Active Disturbance Rejection Control for Air-Breathing Hypersonic Vehicles Based on Prescribed Performance Function

Aiming at the longitudinal motion model of the air-breathing hypersonic vehicles (AHVs) with parameter uncertainties, a new prescribed performance-based active disturbance rejection control (PP-ADRC) method was proposed. First, the AHV model was divided into a velocity subsystem and altitude system. To guarantee the reliability of the control law, the design process was based on the nonaffine form of the AHV model. Unlike the traditional prescribed performance control (PPC), which requires accurate initial tracking errors, by designing a new performance function that does not depend on the initial tracking error and can ensure the small overshoot convergence of the tracking error, the error convergence process can meet the desired dynamic and steady-state performance. Moreover, the designed controller combined with an active disturbance rejection control (ADRC) and extended state observer (ESO) further enhanced the disturbance rejection capability and robustness of the method. To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, a new tracking differentiator was proposed. Finally, the effectiveness of the proposed method was verified by comparative simulations.


Introduction
Air-breathing hypersonic vehicles (AHVs) are a new type of aircraft, which fly at speeds greater than Mach 5 at near space altitudes.AHVs exhibit fast flying speeds, strong penetration abilities, and long combat distances.It is difficult to detect and intercept AHVs.AHVs have strong survivability, and they have outstanding advantages in strategy, tactics, and cost-effectiveness compared to traditional aerospace vehicles [1][2][3].AHVs have become a priority development direction for all of the aerospace powers competing for air and space rights.However, AHVs are multivariable and strongly coupled nonlinear systems.AHVs have large flight airspaces, and the flight environments are complex and variable, which results in large and fast time-varying characteristics and altitude uncertainty of the AHV model.Thus, the design of the AHV control system involves unprecedented difficulties and challenges [4,5].
Most of the previous research on the modelling and control of AHVs has mainly focused on the AHV's longitudinal motion plane.On the one hand, the longitudinal motion model is complex enough to require flight control.On the other hand, due to the scramjet engine's extreme sensitivity to the flight attitude, AHVs should avoid horizontal manoeuvres during actual flight [6].In a previous study [7], a robust L ∞ gain control method was designed for AHVs, which used the Takagi-Sugeno (T-S) fuzzy system to approximate the unknown state of the model.To guarantee the robustness of the control scheme, a new fuzzy disturbance observer was designed to estimate the disturbance.A similar longitudinal elastic model of the AHV was expressed as a T-S fuzzy system [8], and a H 2 /H ∞ tracking control law was designed to achieve the robust tracking of the speed and altitude reference inputs.To improve the tracking effect, a nonlinear adaptive back-stepping controller was designed for the AHV based on the back-stepping design [9].Based on the traditional back-stepping control method, a nonsingular fast terminal sliding model control was applied to control the pitch angle and pitch rate, which optimized the control structure of the back-stepping method and achieved finite time convergence of the system [10].In one study, an integral sliding model control method was proposed [11].When there are model uncertainties and external disturbances, the control scheme can still guarantee finite time convergence of the velocity and altitude tracking errors.Aiming at the attitude control problem of AHVs, a robust fuzzy control method based on a nonlinear switching system was proposed [12], which used a fuzzy system to approximate the unknown function of the model and guaranteed the robustness of the control scheme.Neural control methods for scenarios where the control input of the AHV contained a dead zone [13] and the actuator of the AHV contained failures [14] were studied.Two new neural back-stepping control methods were put forward [15,16].The altitude subsystem was rewritten in strict and pure feedback forms.Based on the improved back-stepping strategy, the control scheme was designed, and a minimal learning parameter (MLP) algorithm was applied to reduce the online learning parameters.The tracking simulation results of the velocity and altitude reference input showed that the proposed method exhibited better robustness and control effects.To solve the control problem of a strongly nonlinear model, such as an AHV, in addition to the fuzzy system or neural network used in the above literature to estimate the uncertainty of the model, active disturbance rejection control (ADRC) can also be considered.The ADRC term was first used in [17] where Professor J.Q. Han's unique ideas were first systematically introduced into the English literature.In recent years, many scholars have conducted meaningful work on ADRC.In a previous study [18], ADRC was offered as the basis of a paradigm shift, providing the framework, the objectives, and constraints for future control theory development.For a class of multipleinput multiple-output (MIMO) lower-triangular systems which have uncertain dynamics and disturbance, an ADRC method was designed to solve the control problem [19].An overview of the concept, principles, and practice of the ADRC was presented in [20].The key idea of ADRC is to use an extended state observer (ESO) to estimate the total disturbance, which is then compensated in the feedback loop.For a class of nonlinear systems with large uncertainties that come from both internal unknown dynamics and external stochastic disturbance, a novel ESO was designed to estimate both state and total disturbance which included the internal uncertain nonlinear part and the external uncertain stochastic disturbance [21].To improve the performance of ADRC, a fal-based, single-parameter-tuning ESO was proposed, and the convergence of the fal-based ESO and the output tracking were established [22].To improve the performance of the tracking differentiator (TD), an ADRC method based on a radial basis function neural network (RBFNN) was proposed [23], which enhanced the robustness and disturbance rejection ability of the method.To assist in the tuning of the parameters of the linear ADRC controller, a tuning rule was proposed to minimize the load disturbance attenuation performance [24].Based on the existing swarm intelligence algorithms, a parameter tuning optimization design of ADRC was achieved [25].An indepth study on the scheme design of an ADRC was conducted and applied to a three-degree-of-freedom pneumatic motion system subject to actuator saturation [26], which achieve good control.
Although the methods in the above literature achieved certain control effects, the research focus was on the robustness and steady-state performance of the AHV closed-loop control system, neglecting the dynamic performance of the control system.However, AHV's super-manoeuvrable, large-envelope, and hypersonic flight demands better dynamic performance of the control system than any other existing aircraft.In most cases, a small control delay will cause significant errors in the hypersonic flight.Therefore, to guarantee the robustness and steady-state accuracy of the control system, more attention should be paid to the dynamic performance and real-time performance of the control system.To consider both the steady-state and dynamic performances, the concept of prescribed performance control was proposed by Charalampos and George [27].The prescribed performance ensured that the tracking error converged to a prescribed arbitrary small area.Meanwhile, the convergence rate and overshoot met the prescribed conditions.Based on the prescribed performance control, a longitudinal innerloop controller of the hypersonic vehicle was designed [28].However, the vehicle was considered to be a pure rigid body and the elastic problem was not considered, which resulted in the limitations of the proposed method.Aiming at the elastic body of the AHV model, a prescribed performance fuzzy back-stepping control method and performance neural back-stepping control method were proposed [29,30], which guaranteed the steady-state performance and dynamic performance of the control system.However, the above two methods considered the AHV model to be an affine model, which made the reliability of the design method not guaranteed.An AHV adaptive neural control method was designed based on the prescribed performance function, which avoided the cumbersome virtual control law design process [31].Meanwhile, the accuracy and rapidity of the control system were achieved.However, the proposed method relied too much on the initial value of the tracking error, which resulted in the poor practicality and operability.
In view of the deficiencies of the research in the above studies, an elastic hypersonic vehicle was taken as the research object.In this study, a new prescribed performance function was designed based on the hyperbolic cosine function, which avoided the singular control problem caused by the improper initial value setting.Thus, the steady-state performance and dynamic performance of the control system could be guaranteed.Meanwhile, active disturbance rejection control was introduced and an ESO was designed for each unknown nonaffine function in the AHV system [32], which further guaranteed the control accuracy and the robustness of the method.To address the complexity of the derivative of the virtual control law, a track differentiator was applied to estimate the related signals and signal derivatives.The effectiveness and superiority of the proposed method was verified through the simulation and comparison.International Journal of Aerospace Engineering combined with Hooke's law and the Lagrange equation to establish the following control-oriented AHV longitudinal motion parameter fitting model: Equations ( 1)-( 5) describe the rigid body part of the AHV.The five rigid states are the velocity V, altitude h, flight-path angle γ, pitch angle θ, and the pitch rate Q. m is the mass of the AHV, g is the gravitational acceleration constant, and I yy is the moment of inertia of the AHV.The force condition of the AHV is shown in Figure 1.The parameter fitting form of the thrust T, drag D, lift L, pitching moment M, and generalized force N i ði = 1, 2Þ can be expressed as follows [35]: with where the attack angle α = θ − γ, the fuel equivalence ratio Φ, and the elevator angular deflection δ e are control inputs, S and c are the reference area and aerodynamic chord of the AHV, respectively, z T is the thrust moment arm, c e is the coefficient of δ e in M, h 0 and ρ 0 are the nominal altitude and corresponding air density, respectively, 1/h s is the air density decay rate, q and ρ are dynamic pressure and air density at h, respectively, 2 is the contribution of δ e to N 2 , and β i ðh, qÞ is the ith thrust fit parameter.
Equations ( 6) and ( 7) describe the elastic body part of the AHV.The elastic states are η 1 and η 2 .ζ i and ω i ði = 1, 2Þ are the damping ratio and vibrational frequency of the AHV elastic state.k i and ψi ði = 1, 2Þ can be expressed as follows: where L f and L a are the front and rear beam length of the AHV, respectively, mf and ma are the mass distributions of the front and the rear beams, respectively, and ϕ f ð•Þ and ϕ a ð•Þ are the vibration mode functions of the front and rear beam, respectively.The specific values of the model 3 International Journal of Aerospace Engineering parameters and aerodynamic parameters of the abovementioned AHV are shown in [35].
2.2.Prescribed Performance.The AHV prescribed performance control method based on the nonaffine model is studied in this paper.To make the tracking error convergence processes meet the desired dynamic and steady-state performance, the following new performance function pðtÞ is designed to limit the tracking error.
where k p , χ p , p ∞ ∈ R + are parameters to be designed.pðtÞ has following properties: (1) pðtÞ is a positive monotonically decreasing function The prescribed performance is defined as follows: where eðtÞ is the tracking error.By choosing a small enough χ p , it is guaranteed that pðtÞ → +∞ and −pðtÞ → −∞ based on the property (3).Therefore, for any unknown but bounded eð0Þ, the following inequality always holds: The prescribed performance defined by Equation ( 12) is shown in Figure 2. p ∞ is the upper bound of the eðtÞ steady-state value, which means −p ∞ < eð∞Þ < p ∞ .Therefore, the desired steady-state accuracy of eðtÞ can be guaranteed by choosing an appropriate p ∞ .pð0Þ is the maximum overshoot allowed by eðtÞ.k p directly influences the decreasing rate of pðtÞ.With the increase in k p , pðtÞ decreases more rapidly.
Remark 1.For any arbitrary bounded eð0Þ whether it is known or not, as long as the χ p is chosen to be small enough, eð0Þ must be included in the prescribed area defined by Equation (12), which avoids the singular control problem caused by the improper initial value setting of the traditional performance function [36].
Remark 2. When the χ p is chosen to be small, the value of pð0Þ will be very large, which can lead to too large of an overshoot of eðtÞ.However, because the response speed of e ðtÞ is limited, the desired dynamic performance, including the overshoot and setting time, can be guaranteed by choosing a larger k p .This suggests that if the tracking error eðtÞ satisfies Equation ( 12), the overshoot and the adjusting time can be constrained within a certain range, as shown in Figure 2. Thus, by designing the controller which can make the tracking error meet the constraint of Equation ( 12), the desired dynamic performance of the control system can be guaranteed.
The transformed error εðtÞ is defined as follows: where λðtÞ = eðtÞ/pðtÞ.The following theorem can be obtained.
Remark 3. The control law below will be designed based on the transformed error εðtÞ.Theorem 1 shows that as long as εðtÞ is bounded, eðtÞ can be limited to the prescribed area defined by Equation (12).By choosing appropriate design parameters for pðtÞ, the desired dynamic performance and steady-state accuracy of eðtÞ can be guaranteed.

Model Conversion and Control
Objective.On the one hand, since the thrust T is directly influenced by the fuel equivalence ratio Φ, the velocity V of the AHV is mainly 4 International Journal of Aerospace Engineering controlled by Φ.On the other hand, since the elevator angular deflection δ e directly influences the pitch rate Q, it further changes the pitch rate θ and the flight-path angle and finally influences the change of the altitude.The altitude change of the AHV is mainly controlled by the elevator angular deflection δ e [37,38].Therefore, most previous studies on AHV control issues first divide the AHV model into a velocity subsystem controlled by Φ (Equation ( 1)) and an altitude subsystem controlled by δ e (Equations ( 2)-( 5)) and then derived control laws [39,40].
Based on the previous studies [37,38], the velocity subsystem of the AHV was considered to be a nonaffine form with the control input: where y V is the output of the system, and f V ðV, ΦÞ is a completely unknown continuous differentiable function.For the velocity subsystem, the control goal is to design an appropriate control law Φ based on the nonaffine model (Equation ( 18)).Thus, the robustness tracking of the velocity V to the reference input V ref can be achieved, and the velocity tracking error can be limited to the prescribed area.
Similarly, the altitude subsystem of the AHV can be expressed as the following nonaffine pure feedback system: where y h is the output of the system, and x = ½γ, θ, Q T , f γ ðγ, θÞ, and f Q ðx, δ e Þ are completely unknown continuous differentiable function.For the altitude subsystem, the control goal is to design an appropriate control law δ e .Thus, the stable tracking of the altitude h to the reference input h ref can be achieved and the altitude tracking error can be limited to the prescribed area.Also, the desired dynamic performance and steady-state accuracy can be guaranteed.

Remark 4.
Most previous studies on AHV control issues designed the control law based on the affine model.However, the AHV motion model is nonaffine.If the nonaffine model of the AHV is forcibly simplified to an affine model, the loss of certain key dynamics is inevitably.The designed control law has the risk of partial or complete failure.The proposed control law in this paper will be designed based on the nonaffine model (Equations ( 18) and ( 19)), which guarantees the reliability of the control law.

Extend States Observer and Tracking Differentiator.
To design the active disturbance rejection control law, the extended state observer is applied to estimate the uncertainty of the AHV model and the external disturbance.The following system is considered: where HðtÞ is the unknown term, and U is the input of the subsystem.The state z of the system is measurable.Therefore, the state of the system can be expanded to the following system: where GðtÞ is the unknown derivative of the unknown term HðtÞ.Therefore, the ESO can be established as follows: where E is the estimation error of the ESO, Z 1 and Z 2 are the outputs of the ESO, β 01 > 0 and β 02 > 0 are the observer gains, and parametric function f ci ð•Þði = 1, 2Þ is an appropriately constructed nonlinear function, which satisfies ef ci ðeÞ > 0, ∀e ≠ 0, and f ci ð0Þ = 0.The parametric function f ci ð•Þ in this paper is chosen to have the following form: Theorem 2. Aiming at the system specified by Equation ( 20), if the ESO (Equation ( 22)) is applied, there will be gain parameters β 01 , β 02 > 0 and 0 ≤ α 1 ≤ 1, which makes the output of the ESO Z 1 and Z 2 converge to the actual state z and a compact set of the unknown term HðtÞ, respectively.Bounded constants σ 1 , σ 2 > 0 exist, which make: Proof.The proof process is shown in the appendix.
Some signals in the control law design process are often difficult to obtain by the model construction.Many scholars have proposed using the tracking differentiator to estimate the signal [41].The design principle is to achieve the highest precision extraction of the differential signal and to ensure certain robustness to the input noise of the signal.A new TD is proposed in this paper to estimate the differential signal.The specific form of the new TD is as follows: where ν is the input signal to be estimated, υ 1 is the estimated value of ν, and υ i ði = 2, 3, ⋯, nÞ are the estimated values of the i − 1th derivative of ν, respectively; R is the parameter to be designed.
Remark 5. Compared with traditional tracking differentiators [41], the new TD proposed in this paper has two advantages.First, the structure is simple and can estimate the arbitrary derivative of the input signal.Second, the new TD has only one parameter R to be designed, and the parameter adjustment process is easier.

Controller Design
3.1.Velocity Controller Design.The velocity subsystem (Equation ( 18)) is assumed to be affected by external disturbances.According to the idea of self-interference, it can be expressed as follows: where d V ðtÞ is the external disturbance, F V ðV, ΦÞ = f V ðV, ΦÞ + d V ðtÞ − l V Φ is the unknown term, and l V > 0 is the parameter to be designed.The velocity tracking error can be defined as follows: The first derivative of Equation ( 28) with respect to time is obtained, and Equation ( 27) is substituted into the result, yielding the following: According to Equation ( 14), the velocity transformed error can be defined as follows: where p V ðtÞ = coth ðk pV t + χ pV Þ − 1 + p V∞ , and k pV , χ pV , p V∞ ∈ R + are parameters to be designed.
The first derivative of Equation ( 30) with respect to time is obtained, and Equation ( 29) is substituted into the result, yielding the following: where To estimate the unknown term F V ðV, ΦÞ, the following ESO is designed for Equation (31): where β V1 , β V2 > 0, 0 ≤ α V ≤ 1 is the parameter to be designed, Z V1 is the estimated value of ε V , and Z V2 /r V is the estimated value of the unknown term F V ðV, ΦÞ.
The active disturbance rejection control law Φ can be designed as follows: where k V > 0 is the parameter to be designed.

Theorem 4.
Considering the velocity subsystem (Equation ( 18)), if the active disturbance rejection control law (Equation ( 33)) and the ESO (Equation ( 32)) are applied, the closed-loop control system is semiglobally uniformly asymptotically stable, and the speed tracking error is limited to the prescribed area.

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Proof.Substituting Equation (33) into Equation (31) yields: The following Lyapunov function was selected: The first derivative of W V with respect to time is obtained, and Equation ( 34) is substituted into the result, yielding the following: Combined with Equation ( 24), a constant ϖ V exists such that: Considering that r V > 0, Equation ( 37) is substituted into Equation (36), yielding: With k V > 1/2, the following compact set is defined: Combining Equations ( 38) and (39), if ε V ∉ Ω ε V , then _ W V < 0. Therefore, the closed-loop control system is semiglobally uniformly asymptotically stable, and the velocity transformed error will finally converge to the compact set Ω ε V .If the estimated accuracy of the ESO is high enough (ϖ V is small enough) and the k V is sufficiently large, the radius of the compact Ω ε V and the ε V can be sufficiently small.
The above proves that the transformed error is bounded.According to Theorem 1, −p V ðtÞ < Ṽ < p V ðtÞ.The velocity transformed error is limited to the prescribed area.

Altitude Controller Design.
Considering the altitude subsystem (Equation ( 19)), the altitude tracking error can be defined as h = h − h ref .The altitude transformed error can be defined as follows: where p h ðtÞ = coth ðk ph t + χ ph Þ − 1 + p h∞ , and k ph , χ ph , p h∞ ∈ R + are parameters to be designed.Furthermore, the first derivative of ε h ðtÞ with respect to time yields the following: where The flight-path angle command is defined as follows: where k h1 , k h2 > 0 are parameters to be designed.
If γ → γ d , γ in Equation ( 41) is replaced with γ d and a Laplace transform is applied, yielding: The two characteristics roots of Equation ( 43), Þ/2, are negative real numbers.Therefore, the altitude transformed error ε h ðtÞ is convergent and bounded.For the altitude subsystem (Equation ( 19)), the control goal can be achieved through the control law δ e design which makes γ → γ d [43].
For the rest of the altitude subsystem (Equation ( 19)), the active disturbance rejection control law of δ e is designed in three steps.
Combined with the concept of active disturbance rejection, _ γ = f γ ðγ, θÞ can be expressed as follows: where d γ ðtÞ is the external disturbance, F γ ðγ, θÞ = f γ ðγ, θÞ + d γ ðtÞ − l γ θ, and l γ > 0 is a parameter to be designed.The flight-path angle tracking error is defined as follows: The first derivative of Equation ( 45) is obtained, and Equation ( 44) is substituted into the result, yielding: The mathematical expression of _ γ d is complicated.Thus, to avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, the new TD proposed in this paper is applied to estimate _ γ d .
where R γ > 0 is the parameter to be designed, and υ γ1 and υ γ2 are the estimated values of γ d and _ γ d , respectively.To estimate the unknown term F γ ðγ, θÞ, the following ESO is designed for Equation ( 46): where β γ1 , β γ2 > 0, 0 ≤ α γ ≤ 1 is the parameter to be designed, and Z γ1 and Z γ2 are the estimated values of γ and F γ ðγ, θÞ, respectively.The pitch angle command θ d can be designed as follows: where k γ > 0 is the parameter to be designed.
Step 2. For _ θ = Q, the pitch rate command Q d is designed.
The pitch angle tracking error is defined as follows: If the AHV is affected by an external disturbance, _ θ = Q can be expressed as follows: where d θ ðtÞ is the unknown external disturbance.
The first derivative of θ with respect to time is obtained, and Equation ( 51) is substituted into the result, yielding: To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, the new TD proposed in this paper is applied to estimate _ θ d : where R θ > 0 is the parameter to be designed, and υ θ1 and υ θ2 are the estimated values of θ d and _ θ d , respectively.To estimate the unknown term d θ ðtÞ, the following ESO is designed for Equation ( 52): where β θ1 , β θ2 > 0, 0 ≤ α θ ≤ 1 is the parameter to be designed, Z θ1 is the estimated value of θ, and Z θ2 is the estimated value of the unknown term d θ ðtÞ.
The pitch rate command Q d can be designed as follows: where k θ > 0 is the parameter to be designed.
The pitch rate tracking error is defined as follows: If the AHV is affected by an external disturbance, combined with the concept of active disturbance rejection, the equation _ Q = f Q ðx, δ e Þ can be expressed as follows: where is the unknown term, and l Q > 0 is the parameter to be designed.The first derivative of Q with respect to time is obtained, and Equation ( 57) is substituted into the result, yielding: The following new TD is applied to estimate _ Q d : where R Q > 0 is the parameter to be designed, and υ Q1 and υ Q2 are the estimated values of Q d and _ Q d , respectively.

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To estimate the unknown term F Q ðx, δ e Þ, the following ESO is designed for Equation ( 58): where β Q1 , β Q2 > 0, 0 ≤ α Q ≤ 1 is the parameter to be designed, Z Q1 is the estimated value of Q, and Z Q2 is the estimated value of the unknown term The control law δ e can be designed as follows: where l Q > 0 is the parameter to be designed.
Theorem 5. Considering the altitude subsystem (Equation ( 19)), if the active disturbance rejection control law δ e (Equation (61)), the ESO (Equation ( 60)), and the new TD are applied, the closed-loop control system is semiglobally uniformly asymptotically stable.The tracking errors ε h , γ, θ, and Q are bounded, and the altitude tracking error h is limited to the prescribed area.The following inequality holds −p h ðtÞ < h < p h ðtÞ.

Simulation Results
Taking the longitudinal motion model of the AHV (Equations (1)-( 10)) as the controlled object, the tracking simulations of speed and altitude reference input were carried out.The simulations were solved using the fourth-order Runge-Kutta method, and the simulation step was taken to 9 International Journal of Aerospace Engineering be 0.01 s.The initial values of the AHV state variables are shown in Table 1.
The velocity reference command V ref and the altitude reference command h ref are specified by the following second-order system: where the damping ratio ζ n = 0:9, the natural frequency ω n = 0:1rad/s, and V I and h I are the input signals of the second-order system.When the control algorithm proposed in this paper was used for the simulation, the prescribed performance parameters were selected as follows: k pV = 0:1, χ pV = 0:35, p V∞ = 0:08, k ph = 0:1, χ ph = 1, and p h∞ = 0:01.The ADRC controller parameters were selected as follows: The designed parameters of the new TD were selected as follows: R γ = 5, R θ = 2, and R Q = 2.The designed parameters of the ESO were selected as follows: , and α Q = 0:4.Simulations were carried out in the following two scenarios.Scenario 1.The prescribed performance-based active disturbance rejection control (PP-ADRC) method proposed in this paper with the robust back-stepping control method (RBC) from a previous study [44] was compared.The simulation time was set to 100 s.V I and h I were selected as a step signal with an amplitude of 200 m/s.To verify the robustness of the proposed method, it was assumed that a 40% perturbation of the aerodynamic parameters existed in the AHV model, which is expressed as C = C 0 ½1 + 0:4 sin ð0:1πtÞ.C 0 represents the nominal value of the aerodynamic parameter and C represents the value of the aerodynamic parameter in the simulation.In addition, an external disturbance was added after 50 s of simulation: The simulation results for Scenario 1 are shown in Figure 3. Figures 3(a)-3(d) show that the velocity and altitude tracking errors were limited to the prescribed area when using the PP-ADRC.Compared with the RBC, the PP-ADRC can guarantee better dynamic performances of the velocity and altitude tracking errors.When there was parameter perturbation and external disturbance, the PP-ADRC exhibited a higher control accuracy and stronger robustness.Figure 3(e) shows that the flight-path angle response of the PP-ADRC was smoother than that of the RBC.Moreover, the PP-ADRC proposed in this paper could estimate the unknown term of the model through the ESO (Figures 3(g) and 3(h)).The virtual reference command and the differential signals could be effectively estimated through the new TD (Figure 3(f)).Thus, the control accuracy of the method is further guaranteed.Scenario 2. The PP-ADRC was compared with the neural back-stepping control method (NBC) from a previous study [45].The simulation time was set to 300 s.To better reflect the actual manoeuvre of the AHV, V I was assumed to be a "step" type signal with a step of 150 m/s per 100 s. h I was assumed to be a square wave signal with a amplitude of 200 m and a period of 200 s.Also, it was assumed that the 40% perturbation of the aerodynamic parameters existed in the AHV model.The following definitions were made: where C 0 represents the nominal value of the aerodynamic parameter, and C represents the value of the aerodynamic parameter in the simulation.In addition, external disturbances were added after 150 s of simulation: The simulation results of the Scenario 2 are shown in Figure 4. Figures 4(a)-4(d) show that the dynamic performances and control accuracy of the velocity and altitude tracking errors of the PP-ADRC were better than those of the NBC. Figure 4(e) shows that the flight-path angle response of the PP-ADRC was smoother than that of the NBC.Meanwhile, the PP-ADRC proposed in this paper could estimate the unknown term of the model through the ESO (Figures 4(g) and 4(h)).The virtual reference command and the differential signals could be effectively estimated through the new TD (Figure 4(f)).Thus, the control accuracy of the method was further guaranteed.Figures 4(i)-4(l) show that both of the control methods could achieve the effective suppression of the elastic vibrations.

Conclusions
(1) An active disturbance rejection control for an AHV based on the prescribed performance function is       16 International Journal of Aerospace Engineering proposed in this paper.The proposed method guarantees the stability of the AHV closed-loop control system.The desired dynamic and steady-state performances of the convergence process of the tracking error were ensured (2) In the controller design process, the adopted active disturbance rejection method and extended state observer further enhanced the capacity to resist the disturbances, which guaranteed the robustness of the method (3) The simulation results in the paper proved the effectiveness of the proposed method.The comparison with the related publications showed that the dynamic and steady-state performances of the proposed method were superior Appendix A.Proof of Theorem 2 Based on Equations ( 20)-( 22), we set E * = Z 2 − HðtÞ = Z 2 − z 0 .The first derivatives of E and E * with respect to time can be expressed as follows: The following Lyapunov function was selected: ðA:3Þ V is positive semidefinite.Below, _ V is studied.The partial derivative of V with respect to E and E * can be expressed as follows: The first derivative of V with respect to time was obtained, and with Equations (70)-(A.1),this yields the following: ðA:6Þ The following compact set is defined as follows: According to Equation (A.4), if E ∉ Ω E or E * ∉ Ω E * , then _ V < 0. V will decrease until E and E * converge to the compact sets Ω E and Ω E * , respectively.Thus, the error system (Equation (70)) is semiglobally uniformly asymptotically stable.E and E * are uniformly asymptotically bounded.However, if the radii of the compact sets Ω E and Ω E * can be made sufficiently small by selecting appropriate values of β 01 , β 02 , and α 1 , the errors E and E * will be sufficiently small, and there must be bounded constants σ 1 , σ 2 > 0 such that: ( ðA: 8Þ ,

Table 1 :
Initial values of the AHV state variables.