© The Author(s) 2023. Published by EDP Sciences and China Science Publishing & Media Ltd.

1. Introduction

The unmanned air vehicle (UAV) represents a new type of aircraft weapon system that can be autonomously controlled and perform air-to-air and air-to-ground military missions [1]. Using UAVs to perform dangerous missions instead of pilot-piloted aircraft is one of the essential trends in future warfare [2]. Meanwhile, due to the advantages of UAVs, a large number of higher requirements have been proposed for the future applications of UAVs, such as intelligence, autonomy, large overload, and long-range, which can further promote the developments of UAVs.

Together with the wide applications of UAVs, flight safety issues have attracted much research attention from many scholars [3–6]. Normally, flight safety refers to the fact that during the operations of the aircraft, there are no incidents of improper operations or external factors (such as disturbances) that cause casualties or damage to the aircraft [7]. Compared with other traffic modes, the accident rate of aviation is low, while its destructiveness is enormous. Based on the results in [8], the leading cause of flight accidents was loss of control (LOC). Unfortunately, there is still no unified concept of the LOC far, and the causes of LOC are unclear [9]. However, it is widely recognized that the flight states exceeding the specific SFE can be considered LOC [10]. Though the LOC does not necessarily induce flight accidents, the accidents must occur sometime if it is undetected or left unchecked [11]. On the other hand, since UAVs are complex nonlinear systems and sensitive to disturbances, it is straightforward to cross the SFE during maneuvering and lead to LOC. Thus, designing SFE protection control systems is vital to prevent the LOC and ensure flight safety. Because of the great potential of the SFE protection system in enhancing flight safety, it has received extensive attention from researchers over two past decades [12]. Current researches on SFE protection systems mainly focus on two main fields: SFE computation and SFE-based controller design [13]. The former uses neural networks, dynamic trim, and reachability set to calculate the SFE, while the latter uses dynamic inverse, robust control, and intelligent control to prevent flight states from exceeding the SFE.

The premise of designing the SFE protection control system aims to obtain the SFE. The traditional SFE refers to the speed and altitude level flight envelope. However, even if the aircraft remains within this flight envelope, the outside disturbances will still induce the LOC problem, so a new concept of the SFE or an extended one need to be presented. It is widely accepted that the SFE comprises an environmental envelope, a structural envelope, and a dynamic envelope. The determination of dynamic envelope has caught many research interests since it takes the aircraft’s dynamic characteristics into account [14, 15]. Reference [16] completed the quantitative analysis of the LOC problem and determined the LOC quantitative criterion, which was composed of five different SFEs derived from the critical factors, such as aircraft’s dynamic characteristics, aerodynamic characteristics, structural integrity, and control variables. The LOC quantification criterion held that if the flight states exceed any three of the five envelopes, it can determine the LOC phenomenon that occurred. In addition, AES-based SFE was widely used in the field of flight safety [17, 18]. In [17], a calculation method of the SFE based on the AES was proposed to analyze the performance and maneuverability of aircraft in steady flight conditions, involving the calculation of two-dimensional AESs for various aircraft and subsequent analysis of maneuvering characteristics. In [18], the AES of a UAV was determined using the discrete-time Newton iteration approach, and a prediction technique for AES was developed in conjunction with the Unscented Kalman Filter, providing the benefits of high accuracy and exceptional stability. Currently, the calculation of the AES-based SFE is mainly built on the nominal systems, that is, some unfavorable factors such as disturbance and model uncertainties are not involved, which remains challenging and important. Especially, in actual flight, the UAVs are unavoidably affected by various outside disturbances, which means that an extension of the AES concept waits to be proposed.

Once the SFE is obtained, the SFE protection control system needs to be established. For nominal systems, some research results have emerged on the SFE protection control. In [7], an L₁ adaptive control scheme was proposed and implemented in a flight envelope protection system for a transport-class aircraft, which enhanced aircraft performance by restoring the nominal performance of the baseline controller under faults and disturbances. An SFE protection system was developed in [19], and it was based on command-limiting architecture, which could enhance the standard gain-scheduling control law by considering the LOC quantitative criterion. A safety control scheme integrating SFE protection and reference command regeneration was proposed in [20] to address the issue of performance degradation and SFE shrinkage resulting from aircraft actuator faults, which involved the calculation of AES-based SFE and a sliding mode control scheme incorporating prescribed performance functions. In [21], an effective convex optimization-based algorithm was developed to solve AES by representing it as a half-space, and a protection control law based on model predictive control was developed by incorporating the AES constraints. In [22], a sparse recurrent fuzzy neural network was utilized to propose an SFE protection system, where a fault-tolerant controller served as the baseline controller to handle model uncertainties and a dynamic SFE protection controller was implemented to prevent the compromised aircraft from breaching the SFE. A command-limiting SFE protection system was developed in [23] and incorporated an exponential potential function defined based on the SFE, where the controller design utilized the gradient of the exponential potential function. To enhance aircraft safety, an SFE protection system was developed in [24], where the SFE protection algorithm was designed by incorporating an adaptive neural network, the least square method, and an online linearization algorithm to predict the states based on the current control inputs and states. In [25], an online SFE protection system including system identification, damage classification, flight-envelope prediction, and fault-tolerant control was proposed for impaired aircraft, and with the designed envelope protection, the LOC accidents were more likely to be prevented. However, as for the application of SFE protection control, it is not enough to only consider these ideal situations. In practice, the SFE is always affected by various factors, such as outside disturbances. Then, for a class of uncertain nonlinear systems with output constraints and external disturbances, an adaptive neural safety boundary protection algorithm was developed in [26] based on disturbance observer, which could effectively limit the output trajectory within the output constraints, ensuring flight safety. For unmanned helicopters under outside disturbances, model uncertainties, and output constraints, an adaptive fuzzy safety tracking control scheme was developed [27], where the safety margin was used to limit the desired output within a given range, and a tracking protection controller was designed by combining a fuzzy system. In summary, the existence of the disturbance will impose negative effects on the SFE of the UAV system, and deteriorate the performance of the SFE protection controller, which needs to be investigated and constitutes the focus of this paper.

Motivated by the above discussions on SFE computation and SFE-based protection control, this paper proposes a co-design of a disturbance-observer-based AES calculation method and an SFE protection control scheme. The main contributions of this paper are listed as follows:

• The scope of the applications of existent AES is extended, and the definition of the AES-based SFE under disturbances is presented. That is to say, a HODO is developed to estimate external disturbances, and the disturbance observations are used for the calculation of the SFE;

• Based on the derived SFE, a time-varying safety margin function is proposed based on the prescribed performance function and safety margin constant. Then, by utilizing the safety margin function, a co-design method for the desired safety trajectory and SFE protection controller is proposed.

Five sections make up the whole structure of this paper. In Section 2, the UAV attitude motion model is given. The definition and computation of AES under disturbance are studied based on the HODO in Section 3. Section 4 proposes a method for generating the safety desired trajectory. Also, an SFE protection control law is developed in Section 4 based on the HODO, backstepping method, and PP control. Section 5 designs and completes simulations to verify the effectiveness of the SFE protection controller. The entire work is summarized in Section 6.

2. Problem statement

This section gives the UAV attitude dynamic model under external disturbances, the control targets, and some related preliminary knowledge. Initially, the disturbances are taken into account in the fast-loop of the UAV’s attitude dynamics, which is given as follows [28]:

(1)

where x₁ = [α, β, μ]^T ∈ ℝ³ denotes the attitude angle vector including the attack angle, sideslip angle, and roll angle, x₂ = [p, q, r]^T ∈ ℝ³ represents the attitude angle rate vector of roll angle rate, pitch angle rate, and yaw angle rate, , u = [δ_a,δ_c,δ_r,δ_y,δ_z]^T ∈ ℝ⁵ is the control input vector that, respectively, represents the aileron, canard, rudder, lateral thrust vectoring, and normal thrust vectoring deflection angles; f₁ : ℝ³ → ℝ³ and f₂ : ℝ⁶ → ℝ³ are the nonlinear vector-valued functions, g₁ : ℝ³ → ℝ^3 × 3 and g₂ : ℝ⁶ → ℝ^3 × 5 are the nonlinear matrix-valued functions. The expressions of f₁,  g₁,  f₂,  g₂ can be found in [29]. Here, d(t)=[d_p(t),d_q(t),d_r(t)]^T ∈ ℝ³ means the external time-varying disturbance, and y ∈ ℝ³ denotes the output vector.

The control objective of this paper is to design a HODO-based SFE protection controller to keep the state x₁ of the system (1) within the SFE constraint. For this reason, the calculation of AES under disturbance is initially studied. Then, a method of obtaining the safety desired trajectory from the desired trajectory is proposed by combining the time-varying safety margin function and the first-order filter. Finally, SFE protection control controller is designed based on the backstepping method, the HODO, and the PP control.

The following assumptions and lemma are given to facilitate the subsequent calculation of the SFE and the design of the SFE protection control scheme.

3. Safety flight envelope calculation under disturbance

According to the basic concept of dynamic trim, the UAV system with twelve states can be divided into four loops: the position loop, the velocity loop, the attitude angle loop, and the attitude angle rate loop. The attitude angle loop and the attitude angle rate loop belong to the fast loop, while the position loop and the speed loop belong to the slow loop. In the UAV system, the states of the fast loop first reach the equilibrium point. That is to say, the derivatives of the fast states are normally zero, while the slow states remain constants. The AES represents the set of all dynamic trim points of the UAV and is defined as the SFE of the UAV. As an extension of the AES definition, this section introduces a new type of AES under disturbance and gives its computation method.

Considering the system (1) without the disturbance d(t), the dynamic model is rewritten as

(4)

The deflections of the rudders of the UAV system are limited within a specific range, which can generally be obtained as

(5)

where δ_i, max, δ_i, min represent the upper bound and lower bound of δ_i,  i ∈ {a, c, r, y, z}, respectively. Then, considering the definition of AES in [17], the AES of system (5) can be written as

(6)

where S ⊂ ℝ⁶ denotes the AES. Once the parameters of system (5) are given, S is uniquely determined and does not change with time on.

Due to the introduction of external disturbance, the AES in (7) will become altered and unpredictable. By combining the idea of dynamic trim and the definition of AES, we define a time-varying AES of equation (1) as:

(7)

where S_d(t)⊂ℝ⁶ denotes the AES under disturbance. It is worth noting that even though the AES in (8) has its rationality and application value, equation (8) contains a time-varying disturbance, usually unknown, which makes the AES uncomputable.

The approximate set of the AES S_d(t) under disturbance can be further extended as:

(8)

In order to obtain the estimation of the disturbance, the following HODO is introduced [30]

(9)

where z_m ∈ ℝ³,  m ∈ {1, 2, …, r} are the internal states of the HODO and are the estimated values of , respectively; is the output of the HODO, k_m ∈ ℝ^3 × 3,  m ∈ {1, 2, …, r} are positive definite matrices to be designed. In what follows, define the qth order derivative estimation error of the disturbance d as:

(10)

where q ∈ {0, 1, …, r − 1} and . Taking the derivative of and invoking equations (1) and (10), one can obtain

(11)

Denoting , then the error dynamic in equation (12) can be rewritten in a compact form as

(12)

where

(13)

(14)

Constructing the Lyapunov function for equation (13), taking the derivative of , and applying Young’s inequality, one yields

(15)

where I is the identity matrix with the appropriate dimension. From Assumption ², the following inequality is established

(16)

Considering equations (17) and (16) can be rewritten as

(17)

If the parameter K to be designed satisfies K − 0.5I >  0, then according to equation (18) and Lemma ¹, the estimated error of the HODO is uniformly bounded.

The basic idea of the AES under disturbance aims to estimate the disturbance online and replace the disturbance with its observed values to calculate AES. In practical applications, due to the limitation of the computing power of the flight control computer, it can be considered to store offline AES under disturbances. When flying online, according to the observed value of the disturbance, the AES can be generated from the stored AES through interpolation. In this paper, we only consider the offline computation of the AES under external disturbance. Thus, based on the optimization method, the calculation of the AES with disturbance can be transformed into an optimization problem, which is described as [20]

(18)

where S₁ ∈ ℝ^3 × 3 and S₂ ∈ ℝ^3 × 3 are positive definite matrices to be predefined. J : ℝ⁶ → ℝ is the objective function.

4. PP-based flight envelope protection control scheme

Since flight safety is the first issue for UAVs, it is essential for the flight control law to conduct maneuvering flights and complete specified tasks, which needs SFE protection control to realize flight safety. Then, for the attitude dynamic model of the UAV system under disturbance, this section proposes a new SFE control method by integrating the HODO, the calculation of SFE, the filtering technology, and the prescribed performance control. The SFE protection control involves both the SFE constraints and the performance requirements of tracking control, such as steady-state and transient performances.

Since the SFE of the system (1) is altered due to time-varying disturbance, there must exist the SFE constraint upper bound x_{u p}(t)=[α_{u p}(t),β_{u p}(t),μ_{u p}(t)]^T and SFE constraint lower bound x_{l o w}(t)=[α_{l o w}(t),β_{l o w}(t),μ_{l o w}(t)]^T such that α_{u p}(t)> α_{l o w}(t), β_{u p}(t)> β_{l o w}(t), and μ_{u p}(t)> μ_{l o w}(t). Yet, the conflicts between desired trajectory x_1d = [α_d, β_d, μ_d]^T and SFE constraints may occur in actual flight. Then, it is necessary to adjust the desired trajectory x_1d to obtain the constraint desired trajectory x_1c = [α_c, β_c, μ_c]^T and the safety desired trajectory x_1s = [α_s, β_s, μ_s]^T to ensure flight safety.

This section proposes a new generation method of constraint and desired safety trajectories based on the PP functions. The procedure is mainly divided into three following steps [27]:

Step 1: The first step is to obtain the SFE constraints x_up(t) and x_low(t). According to the calculation of the SFE proposed in the previous section, x_up(t_i), x_low(t_i), i ∈ {1, 2, …, n} at a given moment can be obtained. Piecewise Hermit interpolation functions x_up(t) and x_low(t) can be constructed by using x_up(t_i), x_low(t_i), i ∈ {1, 2, …, n}. Both x_up(t) and x_low(t) are piecewise continuously differentiable.

Step 2: The second step is to generate the desired constraint trajectory x_1c. Let ρ_α(t), ρ_β(t), and ρ_μ(t) be the performance functions of the angle-of-attack tracking error e_α = α − α_s, sideslip angle tracking error e_β = β − β_s, and roll angle tracking error e_μ = μ − μ_s, respectively. It is necessary to design the SFE protection control law such that the errors e_i,  i ∈ {α, β, μ} satisfy [32]

(19)

where 0 <  δ_i <  1,  i ∈ {α, β, μ} are the parameters to be designed. Here, ρ_i(t), i ∈ {α, β, μ} are selected as [32]

(20)

where ρ_i0 and ρ_i∞ denote the initial and limit values of performance functions with ρ_i0 >  ρ_i∞ >  0, l_i >  0.

Define that τ_α, τ_β, and τ_μ are the safety margin constants to be designed later. The time-varying safety margin functions are constructed as min{ρ_i(t),τ_i}, i ∈ {α, β, μ}. Figure 1 shows a schematic diagram of the time-varying safety margin functions. Moreover, the schematic diagram of generating the constrained desired trajectory according to the time-varying safety margin function, the desired trajectory, and the upper and lower bounds of the SFE are illustrated in Figure 2.

Figure 1. The time-varying safety margin function changes over time

Figure 2. Constrained desired trajectory generation process

Let’s take α_c as an example to illustrate this analytic process. When α_d(t)≥α_up(t)−min{τ_α, ρ_α(t)}, the desired trajectory has the risk of crossing the upper bound of the SFE. Define the desired constraint trajectory as

(21)

As for α_d(t)≤α_low(t)+min{τ_α, ρ_α(t)}, the desired trajectory has the risk of crossing the lower bound of the SFE. Define the constrained desired trajectory as

(22)

For α_low(t)+min{τ_α, ρ_α(t)} <  α_d(t)< α_up(t)−min{τ_α, ρ_α(t)}, the desired trajectory has margins from the upper bound and lower bound of the SFE. The constrained desired trajectory is defined as

(23)

In summary, α_c can be further written as

(24)

Similarly, β_c and μ_c can be accordingly obtained as

(25)

(26)

Based on the above discussions, we can define the desired constraint trajectory x_1c = [α_c, β_c, μ_c]^T. Since min{ρ_i(t),τ_i}, i ∈ {α, β, μ}, x_up(t), and x_low(t) are all piecewise continuously differentiable and bounded, then x_1c is also a piecewise continuously differentiable bounded function. That is, in each differentiable interval, there must exist a constant Q >  0 such that

(27)

Step 3: The desired constraint trajectory defined in the above step is piecewise continuously differentiable, which is inconvenient for the subsequent tracking control law design. Then, a first-order filter is introduced for the smoothing, and it is designed as follows [33]

(28)

where x_1s ∈ ℝ³ is the estimate of the constrained desired trajectory x_1c, called the safety desired trajectory. is a positive definite matrix that needs to be designed later.

Defining the filter error and taking the derivative of with respect to time, we have

(29)

For equation (30), the Lyapunov function is selected as

(30)

Now, differentiating with respect to time and invoking equation (30), one yields

(31)

Then, considering equations (28) and (32) can be rewritten as

(32)

One can check that if the parameter w₁ to be designed satisfies w₁ − 0.5I >  0, then according to equation (33) and Lemma ¹, the filter error is uniformly bounded.

Based on the safety-designed trajectory generated above, the control law is designed with the PP control method to make the output y = x₁ of equation (1) track x_1s in order to ensure flight safety. The attitude angle tracking error e_Ω ∈ ℝ³ and attitude angle rate tracking error e_w ∈ ℝ³ are defined as

(33)

where x₂ ^* ∈ ℝ³ is the virtual control law to be designed. Taking the derivative of e_Ω with respect to time and invoking equations (1), (35), one yields

(34)

Moreover, according to the error transformation in [32], the transformed error can be obtained as

(35)

where ε_i ∈ ℝ is the transformed error of the error variable e_i, z_i = e_i(t)/ρ_i(t), i ∈ {α, β, μ}. S_i : ℝ → ℝ are the error transformation functions and can be written as

(36)

Taking the derivative of ε_i with respect to time yields

(37)

where r_i = (∂S_i ⁻¹/∂z_i)(1/ρ_i), i ∈ {α, β, μ}. Since both ∂S_i ⁻¹/∂z_i and ρ_i are greater than 0, r_i >  0.

Defining R(t)=diag(r_α,r_β,r_μ), , N(t)=[N_α(t),N_β(t),N_μ(t)]^T, i ∈ {α, β, μ}, equation (39) can be written in the compact form as

(38)

where ε_Ω = [ε_α, ε_β, ε_μ]^T. By substituting equations (36) into (40), then can be rewritten as

(39)

Thus, the virtual control law x₂ ^* is designed as

(40)

where is the positive definite matrix to be designed, that is, . Substituting equations (42) into (41), can be obtained as

(41)

For equations (43), the Lyapunov function V_Ω is constructed as

(42)

Differentiating V_Ω with respect to time and invoking equation (43) yield

(43)

Meanwhile, by considering equation (1), the derivative of e_w can be obtained as

(44)

Then, the control law is designed as

(45)

where is the positive definite matrix to be designed, that is, . Thus, substituting equations (47) into (46) yields

(46)

For error dynamic (48), the Lyapunov function V_w is selected as

(47)

Taking the derivative of V_w with respect to time and invoking equation (18), can be written as

(48)

Motivated by the above discussions, as for the system (1), the stability analysis of the SFE protection control law designed based on the HODO and the backstepping method can be summarized as the following theorem.

Taking the derivative of V with respect to time, invoking equations (33), (45), (50), and noticing the fact that , we have

(51)

where ϖ = min{σ_min(w₁−0.5I),σ_min(K_Ω),σ_min(K_ω−0.5I),σ_min(K₂−I)} >  0, σ_min(L) represents the smallest eigenvalue of matrix L, c₀ = 0.5η_r ² + 0.5Q. According to Lemma ¹, the signals of the closed-loop systems are uniformly bounded, and the proof of Theorem ¹ is completed.▫

5. Numerical simulations

In this section, simulations are presented to calculate AES under disturbance, based on which the effectiveness of the proposed SFE protection controller is verified.

Set the initial conditions for the simulation: flight altitude H(0)=3000 m, flight velocity V(0)=150 m/s, engine thrust T = 20, 000N, attitude angle α(0)=β(0)=μ(0)=5°, attitude angle rate p(0)=q(0)=r(0)=5 ° /s. In the simulation, the trust region algorithm is used to solve equation (19), and the parameters are selected as S₁ = diag(3, 3, 3),S₂ = diag(1, 1, 1). Moreover, the disturbance d(t) is assumed as d(t)=[0.25sin(t),0.3sin(1.5t),0.25sin(2t)]^T. The desired trajectories are taken as:

(52)

By exploiting Theorem ¹ and taking the order of HODO as 2, the gain matrices of the HODO are taken as k₁ = diag(10, 10, 10), k₂ = diag(3, 3, 3). The parameters of the prescribed performance functions are taken as: ρ_α0 = ρ_β0 = ρ_μ0 = 0.2 rad, ρ_α∞ = ρ_β∞ = ρ_μ∞ = 0.01 rad, l_α = l_β = l_μ = 1, δ_α = δ_β = δ_μ = 0.8. The safety margin constants are τ_α = τ_β = τ_μ = 0.05 rad. The parameter of the first-order filter is selected as w₁ = diag(12.5, 12.5, 12.5). The controller gain matrices are taken as K_Ω = K_ω = diag(5, 5, 5). Based on Theorem ¹, the selection of the designed parameters is mainly based on equation (51), and the parameters satisfying equation (51) can meet the desired control target. Moreover, the initial values of the internal states z₁ and z₂ of the HODO are taken as z₁ = z₂ = [0, 0, 0]^T. The simulation is mainly from the SFE calculation and the requirements of the safety tracking accuracy to debug the parameters. Then, a series of parameter values are selected, and the optimal parameters are obtained according to the disturbance tracking error and SFE tracking accuracy.

Figure 3 shows the comparisons of the SFE under disturbance and the one without disturbance. It can be seen from Figure 3 that compared with the case without disturbance, the SFE under disturbance can shrink significantly, which means that when designing a boundary protection controller, the impact of disturbance should be taken into account. Otherwise, the performance loss of flight control may occur. The safety tracking curves of the attack, sideslip, and roll angles are illustrated in Figures 4–6. Figure 4 indicates that the attitude protection control law of this paper can achieve adequate protection against the angle of attack. The desired signal of the attack angle does not violate the lower bound of the safety boundary in the whole control process. However, it violates the upper bound of the safety boundary for many times. Due to the time-varying safety margin, when the reference signal of the attack angle is close to the upper bound of the SFE, the SFE protection control system makes corresponding modifications to the reference signal of the attack angle, which can ensure flight safety and avoid LOC. It follows from Figures 5 and 6 that the designed attitude protection control law can realize the protection of sideslip angle and roll angle. During the entire control process, the sideslip angle reference signal and the roll angle reference signal violate the upper and lower bounds of the SFE many times, and the protection control system successfully adjusts the signal to violate the boundary to ensure flight safety.

Figure 3. Upper and lower bounds of SFE. (a) Upper and lower bounds of attack angle, (b) Upper and lower bounds of slide angle and (c) Upper and lower bounds of roll angle

Figure 4. The curve of attack angle during protection control

Figure 5. The curve of slideslip angle during protection control

Figure 6. The curve of roll angle during protection control

The attitude angle tracking errors in the protection control process are shown in Figure 7. Figure 7 shows that the tracking errors are strictly limited in the upper and lower bounds composed of prescribed performance functions. With time on, the upper and lower bounds of the preset performance gradually decrease, the attitude angle tracking error converges, and the prescribed performance conditions are always satisfied. Moreover, we give detailed diagrams of attack angle SFE, desired trajectory, constraint desired trajectory, safety desired trajectory, and actual tracking trajectory in the protection control process, as shown in Figure 8. It can be seen from Figure 8 that the desired constraint trajectory, the safety desired trajectory, and the actual tracking trajectory are strictly limited within the SFE constraints, which ensures flight safety. Finally, each channel’s disturbance and estimated value are presented, as shown in Figure 9. Figure 9 indicates that the HODO designed in this paper can quickly and accurately estimate unknown time-varying disturbances.

Figure 7. The tracking errors of attitude angles during simulation. (a) The tracking error of attack angle, (b) the tracking error of slideslip angle and (c) The tracking error of roll angle

Figure 8. Simulation details of attack angle during protection control

Figure 9. Disturbances and their estimations. (a) dp and , (b) dq and and (c) dr and

6. Conclusions

In this paper, as for the attitude dynamic model of the UAV system under disturbance, the extension and calculation of the SFE based on HODO and AES were first investigated. Secondly, a time-varying safety margin function was constructed, and a desired safety trajectory generation method was proposed. Then, by combining the prescribed performance control, the HODO, and the backstepping method, a SFE protection controller was designed. Finally, simulation results showed that the SFE under disturbance could shrink obviously, and the SFE protection control scheme could achieve the protection of the attitude angles during tracking control. In this article, we mainly focused on the theory and the simulation rather than the experiment, which was mainly due to the lack of supporting conditions for doing the experiment. If possible in the future, we will make up for the experiment.

The main advantages of this paper are summarized in what follows. Firstly, the disturbances are considered in the computation of the SFE, and the influence of the disturbances on the SFE is analyzed. Secondly, a method for generating the safety desired trajectory is constructed based on the time-varying safety margin function and first-order filter, which guarantees that the desired trajectory remains within the SFE. Finally, an SFE protection controller is deduced by combining the HODO, prescribed performance control, and the backstepping method. In future studies, both model uncertainties and outside disturbance in the UAV system will be jointly considered. Moreover, the protection control inside the SFE with the one outside the SFE will need to be considered simultaneously to ensure flight safety in the case of state crossing caused by a sudden change of state.

Conflict of Interest

The author declares no conflict of interest.

Data Availability

No data are associated with this article.

Authors’ Contributions

Biao Ma contributed to the investigation, writing, and validation. Mou Chen contributed to the methodology, project administration, and conceptualization

Acknowledgments

Thanks to the anonymous reviewers for their helpful comments and suggestions.

Funding

This work was supported in part by the National Science Fund for Distinguished Young Scholars 61825302; in part by the National Natural Science Foundation of China under Grant U2013201 and in part by the Key R & D projects (Social Development) in Jiangsu Province of China under Grant BE2020704.

References

Biao Ma received his B.S. degree in Aircraft Design and Engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Jiangsu, China, in 2020. He is currently pursuing his M.S. degree in Control Theory and Control Engineering from the NUAA. His current research interests include flight control and guidance.

Mou Chen received the B.S. degree in material science and engineering and Ph.D. degree in control theory and control engineering from the Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 1998 and 2004, respectively. He is currently a Full Professor with the College of Automation Engineering, NUAA. He was an Academic Visitor with the Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, U.K., from November 2007 to February 2008. From June 2008 to September 2009, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He was a Senior Academic Visitor with the School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, SA, Australia, in 2014 for six months. His current research interests include nonlinear system control, intelligent control, and flight control.

All Figures

	Figure 1. The time-varying safety margin function changes over time
In the text

	Figure 2. Constrained desired trajectory generation process
In the text

	Figure 3. Upper and lower bounds of SFE. (a) Upper and lower bounds of attack angle, (b) Upper and lower bounds of slide angle and (c) Upper and lower bounds of roll angle
In the text

	Figure 4. The curve of attack angle during protection control
In the text

	Figure 5. The curve of slideslip angle during protection control
In the text

	Figure 6. The curve of roll angle during protection control
In the text

	Figure 7. The tracking errors of attitude angles during simulation. (a) The tracking error of attack angle, (b) the tracking error of slideslip angle and (c) The tracking error of roll angle
In the text

	Figure 8. Simulation details of attack angle during protection control
In the text

	Figure 9. Disturbances and their estimations. (a) dp and , (b) dq and and (c) dr and
In the text