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

Authors: Chenyang Xu; Humin Lei; Na Lu
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

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.


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## Body

## 1. 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 [1–3]. 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 H2/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 multiple-input 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 in-depth 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 inner-loop 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.


## 2. AHV Model and Preliminaries

### 2.1. Model Description

To better describe the longitudinal motion of the AHV, American scholar Parker used the research conclusions of Bolender and Doman [33, 34] combined with Hooke’s law and the Lagrange equation to establish the following control-oriented AHV longitudinal motion parameter fitting model:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy.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 Iyy 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 Nii=1,2 can be expressed as follows [35]:
(8)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,with
(9)CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=ρ¯V22,ρ¯=ρ¯0exph0−hhs,where the attack angle α=θ−γ, the fuel equivalence ratio Φ, and the elevator angular deflection δe are control inputs, S and c¯ are the reference area and aerodynamic chord of the AHV, respectively, zT is the thrust moment arm, ce is the coefficient of δe in M, h0 and ρ0 are the nominal altitude and corresponding air density, respectively, 1/hs is the air density decay rate, q¯ and ρ¯ are dynamic pressure and air density at h, respectively, C∗•i·=α,δe;∗=T,D is the ith order coefficient of · in ∗, C∗0∗=T,D,L is the constant coefficient in ∗, CL∗∗=α,δe is the contribution of the coefficient of ∗ in L, CM,ααi is the ith order coefficient of α in M, CM,α0 is the constant coefficient in M, Njαi is the jth order contribution of α to Nj, Ni0 is the constant term in Ni, N2δe is the contribution of δe to N2, and βih,q¯ is the ith thrust fit parameter.Figure 1
AHV force diagram.Equations (6) and (7) describe the elastic body part of the AHV. The elastic states are η1 and η2. ζi and ωii=1,2 are the damping ratio and vibrational frequency of the AHV elastic state. ki and ψ~ii=1,2 can be expressed as follows:
(10)k1=1+ψ~1Iyy,k2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ,where Lf and La are the front and rear beam length of the AHV, respectively, m^f and m^a are the mass distributions of the front and the rear beams, respectively, and ϕf· and ϕa· are the vibration mode functions of the front and rear beam, respectively. The specific values of the model 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 functionpt is designed to limit the tracking error.
(11)pt=cothkpt+χp−1+p∞,where kp,χp,p∞∈R+ are parameters to be designed.pt has following properties:
(1)
pt is a positive monotonically decreasing function
(2)
p0=cothχp−1+p∞=e2χp+1/e2χp−1−1+p∞>p∞
(3)
limχp→0p0→+∞
(4)
limt→+∞pt=p∞The prescribed performance is defined as follows:(12)−pt<et<pt,where et is the tracking error. By choosing a small enough χp, it is guaranteed that pt→+∞ and −pt→−∞ based on the property (3). Therefore, for any unknown but bounded e0, the following inequality always holds:
(13)−p0<e0<p0.The prescribed performance defined by Equation (12) is shown in Figure 2. p∞ is the upper bound of the et steady-state value, which means −p∞<e∞<p∞. Therefore, the desired steady-state accuracy of et can be guaranteed by choosing an appropriate p∞. p0 is the maximum overshoot allowed by et. kp directly influences the decreasing rate of pt. With the increase in kp, pt decreases more rapidly.Remark 1.
For any arbitrary boundede0 whether it is known or not, as long as the χp is chosen to be small enough, e0 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 theχp is chosen to be small, the value of p0 will be very large, which can lead to too large of an overshoot of et. However, because the response speed of et is limited, the desired dynamic performance, including the overshoot and setting time, can be guaranteed by choosing a larger kp. This suggests that if the tracking error et 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.Figure 2
Prescribed performance defined by Equation (12).The transformed errorεt is defined as follows:
(14)εt=lnλt+11−λt,where λt=et/pt. The following theorem can be obtained.Theorem 1.
Ifεt is bounded, then −pt<et<pt.Proof.
Becauseεt is bounded, there must be a bounded constant εM∈R+ that makes εt≤εM. Furthermore, the inverse transformation of Equation (14) is as follows:
(15)eεt=λt+11−λt.
Based on Equation (15),
(16)−1<e−εM−11+e−εM≤λt≤eεM−11+eεM<1.
Substitutingλt=et/pt into Equation (16) yields
(17)−pt<et<pt.Therefore, Theorem1 is established.Remark 3.
The control law below will be designed based on the transformed errorεt. Theorem 1 shows that as long as εt is bounded, et can be limited to the prescribed area defined by Equation (12). By choosing appropriate design parameters for pt, the desired dynamic performance and steady-state accuracy of et can be guaranteed.


### 2.3. Model Conversion and Control Objective

On the one hand, since the thrustT is directly influenced by the fuel equivalence ratio Φ, the velocity V of the AHV is mainly controlled by Φ. On the other hand, since the elevator angular deflection δe 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 δe [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 δe (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:
(18)V̇=fVV,Φ,yV=V,where yV is the output of the system, and fVV,Φ 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 Vref 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:(19)ḣ=Vsinγ,γ̇=fγγ,θ,θ̇=Q,Q̇=fQx,δe,yh=h,where yh is the output of the system, and x=γ,θ,QT, fγγ,θ, and fQx,δe are completely unknown continuous differentiable function. For the altitude subsystem, the control goal is to design an appropriate control law δe. Thus, the stable tracking of the altitude h to the reference input href 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.


### 2.4. 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:(20)ż=Ht+BU,where Ht 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:
(21)ż=z0+BU,ż0=Gt,where Gt is the unknown derivative of the unknown term Ht. Therefore, the ESO can be established as follows:
(22)E=Z1−z,Ż1=Z2−β01fc1E+BU,Ż2=−β02fc2E,where E is the estimation error of the ESO, Z1 and Z2 are the outputs of the ESO, β01>0 and β02>0 are the observer gains, and parametric function fci·i=1,2 is an appropriately constructed nonlinear function, which satisfies efcie>0, ∀e≠0, and fci0=0. The parametric function fci· in this paper is chosen to have the following form:
(23)fc1E=E,fc2E=Eα1sgnE,where 0≤α1≤1.Theorem 2.
Aiming at the system specified by Equation (20), if the ESO (Equation (22)) is applied, there will be gain parameters β01,β02>0 and 0≤α1≤1, which makes the output of the ESO Z1 and Z2 converge to the actual state z and a compact set of the unknown term Ht, respectively. Bounded constants σ1,σ2>0 exist, which make:
(24)Z1−z≤σ1,Z2−Ht≤σ2.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:
(25)υ̇1=υ2,υ̇2=υ3,⋮υ̇n=Rnν−υ1−υ2R−υ3R2−⋯−υnRn−1,where ν is the input signal to be estimated, υ1 is the estimated value of ν, and υii=2,3,⋯,n are the estimated values of the i−1th derivative of ν, respectively; R is the parameter to be designed.Theorem 3.
If the new TD (Equation (25)) is applied to estimate the input signal ν and the derivatives, there exists an R>0 that makes υii=1,2,⋯,n converge to a compact set of νnn=0,1,⋯,n−1, respectively. Bounded constants λ¯i>0i=1,2,⋯,n exist, which make:
(26)υ1−ν≤λ¯1,υ2−ν̇≤λ¯2,⋮υn−νn−1≤λ¯n.Proof.
The certification process is detailed elsewhere [42].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.


## 2.1. Model Description

To better describe the longitudinal motion of the AHV, American scholar Parker used the research conclusions of Bolender and Doman [33, 34] combined with Hooke’s law and the Lagrange equation to establish the following control-oriented AHV longitudinal motion parameter fitting model:
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gVcosγ,(4)θ̇=Q,(5)Q̇=M+ψ~1η¨1+ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−ψ~1MIyy−ψ~1ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−ψ~2MIyy−ψ~2ψ~1η¨1Iyy.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 Iyy 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 Nii=1,2 can be expressed as follows [35]:
(8)T≈CTα3α3+CTα2α2+CTαα+CT0,D≈q¯SCDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0,L≈q¯SCLαα+CLδeδe+CL0,M≈zTT+q¯Sc¯CM,αα2α2+CM,ααα+CM,α0+ceδe,N1≈N1α2α2+N1αα+N10,N2≈N2α2α2+N2αα+N2δeδe+N20,with
(9)CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯,q¯=ρ¯V22,ρ¯=ρ¯0exph0−hhs,where the attack angle α=θ−γ, the fuel equivalence ratio Φ, and the elevator angular deflection δe are control inputs, S and c¯ are the reference area and aerodynamic chord of the AHV, respectively, zT is the thrust moment arm, ce is the coefficient of δe in M, h0 and ρ0 are the nominal altitude and corresponding air density, respectively, 1/hs is the air density decay rate, q¯ and ρ¯ are dynamic pressure and air density at h, respectively, C∗•i·=α,δe;∗=T,D is the ith order coefficient of · in ∗, C∗0∗=T,D,L is the constant coefficient in ∗, CL∗∗=α,δe is the contribution of the coefficient of ∗ in L, CM,ααi is the ith order coefficient of α in M, CM,α0 is the constant coefficient in M, Njαi is the jth order contribution of α to Nj, Ni0 is the constant term in Ni, N2δe is the contribution of δe to N2, and βih,q¯ is the ith thrust fit parameter.Figure 1
AHV force diagram.Equations (6) and (7) describe the elastic body part of the AHV. The elastic states are η1 and η2. ζi and ωii=1,2 are the damping ratio and vibrational frequency of the AHV elastic state. ki and ψ~ii=1,2 can be expressed as follows:
(10)k1=1+ψ~1Iyy,k2=1+ψ~2Iyy,ψ~1=∫−Lf0m^fξϕfξdξ,ψ~2=∫0Lam^aξϕaξdξ,where Lf and La are the front and rear beam length of the AHV, respectively, m^f and m^a are the mass distributions of the front and the rear beams, respectively, and ϕf· and ϕa· are the vibration mode functions of the front and rear beam, respectively. The specific values of the model 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 functionpt is designed to limit the tracking error.
(11)pt=cothkpt+χp−1+p∞,where kp,χp,p∞∈R+ are parameters to be designed.pt has following properties:
(1)
pt is a positive monotonically decreasing function
(2)
p0=cothχp−1+p∞=e2χp+1/e2χp−1−1+p∞>p∞
(3)
limχp→0p0→+∞
(4)
limt→+∞pt=p∞The prescribed performance is defined as follows:(12)−pt<et<pt,where et is the tracking error. By choosing a small enough χp, it is guaranteed that pt→+∞ and −pt→−∞ based on the property (3). Therefore, for any unknown but bounded e0, the following inequality always holds:
(13)−p0<e0<p0.The prescribed performance defined by Equation (12) is shown in Figure 2. p∞ is the upper bound of the et steady-state value, which means −p∞<e∞<p∞. Therefore, the desired steady-state accuracy of et can be guaranteed by choosing an appropriate p∞. p0 is the maximum overshoot allowed by et. kp directly influences the decreasing rate of pt. With the increase in kp, pt decreases more rapidly.Remark 1.
For any arbitrary boundede0 whether it is known or not, as long as the χp is chosen to be small enough, e0 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 theχp is chosen to be small, the value of p0 will be very large, which can lead to too large of an overshoot of et. However, because the response speed of et is limited, the desired dynamic performance, including the overshoot and setting time, can be guaranteed by choosing a larger kp. This suggests that if the tracking error et 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.Figure 2
Prescribed performance defined by Equation (12).The transformed errorεt is defined as follows:
(14)εt=lnλt+11−λt,where λt=et/pt. The following theorem can be obtained.Theorem 1.
Ifεt is bounded, then −pt<et<pt.Proof.
Becauseεt is bounded, there must be a bounded constant εM∈R+ that makes εt≤εM. Furthermore, the inverse transformation of Equation (14) is as follows:
(15)eεt=λt+11−λt.
Based on Equation (15),
(16)−1<e−εM−11+e−εM≤λt≤eεM−11+eεM<1.
Substitutingλt=et/pt into Equation (16) yields
(17)−pt<et<pt.Therefore, Theorem1 is established.Remark 3.
The control law below will be designed based on the transformed errorεt. Theorem 1 shows that as long as εt is bounded, et can be limited to the prescribed area defined by Equation (12). By choosing appropriate design parameters for pt, the desired dynamic performance and steady-state accuracy of et can be guaranteed.


## 2.3. Model Conversion and Control Objective

On the one hand, since the thrustT is directly influenced by the fuel equivalence ratio Φ, the velocity V of the AHV is mainly controlled by Φ. On the other hand, since the elevator angular deflection δe 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 δe [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 δe (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:
(18)V̇=fVV,Φ,yV=V,where yV is the output of the system, and fVV,Φ 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 Vref 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:(19)ḣ=Vsinγ,γ̇=fγγ,θ,θ̇=Q,Q̇=fQx,δe,yh=h,where yh is the output of the system, and x=γ,θ,QT, fγγ,θ, and fQx,δe are completely unknown continuous differentiable function. For the altitude subsystem, the control goal is to design an appropriate control law δe. Thus, the stable tracking of the altitude h to the reference input href 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.


## 2.4. 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:(20)ż=Ht+BU,where Ht 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:
(21)ż=z0+BU,ż0=Gt,where Gt is the unknown derivative of the unknown term Ht. Therefore, the ESO can be established as follows:
(22)E=Z1−z,Ż1=Z2−β01fc1E+BU,Ż2=−β02fc2E,where E is the estimation error of the ESO, Z1 and Z2 are the outputs of the ESO, β01>0 and β02>0 are the observer gains, and parametric function fci·i=1,2 is an appropriately constructed nonlinear function, which satisfies efcie>0, ∀e≠0, and fci0=0. The parametric function fci· in this paper is chosen to have the following form:
(23)fc1E=E,fc2E=Eα1sgnE,where 0≤α1≤1.Theorem 2.
Aiming at the system specified by Equation (20), if the ESO (Equation (22)) is applied, there will be gain parameters β01,β02>0 and 0≤α1≤1, which makes the output of the ESO Z1 and Z2 converge to the actual state z and a compact set of the unknown term Ht, respectively. Bounded constants σ1,σ2>0 exist, which make:
(24)Z1−z≤σ1,Z2−Ht≤σ2.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:
(25)υ̇1=υ2,υ̇2=υ3,⋮υ̇n=Rnν−υ1−υ2R−υ3R2−⋯−υnRn−1,where ν is the input signal to be estimated, υ1 is the estimated value of ν, and υii=2,3,⋯,n are the estimated values of the i−1th derivative of ν, respectively; R is the parameter to be designed.Theorem 3.
If the new TD (Equation (25)) is applied to estimate the input signal ν and the derivatives, there exists an R>0 that makes υii=1,2,⋯,n converge to a compact set of νnn=0,1,⋯,n−1, respectively. Bounded constants λ¯i>0i=1,2,⋯,n exist, which make:
(26)υ1−ν≤λ¯1,υ2−ν̇≤λ¯2,⋮υn−νn−1≤λ¯n.Proof.
The certification process is detailed elsewhere [42].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.


## 3. 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:
(27)V̇=fVV,Φ+dVt−lVΦ+lVΦ=FVV,Φ+lVΦ,yV=V,where dVt is the external disturbance, FVV,Φ=fVV,Φ+dVt−lVΦ is the unknown term, and lV>0 is the parameter to be designed.The velocity tracking error can be defined as follows:(28)V~=V−Vref.The first derivative of Equation (28) with respect to time is obtained, and Equation (27) is substituted into the result, yielding the following:
(29)V~̇=FVV,Φ−V̇ref+lVΦ.According to Equation (14), the velocity transformed error can be defined as follows:
(30)εVt=lnV~/pVt+11−V~/pVt,where pVt=cothkpVt+χpV−1+pV∞, and kpV,χpV,pV∞∈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:
(31)ε̇Vt=rVV~̇−ṗVtpVtV~=rVFVV,Φ−V̇ref+lVΦ−ṗVtpVtV~,where
(32)rV=1pVt1V~/pVt+1−1V~/pVt−1>0,ṗVt=kpV1−coth2kpVt+χpV.To estimate the unknown termFVV,Φ, the following ESO is designed for Equation (31):
(33)EV=ZV1−εV,ŻV1=ZV2−βV1EV+rVlVΦ−V̇ref−ṗVtpVtV~,ŻV2=−βV2EVαVsgnEV,where βV1,βV2>0, 0≤αV≤1 is the parameter to be designed, ZV1 is the estimated value of εV, and ZV2/rV is the estimated value of the unknown term FVV,Φ.The active disturbance rejection control lawΦ can be designed as follows:
(34)Φ=−1lVZV2rV+kVεV−V̇ref−ṗVtpVtV~,where kV>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. The following inequality holds −pVt<V~<pVt.Proof.
Substituting Equation (33) into Equation (31) yields:
(35)ε̇V=rVFVV,Φ−ZV2rV−kVεV.The following Lyapunov function was selected:(36)WV=12εV2.The first derivative ofWV with respect to time is obtained, and Equation (34) is substituted into the result, yielding the following:
(37)ẆV=εVrVFVV,Φ−ZV2rV−kVεV=rVεVFVV,Φ−ZV2rV−kVεV2.Combined with Equation (24), a constant ϖV exists such that:
(38)FVV,Φ−ZV2rV≤ϖV.Considering thatrV>0, Equation (37) is substituted into Equation (36), yielding:
(39)ẆV≤rV−kVεV2+εVϖV≤rV−kV−12εV2+12ϖV2.WithkV>1/2, the following compact set is defined:
(40)ΩεV=εVεV≤1/2ϖV2kV−1/2.Combining Equations (38) and (39), if εV∉ΩεV, then ẆV<0. Therefore, the closed-loop control system is semiglobally uniformly asymptotically stable, and the velocity transformed error will finally converge to the compact set ΩεV. If the estimated accuracy of the ESO is high enough (ϖV is small enough) and the kV is sufficiently large, the radius of the compact ΩεV and the εV can be sufficiently small.The above proves that the transformed error is bounded. According to Theorem1, −pVt<V~<pVt. The velocity transformed error is limited to the prescribed area.


### 3.2. Altitude Controller Design

Considering the altitude subsystem (Equation (19)), the altitude tracking error can be defined as h~=h−href. The altitude transformed error can be defined as follows:
(41)εht=lnh~/pht+11−h~/pht,where pht=cothkpht+χph−1+ph∞, and kph,χph,ph∞∈R+ are parameters to be designed.Furthermore, the first derivative ofεht with respect to time yields the following:
(42)ε̇ht=rhh~̇−ṗhtphth~=rhVsinγ−ḣref−ṗhtphth~,where
(43)rh=1pht1h~/pht+1−1h~/pht−1>0,ṗht=kph1−coth2kpht+χph.The flight-path angle command is defined as follows:(44)γd=arcsin−kh1εh−kh2∫0tεhdτ+ḣref+ṗhth~/phtV,where kh1,kh2>0 are parameters to be designed.Ifγ→γd, γ in Equation (41) is replaced with γd and a Laplace transform is applied, yielding:
(45)s2+kh1s+kh2=0.The two characteristics roots of Equation (43), −kh1−kh12−4kh2/2 and −kh1+kh12−4kh2/2, are negative real numbers. Therefore, the altitude transformed error εht is convergent and bounded. For the altitude subsystem (Equation (19)), the control goal can be achieved through the control law δe design which makes γ→γd [43].For the rest of the altitude subsystem (Equation (19)), the active disturbance rejection control law of δe is designed in three steps.Step 1.
Forγ̇=fγγ,θ, the pitch angle command θd is designed.Combined with the concept of active disturbance rejection,γ̇=fγγ,θ can be expressed as follows:
(46)γ̇=fγγ,θ+dγt−lγθ+lγθ=Fγγ,θ+lγθ,where dγt is the external disturbance, Fγγ,θ=fγγ,θ+dγt−lγθ, and lγ>0 is a parameter to be designed.The flight-path angle tracking error is defined as follows:(47)γ~=γ−γd.The first derivative of Equation (45) is obtained, and Equation (44) is substituted into the result, yielding:
(48)γ~̇=γ̇−γ̇d=Fγγ,θ+lγθ−γ̇d.The mathematical expression ofγ̇d 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 γ̇d.
(49)υ̇γ1=υγ2,υ̇γ2=Rγ2γd−υγ1−υγ2Rγ,where Rγ>0 is the parameter to be designed, and υγ1 and υγ2 are the estimated values of γd and γ̇d, respectively.To estimate the unknown termFγγ,θ, the following ESO is designed for Equation (46):
(50)Eγ=Zγ1−γ~,Żγ1=Zγ2−βγ1eγ−υγ2+lγθ,Żγ2=−βγ2EγαγsgnEγ,where βγ1,βγ2>0, 0≤αγ≤1 is the parameter to be designed, and Zγ1 and Zγ2 are the estimated values of γ~ and Fγγ,θ, respectively.The pitch angle commandθd can be designed as follows:
(51)θd=−1lγZγ2+kγγ~−υγ2,where kγ>0 is the parameter to be designed.Step 2.
Forθ̇=Q, the pitch rate command Qd is designed.The pitch angle tracking error is defined as follows:(52)θ~=θ−θd.If the AHV is affected by an external disturbance,θ̇=Q can be expressed as follows:
(53)θ̇=Q+dθt,where dθt is the unknown external disturbance.The first derivative ofθ~ with respect to time is obtained, and Equation (51) is substituted into the result, yielding:
(54)θ~̇=θ̇−θ̇d=Q+dθt−θ̇d.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 estimateθ̇d:
(55)υ̇θ1=υθ2υ̇θ2=Rθ2θd−υθ1−υθ2Rθ,where Rθ>0 is the parameter to be designed, and υθ1 and υθ2 are the estimated values of θd and θ̇d, respectively.To estimate the unknown termdθt, the following ESO is designed for Equation (52):
(56)Eθ=Zθ1−θ~,Żθ1=Zθ2−βθ1Eθ−υθ2+Q,Żθ2=−βθ2EθαθsgnEθ,where βθ1,βθ2>0, 0≤αθ≤1 is the parameter to be designed, Zθ1 is the estimated value of θ~, and Zθ2 is the estimated value of the unknown term dθt.The pitch rate commandQd can be designed as follows:
(57)Qd=−Zθ2+kθθ~+lγγ~−υθ2,where kθ>0 is the parameter to be designed.Step 3.
ForQ̇=fQx,δe, the control law δe is designed.The pitch rate tracking error is defined as follows:(58)Q~=Q−Qd.If the AHV is affected by an external disturbance, combined with the concept of active disturbance rejection, the equationQ̇=fQx,δe can be expressed as follows:
(59)Q̇=fQx,δe+dQt−lQδe+lQδe=FQx,δe+lQδe,where dQt is the external disturbance, FQx,δe=fQx,δe+dQt−lQδe is the unknown term, and lQ>0 is the parameter to be designed.The first derivative ofQ~ with respect to time is obtained, and Equation (57) is substituted into the result, yielding:
(60)Q~̇=Q̇−Q̇d=FQx,δe+lQδe−Q̇d.The following new TD is applied to estimateQ̇d:
(61)υ̇Q1=υQ2,υ̇Q2=RQ2Qd−υQ1−υQ2RQ,where RQ>0 is the parameter to be designed, and υQ1 and υQ2 are the estimated values of Qd and Q̇d, respectively.To estimate the unknown termFQx,δe, the following ESO is designed for Equation (58):
(62)EQ=ZQ1−Q~,ŻQ1=ZQ2−βQ1EQ−υQ2+lQδe,ŻQ2=−βQ2EQαQsgnEQ,where βQ1,βQ2>0, 0≤αQ≤1 is the parameter to be designed, ZQ1 is the estimated value of Q~, and ZQ2 is the estimated value of the unknown term FQx,δe.The control lawδe can be designed as follows:
(63)δe=−1lQZQ2+kQQ~−υQ2+θ~,where lQ>0 is the parameter to be designed.Theorem 5.
Considering the altitude subsystem (Equation (19)), if the active disturbance rejection control law δe (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 εh, γ~, θ~, and Q~ are bounded, and the altitude tracking error h~ is limited to the prescribed area. The following inequality holds −pht<h~<pht.Proof.
The following Lyapunov function is selected:(64)Wh=12γ~2+12θ~2+12Q~2.
The first derivative ofWh with respect to time is obtained, and Equations (46), (49), (50), (52), (55), (56), (58), and (61) are substituted into the result, yielding:
(65)Ẇh=γ~γ~̇+θ~θ~̇+Q~Q~̇=γ~Fγγ,θ+lγθ~+θd−γ̇d+θ~Q~+Qd+dθt−θ̇d+Q~FQx,δe+lQδe−Q̇d=−kγγ~2+γ~Fγγ,θ−Zγ2+γ~υγ2−γ̇d+lγθ~γ~−kθθ~2−lγθ~γ~+θ~dθt−Zθ2+θ~υθ2−θ̇d+θ~Q~−kQQ~2−θ~Q~+Q~FQx,δe−ZQ2+Q~υQ2−Q̇d.
According to Equations (24) and (26), bounded constants σγ, σθ, σQ, λ¯γ, λ¯θ, and λ¯Q exist, which make:
(66)Fγγ,θ−Zγ2≤σγ,υγ2−γ̇d≤λ¯γ,dθt−Zθ2≤σθ,υθ2−θ̇d≤λ¯θ,FQx,δe−ZQ2≤σQ,υQ2−Q̇d≤λ¯Q,where
(67)γ~σγ≤γ~22+σγ22,γ~λ¯γ≤γ~22+λ¯γ22,θ~σθ≤θ~22+σθ22,θ~λ¯θ≤θ~22+λ¯θ22,Q~σQ≤Q~22+σQ22,Q~λ¯Q≤Q~22+λ¯Q22.
Combining Equations (63), (64), and (65) yields the following:
(68)Ẇh≤−kγ−1γ~2−kθ−1θ~2−kQ−1Q~2+12σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q2.
Withkγ>1, kθ>1, and kQ>1, the following compact sets are defined:
(69)Ωγ~=γ~γ~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kγ−1,Ωθ~=θ~θ~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kθ−1,ΩQ~=Q~Q~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kQ−1.According to Equation (67), if γ~∉Ωγ~, θ~∉Ωθ~, or Q~∉ΩQ~, Ẇh<0. Wh decreases until γ~, θ~, and Q~ converge to the compact sets Ωγ~, Ωθ~, and ΩQ~, respectively. Therefore, the closed-loop control system is semiglobally uniformly asymptotically stable. γ~, θ~, and Q~ are ultimately bounded. However, if the estimated error of the ESO and new TD are sufficiently small (σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q2 is sufficiently small), the radii of the compact sets Ωγ~, Ωθ~, and ΩQ~ can be sufficiently small, and γ~, θ~, and Q~ can be sufficiently small. If the γ~ is sufficiently small, then γ→γd. According to Equations (40)–(43), the altitude transformed error is bounded. Thus, by Theorem 1, −pht<h~<pht, which means the altitude transformed error is limited to the prescribed area.


## 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:
(27)V̇=fVV,Φ+dVt−lVΦ+lVΦ=FVV,Φ+lVΦ,yV=V,where dVt is the external disturbance, FVV,Φ=fVV,Φ+dVt−lVΦ is the unknown term, and lV>0 is the parameter to be designed.The velocity tracking error can be defined as follows:(28)V~=V−Vref.The first derivative of Equation (28) with respect to time is obtained, and Equation (27) is substituted into the result, yielding the following:
(29)V~̇=FVV,Φ−V̇ref+lVΦ.According to Equation (14), the velocity transformed error can be defined as follows:
(30)εVt=lnV~/pVt+11−V~/pVt,where pVt=cothkpVt+χpV−1+pV∞, and kpV,χpV,pV∞∈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:
(31)ε̇Vt=rVV~̇−ṗVtpVtV~=rVFVV,Φ−V̇ref+lVΦ−ṗVtpVtV~,where
(32)rV=1pVt1V~/pVt+1−1V~/pVt−1>0,ṗVt=kpV1−coth2kpVt+χpV.To estimate the unknown termFVV,Φ, the following ESO is designed for Equation (31):
(33)EV=ZV1−εV,ŻV1=ZV2−βV1EV+rVlVΦ−V̇ref−ṗVtpVtV~,ŻV2=−βV2EVαVsgnEV,where βV1,βV2>0, 0≤αV≤1 is the parameter to be designed, ZV1 is the estimated value of εV, and ZV2/rV is the estimated value of the unknown term FVV,Φ.The active disturbance rejection control lawΦ can be designed as follows:
(34)Φ=−1lVZV2rV+kVεV−V̇ref−ṗVtpVtV~,where kV>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. The following inequality holds −pVt<V~<pVt.Proof.
Substituting Equation (33) into Equation (31) yields:
(35)ε̇V=rVFVV,Φ−ZV2rV−kVεV.The following Lyapunov function was selected:(36)WV=12εV2.The first derivative ofWV with respect to time is obtained, and Equation (34) is substituted into the result, yielding the following:
(37)ẆV=εVrVFVV,Φ−ZV2rV−kVεV=rVεVFVV,Φ−ZV2rV−kVεV2.Combined with Equation (24), a constant ϖV exists such that:
(38)FVV,Φ−ZV2rV≤ϖV.Considering thatrV>0, Equation (37) is substituted into Equation (36), yielding:
(39)ẆV≤rV−kVεV2+εVϖV≤rV−kV−12εV2+12ϖV2.WithkV>1/2, the following compact set is defined:
(40)ΩεV=εVεV≤1/2ϖV2kV−1/2.Combining Equations (38) and (39), if εV∉ΩεV, then ẆV<0. Therefore, the closed-loop control system is semiglobally uniformly asymptotically stable, and the velocity transformed error will finally converge to the compact set ΩεV. If the estimated accuracy of the ESO is high enough (ϖV is small enough) and the kV is sufficiently large, the radius of the compact ΩεV and the εV can be sufficiently small.The above proves that the transformed error is bounded. According to Theorem1, −pVt<V~<pVt. The velocity transformed error is limited to the prescribed area.


## 3.2. Altitude Controller Design

Considering the altitude subsystem (Equation (19)), the altitude tracking error can be defined as h~=h−href. The altitude transformed error can be defined as follows:
(41)εht=lnh~/pht+11−h~/pht,where pht=cothkpht+χph−1+ph∞, and kph,χph,ph∞∈R+ are parameters to be designed.Furthermore, the first derivative ofεht with respect to time yields the following:
(42)ε̇ht=rhh~̇−ṗhtphth~=rhVsinγ−ḣref−ṗhtphth~,where
(43)rh=1pht1h~/pht+1−1h~/pht−1>0,ṗht=kph1−coth2kpht+χph.The flight-path angle command is defined as follows:(44)γd=arcsin−kh1εh−kh2∫0tεhdτ+ḣref+ṗhth~/phtV,where kh1,kh2>0 are parameters to be designed.Ifγ→γd, γ in Equation (41) is replaced with γd and a Laplace transform is applied, yielding:
(45)s2+kh1s+kh2=0.The two characteristics roots of Equation (43), −kh1−kh12−4kh2/2 and −kh1+kh12−4kh2/2, are negative real numbers. Therefore, the altitude transformed error εht is convergent and bounded. For the altitude subsystem (Equation (19)), the control goal can be achieved through the control law δe design which makes γ→γd [43].For the rest of the altitude subsystem (Equation (19)), the active disturbance rejection control law of δe is designed in three steps.Step 1.
Forγ̇=fγγ,θ, the pitch angle command θd is designed.Combined with the concept of active disturbance rejection,γ̇=fγγ,θ can be expressed as follows:
(46)γ̇=fγγ,θ+dγt−lγθ+lγθ=Fγγ,θ+lγθ,where dγt is the external disturbance, Fγγ,θ=fγγ,θ+dγt−lγθ, and lγ>0 is a parameter to be designed.The flight-path angle tracking error is defined as follows:(47)γ~=γ−γd.The first derivative of Equation (45) is obtained, and Equation (44) is substituted into the result, yielding:
(48)γ~̇=γ̇−γ̇d=Fγγ,θ+lγθ−γ̇d.The mathematical expression ofγ̇d 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 γ̇d.
(49)υ̇γ1=υγ2,υ̇γ2=Rγ2γd−υγ1−υγ2Rγ,where Rγ>0 is the parameter to be designed, and υγ1 and υγ2 are the estimated values of γd and γ̇d, respectively.To estimate the unknown termFγγ,θ, the following ESO is designed for Equation (46):
(50)Eγ=Zγ1−γ~,Żγ1=Zγ2−βγ1eγ−υγ2+lγθ,Żγ2=−βγ2EγαγsgnEγ,where βγ1,βγ2>0, 0≤αγ≤1 is the parameter to be designed, and Zγ1 and Zγ2 are the estimated values of γ~ and Fγγ,θ, respectively.The pitch angle commandθd can be designed as follows:
(51)θd=−1lγZγ2+kγγ~−υγ2,where kγ>0 is the parameter to be designed.Step 2.
Forθ̇=Q, the pitch rate command Qd is designed.The pitch angle tracking error is defined as follows:(52)θ~=θ−θd.If the AHV is affected by an external disturbance,θ̇=Q can be expressed as follows:
(53)θ̇=Q+dθt,where dθt is the unknown external disturbance.The first derivative ofθ~ with respect to time is obtained, and Equation (51) is substituted into the result, yielding:
(54)θ~̇=θ̇−θ̇d=Q+dθt−θ̇d.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 estimateθ̇d:
(55)υ̇θ1=υθ2υ̇θ2=Rθ2θd−υθ1−υθ2Rθ,where Rθ>0 is the parameter to be designed, and υθ1 and υθ2 are the estimated values of θd and θ̇d, respectively.To estimate the unknown termdθt, the following ESO is designed for Equation (52):
(56)Eθ=Zθ1−θ~,Żθ1=Zθ2−βθ1Eθ−υθ2+Q,Żθ2=−βθ2EθαθsgnEθ,where βθ1,βθ2>0, 0≤αθ≤1 is the parameter to be designed, Zθ1 is the estimated value of θ~, and Zθ2 is the estimated value of the unknown term dθt.The pitch rate commandQd can be designed as follows:
(57)Qd=−Zθ2+kθθ~+lγγ~−υθ2,where kθ>0 is the parameter to be designed.Step 3.
ForQ̇=fQx,δe, the control law δe is designed.The pitch rate tracking error is defined as follows:(58)Q~=Q−Qd.If the AHV is affected by an external disturbance, combined with the concept of active disturbance rejection, the equationQ̇=fQx,δe can be expressed as follows:
(59)Q̇=fQx,δe+dQt−lQδe+lQδe=FQx,δe+lQδe,where dQt is the external disturbance, FQx,δe=fQx,δe+dQt−lQδe is the unknown term, and lQ>0 is the parameter to be designed.The first derivative ofQ~ with respect to time is obtained, and Equation (57) is substituted into the result, yielding:
(60)Q~̇=Q̇−Q̇d=FQx,δe+lQδe−Q̇d.The following new TD is applied to estimateQ̇d:
(61)υ̇Q1=υQ2,υ̇Q2=RQ2Qd−υQ1−υQ2RQ,where RQ>0 is the parameter to be designed, and υQ1 and υQ2 are the estimated values of Qd and Q̇d, respectively.To estimate the unknown termFQx,δe, the following ESO is designed for Equation (58):
(62)EQ=ZQ1−Q~,ŻQ1=ZQ2−βQ1EQ−υQ2+lQδe,ŻQ2=−βQ2EQαQsgnEQ,where βQ1,βQ2>0, 0≤αQ≤1 is the parameter to be designed, ZQ1 is the estimated value of Q~, and ZQ2 is the estimated value of the unknown term FQx,δe.The control lawδe can be designed as follows:
(63)δe=−1lQZQ2+kQQ~−υQ2+θ~,where lQ>0 is the parameter to be designed.Theorem 5.
Considering the altitude subsystem (Equation (19)), if the active disturbance rejection control law δe (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 εh, γ~, θ~, and Q~ are bounded, and the altitude tracking error h~ is limited to the prescribed area. The following inequality holds −pht<h~<pht.Proof.
The following Lyapunov function is selected:(64)Wh=12γ~2+12θ~2+12Q~2.
The first derivative ofWh with respect to time is obtained, and Equations (46), (49), (50), (52), (55), (56), (58), and (61) are substituted into the result, yielding:
(65)Ẇh=γ~γ~̇+θ~θ~̇+Q~Q~̇=γ~Fγγ,θ+lγθ~+θd−γ̇d+θ~Q~+Qd+dθt−θ̇d+Q~FQx,δe+lQδe−Q̇d=−kγγ~2+γ~Fγγ,θ−Zγ2+γ~υγ2−γ̇d+lγθ~γ~−kθθ~2−lγθ~γ~+θ~dθt−Zθ2+θ~υθ2−θ̇d+θ~Q~−kQQ~2−θ~Q~+Q~FQx,δe−ZQ2+Q~υQ2−Q̇d.
According to Equations (24) and (26), bounded constants σγ, σθ, σQ, λ¯γ, λ¯θ, and λ¯Q exist, which make:
(66)Fγγ,θ−Zγ2≤σγ,υγ2−γ̇d≤λ¯γ,dθt−Zθ2≤σθ,υθ2−θ̇d≤λ¯θ,FQx,δe−ZQ2≤σQ,υQ2−Q̇d≤λ¯Q,where
(67)γ~σγ≤γ~22+σγ22,γ~λ¯γ≤γ~22+λ¯γ22,θ~σθ≤θ~22+σθ22,θ~λ¯θ≤θ~22+λ¯θ22,Q~σQ≤Q~22+σQ22,Q~λ¯Q≤Q~22+λ¯Q22.
Combining Equations (63), (64), and (65) yields the following:
(68)Ẇh≤−kγ−1γ~2−kθ−1θ~2−kQ−1Q~2+12σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q2.
Withkγ>1, kθ>1, and kQ>1, the following compact sets are defined:
(69)Ωγ~=γ~γ~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kγ−1,Ωθ~=θ~θ~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kθ−1,ΩQ~=Q~Q~≤σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q22kQ−1.According to Equation (67), if γ~∉Ωγ~, θ~∉Ωθ~, or Q~∉ΩQ~, Ẇh<0. Wh decreases until γ~, θ~, and Q~ converge to the compact sets Ωγ~, Ωθ~, and ΩQ~, respectively. Therefore, the closed-loop control system is semiglobally uniformly asymptotically stable. γ~, θ~, and Q~ are ultimately bounded. However, if the estimated error of the ESO and new TD are sufficiently small (σγ2+λ¯γ2+σθ2+λ¯θ2+σQ2+λ¯Q2 is sufficiently small), the radii of the compact sets Ωγ~, Ωθ~, and ΩQ~ can be sufficiently small, and γ~, θ~, and Q~ can be sufficiently small. If the γ~ is sufficiently small, then γ→γd. According to Equations (40)–(43), the altitude transformed error is bounded. Thus, by Theorem 1, −pht<h~<pht, which means the altitude transformed error is limited to the prescribed area.


## 4. 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 be 0.01 s. The initial values of the AHV state variables are shown in Table 1.Table 1
Initial values of the AHV state variables.
Parameter
Value
Unit
V
2500
m/s
h
27000
m
γ
0
°
θ
1.5
°
Q
0
°/s
η1
0.29
—
η2
0.26
—The velocity reference commandVref and the altitude reference command href are specified by the following second-order system:
(70)VrefsVIs=hrefshIs=ωn2s2+2ζnωns+ωn2,where the damping ratio ζn=0.9, the natural frequency ωn=0.1rad/s, and VI and hI 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:kpV=0.1, χpV=0.35, pV∞=0.08, kph=0.1, χph=1, and ph∞=0.01. The ADRC controller parameters were selected as follows: lV=10, kV=10, kh1=3.5, kh2=0.1, lγ=2, kγ=50, kθ=50, lQ=10, and kQ=20. The designed parameters of the new TD were selected as follows: Rγ=5, Rθ=2, and RQ=2. The designed parameters of the ESO were selected as follows: βV1=2, βV2=2, αV=0.4, βγ1=2, βγ2=2, αγ=0.4, βθ1=25, βθ2=25, αθ=0.4, βQ1=25, βQ2=25, and αQ=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. VI and hI 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=C01+0.4sin0.1πt. C0 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:
(71)dV=dγ=dθ=dQ=2sin0.1πt.The simulation results for Scenario1 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, VI was assumed to be a “step” type signal with a step of 150 m/s per 100 s. hI 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:
(72)C=C0,0s≤t<50s,C01+0.4sin0.1πt,50s≤t<100s,C0,100s≤t<150s,C01+0.4sin0.1πt,150s≤t<200s,C0,200s≤t<250s,C01+0.4sin0.1πt,250s≤t≤300s,where C0 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: dV=dγ=dθ=dQ=2sin0.1πt.Figure 3
Simulation results for Scenario1.
(a)
Velocity tracking performance
(b)
Velocity tracking error
(c)
Altitude tracking performance
(d)
Altitude tracking error
(e)
Flight-path angle
(f)
Estimation ofγd and γ̇d
(g)
FVV,Φ
(h)
FQx,δe
(i)
Elastic stateη1
(j)
Elastic stateη2
(k)
Fuel equivalence ratio
(l)
Elevator angular deflectionThe simulation results of the Scenario2 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.Figure 4
Simulation results for Scenario2.
(a)
Velocity tracking performance
(b)
Velocity tracking error
(c)
Altitude tracking performance
(d)
Altitude tracking error
(e)
Flight-path angle
(f)
Estimation ofγd and γ̇d
(g)
FVV,Φ
(h)
FQx,δe
(i)
Elastic stateη1
(j)
Elastic stateη2
(k)
Fuel equivalence ratio
(l)
Elevator angular deflection


## 5. Conclusions

(1)
An active disturbance rejection control for an AHV based on the prescribed performance function is 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



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*Source: 4129136-2019-11-22.xml*