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Research ArticleActive Disturbance Rejection Control for Air-BreathingHypersonic Vehicles Based on Prescribed Performance Function

Chenyang Xu ,1 Humin Lei,1 and Na Lu2

1Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China2Unit 93142, People’s Liberation Army, Chengdu 610044, China

Correspondence should be addressed to Chenyang Xu; [email protected]

Received 19 July 2019; Revised 9 October 2019; Accepted 29 October 2019; Published 22 November 2019

Academic Editor: Antonio Concilio

Copyright © 2019 Chenyang Xu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Aiming at the longitudinal motion model of the air-breathing hypersonic vehicles (AHVs) with parameter uncertainties, a newprescribed performance-based active disturbance rejection control (PP-ADRC) method was proposed. First, the AHV modelwas divided into a velocity subsystem and altitude system. To guarantee the reliability of the control law, the design process wasbased on the nonaffine form of the AHV model. Unlike the traditional prescribed performance control (PPC), which requiresaccurate initial tracking errors, by designing a new performance function that does not depend on the initial tracking error andcan ensure the small overshoot convergence of the tracking error, the error convergence process can meet the desired dynamicand 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 avoidthe differential expansion problem and effectively filter out the effects of input noise in the differential signals, a new trackingdifferentiator was proposed. Finally, the effectiveness of the proposed method was verified by comparative simulations.

1. Introduction

Air-breathing hypersonic vehicles (AHVs) are a new type ofaircraft, which fly at speeds greater than Mach 5 at near spacealtitudes. AHVs exhibit fast flying speeds, strong penetrationabilities, and long combat distances. It is difficult to detectand intercept AHVs. AHVs have strong survivability, andthey have outstanding advantages in strategy, tactics, andcost-effectiveness compared to traditional aerospace vehicles[1–3]. AHVs have become a priority development directionfor all of the aerospace powers competing for air and spacerights. However, AHVs are multivariable and stronglycoupled nonlinear systems. AHVs have large flight airspaces,and the flight environments are complex and variable, whichresults in large and fast time-varying characteristics and alti-tude uncertainty of the AHV model. Thus, the design of theAHV control system involves unprecedented difficultiesand challenges [4, 5].

Most of the previous research on the modelling and con-trol of AHVs has mainly focused on the AHV’s longitudinalmotion plane. On the one hand, the longitudinal motion

model is complex enough to require flight control. On theother hand, due to the scramjet engine’s extreme sensitivityto the flight attitude, AHVs should avoid horizontal manoeu-vres during actual flight [6]. In a previous study [7], a robustL∞ gain control method was designed for AHVs, which usedthe Takagi-Sugeno (T-S) fuzzy system to approximate theunknown state of the model. To guarantee the robustnessof the control scheme, a new fuzzy disturbance observerwas designed to estimate the disturbance. A similar longitu-dinal elastic model of the AHV was expressed as a T-S fuzzysystem [8], and a H2/H∞ tracking control law was designedto achieve the robust tracking of the speed and altitude refer-ence inputs. To improve the tracking effect, a nonlinearadaptive back-stepping controller was designed for theAHV based on the back-stepping design [9]. Based on thetraditional back-stepping control method, a nonsingular fastterminal sliding model control was applied to control thepitch angle and pitch rate, which optimized the control struc-ture of the back-stepping method and achieved finite timeconvergence of the system [10]. In one study, an integral slid-ing model control method was proposed [11]. When there

HindawiInternational Journal of Aerospace EngineeringVolume 2019, Article ID 4129136, 20 pageshttps://doi.org/10.1155/2019/4129136

are model uncertainties and external disturbances, the con-trol scheme can still guarantee finite time convergence ofthe velocity and altitude tracking errors. Aiming at the atti-tude control problem of AHVs, a robust fuzzy controlmethod based on a nonlinear switching system was proposed[12], which used a fuzzy system to approximate the unknownfunction of the model and guaranteed the robustness of thecontrol scheme. Neural control methods for scenarios wherethe control input of the AHV contained a dead zone [13] andthe actuator of the AHV contained failures [14] were studied.Two new neural back-stepping control methods were putforward [15, 16]. The altitude subsystem was rewritten instrict and pure feedback forms. Based on the improvedback-stepping strategy, the control scheme was designed,and a minimal learning parameter (MLP) algorithm wasapplied to reduce the online learning parameters. The track-ing simulation results of the velocity and altitude referenceinput showed that the proposed method exhibited betterrobustness and control effects. To solve the control problemof a strongly nonlinear model, such as an AHV, in additionto the fuzzy system or neural network used in the above lit-erature to estimate the uncertainty of the model, active dis-turbance rejection control (ADRC) can also be considered.The ADRC term was first used in [17] where ProfessorJ.Q. Han’s unique ideas were first systematically introducedinto the English literature. In recent years, many scholarshave conducted meaningful work on ADRC. In a previousstudy [18], ADRCwas offered as the basis of a paradigm shift,providing the framework, the objectives, and constraints forfuture control theory development. For a class of multiple-input multiple-output (MIMO) lower-triangular systemswhich have uncertain dynamics and disturbance, an ADRCmethod was designed to solve the control problem [19]. Anoverview of the concept, principles, and practice of theADRC was presented in [20]. The key idea of ADRC is touse an extended state observer (ESO) to estimate the totaldisturbance, which is then compensated in the feedback loop.For a class of nonlinear systems with large uncertainties thatcome from both internal unknown dynamics and externalstochastic disturbance, a novel ESO was designed to estimateboth state and total disturbance which included the internaluncertain nonlinear part and the external uncertain sto-chastic disturbance [21]. To improve the performance ofADRC, a fal-based, single-parameter-tuning ESO was pro-posed, and the convergence of the fal-based ESO and theoutput tracking were established [22]. To improve the per-formance of the tracking differentiator (TD), an ADRCmethod based on a radial basis function neural network(RBFNN) was proposed [23], which enhanced the robust-ness and disturbance rejection ability of the method. Toassist in the tuning of the parameters of the linear ADRCcontroller, a tuning rule was proposed to minimize the loaddisturbance attenuation performance [24]. Based on theexisting swarm intelligence algorithms, a parameter tuningoptimization design of ADRC was achieved [25]. An in-depth study on the scheme design of an ADRC was con-ducted and applied to a three-degree-of-freedom pneumaticmotion system subject to actuator saturation [26], whichachieve good control.

Although the methods in the above literature achievedcertain control effects, the research focus was on the robust-ness and steady-state performance of the AHV closed-loopcontrol system, neglecting the dynamic performance ofthe control system. However, AHV’s super-manoeuvrable,large-envelope, and hypersonic flight demands better dynamicperformance of the control system than any other existingaircraft. In most cases, a small control delay will cause signif-icant errors in the hypersonic flight. Therefore, to guaranteethe robustness and steady-state accuracy of the control sys-tem, more attention should be paid to the dynamic perfor-mance and real-time performance of the control system. Toconsider both the steady-state and dynamic performances,the concept of prescribed performance control was proposedby Charalampos and George [27]. The prescribed perfor-mance ensured that the tracking error converged to a pre-scribed arbitrary small area. Meanwhile, the convergencerate and overshoot met the prescribed conditions. Based onthe 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 bodyand the elastic problem was not considered, which resulted inthe limitations of the proposed method. Aiming at the elasticbody of the AHV model, a prescribed performance fuzzyback-stepping control method and performance neuralback-stepping control method were proposed [29, 30], whichguaranteed the steady-state performance and dynamic per-formance of the control system. However, the above twomethods considered the AHV model to be an affine model,which made the reliability of the design method not guaran-teed. An AHV adaptive neural control method was designedbased on the prescribed performance function, whichavoided the cumbersome virtual control law design process[31]. Meanwhile, the accuracy and rapidity of the control sys-tem were achieved. However, the proposed method relied toomuch on the initial value of the tracking error, which resultedin the poor practicality and operability.

In view of the deficiencies of the research in the abovestudies, an elastic hypersonic vehicle was taken as theresearch object. In this study, a new prescribed performancefunction was designed based on the hyperbolic cosine func-tion, which avoided the singular control problem caused bythe improper initial value setting. Thus, the steady-state per-formance and dynamic performance of the control systemcould be guaranteed. Meanwhile, active disturbance rejectioncontrol was introduced and an ESO was designed for eachunknown nonaffine function in the AHV system [32], whichfurther guaranteed the control accuracy and the robustnessof the method. To address the complexity of the derivativeof the virtual control law, a track differentiator was appliedto estimate the related signals and signal derivatives. Theeffectiveness and superiority of the proposed method wasverified through the simulation and comparison.

2. AHV Model and Preliminaries

2.1. Model Description. To better describe the longitudinalmotion of the AHV, American scholar Parker used theresearch conclusions of Bolender and Doman [33, 34]

2 International Journal of Aerospace Engineering

combined with Hooke’s law and the Lagrange equation toestablish the following control-oriented AHV longitudinalmotion parameter fitting model:

_V =T cos θ − γð Þ −D

m− g sin γ, ð1Þ

_h =V sin γ, ð2Þ

_γ = L + T sin θ − γð ÞmV

−gV

cos γ, ð3Þ

_θ =Q, ð4Þ

_Q =M + ~ψ1€η1 + ~ψ2€η2

Iyy, ð5Þ

k1€η1 = −2ζ1ω1 _η1 − ω21η1 +N1 − ~ψ1

MIyy

−~ψ1~ψ2€η2Iyy

, ð6Þ

k2€η2 = −2ζ2ω2 _η2 − ω22η2 +N2 − ~ψ2

MIyy

−~ψ2~ψ1€η1Iyy

: ð7Þ

Equations (1)–(5) describe the rigid body part of theAHV. The five rigid states are the velocity V , altitude h,flight-path angle γ, pitch angle θ, and the pitch rate Q. m isthe mass of the AHV, g is the gravitational acceleration con-stant, and Iyy is the moment of inertia of the AHV. The forcecondition of the AHV is shown in Figure 1. The parameterfitting form of the thrust T , drag D, lift L, pitching momentM, and generalized force Niði = 1, 2Þ can be expressed as fol-lows [35]:

T ≈ Cα3

T α3 + Cα2

T α2 + Cα

Tα + C0T ,

D ≈ �qS Cα2

D α2 + CαDα + Cδ2e

D δ2e + Cδe

D δe + C0D

� �,

L ≈ �qS CαLα + Cδe

L δe + C0L

� �,

M ≈ zTT + �qS�c Cα2

M,αα2 + Cα

M,αα + C0M,α + ceδe

h i,

N1 ≈Nα2

1 α2 +Nα

1α +N01,

N2 ≈Nα2

2 α2 +Nα

2α +Nδe2 δe +N0

2,

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð8Þ

with

Cα3

T = β1 h, �qð ÞΦ + β2 h, �qð Þ,Cα2

T = β3 h, �qð ÞΦ + β4 h, �qð Þ,CαT = β5 h, �qð ÞΦ + β6 h, �qð Þ,

C0T = β7 h, �qð ÞΦ + β8 h, �qð Þ,

�q =�ρV2

2, �ρ = �ρ0 exp

h0 − hhs

� �,

8>>>>>>>>>>><>>>>>>>>>>>:

ð9Þ

where the attack angle α = θ − γ, the fuel equivalence ratio Φ,and the elevator angular deflection δe are control inputs, Sand �c are the reference area and aerodynamic chord of theAHV, respectively, zT is the thrust moment arm, ce is thecoefficient of δe in M, h0 and ρ0 are the nominal altitudeand corresponding air density, respectively, 1/hs is the airdensity decay rate, �q and �ρ are dynamic pressure and air den-sity at h, respectively, C•i

∗ ð· = α, δe;∗ = T ,DÞ is the ith ordercoefficient of · in ∗, C0

∗ð∗ = T ,D, LÞ is the constant coefficientin ∗, C∗

Lð∗ = α, δeÞ is the contribution of the coefficient of ∗in L, Cαi

M,α is the ith order coefficient of α in M, C0M,α is the

constant coefficient in M, Nαij is the jth order contribution

of α to Nj, N0i is the constant term in Ni, N

δe2 is the contribu-

tion of δe to N2, and βiðh, �qÞ is the ith thrust fit parameter.Equations (6) and (7) describe the elastic body part of the

AHV. The elastic states are η1 and η2. ζi and ωiði = 1, 2Þ arethe damping ratio and vibrational frequency of the AHVelastic state. ki and ~ψiði = 1, 2Þ can be expressed as follows:

k1 = 1 +~ψ1Iyy

, k2 = 1 +~ψ2Iyy

,

~ψ1 =ð0−Lf

m̂f ξϕf ξð Þdξ,

~ψ2 =ðLa0m̂aξϕa ξð Þdξ,

8>>>>>>>>><>>>>>>>>>:

ð10Þ

where Lf and La are the front and rear beam length of theAHV, respectively, m̂f and m̂a are the mass distributionsof the front and the rear beams, respectively, and ϕf ð·Þ andϕað·Þ are the vibration mode functions of the front andrear beam, respectively. The specific values of the model

V

�훾

�훼

�휃=�훼+�훾

�훿eTM

G

L

D

Elevator

Figure 1: AHV force diagram.

3International Journal of Aerospace Engineering

parameters and aerodynamic parameters of the abovemen-tioned AHV are shown in [35].

2.2. Prescribed Performance. The AHV prescribed perfor-mance control method based on the nonaffine model is stud-ied in this paper. To make the tracking error convergenceprocesses meet the desired dynamic and steady-state perfor-mance, the following new performance function pðtÞ isdesigned to limit the tracking error.

p tð Þ = coth kpt + χp

� �− 1 + p∞, ð11Þ

where kp, χp, p∞ ∈ R+ are parameters to be designed.pðtÞ has following properties:

(1) pðtÞ is a positive monotonically decreasing function

(2) pð0Þ = coth ðχpÞ − 1 + p∞ = e2χp + 1/e2χp − 1 − 1 + p∞> p∞

(3) limχp→0

pð0Þ→ +∞

(4) limt→+∞

pðtÞ = p∞

The prescribed performance is defined as follows:

−p tð Þ < e tð Þ < p tð Þ, ð12Þ

where eðtÞ is the tracking error. By choosing a smallenough χp, it is guaranteed that pðtÞ→ +∞ and −pðtÞ→−∞ based on the property (3). Therefore, for any unknownbut bounded eð0Þ, the following inequality always holds:

−p 0ð Þ < e 0ð Þ < p 0ð Þ: ð13Þ

The prescribed performance defined by Equation (12)is shown in Figure 2. p∞ is the upper bound of the eðtÞsteady-state value, which means −p∞ < eð∞Þ < p∞. There-fore, the desired steady-state accuracy of eðtÞ can be guaran-teed by choosing an appropriate p∞. pð0Þ is the maximumovershoot allowed by eðtÞ. kp directly influences the decreas-ing rate of pðtÞ. With the increase in kp, pðtÞ decreasesmore rapidly.

Remark 1. For any arbitrary bounded eð0Þ whether it isknown or not, as long as the χp is chosen to be small enough,eð0Þmust be included in the prescribed area defined by Equa-tion (12), which avoids the singular control problem causedby the improper initial value setting of the traditional perfor-mance function [36].

Remark 2.When the χp is chosen to be small, the value ofpð0Þ will be very large, which can lead to too large of anovershoot of eðtÞ. However, because the response speed of eðtÞ is limited, the desired dynamic performance, includingthe overshoot and setting time, can be guaranteed by choos-ing a larger kp. This suggests that if the tracking error eðtÞsatisfies Equation (12), the overshoot and the adjustingtime can be constrained within a certain range, as shownin Figure 2. Thus, by designing the controller which canmake the tracking error meet the constraint of Equation(12), the desired dynamic performance of the control systemcan be guaranteed.

The transformed error εðtÞ is defined as follows:

ε tð Þ = lnλ tð Þ + 11 − λ tð Þ� �

, ð14Þ

where λðtÞ = eðtÞ/pðtÞ. The following theorem can beobtained.

Theorem 1. If εðtÞ is bounded, then −pðtÞ < eðtÞ < pðtÞ.

Proof. Because εðtÞ is bounded, there must be a boundedconstant εM ∈ R+ that makes jεðtÞj ≤ εM . Furthermore, theinverse transformation of Equation (14) is as follows:

eε tð Þ =λ tð Þ + 11 − λ tð Þ : ð15Þ

Based on Equation (15),

−1 <e−εM − 11 + e−εM

≤ λ tð Þ ≤ eεM − 11 + eεM

< 1: ð16Þ

Substituting λðtÞ = eðtÞ/pðtÞ into Equation (16) yields

−p tð Þ < e tð Þ < p tð Þ: ð17Þ

Therefore, Theorem 1 is established.

Remark 3. The control law below will be designed based onthe transformed error εðtÞ. Theorem 1 shows that as longas εðtÞ is bounded, eðtÞ can be limited to the prescribed areadefined by Equation (12). By choosing appropriate designparameters for pðtÞ, the desired dynamic performance andsteady-state accuracy of eðtÞ can be guaranteed.

2.3. Model Conversion and Control Objective. On the onehand, since the thrust T is directly influenced by the fuelequivalence ratio Φ, the velocity V of the AHV is mainly

P∞

p(0)

e(0)

−P∞

–p(0)–p(�푡)

e(�푡)p(�푡)

0

�푡/�푠

Figure 2: Prescribed performance defined by Equation (12).

4 International Journal of Aerospace Engineering

controlled by Φ. On the other hand, since the elevator angu-lar deflection δe directly influences the pitch rate Q, it furtherchanges the pitch rate θ and the flight-path angle and finallyinfluences the change of the altitude. The altitude changeof the AHV is mainly controlled by the elevator angulardeflection δe [37, 38]. Therefore, most previous studies onAHV control issues first divide the AHV model into a veloc-ity subsystem controlled by Φ (Equation (1)) and an altitudesubsystem controlled by δe (Equations (2)–(5)) and thenderived control laws [39, 40].

Based on the previous studies [37, 38], the velocity sub-system of the AHV was considered to be a nonaffine formwith the control input:

_V = f V V ,Φð Þ,yV = V ,

(ð18Þ

where yV is the output of the system, and f VðV ,ΦÞ is acompletely unknown continuous differentiable function. Forthe velocity subsystem, the control goal is to design an appro-priate control law Φ based on the nonaffine model (Equation(18)). Thus, the robustness tracking of the velocity V to thereference input V ref can be achieved, and the velocity track-ing error can be limited to the prescribed area.

Similarly, the altitude subsystem of the AHV can beexpressed as the following nonaffine pure feedback system:

_h =V sin γ,

_γ = f γ γ, θð Þ,_θ =Q,_Q = f Q x, δeð Þ,yh = h,

8>>>>>>>>><>>>>>>>>>:

ð19Þ

where yh is the output of the system, and x = ½γ, θ,Q�T,f γðγ, θÞ, and f Qðx, δeÞ are completely unknown continuousdifferentiable function. For the altitude subsystem, the con-trol goal is to design an appropriate control law δe. Thus,the stable tracking of the altitude h to the reference inputhref can be achieved and the altitude tracking error can belimited to the prescribed area. Also, the desired dynamic per-formance and steady-state accuracy can be guaranteed.

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

2.4. Extend States Observer and Tracking Differentiator. Todesign the active disturbance rejection control law, theextended state observer is applied to estimate the uncertainty

of the AHV model and the external disturbance. The follow-ing system is considered:

_z =H tð Þ + BU , ð20Þ

where HðtÞ is the unknown term, and U is the input of thesubsystem. The state z of the system is measurable. There-fore, the state of the system can be expanded to the followingsystem:

_z = z0 + BU ,

_z0 =G tð Þ,

(ð21Þ

where GðtÞ is the unknown derivative of the unknown termHðtÞ. Therefore, the ESO can be established as follows:

E = Z1 − z,_Z1 = Z2 − β01 f c1 Eð Þ + BU ,_Z2 = −β02 f c2 Eð Þ,

8>><>>: ð22Þ

where E is the estimation error of the ESO, Z1 and Z2 are theoutputs of the ESO, β01 > 0 and β02 > 0 are the observergains, and parametric function f cið·Þði = 1, 2Þ is an appropri-ately constructed nonlinear function, which satisfies ef ciðeÞ> 0, ∀e ≠ 0, and f cið0Þ = 0. The parametric function f cið·Þ inthis paper is chosen to have the following form:

f c1 Eð Þ = E,

f c2 Eð Þ = Ej jα1 sgn Eð Þ,

(ð23Þ

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 param-eters β01, β02 > 0 and 0 ≤ α1 ≤ 1, which makes the output of theESO Z1 and Z2 converge to the actual state z and a compact setof the unknown term HðtÞ, respectively. Bounded constantsσ1, σ2 > 0 exist, which make:

Z1 − zj j ≤ σ1,

Z2 −H tð Þj j ≤ σ2:

(ð24Þ

Proof. The proof process is shown in the appendix.

Some signals in the control law design process are oftendifficult to obtain by the model construction. Many scholarshave proposed using the tracking differentiator to estimatethe signal [41]. The design principle is to achieve the highestprecision extraction of the differential signal and to ensurecertain robustness to the input noise of the signal. A newTD is proposed in this paper to estimate the differential sig-nal. The specific form of the new TD is as follows:

5International Journal of Aerospace Engineering

_υ1 = υ2,

_υ2 = υ3,

⋮

_υn = Rn ν − υ1 −υ2R

−υ3R2 −⋯−

υnRn−1

� �,

8>>>>>><>>>>>>:

ð25Þ

where ν is the input signal to be estimated, υ1 is the estimatedvalue of ν, and υiði = 2, 3,⋯, nÞ are the estimated values ofthe i − 1th derivative of ν, respectively; R is the parameterto be designed.

Theorem 3. If the new TD (Equation (25)) is applied to esti-mate the input signal ν and the derivatives, there exists an R> 0 that makes υiði = 1, 2,⋯, nÞ converge to a compact set ofνðnÞðn = 0, 1,⋯, n − 1Þ, respectively. Bounded constants �λi >0ði = 1, 2,⋯, nÞ exist, which make:

υ1 − νj j ≤ �λ1,

υ2 − _νj j ≤ �λ2,

⋮

υn − ν n−1ð Þ��� ��� ≤ �λn:

8>>>>>><>>>>>>:

ð26Þ

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 arbitraryderivative of the input signal. Second, the new TD has onlyone parameter R to be designed, and the parameter adjust-ment process is easier.

3. Controller Design

3.1. Velocity Controller Design. The velocity subsystem(Equation (18)) is assumed to be affected by external distur-bances. According to the idea of self-interference, it can beexpressed as follows:

_V = f V V ,Φð Þ + dV tð Þ − lVΦ + lVΦ

= FV V ,Φð Þ + lVΦ,

yV =V ,

8>><>>: ð27Þ

where dVðtÞ is the external disturbance, FVðV ,ΦÞ = f VðV ,ΦÞ + dVðtÞ − lVΦ is the unknown term, and lV > 0 is theparameter to be designed.

The velocity tracking error can be defined as follows:

~V =V −V ref : ð28Þ

The first derivative of Equation (28) with respect to timeis obtained, and Equation (27) is substituted into the result,yielding the following:

_~V = FV V ,Φð Þ − _V ref + lVΦ: ð29Þ

According to Equation (14), the velocity transformederror can be defined as follows:

εV tð Þ = ln~V/pV tð Þ + 11 − ~V/pV tð Þ

!, ð30Þ

where pVðtÞ = coth ðkpVt + χpVÞ − 1 + pV∞, and kpV , χpV ,pV∞ ∈ R+ are parameters to be designed.

The first derivative of Equation (30) with respect to timeis obtained, and Equation (29) is substituted into the result,yielding the following:

_εV tð Þ = rV_~V −

_pV tð ÞpV tð Þ

~V� �

= rV FV V ,Φð Þ − _V ref + lVΦ −_pV tð ÞpV tð Þ

~V� �

,ð31Þ

where

rV =1

pV tð Þ1

~V/pV tð Þ + 1−

1~V/pV tð Þ − 1

!> 0,

_pV tð Þ = kpV 1 − coth2 kpV t + χpV

� �� �:

ð32Þ

To estimate the unknown term FVðV ,ΦÞ, the followingESO is designed for Equation (31):

EV = ZV1 − εV ,

_ZV1 = ZV2 − βV1EV + rV lVΦ − _V ref −_pV tð ÞpV tð Þ

~V� �

,

_ZV2 = −βV2 EVj jαV sgn EVð Þ,

8>>>><>>>>:

ð33Þ

where βV1, βV2 > 0, 0 ≤ αV ≤ 1 is the parameter to bedesigned, ZV1 is the estimated value of εV , and ZV2/rV isthe estimated value of the unknown term FVðV ,ΦÞ.

The active disturbance rejection control law Φ can bedesigned as follows:

Φ = −1lV

ZV2rV

+ kVεV − _V ref −_pV tð ÞpV tð Þ

~V� �

, ð34Þ

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-loopcontrol system is semiglobally uniformly asymptotically stable,and the speed tracking error is limited to the prescribed area.The following inequality holds −pVðtÞ < ~V < pVðtÞ.

6 International Journal of Aerospace Engineering

Proof. Substituting Equation (33) into Equation (31) yields:

_εV = rV FV V ,Φð Þ − ZV2rV

− kVεV

� �: ð35Þ

The following Lyapunov function was selected:

WV =12ε2V : ð36Þ

The first derivative of WV with respect to time isobtained, and Equation (34) is substituted into the result,yielding the following:

_WV = εVrV FV V ,Φð Þ − ZV2rV

− kVεV

� �

= rV εV FV V ,Φð Þ − ZV2rV

� �− kVε

2V

� �:

ð37Þ

Combined with Equation (24), a constant ϖV exists suchthat:

FV V ,Φð Þ − ZV2rV

�������� ≤ ϖV : ð38Þ

Considering that rV > 0, Equation (37) is substituted intoEquation (36), yielding:

_WV ≤ rV −kVε2V + εVj jϖV

� �≤ rV − kV −

12

� �ε2V +

12ϖ2V

� �:

ð39Þ

With kV > 1/2, the following compact set is defined:

ΩεV= εV εVj j ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1/2ð Þϖ2

V

kV − 1/2

s������8<:

9=;: ð40Þ

Combining Equations (38) and (39), if εV ∉ΩεV, then

_WV < 0. Therefore, the closed-loop control system is semi-globally uniformly asymptotically stable, and the velocitytransformed error will finally converge to the compact setΩεV

. If the estimated accuracy of the ESO is high enough(ϖV is small enough) and the kV is sufficiently large, the radiusof the compact ΩεV

and the εV can be sufficiently small.The above proves that the transformed error is bounded.

According to Theorem 1, −pVðtÞ < ~V < pVðtÞ. The velocitytransformed error is limited to the prescribed area.

3.2. Altitude Controller Design. Considering the altitude sub-system (Equation (19)), the altitude tracking error can bedefined as ~h = h − href . The altitude transformed error canbe defined as follows:

εh tð Þ = ln~h/ph tð Þ + 11 − ~h/ph tð Þ

!, ð41Þ

where phðtÞ = coth ðkpht + χphÞ − 1 + ph∞, and kph, χph, ph∞∈ R+ are parameters to be designed.

Furthermore, the first derivative of εhðtÞ with respect totime yields the following:

_εh tð Þ = rh_~h −

_ph tð Þph tð Þ

~h� �

= rh V sin γ − _href −_ph tð Þph tð Þ

~h� �

,

ð42Þ

where

rh =1

ph tð Þ1

~h/ph tð Þ + 1−

1~h/ph tð Þ − 1

!> 0,

_ph tð Þ = kph 1 − coth2 kpht + χph

� �� �:

ð43Þ

The flight-path angle command is defined as follows:

γd = arcsin−kh1εh − kh2

Ð t0εhdτ + _href + _ph tð Þ~h/ph tð Þ

V

!,

ð44Þ

where kh1, kh2 > 0 are parameters to be designed.If γ→ γd, γ in Equation (41) is replaced with γd and a

Laplace transform is applied, yielding:

s2 + kh1s + kh2 = 0: ð45Þ

The two characteristics roots of Equation (43), ð−kh1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikh1

2 − 4kh2p

Þ/2 and ð−kh1 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikh1

2 − 4kh2p

Þ/2, are nega-tive real numbers. Therefore, the altitude transformed errorεhðtÞ is convergent and bounded. For the altitude subsystem(Equation (19)), the control goal can be achieved through thecontrol law δe design which makes γ→ γd [43].

For the rest of the altitude subsystem (Equation (19)), theactive disturbance rejection control law of δe is designed inthree steps.

Step 1. For _γ = f γðγ, θÞ, the pitch angle command θd isdesigned.

Combined with the concept of active disturbance rejec-tion, _γ = f γðγ, θÞ can be expressed as follows:

_γ = f γ γ, θð Þ + dγ tð Þ − lγθ + lγθ = Fγ γ, θð Þ + lγθ, ð46Þ

where dγðtÞ is the external disturbance, Fγðγ, θÞ = f γðγ, θÞ +dγðtÞ − lγθ, and lγ > 0 is a parameter to be designed.

The flight-path angle tracking error is defined as follows:

~γ = γ − γd: ð47Þ

7International Journal of Aerospace Engineering

The first derivative of Equation (45) is obtained, andEquation (44) is substituted into the result, yielding:

_~γ = _γ − _γd = Fγ γ, θð Þ + lγθ − _γd: ð48Þ

The mathematical expression of _γd is complicated. Thus,to avoid the differential expansion problem and effectively fil-ter out the effects of input noise in the differential signals, thenew TD proposed in this paper is applied to estimate _γd.

_υγ1 = υγ2,

_υγ2 = R2γ γd − υγ1 −

υγ2Rγ

!,

8>><>>: ð49Þ

where Rγ > 0 is the parameter to be designed, and υγ1 and υγ2are the estimated values of γd and _γd, respectively.

To estimate the unknown term Fγðγ, θÞ, the followingESO is designed for Equation (46):

Eγ = Zγ1 − ~γ,

_Zγ1 = Zγ2 − βγ1eγ − υγ2 + lγθ,

_Zγ2 = −βγ2 Eγ

�� ��αγ sgn Eγ

� �,

8>>><>>>:

ð50Þ

where βγ1, βγ2 > 0, 0 ≤ αγ ≤ 1 is the parameter to be designed,and Zγ1 and Zγ2 are the estimated values of ~γ and Fγðγ, θÞ,respectively.

The pitch angle command θd can be designed as follows:

θd = −1lγ

Zγ2 + kγ~γ − υγ2� �

, ð51Þ

where kγ > 0 is the parameter to be designed.

Step 2. For _θ =Q, the pitch rate command Qd is designed.

The pitch angle tracking error is defined as follows:

~θ = θ − θd: ð52Þ

If the AHV is affected by an external disturbance, _θ =Qcan be expressed as follows:

_θ =Q + dθ tð Þ, ð53Þ

where dθðtÞ 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:

_~θ = _θ − _θd =Q + dθ tð Þ − _θd: ð54Þ

To avoid the differential expansion problem and effec-tively filter out the effects of input noise in the differential

signals, the new TD proposed in this paper is applied toestimate _θd:

_υθ1 = υθ2

_υθ2 = R2θ θd − υθ1 −

υθ2Rθ

� �,

8><>: ð55Þ

where Rθ > 0 is the parameter to be designed, and υθ1 andυθ2 are the estimated values of θd and _θd, respectively.

To estimate the unknown term dθðtÞ, the following ESOis designed for Equation (52):

Eθ = Zθ1 − ~θ,_Zθ1 = Zθ2 − βθ1Eθ − υθ2 +Q,_Zθ2 = −βθ2 Eθj jαθ sgn Eθð Þ,

8>><>>: ð56Þ

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 valueof the unknown term dθðtÞ.

The pitch rate command Qd can be designed as follows:

Qd = − Zθ2 + kθ~θ + lγ~γ − υθ2� �

, ð57Þ

where kθ > 0 is the parameter to be designed.

Step 3. For _Q = f Qðx, δeÞ, the control law δe is designed.

The pitch rate tracking error is defined as follows:

~Q =Q −Qd: ð58Þ

If the AHV is affected by an external disturbance, com-bined with the concept of active disturbance rejection, theequation _Q = f Qðx, δeÞ can be expressed as follows:

_Q = f Q x, δeð Þ + dQ tð Þ − lQδe + lQδe = FQ x, δeð Þ + lQδe, ð59Þ

where dQðtÞ is the external disturbance, FQðx, δeÞ = f Qðx,δeÞ + dQðtÞ − lQδe is the unknown term, and lQ > 0 is theparameter to be designed.

The first derivative of ~Q with respect to time is obtained,and Equation (57) is substituted into the result, yielding:

_~Q = _Q − _Qd = FQ x, δeð Þ + lQδe − _Qd: ð60Þ

The following new TD is applied to estimate _Qd:

_υQ1 = υQ2,

_υQ2 = R2Q Qd − υQ1 −

υQ2RQ

� �,

8><>: ð61Þ

where RQ > 0 is the parameter to be designed, and υQ1 and

υQ2 are the estimated values of Qd and _Qd, respectively.

8 International Journal of Aerospace Engineering

To estimate the unknown term FQðx, δeÞ, the followingESO is designed for Equation (58):

EQ = ZQ1 − ~Q,

_ZQ1 = ZQ2 − βQ1EQ − υQ2 + lQδe,

_ZQ2 = −βQ2 EQ

�� ��αQ sgn EQ

� �,

8>>><>>>:

ð62Þ

where βQ1, βQ2 > 0, 0 ≤ αQ ≤ 1 is the parameter to be

designed, ZQ1 is the estimated value of ~Q, and ZQ2 is the esti-mated value of the unknown term FQðx, δeÞ.

The control law δe can be designed as follows:

δe = −1lQ

ZQ2 + kQ ~Q − υQ2 + ~θ� �

, ð63Þ

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 (Equa-tion (61)), the ESO (Equation (60)), and the new TD areapplied, the closed-loop control system is semiglobally uniformlyasymptotically stable. The tracking errors εh, ~γ, ~θ, and ~Q arebounded, and the altitude tracking error ~h is limited to the pre-scribed area. The following inequality holds −phðtÞ < ~h < phðtÞ.

Proof. The following Lyapunov function is selected:

Wh =12~γ2 +

12~θ2 +

12~Q2: ð64Þ

The first derivative of Wh with respect to time isobtained, and Equations (46), (49), (50), (52), (55), (56),(58), and (61) are substituted into the result, yielding:

_Wh = ~γ _~γ + ~θ _~θ + ~Q _~Q = ~γ Fγ γ, θð Þ + lγ ~θ + θd� �

− _γd

� �+ ~θ ~Q +Qd + dθ tð Þ − _θd� �

+ ~Q FQ x, δeð Þ + lQδe− _Qd

� �= −kγ~γ

2 + ~γ Fγ γ, θð Þ − Zγ2� �

+ ~γ υγ2 − _γd� �

+ lγ~θ~γ

− kθ~θ2− lγ~θ~γ + ~θ dθ tð Þ − Zθ2ð Þ + ~θ υθ2 − _θd

� �+ ~θ~Q − kQ ~Q

2 − ~θ~Q + ~Q FQ x, δeð Þ − ZQ2� �

+ ~Q υQ2 − _Qd

� �:

ð65Þ

According to Equations (24) and (26), bounded con-stants σγ, σθ, σQ, �λγ, �λθ, and �λQ exist, which make:

Fγ γ, θð Þ − Zγ2�� �� ≤ σγ, υγ2 − _γd

�� �� ≤ �λγ,

dθ tð Þ − Zθ2j j ≤ σθ, υθ2 − _θd

��� ��� ≤ �λθ,

FQ x, δeð Þ − ZQ2�� �� ≤ σQ, υQ2 − _Qd

�� �� ≤ �λQ,

8>>>><>>>>:

ð66Þ

where

~γσγ ≤~γ2

2+σ2γ

2, ~γ�λγ ≤

~γ2

2+�λ2γ

2,

~θσθ ≤~θ2

2+σ2θ

2, ~θ�λθ ≤

~θ2

2+�λ2θ

2,

~QσQ ≤~Q2

2+σ2Q2, ~Q�λQ ≤

~Q2

2+�λ2Q

2:

8>>>>>>>>><>>>>>>>>>:

ð67Þ

Combining Equations (63), (64), and (65) yields the fol-lowing:

_Wh ≤ − kγ − 1� �

~γ2 − kθ − 1ð Þ~θ2 − kQ − 1� �

~Q2

+12

σ2γ + �λ2γ + σ2θ + �λ

2θ + σ2Q + �λ

2Q

� �:

ð68Þ

With kγ > 1, kθ > 1, and kQ > 1, the following compactsets are defined:

Ω~γ = ~γ ~γj j ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2γ + �λ

2γ + σ2θ + �λ

2θ + σ2

Q + �λ2Q

2 kγ − 1� �

vuut�������

8><>:

9>=>;,

Ω~θ = ~θ ~θ��� ��� ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2γ + �λ

2γ + σ2

θ + �λ2θ + σ2Q + �λ

2Q

2 kθ − 1ð Þ

vuut�������

8><>:

9>=>;,

Ω~Q = ~Q ~Q�� �� ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2γ + �λ

2γ + σ2θ + �λ

2θ + σ2Q + �λ

2Q

2 kQ − 1� �

vuut�������

8><>:

9>=>;:

ð69Þ

According to Equation (67), if ~γ ∉Ω~γ, ~θ ∉Ω~θ, or ~Q ∉Ω~Q,_Wh < 0.Wh decreases until ~γ, ~θ, and ~Q converge to the com-pact sets Ω~γ,Ω~θ, and Ω~Q, respectively. Therefore, the closed-loop control system is semiglobally uniformly asymptoticallystable. ~γ, ~θ, and ~Q are ultimately bounded. However, if theestimated error of the ESO and new TD are sufficiently small

(σ2γ + �λ2γ + σ2

θ + �λ2θ + σ2Q + �λ

2Q is sufficiently small), the radii of

the compact sets Ω~γ, Ω~θ, and Ω~Q can be sufficiently small,

and ~γ, ~θ, and ~Q can be sufficiently small. If the ~γ is sufficientlysmall, then γ→ γd. According to Equations (40)–(43), thealtitude transformed error is bounded. Thus, by Theorem 1,−phðtÞ < ~h < phðtÞ, which means the altitude transformederror is limited to the prescribed area.

4. Simulation Results

Taking the longitudinal motion model of the AHV (Equa-tions (1)–(10)) as the controlled object, the tracking simu-lations of speed and altitude reference input were carriedout. The simulations were solved using the fourth-orderRunge-Kutta method, and the simulation step was taken to

9International Journal of Aerospace Engineering

be 0.01 s. The initial values of the AHV state variables areshown in Table 1.

The velocity reference command V ref and the altitudereference command href are specified by the followingsecond-order system:

V ref sð ÞVI sð Þ

=href sð ÞhI sð Þ

=ω2n

s2 + 2ζnωns + ω2n, ð70Þ

where the damping ratio ζn = 0:9, the natural frequencyωn = 0:1rad/s, and VI and hI are the input signals of thesecond-order system.

When the control algorithm proposed in this paper wasused for the simulation, the prescribed performance parame-ters were selected as follows: kpV = 0:1, χpV = 0:35, pV∞ =0:08, kph = 0:1, χph = 1, and ph∞ = 0:01. The ADRC controllerparameters 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 fol-lows: Rγ = 5, Rθ = 2, and RQ = 2. The designed parameters ofthe 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 inthe following two scenarios.

Scenario 1. The prescribed performance-based active distur-bance rejection control (PP-ADRC) method proposed in thispaper with the robust back-stepping control method (RBC)from a previous study [44] was compared. The simulationtime was set to 100 s. VI and hI were selected as a step signalwith an amplitude of 200m/s. To verify the robustness of theproposed method, it was assumed that a 40% perturbation ofthe aerodynamic parameters existed in the AHV model,which is expressed as C = C0½1 + 0:4 sin ð0:1πtÞ�. C0 repre-sents the nominal value of the aerodynamic parameter andC represents the value of the aerodynamic parameter in thesimulation. In addition, an external disturbance was addedafter 50 s of simulation:

dV = dγ = dθ = dQ = 2 sin 0:1πtð Þ: ð71Þ

The simulation results for Scenario 1 are shown inFigure 3. Figures 3(a)–3(d) show that the velocity and alti-tude tracking errors were limited to the prescribed areawhen using the PP-ADRC. Compared with the RBC, thePP-ADRC can guarantee better dynamic performances ofthe velocity and altitude tracking errors. When there wasparameter perturbation and external disturbance, the PP-ADRC exhibited a higher control accuracy and strongerrobustness. Figure 3(e) shows that the flight-path angleresponse of the PP-ADRC was smoother than that of theRBC. Moreover, the PP-ADRC proposed in this paper couldestimate the unknown term of the model through the ESO(Figures 3(g) and 3(h)). The virtual reference commandand the differential signals could be effectively estimatedthrough the new TD (Figure 3(f)). Thus, the control accuracyof the method is further guaranteed.

Scenario 2. The PP-ADRC was compared with the neuralback-stepping control method (NBC) from a previous study[45]. The simulation time was set to 300 s. To better reflectthe actual manoeuvre of the AHV, VI was assumed to be a“step” type signal with a step of 150m/s per 100 s. hI wasassumed to be a square wave signal with a amplitude of200m and a period of 200 s. Also, it was assumed that the40% perturbation of the aerodynamic parameters existed inthe AHV model. The following definitions were made:

C =

C0, 0 s ≤ t < 50 s,

C0 1 + 0:4 sin 0:1πtð Þ½ �, 50 s ≤ t < 100 s,

C0, 100 s ≤ t < 150 s,

C0 1 + 0:4 sin 0:1πtð Þ½ �, 150 s ≤ t < 200 s,

C0, 200 s ≤ t < 250 s,

C0 1 + 0:4 sin 0:1πtð Þ½ �, 250 s ≤ t ≤ 300 s,

8>>>>>>>>>>><>>>>>>>>>>>:

ð72Þ

where C0 represents the nominal value of the aerodynamicparameter, and C represents the value of the aerodynamicparameter in the simulation. In addition, external distur-bances were added after 150 s of simulation: dV = dγ = dθ =dQ = 2 sin ð0:1πtÞ.

The simulation results of the Scenario 2 are shown inFigure 4. Figures 4(a)–4(d) show that the dynamic perfor-mances and control accuracy of the velocity and altitudetracking errors of the PP-ADRC were better than those ofthe NBC. Figure 4(e) shows that the flight-path angleresponse of the PP-ADRC was smoother than that of theNBC. Meanwhile, the PP-ADRC proposed in this papercould estimate the unknown term of the model through theESO (Figures 4(g) and 4(h)). The virtual reference commandand the differential signals could be effectively estimatedthrough the new TD (Figure 4(f)). Thus, the control accuracyof the method was further guaranteed. Figures 4(i)–4(l) showthat both of the control methods could achieve the effectivesuppression of the elastic vibrations.

5. Conclusions

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

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 —

10 International Journal of Aerospace Engineering

2750

2700

2650

2600

2550

2500

2450

�푉/(

m/s

)

0 2010 30 40 50�푡/�푠

60 70 80 90 100

VrefPP-ADRCRBC

(a) Velocity tracking performance

3

2

1

0

−1

−2

−3

-0.4-0.2

0.20

30 35 40

-pv(�푡)

pv(�푡)

�푉-�푉

ref/

(m/s

)

0 2010 30 40 50�푡/�푠

60 70 80 90 100

PP-ADRCRBC

(b) Velocity tracking error

2.725×104

2.72

2.715

2.71

2.705

2.7

2.695

h/m

0 2010 30 40 50�푡/�푠

60 70 80 90 100

href

PP-ADRCRBC

(c) Altitude tracking performance

15 0.04

−0.0455 60 65

0.02

−0.02010

5

0

−5

−10

−15

h-h

ref/

(m/s

)p

h(�푡)

-ph(�푡)

0 2010 30 40 50�푡/�푠

60 70 80 90 100

PP-ADRCRBC

(d) Altitude tracking error

0.25

0.2

0.15

0.05

0.1

0

−0.05

−0.1

/(

�허)

0 2010 30 40 50�푡/�푠

60 70 80 90 100

d

PP-ADRCRBC

(e) Flight-path angle

d�푣1

�푣2

0.2

0.15

0.1

0.05

0

−0.050 4020

�푡/�푠

60 80 100

The e

stim

atio

n of

d

and

d

(f) Estimation of γd and _γd

Figure 3: Continued.

11International Journal of Aerospace Engineering

50

40

30

20

10

0

−10

−20

−30

Zv2

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(g) FVðV ,ΦÞ

1

0

−1

−2

−3

−4

−5

ZQ

2

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(h) FQðx, δeÞ0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

�휂1

PP-ADRCRBC

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(i) Elastic state η1

1

0.8

0.6

0.4

0.2

0

−0.2

−0.6

0.4

−0.8

−1

�휂2

PP-ADRCRBC

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(j) Elastic state η2

3

2.5

2

�훷 1.5

1

0.5

0

PP-ADRCRBC

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(k) Fuel equivalence ratio

60

40

20

�훿e/(�허)

−20

0

−40

−60

−80

PP-ADRCRBC

0 2010 30 40 50 60 70 80 90 100�푡/�푠

(l) Elevator angular deflection

Figure 3: Simulation results for Scenario 1.

12 International Journal of Aerospace Engineering

3000

2900

2800

2700

2600

2500

24000 50 100 150 200 250 300

�푉/(

m/s

)

�푡/�푠

Vref

PP-ADRCNBC

(a) Velocity tracking performance

15

20

−2

0.20

−0.2

0.20

−0.2

5

10

0

−50

0 2 4 6 8 10 95 100 105 290 295 300

50 100 150 200 250 300

−pv(�푡)

pv(�푡)

�푉-�푉

ref/

(m/s

)

�푡/�푠

PP-ADRCNBC

(b) Velocity tracking error

2.725

2.72

2.715

2.71

2.705

2.7

2.6950 50 100 150 200 250 300

×104

h/m

�푡/�푠

href

PP-ADRCNBC

(c) Altitude tracking performance

Figure 4: Continued.

13International Journal of Aerospace Engineering

0 50 100 150 200 250 300

h-h

ref/

(m/s

) ph(�푡)

-ph(�푡)

�푡/�푠

PP-ADRCNBC

8

6

4

2

0

−2

−4

−6

−8

0.50

-0.5

0.020

−0.02

0.020

−0.02

0 2 4 6 8 10 95

290 292 294 296 298 300

100 105

(d) Altitude tracking error

0 50 100 150 200 250 300

0.2

0.15

0.1

0.05

0

−0.05

−0.1

−0.15

−0.2

/(

�허)

�푡/�푠

d

PP-ADRCNBC

(e) Flight-path angle

0 50 100 150 200 250 300

0.2

0.15

0.1

0.05

0

−0.05

−0.1

−0.15

−0.2

d

�푣1

�푣2

�푡/�푠

The e

stim

atio

n of

d

and

d

(f) Estimation of γd and _γd

Figure 4: Continued.

14 International Journal of Aerospace Engineering

0 50 100 150 200 250 300

30

2025

1015

05

−10

−5

−20

−15

Zv2

�푡/�푠

(g) FV ðV ,ΦÞ1

0

−1

−2

−3

−4

−50 50 100 150 200 250 300

ZQ

2

�푡/�푠

(h) FQðx, δeÞ0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.10 50 100 150 200 250 300

�휂1

PP-ADRCNBC

�푡/�푠

(i) Elastic state η1

Figure 4: Continued.

15International Journal of Aerospace Engineering

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.80 50 100 150 200 250 300

�휂2

PP-ADRCNBC

�푡/�푠

(j) Elastic state η2

3

2.5

2

1.5

1

0.5

00 50 100 150 200 250 300

�훷

PP-ADRCNBC

�푡/�푠

(k) Fuel equivalence ratio

60

40

20

0

−20

−40

−60

−800 50 100 150 200 250 300

�훿e/(�허)

PP-ADRCNBC

�푡/�푠

(l) Elevator angular deflection

Figure 4: Simulation results for Scenario 2.

16 International Journal of Aerospace Engineering

proposed in this paper. The proposed method guar-antees the stability of the AHV closed-loop controlsystem. The desired dynamic and steady-state perfor-mances of the convergence process of the trackingerror were ensured

(2) In the controller design process, the adopted activedisturbance rejection method and extended stateobserver further enhanced the capacity to resist thedisturbances, which guaranteed the robustness ofthe method

(3) The simulation results in the paper proved the effec-tiveness of the proposed method. The comparisonwith the related publications showed that thedynamic and steady-state performances of the pro-posed method were superior

Appendix

A.Proof of Theorem 2

Based on Equations (20)–(22), we set E∗ = Z2 −HðtÞ = Z2 −z0. The first derivatives of E and E∗ with respect to timecan be expressed as follows:

E = Z1 − z, E∗ = Z2 − z0,_E = _Z1 − _z = Z2 − z0 − β01E = E∗ − β01E,

_E∗ = _Z2 − _z0 = −G tð Þ − β02 Ej jα1 sgn Eð Þ:

8>><>>: ðA:1Þ

The following Lyapunov function was selected:

V =β02

α1 + 1⋅ Ej jα1+1 −

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s

⋅ Ej j α1+1ð Þ/2 sgn Eð Þ ⋅ E∗ + 12E∗2 ,

ðA:2Þ

because

V =β02

α1 + 1⋅ Ej jα1+1 −

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej j α1+1ð Þ/2 sgn Eð Þ ⋅ E∗ +

12E∗2

=

ffiffiffiffiffiffiffiffiffiffiffiffiβ02

α1 + 1

s⋅ Ej j α1+1ð Þ/2 sgn Eð Þ −

ffiffiffi12

rE∗

!2

≥ 0:

ðA:3Þ

V is positive semidefinite. Below, _V is studied. The partialderivative of V with respect to E and E∗ can be expressed asfollows:

∂V∂E

= β02 Ej jα1 sgn Eð Þ −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2E∗,

∂V∂E∗ = −

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej j α1+1ð Þ/2 sgn Eð Þ + E∗:

ðA:4Þ

The first derivative ofV with respect to time was obtained,and with Equations (70)–(A.1), this yields the following:

_V =∂V∂E

_E +∂V∂E∗

_E∗ = β02 Ej jα1 sgn Eð Þ

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2E∗

!E∗ − β01Eð Þ

+ −

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej j α1+1ð Þ/2 sgn Eð Þ + E∗

!

� −G tð Þ − β02 Ej jα1 sgn Eð Þð Þ

= −β01β02 Ej jα1+1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2E∗2

+ β01

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2E ⋅ E∗ +

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s

⋅ Ej j α1+1ð Þ/2 sgn Eð ÞG tð Þ + β02

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej j α1+1ð Þ/2 Ej jα1

= −β01β02 Ej jα1+1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2E∗2

+ β01

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1+1ð Þ/2 sgn Eð ÞE∗

+

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej j α1+1ð Þ/2 sgn Eð ÞG tð Þ

+ β02

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s⋅ Ej jα1+1 Ej j α1−1ð Þ/2

≤ − β01β02 − β02

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

sEj j α1−1ð Þ/2

!Ej jα1+1

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2

!E∗2

+β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rE∗2 +

β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj jα1+1

+12

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

sEj jα1+1 + 1

2

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

sG2 tð Þ + 1

2E∗2 +

12G2 tð Þ

= −

β01β02 − β02

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

sEj j α1−1ð Þ/2

−β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

r−12

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s !Ej jα1+1

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2 −

β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

r−12

!E∗2

+12

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s+12

!G2 tð Þ:

ðA:5Þ

17International Journal of Aerospace Engineering

Values of β01, β02, and α1 are selected, yielding:

β01β02 − β02

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

sEj j α1−1ð Þ/2

−β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

r−12

ffiffiffiffiffiffiffiffiffiffiffiffi2β02α1 + 1

s> 0,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

rEj j α1−1ð Þ/2

−β012

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ02 α1 + 1ð Þ

2

r−12> 0:

ðA:6Þ

The following compact set is defined as follows:

According to Equation (A.4), if E ∉ΩE or E∗ ∉ΩE∗ , then

_V < 0.V will decrease until E and E∗ converge to the compactsets ΩE and ΩE∗ , respectively. Thus, the error system (Equa-tion (70)) is semiglobally uniformly asymptotically stable. Eand E∗ are uniformly asymptotically bounded. However, ifthe radii of the compact sets ΩE and ΩE∗ can be made suffi-ciently small by selecting appropriate values of β01, β02, andα1, the errors E and E∗ will be sufficiently small, and theremust be bounded constants σ1, σ2 > 0 such that:

Z1 − zj j ≤ σ1,

Z2 −H tð Þj j ≤ σ2:

(ðA:8Þ

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural ScienceFoundation of China (Grant no. 61573374 and no.61703421). The funding did not lead to any conflict ofinterests.

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ΩE = E Ej j ≤1/2ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2β02ð Þ/ α1 + 1ð Þp+ 1/2ð Þ

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9>=>;,

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2β02/ α1 + 1ð Þð Þp+ 1/2ð Þ

� �G2 tð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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β02 α1 + 1ð Þ/2ð Þp− 1/2ð Þ

vuut�������

8><>:

9>=>;:

ðA:7Þ

18 International Journal of Aerospace Engineering

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20 International Journal of Aerospace Engineering

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