HHO-Based Cable Tension Control of Tethered UAV with Unknown Input Time Delay

Abstract

A tethered Unmanned Aerial Vehicle (UAV) is a special type of UAV that is powered continuously through a cable, ensuring long-duration flight. However, the pulling interference of the cable significantly affects the UAV’s stability control, limiting its application and development. This paper addresses this issue by first analyzing the effect of cable tension on the UAV’s wind resistance capability and demonstrates the possibility of using cable tension to assist in wind resistance control. Building on this, a robust time-delay compensator is designed to address the problem of unknown external disturbance and unknown time delay in the cable control input. Sufficient conditions for system boundedness are provided using the Lyapunov–Krasovskii functional. Subsequently, to deal with the strong nonlinearity and strong coupling issues of the sufficient conditions, the Harris Hawks Optimization (HHO) algorithm is employed for intelligent optimization of the controller parameters. Simulation results indicate that the HHO-based robust time-delay compensator exhibits excellent robustness and fast response.

1. Introduction

A tethered UAV (TUAV) is a specialized unmanned aerial system composed of a tether cable and a rotorcraft UAV [1]. The UAV is connected to a mobile platform via the tether cable for external power supply, significantly enhancing the endurance of the UAV [2]. Due to this characteristic, TUAVs are widely used in various fields, including disaster relief, border patrol, geological surveying, forest fire prevention, and emergency communication [3,4]. Compared to traditional non-tethered UAVs, TUAVs not only face wind field interference but also experience a coupling effect between the stiffness and flexibility of the tether and the UAV [5]. This results in significant influence of the tether tension on the UAV’s motion, making stable control particularly challenging.

Currently, research on the stable control of TUAVs can be divided into two main categories: adaptive control methods and robust control methods. In the field of adaptive control, Sierra et al. [6] introduced a neural network-based intelligent control strategy, which takes into account variations in UAV mass and wind effects during the mission. They designed a mass estimator and an interference estimator to achieve stable flight of the UAV under various trajectories. Hua et al. [7], based on active disturbance rejection control and radial basis function neural networks, proposed a fault-tolerant flight control method for multi-rotor UAVs in the presence of actuator failures and external wind disturbances, ensuring stable control when the UAV is disturbed. Rodriguez et al. [8] employed the deep deterministic policy gradient algorithm to address the UAV landing problem on a moving platform under wind disturbances.

In the field of robust control, Sun et al. [9] proposed a fuzzy active disturbance rejection control (ADRC) to improve the dynamic performance of the system. Compared to traditional ADRC, this approach exhibits faster response speed and smaller overshoot, providing excellent control capabilities for the attitude stability and disturbance-resistant flight of drones. Eliker K et al. [10] combined a recursive control method with a robust control algorithm to design a finite-time integral backstepping fast terminal sliding mode controller. Hamadi et al. [11] introduced a quadrotor wind compensation strategy, utilizing a second-order sliding mode controller and observer based on the super-twisting algorithm, ensuring robustness against external disturbances, time-varying parameters, and nonlinear uncertainties. Izadi et al. [12] achieved real-time estimation of time-varying disturbances using a high-gain disturbance observer and compensated for these disturbances within the sliding mode controller.

In summary, regardless of the method employed, current research on the stability control of TUAVs is similar to the wind disturbance control methods for traditional rotorcraft UAVs, with the primary difference being the inclusion of tether tension as a disturbance in the dynamic model. In fact, tether tension significantly impacts UAV motion. To address this issue, from a control perspective, Kourani et al. [13] designed a layered control framework for offshore TUAV systems, implementing a precise motion control system for the tethered platform’s buoy design, maintaining the surge velocity within a certain threshold, thereby ensuring the tether remains under low tension and minimizing its impact on UAV motion. Talke et al. [14] developed a tension feedback control based on disturbance estimation for the tether deployment/retraction of a system composed of small surface vessels and UAVs, significantly reducing the tether tension acting on the UAV and enhancing the stability of the tethered UAV. From a planning perspective, Martínez et al. [15] incorporated tether tension and UAV states as constraints, using an optimal rapidly exploring random tree method to jointly plan the motion of both the UAV and the tethered platform, thereby keeping the tether tension within a specific range. However, these methods only limit tether tension within a certain range and essentially treat it as a disturbance, without fully considering the potential of using tether cable tension to enhance wind resistance. Therefore, this paper addresses this issue by exploring the use of tether cable tension as a control input to improve the UAV’s wind resistance capability.

However, providing controllable tether tension is quite challenging. In TUAV systems, the device controlling the retrieving and releasing of the tether is generally located at the ground tethering end. The tether tension is transmitted from the ground tethering end to the UAV, which introduces a certain delay. This results in the tether tension control system being a time-delay system, which cannot be effectively addressed by traditional control methods. For the stabilization of time-delay systems, the most commonly used approach is based on Lyapunov functional stability analysis methods [16], such as the Lyapunov–Razumikhin (L-R) functional [17] and the Lyapunov–Krasovskii (L-K) functional [18].

Building upon this, the current research on control methods for time-delay systems mainly includes predictive-compensatory control and memoryless feedback control. The Smith predictor, proposed in 1957, is a predictive structure used for controlling pure delay systems [19], and over the years, various improvements have been made by scholars based on this approach. Mohanapriya et al. [20] proposed a modified repetitive control scheme based on the Matausek–Micic-modified Smith predictor approach. The integration of the transfer function with the modified Smith predictor block ensures accurate estimation and effective attenuation of external disturbances. Jain et al. [21] introduced a robust compensation control method to solve the tracking control problem for uncertain nonlinear systems with unknown constant input delays, and used the L-K functional to prove the global uniformly ultimate boundedness of the closed-loop system. Sun et al. [22] addressed feedforward systems with unknown parameters and delays and proposed a nested saturation feedback control law with gain-dependent saturation levels, achieving adaptive regulation with global stability simultaneously. Yu et al. [23] proposed an adaptive control of the uncertain systems with unknown delays by memoryless state feedback and the construction of a dynamic-gain-based observer. It can achieve adaptive state regulation with global stability, despite of the presence of unknown parameters as well as unknown delays in the state, input, and output. The problem studied in this paper is the stabilization of the tethered UAV system with unknown input time delay and unknown wind disturbance. The robust compensation control method has the ability of anti-interference and is therefore more suitable for the research object of this paper.

This paper takes the two-dimensional plane in the vertical direction as an example to analyze the capability of the cable to assist in wind resistance for the TUAV. Building upon this, a robust time-delay compensator (RTDC) is designed, which introduces a novel filtering error by simultaneously considering both the error signal and historical input signals within the time-delay interval. Additionally, this approach addresses the issues of unknown external disturbances and unknown input delays within the cable tension control system. The main contributions of this paper are as follows:

• In terms of application, for the TUAV, the influence and characteristics of tether cable tension on the UAV’s wind resistance capability were analyzed through simulation, providing new insights into the wind resistance hovering control of TUAV;
• In terms of method, by introducing error, error integral, error derivative, and historical input signals into the compensator, the issue of unknown input delays in tether cable tension control under unknown disturbances was resolved. Meanwhile, the controller exhibits strong robustness and can effectively resist external disturbances;
• Also in terms of method, for the convenience of controller parameter design, a controller parameter tuning strategy based on the Harris Hawks Optimization algorithm was designed. A new performance index function was introduced, ensuring minimal steady-state error while achieving fast convergence speed.

2. Analysis of the Cable to Assist in Wind Resistance

As described in the Section 1, most of existing research on disturbance rejection control for TUAVs follows the control framework of traditional UAVs, treating tether cable tension merely as an external disturbance. This approach does not fully exploit the tether’s pulling effect. Therefore, this section first analyzes the auxiliary role of tether cable tension in enhancing the wind disturbance resistance capability of the UAV, providing theoretical support for the tether/rotorcraft coordination control framework.

Assumption 1. 

The focus of this study is on the contribution of the horizontal component of cable tension to the wind resistance capability of TUAV. Therefore, without loss of generality, this paper considers the stability control of UAVs within a two-dimensional plane, and assumes that the wind field environment in which the TUAV operates is characterized by uniform horizontal wind.

Firstly, the following coordinate systems are defined: Inertial Coordinate System $Oxy$: The horizontal direction is denoted as the $Ox$ axis, and the vertical direction as the $Oy$ axis. UAV Body Coordinate System: The origin $O_d$ is located at the center of mass of the UAV. The $O{x}_d$ axis is parallel to the plane of the UAV’s rotors, and the forward direction of the UAV is taken as the positive direction along the $O{x}_d$ axis. The direction perpendicular to the rotor plane, upwards, is considered the positive direction along the $O{y}_d$ axis. The coordinate system is shown in Figure 1.

To clarify the auxiliary role of cable tension in improving the wind resistance ability of UAVs, the maximum horizontal wind resistance $D_{\max }$ that a UAV can withstand at different pitch angles $\alpha$ is used to measure its wind resistance capability. First, a force analysis of the UAV in a balanced state is conducted, as shown in Figure 1. The UAV is subjected to gravitational force G, lift $U_1$, horizontal wind resistance D, and cable tension T. For a typical non-tethered quadcopter UAV, the maximum horizontal wind resistance it can withstand can be expressed as

$$\begin{array}{c}\hfill D_{\max }=G\cot \alpha ,\alpha \ge \arcsin \left(G/{\overline{U}}_1\right)\end{array}$$

(1)

where ${\overline{U}}_1$ represents the maximum lift generated by the UAV’s rotors. For TUAV, at a certain tether cable inclination angle $\theta$, the horizontal component of the tether cable tension can be utilized to assist in resisting horizontal wind disturbances. In this case, the UAV’s equilibrium equation can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}_{\max }=U_1\cos \alpha +T\cos \theta \hfill \\ U_1\sin \alpha =G+T\sin \theta \hfill \end{array}\right.\end{array}$$

(2)

Then, the $D_{\max }$ of the TUAV can be calculated as

$$D_{\max }=U_1\cos \alpha +\left(U_1\sin \alpha -G\right)·\cot \theta$$

To visually compare the maximum horizontal wind resistance that non-tethered UAVs and TUAVs can withstand, based on Equations (1) and (2), the relationship between the maximum horizontal wind resistance, UAV inclination angle, and cable inclination angle is obtained, as shown in Figure 2, Figure 3 and Figure 4. The parameters of the UAV and the cable are chosen in

Table 1. UAV parameters.

Parameters	Symbols	Value	Units
Mass of UAV	$m_{UAV}$	1.4	$\mathrm{kg}$
Maximum length of cable	$l_{max}$	20	$\mathrm{m}$
Diameter of cable	d	3	$\mathrm{mm}$
Linear density of cable	$\mu$	0.0125	$\mathrm{kg}/\mathrm{m}$
Atmospheric density	$\rho$	1.2245	$\mathrm{kg}/{\mathrm{m}}^3$
Characteristic area of UAV	S	0.5	${\mathrm{m}}^2$
Drag coefficient of UAV	$C_{Dx}$	0.8	None
Drag coefficient of cable	$C_{LDx}$	0.8	None
Maximum lift of UAV	$U_1$	40	$\mathrm{N}$

.

From Figure 3, it can be found that the maximum wind resistance of a non-tethered UAV is 37.6 N at the critical inclination angle, whereas Figure 2 shows that, in the tethered state, the maximum wind resistance of the UAV is 263 N. This indicates that the wind resistance capacity of the TUAV is significantly greater than that of the non-tethered UAV. Figure 4 demonstrates that the wind resistance of the TUAV is related to the cable inclination angle. When the cable inclination angle is less than 1.18 rad (67.6°), the wind resistance of the TUAV exceeds 37.6 N, which is the maximum wind resistance of the non-tethered UAV. This means that as long as the tether cable inclination angle is smaller than this critical angle, the tether can significantly enhance the wind resistance capability of the TUAV. In fact, for the same cable tension, the smaller the cable inclination angle, the larger the horizontal component of the cable tension, thereby enhancing the wind resistance capability of the TUAV. The simulation results are consistent with this theoretical analysis. The above results indicate that cable tension has a significant effect on the wind resistance capability of UAVs within a certain range, which provides a theoretical foundation for cable-assisted wind resistance control.

Remark 1. 

It is worth noting that the TUAV’s cable examined in this study is utilized solely for power supply, resulting in a relatively low mass. Table 1 shows the parameters of the cable, which yields a gravitational force of approximately 2.45 N. Using the cable drag force formula, we calculated that under a base wind speed of 6 m/s, the horizontal drag force amounts to 2.12 N. Moreover, from the simulation results, the cable tension is typically around 10 N. Therefore, within the framework of tension control established in this study, the tension in the cable predominantly influences its shape. Consequently, we have neglected the effects of aerodynamic loads and gravity on the cable shape, which means that it is assumed that the cable is in a straight line during the control process.

Remark 2. 

Figure 3 shows that under the same wind resistance, the inclination angle of a non-tethered UAV is fixed. However, under the influence of tether cable tension, a TUAV can maintain stability at different inclination angles. This indicates that, with the assistance of the tether, the drone can hover stably in any suitable attitude, which significantly broadens the potential application scenarios of UAVs. However, the focus of this paper is on the wind disturbance rejection control algorithm for TUAVs, and this aspect will not be elaborated further.

3. Robust Time-Delay Compensator Design

3.1. Preliminaries

The discussion in Section 2 indicates that cable tension can significantly improve the wind resistance capability of UAVs. However, the control of cable tension is subject to significant delay effects. In previous research, the author established a tension measurement system using a winch and two tension sensors to investigate the tension transmission process. The results indicated that the time delay in the transmission of cable tension is related to factors such as cable length, magnitude of tension, and cable load. For a cable length of 50 m, the tension acting on the UAV is delayed by more than 0.1 s compared to the tension output at the ground tethering end. Under the influence of external wind disturbances, this time delay in control inputs can significantly complicate the design of the controller. Therefore, this paper addresses the issue of wind-resistant hovering control for TUAVs under input time delay, and designs a robust time-delay compensator. Firstly, the following assumptions are introduced:

Assumption 2. 

As shown in the analysis in Section 2, under the influence of cable tension, the UAV can maintain stable hovering at any suitable attitude. Therefore, for the convenience of controller design, it is assumed that the UAV maintains stable flight with an inclination angle of 0. Furthermore, this paper assumes that the upper end of the cable is connected to the center of mass of the UAV, which means that the tension of the cable will not generate a moment and thereby affect the attitude of the UAV.

Then, the dynamic model of a TUAV in a two-dimensional plane is expressed as shown in Equation (3), where $x,y$ represent the position of the UAV in the horizontal and vertical directions, respectively. $D_x={\displaystyle \frac{1}{2}}{C}_{Dx}\rho S{\dot{x}}^2$ represents the horizontal wind drag force caused by the characteristic area of the UAV, where $C_{Dx}$ is the drag coefficient, $\rho$ is the atmospheric density, S is the characteristic area of the UAV, and $\dot{x}$ is the horizontal velocity of the UAV. The control inputs include cable tension T and UAV lift $U_1$. The cable tension acts on both the horizontal and vertical channels, while under Assumption 2, the UAV lift only affects the vertical channel.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\ddot{x}={\displaystyle \frac{D_x}{m}}+T\left(t+\Delta t\right)\ast {\displaystyle \frac{x}{\sqrt{x^2+y^2}}}\hfill \\ \ddot{y}={\displaystyle \frac{U_1}{m}}-{\displaystyle \frac{D_x}{m}}-T\left(t+\Delta t\right)\ast {\displaystyle \frac{x}{\sqrt{x^2+y^2}}}-G\hfill \end{array}\right.\end{array}$$

(3)

Remark 3. 

It is important to note that in practical scenarios, the response speed and the control bandwidth of the UAV’s rotors are significantly faster than the cable tension control mechanism. Therefore, for the convenience of controller design, the following Assumption 3 is made:

Assumption 3. 

This section focuses on the effect of the cable tension of the TUAV system on the horizontal wind resistance of the UAV. Therefore, it is assumed that the lift control of the UAV is fast enough to ensure that the flight altitude (y-direction position) of the UAV changes within a small range. That is, $\underline{y}<y<\overline{y}$. Considering the position change of the vertical channel of the UAV as a disturbance can simplify the control model while taking into account the coupling effect of the two channels of the UAV.

Under the above assumption, the model can be simplified as

$$\begin{array}{c}\hfill \ddot{x}=f\left(x,\dot{x}\right)+g\left(x\right)u\left(t-\tau \right)+d\left(t\right)\end{array}$$

(4)

where

$$f\left(x,\dot{x}\right)={\displaystyle \frac{1}{2m}}{C}_{Dx}\rho S{\dot{x}}^2,g\left(x\right)={\displaystyle \frac{x}{\sqrt{x^2+y^2}}}$$

and $u\left(t-\tau \right)$ represents the control input with time delay. For $t<\tau$, there is $u\left(t-\tau \right)=0$, where $\tau \in R^{+}$ is the unknown time delay. $d\left(t\right)$ represents the time-varying external disturbance. For the convenience of description, the following assumptions and lemmas are given:

Assumption 4. 

The unknown time delay has a known upper bound ${\tau }_{\max }\in R^{+}$, i.e., $\tau <{\tau }_{\max }$.

Assumption 5. 

According to Assumption 3, the function $g\left(x\right)$ has a known upper bound $\overline{g}$ and a lower bound $\underset{-}{g}$, i.e., $\underset{-}{g}I\le g\left(x\right)\le \overline{g}I$.

Assumption 6. 

The unknown time-varying external disturbance $d\left(t\right)$ has a known upper bound $\overline{d}\in R^{+}$, i.e., $∥d\left(t\right)∥\le \overline{d}$, where $∥d∥$ means the standard Euclidean norm of d.

Lemma 1 

([24]). Let $D\subseteq R^n$ be an open and connected set containing the origin. Let $B_r\subset D$ denote the closed ball of radius $r>0$ centered at the origin and let $f:D\to R^m={\left[f_1,f_2,\cdots ,f_m\right]}^T$ be a differentiable function such that $∥x∥<\infty \Rightarrow ∥f\left(x\right)∥$, $∥\nabla f_i\left(x\right)∥<\infty$ for all $x\in D$. Then, there exists a strictly increasing function $\rho :\left[0,\infty \right)\to \left[0,\infty \right)$ such that $∥f\left(x\right)-f\left(x_d\right)∥\le \rho \left(∥x-x_d∥\right)∥x-x_d∥$ for all $x\in D$ and $x\in B_r$.

Lemma 2 

(Young’s inequality). For $p>1,q>1,{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, then $\forall a,b\in R^{+}$, there is

$$\begin{array}{c}\hfill a·b\le {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}\end{array}$$

(5)

Lemma 3 

(Cauchy–Schwarz inequality) [25]). For the continuous functions $f,g:\left[a,b\right]\to R^n$ to be an integrable function on the interval $\left[a,b\right]$, there is

$$\begin{array}{c}\hfill {\left({\int }_a^{b}f\left(x\right)g\left(x\right)\right)}^2\le {\int }_a^b{f}^2\left(x\right)dx·{\int }_a^b{g}^2\left(x\right)dx\end{array}$$

(6)

Lemma 4. 

Let $f:\left[a,b\right]\to R^n$ be an integrable function on the interval $\left[a,b\right]$; then, the following inequality holds:

$$\begin{array}{c}\hfill {\int }_a^b{\int }_{s^{\prime }}^t\left|h\left(s\right)\right|dsd{s}^{\prime }\le \left(b-a\right){\int }_a^b\left|h\left(s\right)\right|ds\end{array}$$

(7)

3.2. Controller Design

Define the position error variable as $e\left(t\right)=x_d-x\left(t\right)$, where $x_d$ is the desired position instruction. Then, an auxiliary variable representing the filter error is defined as

$$\begin{array}{c}\hfill \lambda \left(t\right)=\dot{e}\left(t\right)+\alpha e\left(t\right)+\beta {\int }_0^{t}e\left(s\right)ds-\gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds-u\left(t\right)\end{array}$$

(8)

where $\alpha ,\beta \in R^{+}$, $\gamma >\underset{-}{g}$. The auxiliary variable above includes the proportional, differential, and integral terms of the error variable, as well as the integral of the input from time ${\tau }_{\max }$ to the current moment t. This approach effectively compensates for the effects of input delays while ensuring robustness. Taking the derivative of the above equation yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\lambda }\left(t\right)& =\omega \left(t\right)+g\left(x\right)\left[u\left(t-{\tau }_{\max }\right)-u\left(t-\tau \right)\right]-\gamma u\left(t\right)-\dot{u}\left(t\right)\hfill \\ \hfill & +\alpha \lambda \left(t\right)+\left(\beta -{\alpha }^2\right)e\left(t\right)-\alpha \beta {\int }_0^{t}e\left(s\right)ds+\alpha \gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\hfill \\ \hfill & +\alpha u\left(t\right)+\left(\gamma -g\right)u\left(t-{\tau }_{\max }\right)\hfill \end{array}\end{array}$$

(9)

where $\omega \left(t\right)={\ddot{x}}_d-d\left(t\right)-f\left(x,\dot{x}\right)+f\left(x_d,{\dot{x}}_d\right)-f\left(x_d,{\dot{x}}_d\right)-\beta e\left(0\right)$

From Lemma 1 and Assumption 6, $\omega \left(t\right)$ is upper-bounded as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill ∥\omega \left(t\right)∥\le ∥{\ddot{x}}_d∥+\overline{d}+{\delta }_1∥e\left(t\right)∥+{\delta }_2+∥f\left(x_d,{\dot{x}}_d\right)∥+∥\beta e\left(0\right)∥\le {\mu }_1∥e\left(t\right)∥+{\mu }_2\end{array}\end{array}$$

(10)

where ${\mu }_1,{\mu }_2\in R^{+}$ are known positive constants. From the mean value theorem, there is

$$\begin{array}{c}\hfill ∥u\left(t-{\tau }_{\max }\right)-u\left(t-\tau \right)∥\le \left|\tau -{\tau }_{\max }\right|∥\dot{u}\left(t-\widehat{\tau }\right)∥\end{array}$$

(11)

where $\widehat{\tau }\in \left(\tau ,{\tau }_{\max }\right)$. The filtering tracking error $\lambda \left(t\right)$ is passed through a first-order filter to obtain the control law as shown in (control law)

$$\begin{array}{c}\hfill u\left(t\right)={\displaystyle \frac{K}{\gamma +s}}\lambda \left(t\right)\end{array}$$

(12)

Then, Equation (9) can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\lambda }\left(t\right)& =\omega \left(t\right)+g\left(x\right)\left[u\left(t-{\tau }_{\max }\right)-u\left(t-\tau \right)\right]-K\lambda \left(t\right)\hfill \\ \hfill & +\alpha \lambda \left(t\right)+\left(\beta -{\alpha }^2\right)e\left(t\right)-\beta e\left(0\right)-\alpha \beta {\int }_0^{t}e\left(s\right)ds+\alpha \gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\hfill \\ \hfill & +\alpha u\left(t\right)+\left(\gamma -g\right)u\left(t-{\tau }_{\max }\right)\hfill \end{array}\end{array}$$

(13)

Then, the following theorem holds:

Theorem 1. 

for given ${\mu }_1,{\mu }_2,\eta >0$, the closed-loop system Equation (4) is global uniformly ultimately bounded (GUUB) under the control law Equation (12) if there exist scalars $\alpha ,\beta ,\gamma ,K>0$ such that

$$\begin{array}{c}\hfill K>3+{\overline{g}}^2{K}^2{\gamma }^2{{\tau }_{\max }}^2+\beta +5\alpha +{\displaystyle \frac{2{\tau }_{\max }+2\widehat{\tau }}{{\tau }_{\max }{\gamma }^2}}\end{array}$$

(14)

$$\begin{array}{c}\hfill {\alpha }^3>{\displaystyle \frac{{{\mu }_1}^2}{2}}+{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{3{\alpha }^2}{2}}+\alpha {\beta }^2\left(1+\alpha \right)t_{\max }\end{array}$$

(15)

$$\begin{array}{c}\hfill \gamma >{\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{{\alpha }^2}{2}}+{\displaystyle \frac{K}{2}}+{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}+{\displaystyle \frac{\eta }{2}}\alpha {\gamma }^2{{\tau }_{\max }}^2+{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{K}^2}}\end{array}$$

(16)

$$\begin{array}{c}\hfill \eta >2+2\alpha \end{array}$$

(17)

The proof process will be presented in the next subsection.

3.3. Stability Analysis

Proof. 

To prove Theorem 1, the L-K function is defined as follows:

$$\begin{array}{c}\hfill V\left(t\right)=V_1\left(t\right)+V_2\left(t\right)+V_3\left(t\right)+V_4\left(t\right)+V_5\left(t\right)+V_6\left(t\right)\end{array}$$

(18)

where

$$V_1\left(t\right)={\displaystyle \frac{1}{2}}{r}^T\left(t\right)r\left(t\right)+{\displaystyle \frac{1}{2}}{\alpha }^2{e}^T\left(t\right)e\left(t\right)+{\displaystyle \frac{1}{2}}{u}^T\left(t\right)u\left(t\right)$$

$$V_2\left(t\right)={\displaystyle \frac{1}{2K^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{∥\dot{u}\left(s\right)∥}^{2}ds$$

$$V_3\left(t\right)={\displaystyle \frac{1}{2{\tau }_{\max }{K}^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{\int }_{s^{\prime }}^t{∥\dot{u}\left(s\right)∥}^{2}dsd{s}^{\prime }$$

$$V_4\left(t\right)={\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}{\int }_{t-{\tau }_{\max }}^t{∥u\left(s\right)∥}^{2}ds$$

$$V_5\left(t\right)={\displaystyle \frac{\eta }{2}}\alpha {\tau }_{\max }{\gamma }^2{\int }_{t-{\tau }_{\max }}^t{\int }_{s^{\prime }}^t{∥u\left(s\right)∥}^{2}dsd{s}^{\prime }$$

$$V_6\left(t\right)=\alpha {\beta }^2\left(1+\alpha \right)t_{\max }{\int }_0^t{\int }_{s^{\prime }}^t{∥e\left(s\right)∥}^{2}dsd{s}^{\prime }$$

where $\eta \in R^{+}$ is an auxiliary parameter. $t_{\max }$ is the maximum time constant of the integration of the error variable. Taking the derivative of the Equation (18) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}\left(t\right)& ={\lambda }^T[\omega \left(t\right)+g\left(x\right)\left[u\left(t-{\tau }_{\max }\right)-u\left(t-\tau \right)\right]-K\lambda \left(t\right)\hfill \\ \hfill & +\alpha \lambda \left(t\right)+\left(\beta -{\alpha }^2\right)e\left(t\right)-\beta e\left(0\right)-\alpha \beta {\int }_0^{t}e\left(s\right)ds+\alpha \gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\hfill \\ \hfill & +\alpha u\left(t\right)+\left(\gamma -g\right)u\left(t-{\tau }_{\max }\right)]\hfill \\ \hfill & +{\alpha }^2{e}^T\left(\lambda \left(t\right)-\alpha e\left(t\right)-\beta {\int }_0^{t}e\left(s\right)ds+\gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds+u\left(t\right)\right)\hfill \\ \hfill & +u^T\left(t\right)\left(-\gamma u\left(t\right)+K\lambda \left(t\right)\right)+{\dot{V}}_2\left(t\right)+{\dot{V}}_3\left(t\right)+{\dot{V}}_4\left(t\right)+{\dot{V}}_5\left(t\right)+{\dot{V}}_6\left(t\right)\hfill \end{array}\end{array}$$

(19)

where

$${\dot{V}}_2\left(t\right)={\displaystyle \frac{1}{2K^2{\gamma }^2}}\left({∥\dot{u}\left(t\right)∥}^2-{∥\dot{u}\left(t-\widehat{\tau }\right)∥}^2\right)$$

$${\dot{V}}_3\left(t\right)={\displaystyle \frac{\widehat{\tau }}{2{\tau }_{\max }{K}^2{\gamma }^2}}{∥\dot{u}\left(t\right)∥}^2-{\displaystyle \frac{1}{2{\tau }_{\max }{K}^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{∥\dot{u}\left(s\right)∥}^{2}ds$$

$${\dot{V}}_4\left(t\right)={\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}\left({∥u\left(t\right)∥}^2-{∥u\left(t-{\tau }_{\max }\right)∥}^2\right)$$

$${\dot{V}}_5\left(t\right)={\displaystyle \frac{\eta }{2}}{\alpha }^2{{\tau }_{\max }}^2{\gamma }^2{∥u\left(t\right)∥}^2-{\displaystyle \frac{\eta }{2}}{\alpha }^2{\tau }_{\max }{\gamma }^2{\int }_{t-{\tau }_{\max }}^t{∥u\left(s\right)∥}^{2}ds$$

$${\dot{V}}_6\left(t\right)={\alpha }^2\beta \left(1+\alpha \right)t_{\max }t{∥e\left(t\right)∥}^2-{\alpha }^2\beta \left(1+\alpha \right)t_{\max }{\int }_0^t{∥e\left(s\right)∥}^{2}ds$$

By organizing the above formula, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}\left(t\right)& ={\mu }_1∥e\left(t\right)∥∥\lambda \left(t\right)∥+{\mu }_2∥\lambda \left(t\right)∥+\overline{g}\left|\tau -{\tau }_{\max }\right|∥\dot{u}\left(t-\widehat{\tau }\right)∥∥\lambda \left(t\right)∥\hfill \\ \hfill & +{\lambda }^T\left(t\right)\left(\alpha -K\right)\lambda \left(t\right)+{\lambda }^T\left(t\right)\beta e\left(t\right)-{\lambda }^T\left(t\right)\alpha \beta {\int }_0^{t}e\left(s\right)ds\hfill \\ \hfill & +{\lambda }^T\left(t\right)\alpha \gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds+{\lambda }^T\left(t\right)\alpha u\left(t\right)\hfill \\ \hfill & +{\lambda }^T\left(t\right)\left(\gamma -g\right)u\left(t-{\tau }_{\max }\right)+e^T\left(t\right){\alpha }^{3}e\left(t\right)-e^T{\alpha }^2\beta {\int }_0^{t}e\left(s\right)ds\hfill \\ \hfill & +e^T{\alpha }^2\gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds+e^T{\alpha }^{2}u\left(t\right)-u^T\left(t\right)\gamma u\left(t\right)+u^T\left(t\right)K\lambda \left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2K^2{\gamma }^2}}\left({∥\dot{u}\left(t\right)∥}^2-{∥\dot{u}\left(t-\widehat{\tau }\right)∥}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{\widehat{\tau }}{2{\tau }_{\max }{K}^2{\gamma }^2}}{∥\dot{u}\left(t\right)∥}^2-{\displaystyle \frac{1}{2{\tau }_{\max }{K}^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{∥\dot{u}\left(s\right)∥}^{2}ds\hfill \\ \hfill & +{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}\left({∥u\left(t\right)∥}^2-{∥u\left(t-{\tau }_{\max }\right)∥}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{\eta }{2}}{\alpha }^2{{\tau }_{\max }}^2{\gamma }^2{∥u\left(t\right)∥}^2-{\displaystyle \frac{\eta }{2}}{\alpha }^2{\tau }_{\max }{\gamma }^2{\int }_{t-{\tau }_{\max }}^t{∥u\left(s\right)∥}^{2}ds\hfill \\ \hfill & +{\alpha }^2\beta \left(1+\alpha \right)t_{\max }t{∥e\left(t\right)∥}^2-{\alpha }^2\beta \left(1+\alpha \right)t_{\max }{\int }_0^t{∥e\left(s\right)∥}^{2}ds\hfill \end{array}\end{array}$$

(20)

From Lemma 2 (Young’s inequality), we can get

$$\begin{array}{c}\hfill {\lambda }^T\left(t\right)\beta e\left(t\right)\le {\displaystyle \frac{\beta }{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{\beta }{2}}{∥e\left(t\right)∥}^2\end{array}$$

(21)

$$\begin{array}{c}\hfill -{\lambda }^T\left(t\right)\alpha \beta {\int }_0^{t}e\left(s\right)ds\le {\displaystyle \frac{\alpha }{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{\alpha {\beta }^2}{2}}{∥{\int }_0^{t}e\left(s\right)ds∥}^2\end{array}$$

(22)

$$\begin{array}{c}\hfill {\lambda }^T\left(t\right)\alpha \gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\le {\displaystyle \frac{\alpha }{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{\alpha {\gamma }^2}{2}}{∥{\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds∥}^2\end{array}$$

(23)

$$\begin{array}{c}\hfill {\lambda }^T\left(t\right)\alpha u\left(t\right)\le {\displaystyle \frac{\alpha }{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{\alpha }{2}}{∥u\left(t\right)∥}^2\end{array}$$

(24)

$$\begin{array}{c}\hfill {\lambda }^T\left(t\right)\left(\gamma -g\right)u\left(t-{\tau }_{\max }\right)\le {\displaystyle \frac{1}{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}{∥u\left(t-{\tau }_{\max }\right)∥}^2\end{array}$$

(25)

$$\begin{array}{c}\hfill -e^T{\alpha }^2\beta {\int }_0^{t}e\left(s\right)ds\le {\displaystyle \frac{{\alpha }^2}{2}}{∥e\left(t\right)∥}^2+{\displaystyle \frac{{\alpha }^2{\beta }^2}{2}}{∥{\int }_0^{t}e\left(s\right)ds∥}^2\end{array}$$

(26)

$$\begin{array}{c}\hfill e^T{\alpha }^2\gamma {\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\le {\displaystyle \frac{{\alpha }^2}{2}}{∥e\left(t\right)∥}^2+{\displaystyle \frac{{\alpha }^2{\gamma }^2}{2}}{∥{\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds∥}^2\end{array}$$

(27)

$$\begin{array}{c}\hfill e^T{\alpha }^{2}u\left(t\right)\le {\displaystyle \frac{{\alpha }^2}{2}}{∥e\left(t\right)∥}^2+{\displaystyle \frac{{\alpha }^2}{2}}{∥u\left(t\right)∥}^2\end{array}$$

(28)

$$\begin{array}{c}\hfill u^T\left(t\right)K\lambda \left(t\right)\le {\displaystyle \frac{K}{2}}{∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{K}{2}}{∥u\left(t\right)∥}^2\end{array}$$

(29)

$$\begin{array}{c}\hfill {\mu }_1∥e\left(t\right)∥∥\lambda \left(t\right)∥\le {\displaystyle \frac{{{\mu }_1}^2}{2}}{∥e\left(t\right)∥}^2+{\displaystyle \frac{1}{2}}{∥\lambda \left(t\right)∥}^2\end{array}$$

(30)

$$\begin{array}{c}\hfill {\mu }_2∥\lambda \left(t\right)∥\le {\displaystyle \frac{{{\mu }_2}^2}{2}}+{\displaystyle \frac{1}{2}}{∥\lambda \left(t\right)∥}^2\end{array}$$

(31)

$$\begin{array}{c}\hfill \overline{g}\left|\tau -{\tau }_{\max }\right|∥\dot{u}\left(t-\widehat{\tau }\right)∥∥\lambda \left(t\right)∥\le {\displaystyle \frac{1}{2K^2{\gamma }^2}}{∥\dot{u}\left(t-\widehat{\tau }\right)∥}^2+{\displaystyle \frac{{\overline{g}}^2{K}^2{\gamma }^2{{\tau }_{\max }}^2}{2}}{∥\lambda \left(t\right)∥}^2\end{array}$$

(32)

According to Lemma 3, the following two terms in Equations (22) and (23) can be upper-bounded as

$$\begin{array}{c}\hfill {\left({\int }_{t-{\tau }_{\max }}^{t}u\left(s\right)ds\right)}^2\le {\tau }_{\max }{\int }_{t-{\tau }_{\max }}^t{∥u\left(s\right)∥}^{2}ds\end{array}$$

(33)

$$\begin{array}{c}\hfill {\left({\int }_0^{t}e\left(s\right)ds\right)}^2\le t{\int }_0^t{∥e\left(s\right)∥}^{2}ds\end{array}$$

(34)

According to Lemma 4, there is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)& \le {\displaystyle \frac{\widehat{\tau }}{2{\tau }_{\max }{K}^2{\gamma }^2}}{∥\dot{u}\left(t\right)∥}^2-{\displaystyle \frac{1}{4{\tau }_{\max }{K}^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{∥\dot{u}\left(s\right)∥}^{2}ds\hfill \\ \hfill & -{\displaystyle \frac{1}{4{\tau }_{\max }\widehat{\tau }{K}^2{\gamma }^2}}{\int }_{t-\widehat{\tau }}^t{\int }_{s^{\prime }}^t{∥\dot{u}\left(s\right)∥}^{2}dsd{s}^{\prime }\hfill \end{array}\end{array}$$

(35)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_5\left(t\right)& \le {\displaystyle \frac{\eta }{2}}\alpha {{\tau }_{\max }}^2{\gamma }^2{∥u\left(t\right)∥}^2-{\displaystyle \frac{\eta }{4}}\alpha {\tau }_{\max }{\gamma }^2{\int }_{t-{\tau }_{\max }}^t{∥u\left(s\right)∥}^{2}ds\hfill \\ \hfill & -{\displaystyle \frac{\eta }{4}}\alpha {\gamma }^2{\int }_{t-{\tau }_{\max }}^t{\int }_{s^{\prime }}^t{∥u\left(s\right)∥}^{2}dsd{s}^{\prime }\hfill \end{array}\end{array}$$

(36)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_6\left(t\right)& \le \alpha {\beta }^2\left(1+\alpha \right)t_{\max }{∥e\left(t\right)∥}^2-{\displaystyle \frac{1}{2}}\alpha {\beta }^2\left(1+\alpha \right)t_{\max }{\int }_0^t{∥e\left(s\right)∥}^{2}ds\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\alpha {\beta }^2\left(1+\alpha \right){\int }_0^t{\int }_{s^{\prime }}^t{∥e\left(s\right)∥}^{2}dsd{s}^{\prime }\hfill \end{array}\end{array}$$

(37)

Substituting Equations (21)–(37) into Equation (20), one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}\left(t\right)& \le -\left({\alpha }^3-{\displaystyle \frac{{{\mu }_1}^2}{2}}-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{3{\alpha }^2}{2}}-\alpha {\beta }^2\left(1+\alpha \right)t_{\max }\right){∥e\left(t\right)∥}^2\hfill \\ \hfill & -\left({\displaystyle \frac{K}{2}}-{\displaystyle \frac{3}{2}}-{\displaystyle \frac{{\overline{g}}^2{K}^2{\gamma }^2{{\tau }_{\max }}^2}{2}}-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{5\alpha }{2}}-{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{\gamma }^2}}\right){∥\lambda \left(t\right)∥}^2+{\displaystyle \frac{{{\mu }_2}^2}{2}}\hfill \\ \hfill & -\left(\gamma -{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{{\alpha }^2}{2}}-{\displaystyle \frac{K}{2}}-{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}-{\displaystyle \frac{\eta }{2}}\alpha {\gamma }^2{{\tau }_{\max }}^2-{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{K}^2}}\right){∥u\left(t\right)∥}^2\hfill \\ \hfill & -{\displaystyle \frac{\alpha {\gamma }^2{\tau }_{\max }}{2{\left(\gamma -\underset{-}{g}\right)}^2}}\left(\eta -2-2\alpha \right)V_4\left(t\right)-{\displaystyle \frac{1}{2{\tau }_{\max }}}{V}_2\left(t\right)-{\displaystyle \frac{1}{2\widehat{\tau }}}V{\left(t\right)}_3\hfill \\ \hfill & -{\displaystyle \frac{1}{2{\tau }_{\max }}}{V}_5\left(t\right)-{\displaystyle \frac{1}{2t_{\max }}}{V}_6\left(t\right)\hfill \end{array}\end{array}$$

(38)

The Equation (38) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \dot{V}\left(t\right)\le -{\vartheta }_{1}V\left(t\right)+{\vartheta }_2\end{array}\end{array}$$

(39)

where

$${\vartheta }_1=\min \left[\begin{array}{c}{\displaystyle \frac{K}{2}}-{\displaystyle \frac{3}{2}}-{\displaystyle \frac{{\overline{g}}^2{K}^2{\gamma }^2{{\tau }_{\max }}^2}{2}}-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{5\alpha }{2}}-{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{\gamma }^2}},\hfill \\ {\alpha }^3-{\displaystyle \frac{{{\mu }_1}^2}{2}}-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{3{\alpha }^2}{2}}-\alpha {\beta }^2\left(1+\alpha \right)t_{\max },\hfill \\ \gamma -{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{{\alpha }^2}{2}}-{\displaystyle \frac{K}{2}}-{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}-{\displaystyle \frac{\eta }{2}}\alpha {\gamma }^2{{\tau }_{\max }}^2-{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{K}^2}},\hfill \\ {\displaystyle \frac{\alpha {\gamma }^2{\tau }_{\max }}{2{\left(\gamma -\underset{-}{g}\right)}^2}}\left(\eta -2-2\alpha \right),-{\displaystyle \frac{1}{2{\tau }_{\max }}},-{\displaystyle \frac{1}{2\widehat{\tau }}},-{\displaystyle \frac{1}{2t_{\max }}}\hfill \end{array}\right]$$

$${\vartheta }_2={\displaystyle \frac{{{\mu }_2}^2}{2}}$$

Therefore, given that the following formulas hold, the closed-loop system Equation (4) is GUUB.

$$K>3+{\overline{g}}^2{K}^2{\gamma }^2{{\tau }_{\max }}^2+\beta +5\alpha +{\displaystyle \frac{2{\tau }_{\max }+2\widehat{\tau }}{{\tau }_{\max }{\gamma }^2}}$$

$${\alpha }^3>{\displaystyle \frac{{{\mu }_1}^2}{2}}+{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{3{\alpha }^2}{2}}+\alpha {\beta }^2\left(1+\alpha \right)t_{\max }$$

$$\gamma >{\displaystyle \frac{\alpha }{2}}+{\displaystyle \frac{{\alpha }^2}{2}}+{\displaystyle \frac{K}{2}}+{\displaystyle \frac{{\left(\gamma -\underset{-}{g}\right)}^2}{2}}+{\displaystyle \frac{\eta }{2}}\alpha {\gamma }^2{{\tau }_{\max }}^2+{\displaystyle \frac{{\tau }_{\max }+\widehat{\tau }}{{\tau }_{\max }{K}^2}}$$

$$\eta >2+2\alpha$$

This completes the proof. □

Remark 4. 

The aforementioned controller is valid under Assumptions 1–6, and external disturbances d generally adhere to Assumption 4 under normal weather conditions. When wind disturbances suddenly increase during extreme weather, d may exceed $\overline{d}$. In this situation, this will only result in an increase in the upper bound of system variable convergence, without significantly affecting the overall convergence of the system.

Remark 5. 

The design of the control law in Equation (12) is motivated by the Lyapunov stability analysis. The parameters can be selected based on the sufficient conditions provided by Equations (14)–(17) to ensure the system’s GUUB. It is important to note that the conditions provided by Equations (14)–(17) are in a highly nonlinear form, making direct calculation challenging. Therefore, this study employs a population optimization algorithm for parameter optimization, using the Equations (14)–(17) expression as a penalty term. After multiple iterations, a set of controller parameters satisfying the condition of Theorem 1 can be obtained.

4. Result

To verify the effectiveness of the proposed robust time-delay controller in addressing the input time delay as well as the external disturbance problem in cable tension control of the TUAV, simulations are conducted in this section. The simulation parameters are provided in Table 1. Time-varying disturbances are composed of two parts: $d\left(t\right)=d_1+d_2$, where $d_1=7+sin\left(2t\right)$ represents low-frequency significant disturbances, used to simulate periodic gust disturbances and low-frequency significant rope swings, while $d_2$ represents high-frequency minor disturbances, used to simulate atmospheric turbulence. In addition, to simulate the coupling effect between the x and y channels, the change in g is set to $g=1+0.2sin\left(5t\right)$. The comparison algorithm selected in this paper is a robust controller that does not consider historical input information, i.e.,

$$\begin{array}{c}\hfill u_{compare}\left(t\right)={\displaystyle \frac{K_1}{1+K_{2}s}}\left(\gamma \dot{e}\left(t\right)+\alpha e\left(t\right)+\beta {\int }_0^{t}e\left(s\right)ds\right)\end{array}$$

(40)

To facilitate the solution of the controller parameters and avoid the influence of parameter selection on the comparison results which would affect the comparison results, this paper utilizes the Harris Hawks Optimizer (HHO) swarm optimization algorithm [26] for controller parameter tuning. HHO is a metaheuristic algorithm proposed in 2019, which simulates the unique group hunting behavior of Harris hawks to perform swarm intelligence optimization. The algorithm consists of three phases: global search, global exploration to local exploitation transition, and local exploitation. The position of the Harris hawk is considered as a candidate solution, and the best candidate solution at each iteration represents the prey. The fitness function is defined as

$$\begin{array}{c}\hfill fitness=\left\{\begin{array}{cc}{J}_{\max }-K_{error}{t}_{error}\hfill & ,iferror\\ {\displaystyle \sum _{i=1}^n}{K}_e{t}_i·{\left|e_i\right|}^2\hfill & ,else\end{array}\right.\end{array}$$

(41)

where $J_{\max }$ is the given upper bound of fitness, $t_{error}$ is the maximum time of system operation error, i is the number of operational steps, $t_i$ is the time at the $i\mathrm{th}$ step, $e_i$ is the error at the $i\mathrm{th}$ step, and n is the total number of simulation steps. $K_{error}$ and $K_e$ are parameters to be designed. The above fitness function ensures that the optimization results exhibit both fast convergence and stability.

The parameter optimization for the RTDC algorithm designed in this paper and the comparison algorithm robust compensator (RC) is performed separately. The RTDC algorithm has four parameters to be optimized, which are $\alpha$, $\beta$, $\gamma$, K, while the RC algorithm has five parameters to be optimized, which are $K_p$, $K_I$, $K_D$, $K_1$, $K_2$. The population size is set to 1000, and the maximum number of iterations is set to 100. The fitness function values for the 1st, 50th, and 100th iterations are shown in the table below (

Table 2. Iteration process.

Iteration Number	1	50	100
RTDC	$0.7207$	$0.0095$	$0.0070$
RC	$1187.8993$	$1.8044$	$0.0749$

):

The evolution of the fitness function and various controller parameters during the optimization process are shown in Figure 5 and Figure 6. It can be observed that after 100 iterations, the fitness function of the RTDC reaches $0.0070$, which is significantly lower than the fitness function of the RC, which is $0.0749$. The performance of the optimized controller is illustrated in Figure 7, Figure 8 and Figure 9.

Figure 7 shows the variation curve of the composite disturbance, which includes both high-frequency and low-frequency disturbance. Figure 8 shows the variation curve of the position error of the UAV under two controllers. It can be observed that the RC algorithm exhibits a lower convergence rate compared to the RTDC. Due to the presence of input delay, the RC algorithm only starts to show a convergence trend after 4 s. Furthermore, due to the time-varying wind disturbance, the delayed control input is unable to counteract the wind disturbance timely, resulting in a fluctuation of 0.5 m even after stabilization. In contrast, under the effect of RTDC, the system converges within 2 s to within 0.1 m, and due to the compensation from historical control inputs, the oscillation amplitude after stabilization is less than 0.01m, demonstrating excellent disturbance rejection capability. Figure 9 shows the variation of the cable tension under RTDC. The tension limit for the cable to break is greater than 200$\mathrm{N}$, so the maximum tension of the cable is set at 150$\mathrm{N}$. As a result, under the condition that the tension is less than 150$\mathrm{N}$, the UAV can maintain stability very well.

5. Discussion

This paper investigates the wind resistance stability control problem of TUAVs. By analyzing the feasibility and characteristics of cable to assist in wind resistance, the following conclusions are drawn: (1) Under a certain cable inclination angle, the horizontal component of the cable tension has a positive effect on the wind resistance capability of the UAV, helping the UAV resist stronger wind disturbances; (2) With cable tension-assisted control, the TUAV can make appropriate attitude adjustments, unlike traditional quadcopters, which can only maintain a fixed attitude under certain wind speeds. This significantly broadens the application scenarios for UAVs and is of great significance to their practical use.

Based on this, a robust time-delay compensator is designed to address the issue of unknown input delays in the cable tension control system. Furthermore, to tackle the challenges of strong nonlinearity and strong coupling in the sufficient conditions for ultimate boundedness, an HHO-based parameter intelligent optimization strategy is proposed. The final simulation results show that the optimized controller parameters can effectively achieve stable control of delay systems under unknown disturbances.

It is important to note that the focus of this paper is on demonstrating the feasibility of the cable tension to assist in wind resistance and addressing the time-delay issue in the tension control system. In fact, several areas require further investigation: (1) Considering more complex and precise TUAV dynamic models. This includes the dynamic characteristics of the cable and the coupling effect between the cable and the UAV. Meanwhile, further analyze the transmission process of cable tension and establish an accurate tension transmission model to assist in the wind resistance in three-dimensional space; (2) Considering the multi-actuator coordination control problem with different frequency bands, amplitudes, and time delays for UAV rotors, cable retraction devices, and other components, and establishing a complete framework for the wind resistance stability control of TUAVs.