Event-Triggered Robust Fuzzy Controller Design for a Quadcopter Under Network Bandwidth Constraints ^†

Abstract

Quadcopter drones have been extensively researched due to their flexibility and suitability for diverse tasks. In this study, a control strategy tailored for scenarios with restricted network bandwidth is developed. An event-triggered control approach was used to minimize network bandwidth load. Also, a robust fuzzy controller was integrated to enhance the system’s resilience and efficiency. The simulation results confirmed that the developed control strategy fosters stable performance, even under constrained network conditions.

1. Introduction

Drones have been widely employed across various applications, including military operations, geospatial surveying, and environmental monitoring [1,2,3,4,5,6,7,8]. Drones mitigate risks to human operators, enhance the accuracy and efficiency of data acquisition, and enable access to otherwise difficult or hazardous environments. The quadcopter has garnered particular attention among various drones due to its superior maneuverability, vertical takeoff and landing (VTOL) capabilities, cost efficiency, and relatively straightforward mechanical architecture. These attributes increase interest in quadcopter platforms in drone system design and control research.

As quadcopters [4] are used in diverse domains, their reliance on wireless communication networks for data transmission becomes increasingly significant. With the growing number of aerial units operating concurrently, the demand for communication bandwidth increases accordingly. However, wireless bandwidth is a finite resource. When multiple quadcopters share the same network, they compete for limited bandwidth, potentially causing congestion, reduced communication quality, and degraded overall system performance.

To mitigate the impact of constrained communication bandwidth, an event-triggered control (ETC, [9,10,11,12,13,14]) is used for adaptive sampling and transmissions based on state-dependent triggering conditions, which are based on low system error thresholds [12,14,15,16,17,18,19,20,21,22]. This event-driven mechanism effectively lowers the control signal update frequency, resulting in reduced network traffic and enhanced bandwidth utilization. A robust fuzzy control is adopted to compensate for parametric uncertainties and exogenous disturbances affecting the quadcopter’s nonlinear dynamics [16,18,19,22]. The system is represented by a Takagi–Sugeno (T-S) fuzzy model, and controller synthesis is performed through parallel distributed compensation (PDC) to guarantee closed-loop stability. Stability and robustness proofs are conducted using Lyapunov function-based analysis [15], ensuring the system’s convergence properties under bounded disturbances.

In this study, the event-triggered robust fuzzy controller maintained satisfactory tracking performance and system stability despite stringent network bandwidth constraints, validating its practical applicability for quadcopter control in resource-limited communication environments.

The remainder of this manuscript is structured as follows. Section 2 introduces the problem formulation and underlying assumptions. Section 3 details the design methodology of the proposed event-triggered robust fuzzy control scheme. In Section 4, simulation studies are conducted to evaluate and verify the effectiveness of the control approach. Finally, Section 5 summarizes the key findings and concludes the study.

2. Problem Formulation

The kinematic and dynamic models of the quadcopter were established in this study. The quadcopter consists of four rotors mounted symmetrically on a cross-shaped frame, as illustrated in Figure 1 [6]. This configuration is a basis for deriving the system’s motion equations and control dynamics. In the figure, the quadcopter’s attitude $\Theta ={\left[\varphi ,\theta ,\psi \right]}^T\in {\mathfrak{R}}^3$, which consists of the roll (ϕ), pitch (θ), and yaw (ψ) angles in the earth-fixed coordinate system (E), the angular velocity of the vehicle in the body-fixed coordinate system (B) is set as ${\omega }_b={\left[p,q,r\right]}^T\in {\mathfrak{R}}^3$, and the inertia in B is $J_b\in {\mathfrak{R}}^{3\times 3}$.

The quadcopter’s rotational dynamics are expressed as follows.

$${\omega }_b=R(\mathrm{\Theta })\dot{\Theta }$$

(1)

$$J_b{\dot{\omega }}_b=-{\omega }_b\times J_b{\omega }_b+\tau +{\tau }_d$$

(2)

where $\tau$ is the notation of the control torque ${\tau }_d$ is the notation of the lumped disturbance, including parameterized uncertainties and external disturbances, acting on the quadcopter. The matrix $R(\mathrm{\Theta })\in {\mathfrak{R}}^{3\times 3}$ is defined to represent the rotation from B to E as follows.

$$R\left(\Theta \right)=\left[\begin{array}{ccc}1& 0& -sin\theta \\ 0& cos\varphi & sin\varphi cos\theta \\ 0& -sin\varphi & cos\varphi cos\theta \end{array}\right]$$

(3)

The control torque vector $\tau ={\left[{\tau }_1,{\tau }_2,{\tau }_3\right]}^T$ is generated by the thrust forces of the four propellers, each driven by an individual motor. The relationship between the motor inputs and the resulting torques is described by the following equations.

$$\begin{gathered} {\tau }_1=l\left(F_2-F_4\right) \\ {\tau }_2=l\left(F_3-F_1\right) \\ {\tau }_3=c_d\left(-F_1+F_2-F_3+F_4\right) \end{gathered}$$

(4)

where $F_i=c_b{\Omega }_i^2$ is the force generated by the ith propeller ($i=1,2,3,4$), $c_b$ denotes the thrust coefficient, ${\Omega }_i$ denotes the propeller rotary speed, $l$ is the length of the quadcopter arm, and $c_b$ is the force-to-torque coefficient.

An altitude control module generates a virtual thrust signal, which is then integrated with the computed virtual torque input to construct the complete control vector. This vector is subsequently translated into the corresponding motor control signals for system actuation. By performing the time derivative of Equation (1), the following expression is obtained.

$${\dot{\omega }}_b=R\left(\Theta \right)\ddot{\mathrm{\Theta }}+\dot{R}(\mathrm{\Theta })\dot{\Theta }$$

(5)

By substituting Equation (1) into (5), the quadcopter dynamic equation is obtained as follows.

$$M\left(\Theta \right)\ddot{\mathrm{\Theta }}+C\left(\Theta ,\dot{\Theta }\right)\dot{\Theta }+G\left(\Theta ,\dot{\Theta }\right)+{\tau }_d=\tau$$

(6)

where $M\left(\mathrm{\Theta }\right)=J_{b}R\left(\mathrm{\Theta }\right)$ and $C\left(\mathrm{\Theta },\dot{\mathrm{\Theta }}\right)=J_b\dot{R}\left(\mathrm{\Theta }\right)+\dot{R}(\mathrm{\Theta })\dot{\mathrm{\Theta }}\times J_{b}R\left(\mathrm{\Theta }\right)$.

In this study, a controller was designed to stabilize the quadcopter’s state trajectory around a predefined equilibrium point, ensuring asymptotic convergence and overall system stability.

3. Event-Triggered Robust Fuzzy Controller

To fulfill the control objective of stabilizing the quadcopter at a designated equilibrium point, a T-S fuzzy model [15] was constructed to approximate the complex nonlinear input–output characteristics of the system. The ith fuzzy rule of the T-S fuzzy model is designed as

$$\begin{gathered} \mathrm{IF}{q}_1\mathrm{is}{\tilde{F}}_{q_1}^i\mathrm{and}\dots \mathrm{and}{q}_k\mathrm{is}{\tilde{F}}_{q_k}^i \\ \mathrm{THEN}\mathrm{\Theta }\left(k+1\right)={\mathit{A}}_i\mathrm{\Theta }\left(k\right)+{\mathit{B}}_i\tau \left(k\right) \end{gathered}$$

(7)

where $i=1,2,\dots L$; $q_i(i=1,2,\dots ,k)$ are the premise variables, $\mathbf{x}(k)$ is the system parameter matrix, $\tau \left(k\right)$ is the control input, ${\mathit{A}}_i\in {\mathfrak{R}}^{n\times n}\mathrm{a}\mathrm{n}\mathrm{d}{\mathit{B}}_i\in {\mathfrak{R}}^{n\times m}$ are the system parameters, and ${\tilde{F}}_{q_h}^i$ is the fuzzy set of the premise variables.

By using the singleton fuzzifier, the product inference engine, the center-average defuzzifier [15], the output of the T-S fuzzy model (Equation (7)) is expressed as

$$\mathrm{\Theta }\left(k+1\right)={\sum }_{i=1}^L{\xi }^i\left({\mathit{A}}_i\Theta \left(k\right)+{\mathit{B}}_i\tau \left(k\right)\right)$$

(8)

where

$${\xi }^i={\displaystyle \frac{\prod _{h=1}^k{\tilde{F}}_{q_h}^i}{\sum _{i=1}^L\prod _{h=1}^k{\tilde{F}}_{q_h}^i}}$$

Using Equation (8), the quadcopter’s dynamic behavior is presented as a weighted combination of local linear models to design a controller while preserving essential nonlinearities inherent in the system dynamics.

Based on Equation (8), a robust fuzzy controller was designed corresponding to each local linear model. Therefore, the fuzzy rules of the robust fuzzy controller were designed as

$$\begin{gathered} \mathrm{IF}{q}_1\mathrm{is}{\tilde{F}}_{q_1}^i\mathrm{and}\dots \mathrm{and}{q}_k\mathrm{is}{\tilde{F}}_{q_k}^i \\ \mathrm{THEN}\tau \left(k\right)={\mathit{K}}_i\mathrm{\Theta }\left(k\right) \end{gathered}$$

(9)

where $i=1,2,\dots L$, and ${\mathit{K}}_i\in {\mathfrak{R}}^{m\times n}$ is the control gain matrix. By using the singleton fuzzifier, the product inference engine and the center-average defuzzifier [15], the output of the robust fuzzy controller (Equation (9)) is expressed as

$$\left(k\right)={\sum }_{i=1}^L{\xi }^i{\mathit{K}}_i\Theta \left(k\right)$$

(10)

Substituting Equation (10) into Equation (8) yields the following equation.

$$\mathrm{\Theta }\left(k+1\right)={\sum }_{i=1}^L{\xi }^i{\sum }_{j=1}^L{\xi }^i\left({\mathit{A}}_i\Theta \left(k\right)+{\mathit{B}}_i{\mathit{K}}_j\Theta \left(k\right)\right)$$

(11)

To reduce network bandwidth requirements, the event-triggered control was adoped. And the triggering instant was set as $k_{s+1}(s=1,2,\dots )$.

$$k_{s+1}=\mathrm{m}\mathrm{i}\mathrm{n}\{k>k_s|{\left(\mathit{x}\left(k\right)-\mathit{x}\left(k_s\right)\right)}^T\mathsf{\Xi }\left(\mathit{x}\left(k\right)-\mathit{x}\left(k_s\right)\right)>\delta {\left(\mathit{x}\left(k_s\right)\right)}^T\mathsf{\Xi }\left(\mathit{x}\left(k_s\right)\right)\}$$

(12)

where $\delta \in [0,1)$ denotes the known proportional threshold and $\mathsf{\Xi }$ denotes the symmetric positive definite weighting matrix. Then, the control law (Equation (11)) was rewritten as

$$\mathrm{\Theta }\left(k+1\right)=\left\{\begin{array}{l}{\sum }_{i=1}^L{\xi }^i{\sum }_{j=1}^L{\xi }^i\left({\mathit{A}}_i\Theta \left(k\right)+{\mathit{B}}_i{\mathit{K}}_j\Theta \left(k\right)\right)\\ \Theta \left(k\right)\end{array}\begin{array}{l}k=k_{s+1}\\ k\in (k_s,k_{s+1})\end{array}\right.$$

(13)

Using Equation (13), the control strategy was discontinued when the feedback input was not updated and the output signal varied within a preset range. Therefore, it was unnecessary to transmit system states and control signals at every time instant. Under the network bandwidth constraints, the developed control strategy in this study showed excellent control performance.

4. Simulation Results

To evaluate the effectiveness of the proposed controller, computer simulations were conducted under the following quadcopter parameter settings: an arm length of 0.295 m, a concentrated mass of 0.08 kg representing the engine and propeller assembly, a spherical body mass of 0.75 kg, and a sampling interval of 0.005 s. The target attitude states were specified as ${\mathrm{\Theta }}_{\mathrm{d}}={\left[{\varphi }_d,{\theta }_d,{\psi }_d\right]}^T=[-\pi /8,\pi /8,0]$. The resulting state trajectories of the quadcopter under control are illustrated in Figure 2, Figure 3 and Figure 4.

The simulation results showed that the developed event-triggered robust fuzzy control strategy stabilized the quadcopter system despite the presence of limited communication resources due to constrained network bandwidth. The controller maintained accurate attitude tracking performance, wherein all state variables converged smoothly to their corresponding reference trajectories without exhibiting noticeable oscillations or delays. The triggering mechanism significantly reduced the frequency of control updates and data transmissions, thereby showing stable control to manage communication constraints efficiently. The consistent tracking behavior under reduced communication highlights both the robustness and the viability of the proposed approach. These results collectively validated the effectiveness of the controller in achieving reliable and stable flight performance in networked control environments with bandwidth limitations.

5. Conclusions

An event-triggered robust fuzzy controller was developed for quadcopter systems subject to network bandwidth limitations. The quadcopter’s nonlinear dynamics were determined using a Takagi–Sugeno fuzzy model to effectively capture the input–output relationship of the system across different operating regions. Building on this model, a fuzzy control strategy based on PDC was developed to stabilize each local linear subsystem to ensure overall system stability. Simulations were performed under constrained network bandwidth scenarios to evaluate the practical performance of the developed controller. The results confirmed that the control scheme reduced communication load through event-triggered mechanisms and guaranteed the robust and stable flight of the quadcopter, validating the effectiveness and applicability of the proposed method in realistic operating environments.