{i 2}}}} = Λ_{i} .

(22)

Summarizing the above two cases, it is ensured that ${\tilde{W}}_{i 2} (t)$ is ultimately bounded for all $t > 0$ .

Remark 1.

The proposed neural-network updating law is an interval-weighted event-triggered mechanism, where each weight correction is scaled by the corresponding inter-event interval. Under this scheme, the neural weights are updated only at triggering instants. When the inter-event interval is relatively long, the weights remain unchanged, which may temporarily weaken the approximation of time-varying uncertainties. The resulting approximation error can then be treated as a bounded time-varying disturbance, whose bound depends on the state evolution and the current inter-event interval. Hence, the closed-loop system still preserves uniform ultimate boundedness. Once the approximation error grows enough to violate the triggering condition, the interval-weighted law is reactivated and the neural correction is adjusted according to the actual triggering interval, thereby restoring the approximation capability in an on-demand manner.

2) Proof of Lemma 4: In event-triggered controllers with discontinuous virtual controls, an explicit bound on the filter error $η_{i} (t)$ is needed for stability analysis. Since backstepping relies on virtual-control derivatives that are not well defined at discontinuities, a first-order filter is introduced to smooth the virtual-control signal and provide well-defined filter dynamics. This treatment introduces jumps in the filter error, and a nonlinear impulsive-dynamics framework is therefore used to handle the resulting discontinuities.

To assess the boundedness of the filter-induced error, a first-order filter is introduced for each virtual-control component, $τ_{i} {\dot{α}}_{i, f} (t) + α_{i, f} (t) = {\tilde{α}}_{i} (t)$ , and the corresponding filter error is defined as $η_{i} (t) = α_{i, f} (t) - {\tilde{α}}_{i} (t)$ . The following Lyapunov function is selected:

V_{η} = \sum_{i = 1}^{6} \frac{1}{2} η_{i}^{2} (t) .

(23)

Differentiation of $η_{i} (t)$ yields ${\dot{η}}_{i} (t) = {\dot{α}}_{i, f} (t) - {\dot{\tilde{α}}}_{i} (t) = - η_{i} (t) / τ_{i} - {\dot{\tilde{α}}}_{i} (t)$ during the flow interval. The discontinuous virtual-control signal is piecewise differentiable between triggering instants, and its derivative satisfies $| {\dot{\tilde{α}}}_{i} (t) | \leq {\bar{α}}_{i}$ , where ${\bar{α}}_{i} > 0$ is a constant.

(i) Flow set ( $t \in (t_{k}, t_{k + 1})$ ): During each continuous interval between triggering instants, the inequality $- η_{i} (t) {\dot{\tilde{α}}}_{i} (t) \leq η_{i}^{2} (t) + {\dot{\tilde{α}}}_{i}^{2} (t) / 4$ is employed. Substituting the filter-error dynamics into the derivative of $V_{η}$ gives

\begin{aligned} {\dot{V}}_{η} & = \sum_{i = 1}^{6} η_{i} (t) (- \frac{η_{i} (t)}{τ_{i}} - {\dot{\tilde{α}}}_{i} (t)) \\ \leq \sum_{i = 1}^{6} [- (\frac{1}{τ_{i}} - 1) η_{i}^{2} (t) + \frac{1}{4} {\dot{\tilde{α}}}_{i}^{2} (t)] \\ \leq - 2 a V_{η} + D . \end{aligned}

(24)

where

a = min_{i} {(1 / τ_{i}) - 1} > 0

and

D = \sum_{i = 1}^{6} {\bar{α}}_{i}^{2} / 4

. Hence,

V_{η}

is exponentially dissipative during the continuous intervals.

(ii) Jump set ( $t = t_{k}$ ): At the triggering instant $t = t_{k}$ , the filtered signal $α_{i, f} (t)$ is continuous, whereas ${\tilde{α}}_{i} (t)$ may jump. Introduce the triggering error $e_{η, i, k} = Δ η_{i} = {\tilde{α}}_{i} (t_{k}^{-}) - {\tilde{α}}_{i} (t_{k}^{+})$ . Since $α_{i, f} (t_{k}^{+}) = α_{i, f} (t_{k}^{-})$ , one has $η_{i} (t_{k}^{+}) = η_{i} (t_{k}^{-}) + e_{η, i, k}$ . In light of the virtual-control design, the discontinuous-control jump is bounded, i.e., $| e_{η, i, k} | \leq M_{i}$ , with $M_{i} > 0$ a constant.

The jump increment of the Lyapunov function satisfies

\begin{aligned} Δ V_{η} & = V_{η} (t_{k}^{+}) - V_{η} (t_{k}^{-}) \\ = \sum_{i = 1}^{6} [η_{i} (t_{k}^{-}) e_{η, i, k} + \frac{1}{2} e_{η, i, k}^{2}] . \end{aligned}

(25)

By Young’s inequality, for any $χ > 0$ , the jump satisfies

V_{η} (t_{k}^{+}) \leq (1 + χ) V_{η} (t_{k}^{-}) + D_{η},

(26)

where

D_{η} = (1 + χ^{- 1}) \sum_{i = 1}^{6} M_{i}^{2} / 2

is a bounded constant.

During the interval between two consecutive triggering instants, the Lyapunov function $V_{η} (t)$ satisfies (24). By the comparison principle, for $T_{k} = t_{k + 1} - t_{k}$ , one has

V_{η} (t_{k + 1}^{-}) \leq V_{η} (t_{k}^{+}) e^{- 2 a T_{k}} + \frac{D}{2 a} (1 - e^{- 2 a T_{k}}) .

(27)

Under the designed event-triggered mechanism, there exists a positive constant $σ_{t} > 0$ such that $T_{k} \geq σ_{t}$ . Combining the flow-phase bound with the jump-phase bound gives

V_{η} (t_{k + 1}^{+}) \leq q V_{η} (t_{k}^{+}) + {\bar{D}}_{η},

(28)

where

q = (1 + χ) e^{- 2 a σ_{t}}

and

{\bar{D}}_{η} = (1 + χ) D / (2 a) + D_{η}

. Choosing

0 < χ < e^{2 a σ_{t}} - 1

gives

0 < q < 1

. Therefore,

V_{η} (t_{k}^{+})

is uniformly ultimately bounded at all triggering instants.

Furthermore, the flow-phase comparison bound implies that the boundedness also holds for all $t \in [t_{k}, t_{k + 1})$ . Therefore, there exists a positive constant ${\bar{η}}_{i}$ such that $| η_{i} (t) | \leq {\bar{η}}_{i}$ for $i = 1, 2, \dots, 6$ .

Thus, the filter error $η_{i} (t)$ is uniformly ultimately bounded.

3) Proof of Theorem 1: For the second-order quadrotor UAV subsystems with external disturbances, neural networks are used to compensate for the unknown disturbance-related dynamics. Since only triggered signals are available to the controller, the closed-loop system contains discontinuous virtual controls and sample-and-hold inputs. More importantly, the asynchronous state-triggering instants ${t_{1, k}}$ and control-triggering instants ${t_{2, ℓ}}$ introduce two different error sources. One is the holding error caused by the control-update channel, and the other is the computation error caused by using triggered states in the controller. The following proof reorganizes these errors before deriving the final Lyapunov bound.

Let $z_{i 1} (t) = ξ_{i 1} - γ_{i d}$ and $z_{i 2} (t) = ξ_{i 2} - α_{i 1, f}$ . Choose the Lyapunov function as

V = \sum_{i = 1}^{6} \frac{1}{2} z_{i 1}^{2} + \sum_{i = 1}^{6} \frac{1}{2} z_{i 2}^{2} .

(29)

Differentiating $V$ along (8) gives

\begin{aligned} \dot{V} = & \sum_{i = 1}^{6} z_{i 1} (z_{i 2} + α_{i 1, f} - v_{i d}) \\ + \sum_{i = 1}^{6} z_{i 2} (b_{i} u_{i} + f_{i} + d_{i} - {\dot{α}}_{i 1, f}) . \end{aligned}

(30)

To separate the nominal stabilizing part from the errors induced by triggering, introduce the auxiliary virtual control and the corresponding continuous control input as

\begin{aligned} α_{i 1}^{*} & = - k_{i 1} z_{i 1} + v_{i d}, \\ w_{i}^{*} & = (- k_{i 2} z_{i 2} - z_{i 1} - {\hat{W}}_{i 2}^{T} Υ_{i} ({\tilde{\bar{ξ}}}_{i}) + \frac{{\tilde{α}}_{i 1} - α_{i 1, f}}{τ_{i 1}} \\ - b_{i} μ_{i} \tanh (\frac{b_{i} μ_{i} z_{i 2}}{c_{i}})) \frac{1}{b_{i}} . \end{aligned}

(31)

Substituting (31) into (30) yields

\begin{aligned} \dot{V} = & \sum_{i = 1}^{6} [- k_{i 1} z_{i 1}^{2} + z_{i 1} (α_{i 1, f} - α_{i 1}^{*}) - k_{i 2} z_{i 2}^{2} \\ + z_{i 2} (f_{i} + d_{i} - {\hat{W}}_{i 2}^{T} Υ_{i} ({\tilde{\bar{ξ}}}_{i})) \\ - b_{i} μ_{i} z_{i 2} \tanh (\frac{b_{i} μ_{i} z_{i 2}}{c_{i}}) + b_{i} z_{i 2} (u_{i} - w_{i}^{*})] . \end{aligned}

(32)

Thus, the derivative is organized into four parts: the virtual-control deviation, the neural approximation residual, the robust hyperbolic-tangent term, and the asynchronous control-deviation term.

According to Lemma 1, Lemma 2, and Lemma 3, for each subsystem, there exist an ideal weight vector $W_{i}$ , an approximation error $ε_{i} ({\bar{ξ}}_{i})$ , and a bounded weight estimation error ${\tilde{W}}_{i} = W_{i} - {\hat{W}}_{i 2}$ such that $f_{i} ({\bar{ξ}}_{i}) + d_{i} = W_{i}^{T} Υ_{i} ({\tilde{\bar{ξ}}}_{i}) + ε_{i} ({\bar{ξ}}_{i})$ , $‖ Υ_{i} ({\tilde{\bar{ξ}}}_{i}) ‖ \leq {\bar{Υ}}_{i}$ , $‖ {\tilde{W}}_{i} ‖ \leq Λ_{i}$ , and $| ε_{i} ({\bar{ξ}}_{i}) | \leq {\bar{ε}}_{i}$ . Hence,

\begin{aligned} z_{i 2} (f_{i} + d_{i} - {\hat{W}}_{i 2}^{T} Υ_{i} ({\tilde{\bar{ξ}}}_{i})) \\ = z_{i 2} {\tilde{W}}_{i}^{T} Υ_{i} ({\tilde{\bar{ξ}}}_{i}) + z_{i 2} ε_{i} ({\bar{ξ}}_{i}) \\ \leq ϵ_{i 1} z_{i 2}^{2} + \frac{{(Λ_{i} {\bar{Υ}}_{i} + {\bar{ε}}_{i})}^{2}}{2 ϵ_{i 1}}, \end{aligned}

(33)

where

ϵ_{i 1} > 0

Next, the virtual-control deviation is estimated directly. Since $α_{i 1, f} - α_{i 1}^{*} = η_{i} - k_{i 1} ({\tilde{z}}_{i 1} - z_{i 1}) + ({\tilde{v}}_{i d} - v_{i d})$ , Lemma 4 gives $| η_{i} | \leq {\bar{η}}_{i}$ , and the triggering conditions imply $| {\tilde{z}}_{i 1} - z_{i 1} | \leq ℏ_{i 1} + ℏ_{i 3}$ and $| {\tilde{v}}_{i d} - v_{i d} | \leq ℏ_{i 2}$ . Therefore,

\begin{aligned} z_{i 1} (α_{i 1, f} - α_{i 1}^{*}) \leq & (ϵ_{i 2} + ϵ_{i 3} + ϵ_{i 4} + ϵ_{i 5}) z_{i 1}^{2} \\ + \frac{{\bar{η}}_{i}^{2}}{2 ϵ_{i 2}} + \frac{k_{i 1}^{2} ℏ_{i 1}^{2}}{2 ϵ_{i 3}} + \frac{k_{i 1}^{2} ℏ_{i 3}^{2}}{2 ϵ_{i 4}} + \frac{ℏ_{i 2}^{2}}{2 ϵ_{i 5}}, \end{aligned}

(34)

where

ϵ_{i 2}

ϵ_{i 3}

ϵ_{i 4}

, and

ϵ_{i 5}

are positive constants.

The last term in (32) contains the main effect of asynchronous dual-channel triggering. To distinguish the two triggering-induced errors, define

\begin{aligned} Δ_{i u} (t) & = u_{i} (t) - w_{i} (t) = e_{i c} (t), \\ Δ_{i x} (t) & = \frac{1}{b_{i}} [- k_{i 2} ({\tilde{z}}_{i 2} - z_{i 2}) - ({\tilde{z}}_{i 1} - z_{i 1})] \\ + μ_{i} [\tanh (\frac{b_{i} μ_{i} z_{i 2}}{c_{i}}) - \tanh (\frac{b_{i} μ_{i} {\tilde{z}}_{i 2}}{c_{i}})], \end{aligned}

(35)

where

Δ_{i u}

denotes the control-channel holding error, and

Δ_{i x}

denotes the state-induced control-computation error. Then, the actual control-deviation term can be decomposed as

b_{i} z_{i 2} (u_{i} - w_{i}^{*}) = b_{i} z_{i 2} (Δ_{i u} + Δ_{i x}) .

(36)

This decomposition shows that the control deviation is affected by both the control-update channel and the state-transmission channel. By applying Young’s inequality to the coupled term $(Δ_{i u} + Δ_{i x})^{2}$ , and using the control-triggering condition $| Δ_{i u} (t) | \leq μ_{i}$ together with the state-triggering bounds, one obtains

| Δ_{i x} (t) | \leq \frac{k_{i 2} ℏ_{i 4} + ℏ_{i 1} + ℏ_{i 3}}{b_{i}} + \frac{b_{i} μ_{i}^{2}}{c_{i}} ℏ_{i 4} = {\bar{Δ}}_{i x} .

(37)

Therefore, the asynchronous control-deviation term is bounded as

\begin{aligned} b_{i} z_{i 2} (u_{i} - w_{i}^{*}) \leq & ϵ_{i 6} z_{i 2}^{2} + \frac{b_{i}^{2}}{4 ϵ_{i 6}} [(1 + λ_{i}^{- 1}) μ_{i}^{2} \\ + (1 + λ_{i}) {\bar{Δ}}_{i x}^{2}], \end{aligned}

(38)

where

ϵ_{i 6} > 0

For the robust hyperbolic-tangent term, let $ρ_{i} = b_{i} μ_{i} z_{i 2}$ . Using $0 \leq | ρ | - ρ \tanh (ρ / c) \leq 0.2785 c$ , one has

- b_{i} μ_{i} z_{i 2} \tanh (\frac{b_{i} μ_{i} z_{i 2}}{c_{i}}) \leq 0.2785 c_{i} .

(39)

Substituting (33), (34), (38), and (39) into (32), the derivative of $V$ satisfies

\begin{aligned} \dot{V} \leq & \sum_{i = 1}^{6} [- k_{i 1} + ϵ_{i 2} + ϵ_{i 3} + ϵ_{i 4} + ϵ_{i 5}] z_{i 1}^{2} \\ + \sum_{i = 1}^{6} [- k_{i 2} + ϵ_{i 1} + ϵ_{i 6}] z_{i 2}^{2} \\ + \sum_{i = 1}^{6} [0.2785 c_{i} + \frac{{(Λ_{i} {\bar{Υ}}_{i} + {\bar{ε}}_{i})}^{2}}{2 ϵ_{i 1}} + \frac{{\bar{η}}_{i}^{2}}{2 ϵ_{i 2}} \\ + \frac{k_{i 1}^{2} ℏ_{i 1}^{2}}{2 ϵ_{i 3}} + \frac{k_{i 1}^{2} ℏ_{i 3}^{2}}{2 ϵ_{i 4}} + \frac{ℏ_{i 2}^{2}}{2 ϵ_{i 5}} \\ + \frac{b_{i}^{2}}{4 ϵ_{i 6}} [(1 + λ_{i}^{- 1}) μ_{i}^{2} + (1 + λ_{i}) {\bar{Δ}}_{i x}^{2}]] \\ \leq & - K_{1} \sum_{i = 1}^{6} \frac{1}{2} z_{i 1}^{2} - K_{2} \sum_{i = 1}^{6} \frac{1}{2} z_{i 2}^{2} + Γ, \end{aligned}

(40)

where

K_{1}

K_{2}

, and

Γ

are given by

\begin{aligned} K_{1} & = 2 min_{i = 1, \dots, 6} {k_{i 1} - ϵ_{i 2} - ϵ_{i 3} - ϵ_{i 4} - ϵ_{i 5}}, \\ K_{2} & = 2 min_{i = 1, \dots, 6} {k_{i 2} - ϵ_{i 1} - ϵ_{i 6}}, \\ Γ & = \sum_{i = 1}^{6} [0.2785 c_{i} + \frac{{(Λ_{i} {\bar{Υ}}_{i} + {\bar{ε}}_{i})}^{2}}{2 ϵ_{i 1}} + \frac{{\bar{η}}_{i}^{2}}{2 ϵ_{i 2}} \\ + \frac{k_{i 1}^{2} ℏ_{i 1}^{2}}{2 ϵ_{i 3}} + \frac{k_{i 1}^{2} ℏ_{i 3}^{2}}{2 ϵ_{i 4}} + \frac{ℏ_{i 2}^{2}}{2 ϵ_{i 5}} \\ + \frac{b_{i}^{2}}{4 ϵ_{i 6}} [(1 + λ_{i}^{- 1}) μ_{i}^{2} + (1 + λ_{i}) {\bar{Δ}}_{i x}^{2}]] . \end{aligned}

(41)

Since $ϵ_{i j}$ $(j = 1, 2, \dots, 6)$ are constants selected according to the inequality conditions, there exist $ϵ_{i j}$ such that the designed parameters $k_{i 1}$ and $k_{i 2}$ satisfy $K_{1} > 0$ , $K_{2} > 0$ , and $Γ > 0$ , which are bounded constants. Therefore, the asynchronous coupling between the state-triggered channel and the control-triggered channel is explicitly modeled, explicitly bounded, and shown not to destroy the uniform ultimate boundedness of the closed-loop system.

Next, to avert the Zeno phenomenon, it is necessary to prove that there exists a positive constant $Δ_{i s}$ for each $i$ and $s$ , such that $t_{s, k + 1} - t_{s, k} \geq Δ_{i s}$ .

The triggering errors are defined as $e_{i 1}, e_{i 2}, e_{i 3}, e_{i 4}, e_{i 5}$ , and $e_{i 6}$ . Within the triggering interval $t \in (t_{1, k}, t_{1, k + 1}]$ , the triggered values remain constant, so the derivatives of the errors satisfy $d | e_{i 1} | / d t \leq | {\dot{γ}}_{i} |$ , $d | e_{i 2} | / d t \leq | {\dot{ξ}}_{i 2} |$ , $d | e_{i 3} | / d t \leq | {\dot{ξ}}_{i 1} |$ , $d | e_{i 4} | / d t \leq | {\dot{ξ}}_{i 2} |$ , $d | e_{i 5} | / d t \leq | {\dot{ξ}}_{i 1} | + | {\dot{γ}}_{i d} |$ , $d | e_{i 6} | / d t \leq | {\dot{ξ}}_{i 2} | + | {\dot{v}}_{i d} |$ .

Due to the boundedness of closed-loop signals, there exist positive constants $ϖ_{i 1}, ϖ_{i 2}, ϖ_{i 3}, ϖ_{i 4}, ϖ_{i 5}, ϖ_{i 6}$ such that $| {\dot{γ}}_{i} | \leq ϖ_{i 1}$ , $| {\dot{ξ}}_{i 2} | \leq ϖ_{i 2}$ , $| {\dot{ξ}}_{i 1} | \leq ϖ_{i 3}$ , $| {\dot{γ}}_{i d} | \leq ϖ_{i 4}$ , $| {\dot{v}}_{i d} | \leq ϖ_{i 5}$ .

By combining the fixed thresholds $ℏ_{i 1}$ , $ℏ_{i 2}$ , $ℏ_{i 3}$ , $ℏ_{i 4}$ , $ℏ_{i 5}$ , $ℏ_{i 6}$ in the triggering conditions, the minimum interval is $t_{1, k + 1} - t_{1, k} \geq$ $min ($ $ℏ_{i 1} / ϖ_{i 1},$ $ℏ_{i 2} / ϖ_{i 2},$ $ℏ_{i 3} / ϖ_{i 3},$ $ℏ_{i 4} / ϖ_{i 4},$ $ℏ_{i 5} / ϖ_{i 5},$ $ℏ_{i 6} / ϖ_{i 6}$ $) = Δ_{i 1} > 0.$

The control triggering error is $e_{i c}$ , and its derivative satisfies $d | e_{i c} | / d t \leq | {\dot{w}}_{i} |$ . Since $w_{i}$ is composed of bounded signals, there exists a positive constant $ϖ_{i c}$ such that $| {\dot{w}}_{i} | \leq ϖ_{i c}$ . Considering the fixed threshold $ℏ_{i c}$ in the control triggering condition, the minimum control triggering interval satisfies $t_{2, k + 1} - t_{2, k} \geq (ℏ_{i c} / ϖ_{i c}) = Δ_{i 2} > 0$ .

Thus, it can be shown that Zeno phenomenon cannot happen.

Simulation

In this section, two sets of simulation studies are carried out under constant and time-varying reference signals to evaluate the tracking performance and communication–computation efficiency of the proposed method. Three schemes are considered for comparison, namely, DET $_{our}$ , DET, and SET, where DET $_{our}$ denotes the proposed dual-event-triggered strategy with trigger-interval-weighted neural updating, DET denotes the conventional dual-event-triggered strategy with fixed-scale neural updating,³⁵ and SET denotes the single-trigger event-triggered strategy with fixed-step neural updating.³⁶ Subsequently, the model parameters of the quadrotor are presented in Table 1.³⁷

Table 1.

Basic parameters of the quadrotor UAV.

Description	Parameter	Value
Roll inertia	$I_{x x}$	$0.1 kg \cdot m^{2}$
Pitch inertia	$I_{y y}$	$0.1 kg \cdot m^{2}$
Yaw inertia	$I_{z z}$	$0.2 kg \cdot m^{2}$
Mass of the quadrotor	$m$	$1 kg$
Gravitational acceleration	$g$	$9.81 {m / s}^{2}$
Arm length	$l$	$0.2 m$
Thrust coefficient	$k_{t}$	$1.2 \times 10^{- 5}$
		$N / (rad / s)^{2}$
Drag moment coefficient	$k_{d}$	$1.36 \times 10^{- 7}$
		$N \cdot m / (rad / s)^{2}$

Simulation 1

This set of simulations compares the performance of DET $_{our}$ , DET, and SET under constant reference signals.

The reference trajectory is specified as $x_{d} (t) = 1$ , $y_{d} (t) = 2$ , $z_{d} (t) = 1$ , and $ψ_{d} (t) = 1$ , with ${\dot{x}}_{d} (t) = {\dot{y}}_{d} (t) = {\dot{z}}_{d} (t) = 0$ and ${\dot{ψ}}_{d} (t) = 0$ . The initial state is set to $x (0) = y (0) = z (0) = 0$ and $ϕ (0) = θ (0) = ψ (0) = 0$ , and $\dot{x} (0) = \dot{y} (0) = \dot{z} (0) = 0$ and $\dot{ϕ} (0) = \dot{θ} (0) = \dot{ψ} (0) = 0$ . The external disturbances are defined as $d_{x} (t) = 0.2 \sin (0.3 t)$ , $d_{y} (t) = 0.2 \sin (0.3 t + 0.4 π)$ , and $d_{z} (t) = 0.2 \cos (0.2 t + 0.1 π)$ . The controller parameters are chosen as $k_{i 1} = 0.5$ , $k_{i 2} = 8$ , $μ_{i} = 0.1$ , $τ_{i 1} = 0.025$ , and $c_{i} = 0.1$ for $i = 1, 2, 3$ , and $k_{i 1} = 4$ , $k_{i 2} = 10$ , $μ_{i} = 0.1$ , $τ_{i 1} = 0.025$ , and $c_{i} = 0.1$ for $i = 4, 5, 6$ . The neural network uses $N = 12$ nodes and $D = 6$ , with widths $B = 4$ , centers in $[- 4, 4]$ , and $\hat{W} (0) = 0$ ; the simulation step is $D_{t} = 0.001$ .

For DET $_{our}$ , trigger-interval-weighted neural updating is adopted at the state-triggering instants, with $g_{i 2} = 0.02$ , $σ_{i 2} = 0.0002$ , and $c_{i 2} = 1$ . For DET, the same dual-trigger framework is employed, but fixed-scale neural updating is used in the constant-reference case, with $g_{i 2} = 8 \times 10^{- 5}$ , $σ_{i 2} = 8 \times 10^{- 7}$ , and $c_{i 2} = 1$ . For SET, fixed-step neural updating is adopted under the single-trigger scheme, with $g_{i 2} = 0.02$ , $σ_{i 2} = 0.0002$ , and $c_{i 2} = 1$ .

The triggering parameters are set to $ℏ_{i 1} = ℏ_{i 2} = ℏ_{i 3} = ℏ_{i 4} = 0.01$ , $ℏ_{i 5} = ℏ_{i 6} = 0.02$ , $ℏ_{i c} = 0.05$ , and $ϱ_{i c} = 0.05$ .

The simulation results under constant reference inputs are shown in Figures 3 to 17. Figures 3 to 5 show that all three schemes achieve stable tracking of the constant reference signals in the X-, Y-, and Z-axes. Under the same dual-trigger framework, DET $_{our}$ exhibits responses that are closer to the desired values than those of DET, indicating that the proposed trigger-interval-weighted neural updating improves the tracking quality. Compared with SET, DET $_{our}$ still maintains sufficiently accurate tracking in all three translational channels. Figures 6 to 8 further show that the tracking errors under DET $_{our}$ remain within an acceptable range and are generally smaller than those under DET, while SET provides slightly tighter tracking due to its more frequent updating. Figures 9 to 11 illustrate the neural-network outputs in the translational channels, which confirm the effectiveness of the adaptive neural compensation in handling unknown nonlinearities and external disturbances. Figures 12 to 14 present the controller outputs and triggering instants along the X-, Y-, and Z-axes. After suppressing the oscillation induced by the desired-attitude inversion, DET $_{our}$ maintains satisfactory control performance with reduced update activity. Finally, Figures 15 to 17 compare the triggering intervals under DET $_{our}$ and DET. The results indicate that DET $_{our}$ preserves satisfactory pose-regulation performance while achieving a more favorable balance between control accuracy and transmission–computation cost.

Figure 3.

Position tracking responses in the X-axis under DET $_{our}$ , DET, and SET.

Figure 4.

Position tracking responses in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 5.

Position tracking responses in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 6.

Position tracking errors in the X-axis under DET $_{our}$ , DET, and SET.

Figure 7.

Position tracking errors in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 8.

Position tracking errors in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 9.

Neural network outputs in the X-axis under DET $_{our}$ , DET, and SET.

Figure 10.

Neural network outputs in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 11.

Neural network outputs in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 12.

Controller outputs and triggering instants in the X-axis under DET $_{our}$ , DET, and SET.

Figure 13.

Controller outputs and triggering instants in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 14.

Controller outputs and triggering instants in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 15.

State-triggering intervals under DET $_{our}$ .

Figure 16.

Control-triggering intervals under DET $_{our}$ .

Figure 17.

Control-triggering intervals under DET.

Table 2 compares the communication and update counts under constant reference signals. For the proposed DET $_{our}$ , the state and control channels are triggered 8638 and 1278 times, respectively, resulting in 9916 total state/control-side operations. For DET, the corresponding counts are 17342 and 1995, which gives 19337 total operations. For SET, although the control channel is triggered only 604 times, the state information and neural updating are executed at all 100001 simulation steps, leading to 100605 total state/control-side operations. These results indicate that, under the same dual-trigger framework, DET $_{our}$ reduces both the state-side and control-side activities relative to DET while maintaining satisfactory pose-regulation performance. Compared with SET, DET $_{our}$ requires more control-channel updates, but it markedly reduces the overall communication–computation burden by avoiding step-by-step state-side operation and fixed-step neural updating. Overall, the proposed trigger-interval-weighted neural updating law improves the resource efficiency of the dual-trigger framework, and DET $_{our}$ provides a more favorable balance between control performance and total update cost in resource-constrained UAV systems.

Table 2.

Triggering counts under constant signals.

Strategy	State channel	Control channel
DET $_{our}$	8638	1278
DET	17342	1995
SET	100001	604

Simulation 2

This set of simulations evaluates the performance differences under time-varying reference signals among the proposed dual-event-triggered strategy DET $_{our}$ , the conventional dual-event-triggered strategy DET, and the single-trigger event-triggered strategy SET.

The reference three-dimensional trajectory is defined as $x_{d} (t) = \sin (0.2 t)$ , $y_{d} (t) = \cos (0.2 t)$ , $z_{d} (t) = \arctan (0.1 t)$ , and $ψ_{d} (t) = 0.1 \sin (0.2 t)$ , with ${\dot{x}}_{d} (t) = 0.2 \cos (0.2 t)$ , ${\dot{y}}_{d} (t) = - 0.2 \sin (0.2 t)$ , ${\dot{z}}_{d} (t) = 0.1 / (1 + 0.01 t^{2})$ , and ${\dot{ψ}}_{d} (t) = 0.02 \cos (0.2 t)$ . All other parameters remain the same as in Simulation 1.

The simulation results of the quadrotor UAV under time-varying reference inputs and external disturbances are shown in Figures 18 to 33. Figure 18 shows that all three schemes are able to follow the desired three-dimensional path, while DET $_{our}$ maintains a trajectory that remains close to the reference under time-varying operating conditions. Figures 19 to 21 further illustrate the position responses along the X-, Y-, and Z-axes. DET $_{our}$ preserves satisfactory tracking of the time-varying references, and under the same dual-trigger framework its responses remain generally closer to the desired signals than those of DET. Compared with SET, DET $_{our}$ still provides sufficiently accurate dynamic tracking in all three translational channels. Figures 22 to 24 show that the corresponding tracking errors remain bounded and within an acceptable range throughout the maneuver. Figures 25 to 27 illustrate the neural-network outputs in the translational channels, confirming that the adaptive neural compensation remains effective in handling the time-varying nonlinearities and disturbances. Figures 28 to 30 present the controller outputs and triggering instants along the X-, Y-, and Z-axes. Under the revised simulation setting, DET $_{our}$ maintains satisfactory control performance with moderated update activity. Finally, Figures 31 to 33 compare the triggering intervals under DET $_{our}$ and DET. The results indicate that DET $_{our}$ preserves satisfactory pose-regulation performance while achieving a more favorable balance between dynamic tracking accuracy and transmission–computation cost under time-varying operating conditions.

Figure 18.

Three-dimensional translational trajectories under DET $_{our}$ , DET, and SET.

Figure 19.

Position tracking responses in the X-axis under DET $_{our}$ , DET, and SET.

Figure 20.

Position tracking responses in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 21.

Position tracking responses in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 22.

Position tracking errors in the X-axis under DET $_{our}$ , DET, and SET.

Figure 23.

Position tracking errors in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 24.

Position tracking errors in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 25.

Neural network outputs in the X-axis under DET $_{our}$ , DET, and SET.

Figure 26.

Neural network outputs in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 27.

Neural network outputs in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 28.

Controller outputs and triggering instants in the X-axis under DET $_{our}$ , DET, and SET.

Figure 29.

Controller outputs and triggering instants in the Y-axis under DET $_{our}$ , DET, and SET.

Figure 30.

Controller outputs and triggering instants in the Z-axis under DET $_{our}$ , DET, and SET.

Figure 31.

State-triggering intervals under DET $_{our}$ .

Figure 32.

Control-triggering intervals under DET $_{our}$ .

Figure 33.

Control-triggering intervals under DET.

Table 3 summarizes the communication and update counts under time-varying reference signals. Under DET $_{our}$ , the state and control channels are triggered 932 and 6993 times, respectively, resulting in 7925 total transmissions and updates. Under DET, the corresponding triggering counts are 1021 and 7004, respectively, yielding 8025 total transmissions and updates. Since DET adopts the same dual-trigger framework as DET $_{our}$ , their triggering counts remain close. In this case, DET $_{our}$ achieves slightly lower state-side activity and a slightly reduced total update burden, showing that the proposed trigger-interval-weighted neural updating does not introduce extra communication cost under time-varying conditions. For SET, the state-related signals remain continuously available at all 100001 sampling instants, while the control channel is updated 520 times, resulting in 100521 total transmissions and updates. Although SET involves fewer control-channel triggering instants, it still requires continuous state-side information and fixed-step-like neural updating, which leads to a much larger overall communication–computation burden. Overall, DET $_{our}$ maintains satisfactory pose-regulation performance under time-varying references while keeping the total update cost close to that of DET and substantially lower than that of SET, thereby providing a favorable balance between dynamic tracking performance and resource consumption.

Table 3.

Triggering counts under time-varying signals.

Strategy	State channel	Control channel
DET $_{our}$	932	6993
DET	1021	7004
SET	100001	520

Conclusion

This paper proposes a dual-event-triggered adaptive neural-network control strategy for resource-constrained quadrotor UAV control. By incorporating asynchronous event-triggering at both the state-transmission and control-update levels, the scheme reduces unnecessary updates and lowers the overall communication–computation burden while preserving closed-loop stability. Moreover, a trigger-interval-weighted neural-network updating law has been introduced to improve the applicability of the adaptive mechanism under different operating conditions while reducing unnecessary online computation. Rigorous Lyapunov-based analysis has established the uniform ultimate boundedness of all closed-loop signals and strictly excluded Zeno behavior. Both theoretical analysis and numerical simulations have verified the effectiveness of the proposed method and the trade-off between control performance and resource consumption for resource-constrained UAV systems. Future research will extend this framework to cooperative multi-UAV systems and investigate more flexible event-triggered mechanisms, such as dynamic thresholds and data-driven strategies, to further enhance system adaptability and robustness.

Footnotes

ORCID iD

Pengxi Ren

Ethical considerations

This study does not involve human participants, human data, human tissue, or animals. Therefore, ethical approval was not required.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data availability statement

Not applicable.

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