Data-based dynamic event-triggered attitude control for helicopter with aperiodic denial-of-service attacks

2.1. MFAC method

The core idea is to build an equivalent dynamic linearized model at each operating point, and to estimate the pseudo-partial derivative (PPD) parameters of the system online from the input and output data of the controlled system. In this case, the PPD parameter is a mathematical concept. It is a parameter that approximates the relationship between the system's input and output data to establish a dynamic linearization model, such as compact form dynamic linearization (CFDL), partial form dynamic linearization (PFDL), and full form dynamic linearization (FFDL). Subsequently, a MFAC method is proposed based on the established dynamic linearization model. The flowchart of the control algorithm is shown in Figure 1^[32]. For complete theoretical foundations (e.g., stability proofs, algorithmic variants), refer to Refs.^[33] and^[34].

Figure 1. Flowchart of the MFAC algorithm. MFAC: Model-free adaptive control.

Considering a single-input single-output nonlinear system^[35] satisfies that

where $$ f(\cdot) $$ is an unknown nonlinear function. $$ y(k) \in R $$ and $$ u(k) \in R $$ denote the system input and output, respectively. $$ {n_y} $$ and $$ {n_u} $$ are unknown positive orders of the output and input, respectively. Besides, we consider the following assumptions.

Assumption 1^[36] The partial derivative of $$ f(\cdot) $$ with respect to $$ u(k) $$ is continuous.

Assumption 2^[37] The system, given in Equation (1), satisfies the generalized Lipschitz condition, if $$ |\Delta u(k)| = |u(k) - u(k - 1)| \ne 0 $$, then it is obtained that $$ |y(k + 1) - y(k)| $$$$ \le b |u(k) - u(k - 1)| $$, where $$ b $$ is a positive constant.

Remark 1 Assumption 1 is a common condition of nonlinear systems. Assumption 2 is a restriction on the upper bound of the system output change rate. More deities of Assumptions 1 and 2 can be found in the reference^[38].

Lemma 1^[39] If Equation (1) satisfies Assumptions 1-2 and $$ \Delta u(k) \ne 0 $$, there exists a time-varying parameter called PPDs $$ \phi (k) $$, such that Equation (1) is transformed into

where $$ \phi (k) $$ is bounded and satisfies $$ |\phi (k)| \le b $$.

Assumption 3^[40] The sign of the PPD parameter remains unchanged for all $$ k $$ and satisfies $$ \phi (k) > \varepsilon > 0 $$ or $$ \phi (k) < -\varepsilon, \forall k $$. Without loss of generality, we assume $$ \phi (k) > \varepsilon $$, where $$ \varepsilon $$ is a small positive number.

Remark 2 Assumption 3 implies that the system's output does not decrease when the input increases, also called a quasi-linearization requirement^[41].

2.2. System descriptions

The three-degree-of-freedom helicopter features two direct current motors, one mounted at each end of a rectangular frame, which drive two propellers. The helicopter's attitude is controlled by the thrust forces $$ F_b $$ and $$ F_f $$ generated by the two propellers. Note that the system is underactuated with only two control forces and three degrees of freedom. Moreover, the helicopter has three degrees of freedom: elevation angle $$ \varepsilon $$, heading angle $$ \psi $$, and pitch angle $$ \theta $$. The nonlinear model of the three-degree-of-freedom helicopter attitude control subsystem is given as

where $$ V_f $$ and $$ V_b $$ are the two voltages of the helicopter's motors. $$ J_x $$, $$ J_y $$, and $$ J_z $$ are the moments of inertia of the elevator, heading, and pitch axes, respectively. $$ T_g $$ is the equivalent gravitational moment of the pitch axis, $$ L_h $$ is the distance from the pitch axis to each motor, $$ L_a $$ is the distance from the heading axis to the helicopter fuselage, and $$ K_f $$ is the propeller thrust constant.

Remark 3 When helicopters are performing smooth hovering or slow attitude adjustment tasks, their pitch angle is usually in a small range. At this point, the Taylor series expansion of $$ \sin\theta $$ is performed: $$ \sin\theta = \theta - \frac{\theta^{3}}{6} + \frac{\theta^{5}}{120} - \cdots $$. When $$ \theta \leq 0.1 \ \mathrm{rad} $$, the absolute value of the higher order terms of the third order and above is much smaller than $$ \theta $$ itself, such as $$ \frac{\theta^3}{6}\leq\frac{0.1^3}{6}\approx1.67\times10^{-4} $$. For large-angle scenarios, the enhanced nonlinear coupling will lead to the failure of the approximation, and this issue will be the main focus of the next phase of research.

Then it is obtained that $$ \sin( \theta(k) ) \approx \theta(k) $$. Let $$ V_f + V_b = V_1 $$ and $$ V_f - V_b = V_2 $$. Hence, Equation (3) is discretized as

where $$ T_s $$ denotes the system sampling time. Moreover, $$ r_\varepsilon(k) $$, $$ r_\psi(k) $$, and $$ r_\theta(k) $$ denote the angular velocities of elevation angle, heading angle, and pitch angle, respectively.

Remark 4 From the discrete-time model of the helicopter, given in Equation (4), it is found that the angular velocity of $$ \psi (k) $$ is positively correlated with $$ \theta (k) $$, which means that $$ \psi (k) $$ will continue to change when $$ \theta (k) \neq 0 $$; that is, $$ \theta (k) $$ must eventually converge to 0 for $$ \psi (k) $$ to stabilize.

Remark 5 Although the dynamics models of the helicopter are discussed in Equations (3) and (4), those models are only uncertainties and are employed to demonstrate the relationship between input data and output data of the controlled helicopter. It should be noted that the mentioned models are no longer required when designing the controller. This paper aims to design a data-driven control method for the helicopter to realize attitude control.

2.3. Aperiodic DoS attacks

DoS attacks aim to break system stability by blocking data transmission. Figure 2 illustrates a schematic diagram of the aperiodic DoS attack behavior. The $$ n $$th DoS attack interval is defined as $$ [T^{\mathrm{on}}_{n}, T^{\mathrm{off}}_{n} ) $$, where $$ T^{\mathrm{on}}_{n} $$ and $$ T^{\mathrm{off}}_{n} \in R^{+} $$ are the start instant and the end instant of DoS attacks, respectively. The joint value of the DoS attacks for $$ k \in R^{+} $$ in the interval $$ [ 0, k ] $$ is

Figure 2. Scheme of DoS attacks. DoS: Denial-of-service.

Then, the time interval without DoS attacks is given by

Assumption 4^[42] Defining $$ |\upsilon_a(0, k)| $$ as the total time interval with DoS attacks in $$ [0, k] $$, it satisfies that

where $$ \tau_a>1 $$ and $$ \upsilon_0 \ge 0 $$ are constants to be determined.

Remark 6 Assumption 4 implies that the total duration of DoS attacks is bounded. In addition, the role of $$ \upsilon_0 $$ is to consider the situation at the beginning of the DoS attacks. $$ \tau_a $$ is a parameter that affects the frequency of the DoS attack. The frequency of the DOS attack is $$ \frac{1}{\tau_a} $$.

2.4. Problem statement

The primary objective of this paper is to propose a data-driven, dynamic event-triggered attitude control method for helicopters to facilitate the implementation of attitude control tasks. To improve the control performance and resource utilization efficiency, it faces the following key challenges:

(1) Quasi-linearization Issue: The relationship between input and output data of the controlled helicopter does not satisfy the quasi-linearization requirement, which is a basic requirement for MFAC theory.

(2) Aperiodic DoS Attack Issue: The controlled helicopter system is subject to aperiodic DoS attacks, leading to the controller being unable to receive the feedback data.

(3) Limited Communication Resources: Generally, due to the restrictions on size and quality, helicopters are equipped with a lightweight communication device, which will cause communication jamming.

To summarize, this paper proposes a data-driven control scheme based on the MFAC theory, addressing the issues of aperiodic DoS attacks and limited communication resources.

3.1. Output redefinition

It should be pointed out that the attitude angles of the helicopter have a characteristic of suddenly changing to the opposite extreme value (for example, $$ 180^\circ $$ jumps to $$ -180^\circ $$) when approaching the critical value. Therefore, the three-degree-of-freedom helicopter attitude control model given in Equation (4) does not satisfy Assumption 3. By analyzing the dynamic characteristics of the helicopter system and incorporating the quasi-linearization requirements of the MFAC theory, a redefined output is designed.

This article reconstructs the output signal using the angles and angular velocities of the three attitude angles and proves that the redefined output and the input satisfy Assumption 3. The output signal is redefined as

where $$ \vartheta (k) $$ and $$ r(k) $$ represent the angle and angular velocity, respectively. $$ K \ge K_{min} $$ is the redefined angular velocity gain, and $$ K_{min} $$ is a certain least positive number.

Redefine the reference signal for the system output as

where $$ \vartheta^{*} (k) $$ and $$ r^{*}(k) $$ represent the desired angle and angular velocity, respectively. At the stabilization point, there exists $$ z^{*}(k) = \vartheta^{*} (k) + K r^{*}(k) = \vartheta^{*} + K r^{*} = \mathrm{const} $$, where $$ \vartheta^{*} $$ represents the desired heading angle, and the desired angular velocity $$ r^{*} $$ is always zero. Assumption 3 is satisfied by choosing an appropriate parameter $$ K $$ such that $$ K*r $$ continues to increase during attitude angle jumps, thus ensuring that the control output $$ z $$ also increases.

As shown in Figure 4, the system introduces the angular velocity signal into the outer loop feedback system by redefining the output. This reflects the combined effect of interference on heading and angular velocity in a prompt manner. Thus, the controller can be adjusted more quickly to improve the system's anti-interference capability and dynamic response speed. Relevant details can be found in^[43].

Figure 4. The schematic diagram of the formulated method.

Theorem 1 The redefined output Equation (8) satisfies Assumption 3 when $$ K \ge K_{min} = \max(K_\varepsilon, K_\lambda, K_\rho) $$ with $$ K_\varepsilon = \frac{2\pi J_x}{T_sK_fL_a\cos(\theta (k))\Delta V_1(k)-T_sT_g\cos(\varepsilon (k)))}-T_s $$, $$ K_\lambda = \frac{2\pi J_y}{T_sT_gL_a\theta(k)} - T_s $$, and $$ K_\rho = \frac{2\pi J_z}{T_sK_fL_hV_2(k)} - T_s $$. At the same time, if set $$ z(k) = z^{*}(k) $$, it is obtained that $$ \vartheta (k) $$ converges asymptotically to $$ \vartheta^{*} (k) $$.

Proof 1 The output increment satisfies:

Supposing $$ \Delta u(k) > 0 $$, take elevation angle $$ \varepsilon (k) $$ as an example. Then, consider an extreme case, the angle jumps from $$ 180^\circ $$ to $$ –180^\circ $$. Under this circumstances, $$ \varepsilon (k) = -2\pi $$. Substituting Equation (4) into $$ \Delta z(k+1) $$ yields

In actual systems, a dead zone exists in the controller's output, allowing the helicopter to take off. This means that when $$ u (k) > u_{min} > 0 $$, $$ \Delta r (k) > 0 $$ must hold. Thus, Equation (11) is transformed into

Obviously, if $$ K \ge K_\varepsilon = \frac{2\pi J_x}{T_sK_fL_a\cos(\theta (k))\Delta V_1(k)-T_sT_g\cos(\varepsilon (k)))}-T_s $$, Equation (11) holds. Similarly, $$ K \ge K_\lambda = \frac{2\pi J_y}{T_sT_gL_a\theta(k)} - T_s $$ and $$ K \ge K_\rho = \frac{2\pi J_z}{T_sK_fL_hV_2(k)} - T_s $$ ensure that Assumption 3 can be satisfied. In summary, if $$ K \ge \max(K_\varepsilon, K_\lambda, K_\rho) $$, it is concluded that Equation (8) satisfies Assumption 3. When $$ \Delta u (k) < 0 $$, the same reasoning holds.

Substituting Equations (8) and (9) into $$ z(k) = z^{*}(k) $$, we have

Considering

where $$ T_s $$ is the sampling period.

Substituting Equation (14) into Equation (13) yields

Defining $$ e_{\vartheta}(k) = \vartheta (k) - \vartheta^{*} (k) $$, Equation (15) becomes

According to Equation (16), $$ e_{\vartheta}(k) $$ converges to $$ 0 $$ as $$ k $$ tends to infinity, that is, $$ \vartheta(k) $$ converges to $$ \vartheta^{*}(k) $$ asymptotically. Based on the above calculation, Theorem 1 is proved.

It should be pointed out that although the control error converges when the design parameter $$ K $$ takes an appropriate value, its value affects the system performance. To solve this problem, we add a differential signal at the input of the system, and the control input becomes

where $$ K_d $$ and $$ K_{pd} $$ is the differential gain and proportional gain, respectively. $$ T_s $$ is a sampling period.

3.2. DoS attack compensation algorithm

Unmanned helicopters rely on network communications to send data to interact with ground terminals or other aircraft. For example, when performing missions, they need to upload flight attitude and position data and receive control commands in real-time. This exposes them to the risk of cyberattacks, especially DoS attacks, which are easy to implement. Hence, designing a DoS attack compensation algorithm for the helicopter is meaningful work.

DoS attacks can cause delays, data loss and interruption of communication between unmanned helicopters and ground control stations. These attacks consume network bandwidth and send large numbers of invalid packets. These issues significantly affect flight control, rendering it impossible to execute control commands promptly or provide accurate feedback on flight attitude data. This paper presents a compensation algorithm based on MFAC that aims to counteract the effects of DoS attacks. This algorithm compensates for the effects of a DoS attack on the communication system by dynamically adjusting control inputs to ensure flight stability and control accuracy.

Take the elevation channel as an example, a DoS attack compensation algorithm^[44] is designed as

where $$ \beta \in (0, 1) $$ is a compensation parameter for DoS attacks. $$ \varrho(k) = 0 $$ when the system is under DoS attack, $$ \varrho(k) = 1 $$ when the system is not under DoS attack. $$ m $$ is the number of consecutive DoS attacks. $$ \Delta u_{a}(k_{cr}) = u_{a}(k_{cr}) - u_{a}(k_{cr} - 1) $$, $$ k_{cr} $$ is the moment of the latest DoS attack, and

3.3. Dynamic event-triggered mechanism

Notably, the developed scheme is based on a continuous network communication condition. It should be pointed out that the helicopter's network resources are limited. To save communication resources, we introduce a dynamic event-triggered mechanism. Defining the set of event-triggered instants is $$ \{ k_{i}, i = 0, 1, \dots \} $$. Then, a dynamic event-triggered function is developed as

where $$ \nu $$ and $$ \omega $$ are positive constants. $$ e(k) = y^{*}(k) - y(k) $$ is the tracking error. $$ y^{*}(k) = \vartheta^{*}(k) + Kr^{*}(k) $$, $$ y(k) = \vartheta(k) + Kr(k) $$. $$ \upsilon (k) $$ represents the dynamic variable satisfying

where $$ \varphi > 0 $$, and $$ \upsilon_{0} $$ is the initial value of $$ v(k) $$.

Then, an event-triggered condition is formulated as

When $$ g(k) \geq 0 $$, that is, $$ |e(k)| \leq \frac{1}{\nu}|\upsilon(k)|+ \omega $$, the control signal stops updating. When $$ g(k) < 0 $$, that is, $$ |e(k)| > \frac{1}{\nu}|\upsilon(k)|+ \omega $$, the system triggers communication and updates the control signal.

Remark 7 As $$ \nu $$ tends to infinity, the event triggering strategy degenerates into a static triggering strategy. Compared with the static event triggering strategy with a fixed threshold, a dynamic triggering strategy can adaptively adjust the triggering threshold according to the system state $$ |e(k)| $$, thus further reducing the number of communications and the communication load.

3.4. Dynamic event-triggered MFAC approach development

Considering that the helicopter is an underactuated system, the first equation in Equation (4) is treated as a subsystem with control input $$ V_1(k) $$, and the second and third equations in Equation (4) are treated as another subsystem with control input $$ V_2(k) $$. Hence, the designed MFAC scheme includes two parts.

Combining Equations (17)-(22) and the DoS attack mechanism, the improved control scheme can be summarized as

where

Moreover, $$ y_{\varepsilon}^{*}(k + 1) = \varepsilon^{*}(k + 1) + K_{\varepsilon} r_{\varepsilon}^{*}(k+1) $$, $$ y_{\varepsilon}(k) = \varepsilon(k) + K_{\varepsilon} r_{\varepsilon}(k) $$, $$ \Delta y_{\varepsilon}(k) = y_{\varepsilon}(k) - y_{\varepsilon}(k - 1) $$, $$ \eta_{a} $$, and $$ \sigma_{a} $$ are step factors. Moreover, $$ \mu_{a} $$ and $$ \alpha_{a} $$ are weighting factors. $$ \varepsilon^{*} $$ and $$ r_{\varepsilon}^{*} $$ are the expected values of the elevation angle and corresponding angular velocity, respectively. $$ K_{\varepsilon} $$ is the angular velocity gain of $$ \varepsilon $$. $$ K_{a} $$ is the differential gain. $$ K_{pa} $$ is the proportional gain.

The heading and pitch channels are single-input and two-output nonlinear systems. Similarly to the designed method in Equation (23), the controller $$ V_2(k) $$ is designed as

where $$ y_{\psi}^{*}(k + 1) = \psi^{*}(k + 1) + K_{\psi} r_{\psi}^{*}(k+1) $$, $$ y_{\psi}(k) = \psi(k) + K_{\psi} r_{\psi}(k) $$, $$ \Delta y_{\psi}(k) = y_{\psi}(k) - y_{\psi}(k - 1) $$, $$ y_{\theta}^{*}(k + 1) = \theta^{*}(k + 1) + K_{\theta} r_{\theta}^{*}(k+1) $$, $$ y_{\theta}(k) = \theta(k) + K_{\theta} r_{\theta}(k) $$, and $$ \Delta y_{\theta}(k) = y_{\theta}(k) - y_{\theta}(k - 1) $$. $$ \eta_{c} $$ and $$ \sigma_{c} $$ are step factors. $$ \mu_{c} $$ and $$ \alpha_{c} $$ are weighting factors. $$ \psi^{*} $$ and $$ r_{\psi}^{*} $$ are the expected values of the heading angle and corresponding angular velocity, respectively. $$ \theta^{*} $$ and $$ r_{\theta}^{*} $$ are the expected values of the pitch angle and angular velocity, respectively. $$ K_{\psi} $$ and $$ K_{\theta} $$ are the angular velocity gains for $$ \psi $$ and $$ \theta $$, respectively. $$ K_{c} $$ is the differential gain, and $$ K_{pc} $$ is the proportional gain.

5.1. Time-invariant angle tracking

Here, the three attitude angles are set as $$ \varepsilon ^{*}=0.6 $$, $$ \psi^{*}=0.6 $$, and $$ \theta ^{*}=0 $$. The designed method is evaluated by comparing it with the existing method^[40]. Figure 6A-C presents the simulation results of elevation, heading, and pitch channel, respectively. Figure 6D is the event-triggered interval for the elevation angle of the developed method. Figure 6E shows the simulation results of the elevation angle under different DoS attack intensities.

Figure 6. Time-invariant angle tracking. DoS: Denial-of-service.

As shown in Figure 6A-C, the proposed method enables the helicopter system to efficiently track the desired angles by adjusting the control inputs in real-time, even under aperiodic DoS attacks. It should be pointed out that the gray bars denote the successive DoS attacks, as shown in Figure 6A-C. The designed method is effective against successive DoS attacks. Figure 6A shows that the dynamic event triggering strategy provides a slight improvement in response speed compared to the static event triggering strategy. Figure 6B and C illustrates that the dynamic event triggering strategy reduces overshooting by a greater amount than the static event triggering strategy. Figure 6D indicates that the dynamic event triggering strategy reduces the communication resources by 63.47%, while the static event triggering strategy reduces them by only 54.8%. This indicates that the dynamic event-triggering strategy can further reduce communication resources and load. Figure 6E demonstrates that the convergence rate slows down as the intensities of DoS attacks gradually increase.

Remark 8 To further validate the real-time potential of the proposed control method, the controller was deployed on a remote computer. The hardware configuration of the remote computer is as follows: it is equipped with an Intel Core i5-9300H processor (base frequency 2.4 GHz, maximum turbo frequency 4.1 GHz, four cores and eight threads), two 8GB DDR4 memories (frequency 2, 666 MHz), and an NVIDIA GeForce GTX 1650 graphics card. By executing a 50, 000-step simulation test, the average runtime per step was found to be 4.416 us.

5.2. Time-varying angle tracking

The desired angle of the elevation channel is set as $$ \varepsilon^{*} =0.6\sin(0.5t) $$, however, other angles are set the same as the time-invariant example. Selecting $$ \beta=0.5 $$, the control results of different DoS compensations are shown in Figure 7. Figure 7A-C presents the simulation results of elevation, heading, and pitch angles, respectively. Figure 7D is the event-triggered interval of the proposed method and the existing method^[44]. From Figure 7A-C, the proposed algorithm enables the helicopter system to efficiently track the time-varying target angle under aperiodic DoS attacks. Meanwhile, it can be seen from Figure 7A that the elevation angle possesses a faster convergence rate, from Figure 7B that the heading angle possesses a smaller overshoot and a faster convergence rate, and from Figure 7C that the pitch angle possesses a faster convergence rate.

Figure 7. Time-varying angle tracking.

5.3. Parametric analysis

The results of the parameter analysis are shown in Figure 8, where the parameters involved in this section are tuned individually, and the other parameters are consistent with the time-invariant angle tracking experiment. First, the parameter $$ w $$ is analyzed, and the results are shown in Figure 8A. As the event-triggered threshold $$ w $$ is changed from $$ 10^{-5} $$ to $$ 10^{-2} $$, the event-triggered frequency decreases obviously. However, once the event-triggered threshold $$ w $$ continues to grow larger than $$ 10^{-2} $$, the system will not be able to track the desired angle accurately, and the event-triggered frequency increases rapidly instead. Therefore, the user can balance communication costs and control performance by selecting the value of the parameter.

Figure 8. Parametric analysis.

Then, as shown in Figure 8B, the parameter $$ K_{pa} $$ is analyzed. When $$ K_{pa} $$ gradually increases from $$ 1.5 $$ to $$ 1.8 $$, the event-triggered interval decreases from $$ 260 $$ to around $$ 210 $$. However, when $$ K_{pa} $$ increases from $$ 1.8 $$ to $$ 2.0 $$, the event-triggered interval oscillates rather than gradually decreases. After that, as $$ K_{pa} $$ continues to increase, the event-triggered interval decreases rapidly.

Authors' contributions

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Xu, Y.; Zhao, H.

Performed data acquisition and provided administrative and technical support: Gao, Y.; Yu, H.; Peng, L.

Availability of data and materials

The original contributions presented in this study are included in the article.

Financial support and sponsorship

This work was supported in part by the National Natural Science Foundation of China (62403216), in part by the Basic Research Program of Jiangsu Province (BK20241608), in part by the Wuxi Science and Technology Development Fund Project (K20231015), in part by the Wuxi Young Science and Technology Talent Support Program (TJXD-2024-114), in part by the 111 Project (B23008), and in part by the EU iMARs Project (HORIZON-MSCA-2023-101182996).

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

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Copyright

© The Author(s) 2025.