Issue 
Security and Safety
Volume 2, 2023
Security and Safety in Unmanned Systems



Article Number  2023020  
Number of page(s)  17  
Section  Other Fields  
DOI  https://doi.org/10.1051/sands/2023020  
Published online  23 August 2023 
Research Article
Safety flight envelope calculation and protection control of UAV based on disturbance observer
College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China
^{*} Corresponding authors (email: chenmou@nuaa.edu.cn)
Received:
23
March
2023
Revised:
13
June
2023
Accepted:
28
June
2023
In this paper, as for the unmanned air vehicle (UAV) under external disturbance, an attainableequilibriumsetbased safety fight envelope (SFE) calculation method is proposed, based on which a prescribed performance protection control scheme is presented. Firstly, the existing definition of the SFE based on attainable equilibrium set (AES) is extended to make it consistent and suitable for the UAV system under disturbance. Secondly, a higherorder disturbance observer (HODO) is developed to estimate the disturbances and the disturbance estimation is applied in the computation of the SFE. Thirdly, by using the calculated SFE, a desired safety trajectory based on the timevarying safety margin function and firstorder filter is developed to prevent the states of the UAV system from exceeding the SFE. Moreover, an SFE protection controller is proposed by combining the desired safety trajectory, backstepping method, HODO design, and prescribed performance (PP) control technique. In particular, the closedloop system is established on the basis of disturbance estimation error, filter error, and tracking error. Finally, the stability of the closedloop system is verified by the Lyapunov stability theory, and the simulations are presented to illustrate the effectiveness of the proposed control scheme.
Key words: Unmanned aerial vehicle (UAV) / safety flight envelope (SFE) / attainable equilibrium set (AES) / higher order disturbance observer (HODO) / backstepping method / prescribed performance control
Citation: Ma B and Chen M. Safety flight envelope calculation and protection control of UAV based on disturbance observer. Security and Safety 2023; 2: 2023020. https://doi.org/10.1051/sands/2023020
© The Author(s) 2023. Published by EDP Sciences and China Science Publishing & Media Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The unmanned air vehicle (UAV) represents a new type of aircraft weapon system that can be autonomously controlled and perform airtoair and airtoground military missions [1]. Using UAVs to perform dangerous missions instead of pilotpiloted 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 longrange, 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 SFEbased 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, AESbased 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 twodimensional AESs for various aircraft and subsequent analysis of maneuvering characteristics. In [18], the AES of a UAV was determined using the discretetime 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 AESbased 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 L_{1} adaptive control scheme was proposed and implemented in a flight envelope protection system for a transportclass 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 commandlimiting architecture, which could enhance the standard gainscheduling 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 AESbased SFE and a sliding mode control scheme incorporating prescribed performance functions. In [21], an effective convex optimizationbased algorithm was developed to solve AES by representing it as a halfspace, 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 faulttolerant 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 commandlimiting 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, flightenvelope prediction, and faulttolerant 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 SFEbased protection control, this paper proposes a codesign of a disturbanceobserverbased 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 AESbased 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 timevarying safety margin function is proposed based on the prescribed performance function and safety margin constant. Then, by utilizing the safety margin function, a codesign 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 fastloop of the UAV’s attitude dynamics, which is given as follows [28]:
where x_{1} = [α, β, μ]^{T} ∈ ℝ^{3} denotes the attitude angle vector including the attack angle, sideslip angle, and roll angle, x_{2} = [p, q, r]^{T} ∈ ℝ^{3} 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} ∈ ℝ^{5} is the control input vector that, respectively, represents the aileron, canard, rudder, lateral thrust vectoring, and normal thrust vectoring deflection angles; f_{1} : ℝ^{3} → ℝ^{3} and f_{2} : ℝ^{6} → ℝ^{3} are the nonlinear vectorvalued functions, g_{1} : ℝ^{3} → ℝ^{3 × 3} and g_{2} : ℝ^{6} → ℝ^{3 × 5} are the nonlinear matrixvalued functions. The expressions of f_{1}, g_{1}, f_{2}, g_{2} can be found in [29]. Here, d(t)=[d_{p}(t),d_{q}(t),d_{r}(t)]^{T} ∈ ℝ^{3} means the external timevarying disturbance, and y ∈ ℝ^{3} denotes the output vector.
The control objective of this paper is to design a HODObased SFE protection controller to keep the state x_{1} 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 timevarying safety margin function and the firstorder 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.
[27] The desired trajectory y_{d} = x_{1d} = [α_{d}, β_{d}, μ_{d}]^{T} with its first and second derivatives, and , are all bounded for t ≥ 0. That is, there exists a constant M > 0 such that
[30]. The unknown external timevarying disturbance d(t) and its first rth derivatives are all bounded for t ≥ 0, that is, ∥d^{(j)}∥_{2} ≤ η_{j}, where η_{j} > 0, j ∈ {0, 1, …, r} are unknown constants.
Assumptions 1 and 2 are the standard assumptions that have been widely adopted in many publications such as [26, 27], [30], and [31]. Here, Assumption 1 implies that the desired trajectory y_{d} is of finite energy. For Assumption 2, the disturbance d(t) is assumed to be smooth and bounded while its bounds are unknown, which is crucial for the design of the disturbance observer.
For a control system with the bounded initial condition, if there exists a C ^{1} continuous and positive definite Lyapunov function V(x): ℝ^{ n } → ℝ such that
where ϑ_{1}, ϑ_{2} : ℝ → ℝ are class K functions, κ_{1} > 0, and κ_{2} > 0, then the state x(t) is uniformly bounded [27].
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
The deflections of the rudders of the UAV system are limited within a specific range, which can generally be obtained as
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
where S ⊂ ℝ^{6} denotes the AES. Once the parameters of system (5) are given, S is uniquely determined and does not change with time on.
Considering equation (1), it can be seen that the outside disturbance will change the dynamic trim points of the UAV system. Then, a trim point without the disturbance may become an untrimmed point, which means that considering the problem of the AES under disturbance is essential and challenging.
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 timevarying AES of equation (1) as:
where S_{d}(t)⊂ℝ^{6} 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 timevarying disturbance, usually unknown, which makes the AES uncomputable.
Considering that in the control law design and the SFE computation, a technique of estimating the disturbance needs to the proposed to observe the timevarying unknown disturbance. Therefore, for the convenience of calculation, the observed value of the disturbance d(t) is used to replace the d(t) described in equation (8).
The approximate set of the AES S_{d}(t) under disturbance can be further extended as:
In order to obtain the estimation of the disturbance, the following HODO is introduced [30]
where z_{m} ∈ ℝ^{3}, 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:
where q ∈ {0, 1, …, r − 1} and . Taking the derivative of and invoking equations (1) and (10), one can obtain
Denoting , then the error dynamic in equation (12) can be rewritten in a compact form as
where
Constructing the Lyapunov function for equation (13), taking the derivative of , and applying Young’s inequality, one yields
where I is the identity matrix with the appropriate dimension. From Assumption 2, the following inequality is established
Considering equations (17) and (16) can be rewritten as
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]
where S_{1} ∈ ℝ^{3 × 3} and S_{2} ∈ ℝ^{3 × 3} are positive definite matrices to be predefined. J : ℝ^{6} → ℝ is the objective function.
From (9), the optimization problem (19) has simple constraints, and its objective function is continuously differentiable. Thus, some firstorder or secondorder methods, such as the gradient, Newton, and Trust Region, can be utilized to solve such problems (19) since they are all based on iteration. The iterative procedure means that it starts from an initial point, looks for the next point that reduces the objective function, until the termination condition is satisfied, then the algorithm stops.
4. PPbased 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 steadystate and transient performances.
Since the SFE of the system (1) is altered due to timevarying 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 angleofattack 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]
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]
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 timevarying safety margin functions are constructed as min{ρ_{i}(t),τ_{i}}, i ∈ {α, β, μ}. Figure 1 shows a schematic diagram of the timevarying safety margin functions. Moreover, the schematic diagram of generating the constrained desired trajectory according to the timevarying safety margin function, the desired trajectory, and the upper and lower bounds of the SFE are illustrated in Figure 2.
Figure 1. The timevarying 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
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
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
In summary, α_{c} can be further written as
Similarly, β_{c} and μ_{c} can be accordingly obtained as
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
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 firstorder filter is introduced for the smoothing, and it is designed as follows [33]
where x_{1s} ∈ ℝ^{3} 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
For equation (30), the Lyapunov function is selected as
Now, differentiating with respect to time and invoking equation (30), one yields
Then, considering equations (28) and (32) can be rewritten as
One can check that if the parameter w_{1} to be designed satisfies w_{1} − 0.5I > 0, then according to equation (33) and Lemma 1, the filter error is uniformly bounded.
Based on the safetydesigned trajectory generated above, the control law is designed with the PP control method to make the output y = x_{1} of equation (1) track x_{1s} in order to ensure flight safety. The attitude angle tracking error e_{Ω} ∈ ℝ^{3} and attitude angle rate tracking error e_{w} ∈ ℝ^{3} are defined as
where x_{2} ^{*} ∈ ℝ^{3} 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
Moreover, according to the error transformation in [32], the transformed error can be obtained as
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
Taking the derivative of ε_{i} with respect to time yields
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
where ε_{Ω} = [ε_{α}, ε_{β}, ε_{μ}]^{T}. By substituting equations (36) into (40), then can be rewritten as
Thus, the virtual control law x_{2} ^{*} is designed as
where is the positive definite matrix to be designed, that is, . Substituting equations (42) into (41), can be obtained as
For equations (43), the Lyapunov function V_{Ω} is constructed as
Differentiating V_{Ω} with respect to time and invoking equation (43) yield
Meanwhile, by considering equation (1), the derivative of e_{w} can be obtained as
Then, the control law is designed as
where is the positive definite matrix to be designed, that is, . Thus, substituting equations (47) into (46) yields
For error dynamic (48), the Lyapunov function V_{w} is selected as
Taking the derivative of V_{w} with respect to time and invoking equation (18), can be written as
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.
For the nonlinear attitude control UAV system (1) under outside disturbance, Assumptions 1 and 2 are satisfied, the design of HODO is shown in (10), and the SFE under external disturbance is defined in (9) and computed by (19), the constrained desired trajectories are designed by (25), (26), and (27), based on which the safety desired trajectories are obtained by (29), and the design of virtual control law x_{2}^{*} and control input u are designed as (42) and (47). Then, as for the performance functions ρ_{i}(t),i = (α, β, μ) taken as (21) with 0 < δ_{i} < 1 and the safety margin constants τ_{i} > 0, the UAV attitude SFE protection tracking errors are bounded convergent, and the signals of all closedloop systems are uniformly bounded if there exist positive definite matrices w_{1}, K_{Ω}, K_{ω}, and K such that
In order to prove Theorem 1, the overall Lyapunov function V is constructed as
Taking the derivative of V with respect to time, invoking equations (33), (45), (50), and noticing the fact that , we have
where ϖ = min{σ_{min}(w_{1}−0.5I),σ_{min}(K_{Ω}),σ_{min}(K_{ω}−0.5I),σ_{min}(K_{2}−I)} > 0, σ_{min}(L) represents the smallest eigenvalue of matrix L, c_{0} = 0.5η_{r} ^{2} + 0.5Q. According to Lemma 1, the signals of the closedloop 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 S_{1} = diag(3, 3, 3),S_{2} = 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:
By exploiting Theorem 1 and taking the order of HODO as 2, the gain matrices of the HODO are taken as k_{1} = diag(10, 10, 10), k_{2} = 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 firstorder filter is selected as w_{1} = 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 z_{1} and z_{2} of the HODO are taken as z_{1} = z_{2} = [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 timevarying 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 timevarying 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) d_{p} and , (b) d_{q} and and (c) d_{r} 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 timevarying 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 timevarying safety margin function and firstorder 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.
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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 timevarying 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) d_{p} and , (b) d_{q} and and (c) d_{r} and 

In the text 
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