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Research ArticleA Novel Linear Active Disturbance Rejection Control Design forAir-Breathing Supersonic Vehicle Attitude System withPrescribed Performance

Chao Ming and Xiaoming Wang

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Chao Ming; [email protected]

Received 1 December 2019; Revised 5 June 2020; Accepted 13 June 2020; Published 1 August 2020

Academic Editor: Franco Bernelli-Zazzera

Copyright © 2020 Chao Ming and Xiaoming Wang. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

This paper investigates the design problem of the attitude controller for air-breathing supersonic vehicle subject to uncertaintiesand disturbances. Firstly, the longitudinal model is established for the attitude controller design which is devised as a strictfeedback formulation, and a transformed tracking error is derived with the prescribed performance control technique such thatit can limit the tracking error to a predefined region. Then, a novel linear active disturbance rejection control scheme isproposed for the attitude system to enhance the steady-state and transient-state performances by incorporating the transformedtracking error. On the basis of the Lyapunov stability theorem, the convergence and stability characteristics are both rigorouslyproved for the closed-loop system. Finally, extensive contrast simulations are conducted to demonstrate the effectiveness,robustness, and advantage of the proposed control strategy.

1. Introduction

Hypersonic technique is a wide development prospect inboth military and civil fields [1, 2]. As a derivative of hyper-sonic technique, air-breathing supersonic vehicle (ASV) is anew type of aircraft with large airspace, super flight velocity,long range, and high precision compared with the traditionalaircraft [3], and flight guidance-control design is a key tech-nology in the ASV system. However, as the special structureand changeable flight conditions, the vehicle dynamics arepeculiar including the fast time varying, nonlinearity, uncer-tainty, and multiple disturbances [4], which lead to more dif-ficulties and complexities in the control design and analysis.In addition, the ASV usually attacks the target at a super-sonic velocity such that the flight states are changed rapidlywhich imposes a higher requirement for the transient char-acteristic of the control system [5]. Therefore, the controlperformance index such as the overshoot, steady-state error,convergence rate, and robustness must be considered themajor designed indexes in the control system design of suchflight vehicle.

In the past few years, the air-breathing vehicle techniquehas attracted the wide attention of domestic and foreign sci-entific research institutions and scholars due to the afore-mentioned superiorities, and various control methods havebeen explored for such vehicles, such as linear parametervarying (LPV) method [6], dynamic inversion [7, 8], trajec-tory linearization control [9, 10], fuzzy control method[11], dynamic surface control [12, 13], back-stepping controlmethod [14, 15], neural network [16, 17], and sliding modecontrol [18–21].

It is worth noting that the above listed control methodachieved an outstanding control effect for air-breathing vehi-cle, but most mainly fasten the attention on the stability,robustness, and accuracy of the control system. However,the situation of autodisturbance rejection is seldom consid-ered in control design. As we all know that the active distur-bance rejection control (ADRC) method can achieve asatisfactory performance for nonlinear systems, where theparameter perturbations and external disturbances can beestimated and rejected actively, it is successfully applied inindustrial control [22, 23]. However, there is an inevitable

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 1676739, 15 pageshttps://doi.org/10.1155/2020/1676739

problem that many parameters are needed to be tuned whichlimits the application of ADRC in practice. In this connec-tion, a linear active disturbance rejection control (LADRC)was firstly designed by Prof. Gao [24] which is extremely sim-ple and easily implementable and it has been widely extendedto various industrial control fields, such as electric erectionsystem [25], hovercraft system [26], servo systems [27], windpower systems [28], and photobioreactor [29]. Meanwhile,[30, 31] adopt the LADRC approach to design the attitudecontrol for a supersonic missile. As far as we know, thereare few literatures which concern with the attitude controlsystem design by adopting the LADRC method. Further-more, in the design process of the above LADRC scheme,both did not consider the transient performance design.Therefore, it is vitally essential for the ASV attitude controlsystem design that can guarantee the system for both thesteady-state performance and transient performance withLADRC method.

Recently, a newly emerging control method called theprescribed performance control (PPC) for the nonlinear sys-tem was firstly proposed by Na et al. [32], where the trackingerror can converge to an arbitrarily predefined small set andthe convergence rate and maximum overshoot can be delim-ited less than a prespecified constant. Owing to the significantadvantages in improving the control performance, the PPC isintroduced to the vehicle suspension system [33], manipula-tor system [34], servo mechanisms [35, 36], unmanned aerialvehicle [37], spacecraft [38–40], etc., and several new ideas ofthe PPC design are emerged; [41] proposes a new error trans-formation method and the performance function so that thelimitation of PPC on the known initial error can be relaxed.[42] investigates a new PPC methodology for the longitudi-nal dynamic model of an air-breathing hypersonic vehiclevia neural approximation that the satisfactory transient per-formance with small overshoot can be achieved. [43] presentsa new error transformation to reduce the complexity of thecontrol system which is caused by conventional error con-straint approaches. Berger [44] proposes a novel prescribedperformance controller which can guarantee the output tostay within a prescribed performance funnel bound by incor-porating the funnel control with the PPC technique. [45]designs a new prescribed performance controller withoutan approximation structure which can avoid the largeamount of calculation and some specific problems of thefuzzy or neural network method in the approximation pro-cess. Although extensive achievements have been yielded intheory and application of the PPC technique, the previouscontroller design methods are mainly neural network con-trol, backstepping control, dynamic surface control, and slid-ing model control. To our best knowledge, no results on theLADRC design with the PPC technique have been reportedfor nonlinear systems.

Motivated by the aforementioned discussions, this studyinvestigates a novel LADRC scheme for the longitudinal atti-tude mode of the air-breathing supersonic vehicle in thepresence of uncertain dynamics and external disturbanceswith a prescribed performance constraint. The main contri-butions of this paper are summarized as follows: (1) A novelLADRC approach design method with a prescribed perfor-

mance constraint is firstly proposed for the ASV attitude sys-tem with multiple disturbances. (2) The proposed controlscheme does not need the knowledge of the flight dynamicmodel, and the uncertainty and disturbance can be activelyestimated and compensated into the control signal. (3) Theproposed controller can improve the steady-state and tran-sient performances of the ASV attitude system comparedwith the traditional LADRC method. (4) The system stabilityand convergence characteristic are both proved strictly.

The rest of this paper is outlined as follows. The consid-ered ASV attitude control model is established and the priorknowledge and problem formulation is given in Section 2.Section 3 presents the LADRC-based prescribed perfor-mance attitude controller design procedures. Comparativesimulation results are provided in Section 4. Finally, someconclusions of this work are presented in Section 5.

2. Problem Statement and Preliminaries

In the section, the considered ASV attitude dynamic model ispresented; then, some basic definitions of the prescribed per-formance control method are provided for the subsequentanalysis, and finally, the objective of controller design isoutlined.

2.1. Vehicle Attitude Model. For the subsequent design, thenonlinear longitudinal model of the ASV attitude system inthis paper is derived on the basis of the model which includesthe altitude, velocity, and attitude motion of the hypersonicvehicle modeled in [46, 47]. In addition, the main purposeis to investigate a novel attitude control method of ASV,and the attitude dynamics belongs to a fast motion comparedwith the change of velocity and altitude. With design simpli-fication and no loss of generality, the changes of altitude andvelocity can be ignored due to the limited influence on atti-tude dynamics. Hence, the ASV attitude system can be sim-plified as follows:

_ϑ = ωz ,

_ωz =Mz

Jz,

α = ϑ − θ,

8>>>><>>>>:

ð1Þ

where the pitch moment Mz is defined as

Mz = �qSL m0 +mαzα +mα2

z α2� �

+ �qSLmδzz δz , ð2Þ

where elevator deflection δz is the control input, and m0,mαz ,

mα2z , andmδz

z are the aerodynamic coefficients.For simplicity, defining the state ½x1, x2� = ½ϑ, ωz� and

combining with (1) and (2), the ASV attitude control systemcan be described with the strict feedback state-space form:

_x1 = x2,

_x2 = f + bu,

(ð3Þ

2 International Journal of Aerospace Engineering

where

f =�qSL m0 +mα

zα +mα2z α2

� �Jz

+ Δ

b =�qSLmδz

z

Jz

u = δz

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

ð4Þ

Here, the term f is described as “total disturbance,”which consists of the unknown dynamic uncertainty andexternal disturbance.

Remark 1. The uncertainty term Δ is mainly the resultof parameter uncertainties and external disturbances,e.g., the aerodynamic parameters perturbations and windinterference.

Assumption 1. The total disturbance f is differentiable, andits derivative _f is described as _f = hðx1, x2, ΔÞ . The distur-bance Δ in system (3) is unknown, but the disturbance andits derivative are all bounded.

2.2. Prescribed Performance Theory. The prescribed perfor-mance denotes that the prescribed transient and steady-state performances can strictly limit the tracking error to apredesigned small residual set. Based on the prescribed per-formance concepts [31], the prescribed performance can beobtained in the condition that the tracking error eðtÞ evolvesin the predefined bounds with a decreasing smooth functionas follows:

−ρ tð Þ < e tð Þ < ρ tð Þ, ∀t > 0: ð5Þ

The prescribed performance function (PPF) ρðtÞ isselected as [38]

ρ tð Þ = ρ0 − ρ∞ð Þ exp −κtð Þ + ρ∞, ð6Þ

where ρ0 > ρ∞ > 0 and κ > 0. The maximum overshoot,steady-state error, and convergence rate will be limited byselecting the appropriate parameters ρ0, ρ∞, and κ, respec-tively. For a more intuitive insight of the concept, Figure 1illustrates the schematic diagram of the aforementioned pre-scribed performance theory.

It is worth to mention that the complexity of the control-ler design will be significantly increased by directly adopting(5) and (6) which correspond to an additional constraint ofthe controlled system. In order to evade this problem, the sys-tem with constrains can be converted to an equivalent disen-gaged one by introducing an error transformation functionTð⋅Þ, i.e.,

e tð Þ = ρ tð ÞT ε tð Þð Þ, ð7Þ

where εðtÞ is the transformed tracking error, and Tð⋅Þ issmooth and strictly increasing which satisfies the conditions−1 < TðεÞ < 1 and lim

ε→+∞TðεÞ = 1, lim

ε→−∞TðεÞ = −1.

Here, we design the error transformation function Tð⋅Þ as

T εð Þ = exp εð Þ − exp −εð Þexp εð Þ + exp −εð Þ : ð8Þ

According to the property of the function Tð⋅Þ, we caninversely derive the error εðtÞ as follows:

ε tð Þ = T−1 λð Þ = 12ln 1 + λ1ð Þ − ln 1 − λ1ð Þ½ �, ð9Þ

where λ = eðtÞ/ρðtÞ is the normalized tracking error.

Remark 2. In the subsequent parts, the transformed error εðtÞwill be used into the controller in place of the tracking erroreðtÞ to deal with the problem of control system design withprescribed performance constraint.

2.3. Control Objective. The control objective is the proposednovel controller u for the ASV attitude system (3) that thestate x1 track the desired command x1c accurately, and thetracking error e1 = x1c − x1 can be limited within a predefinedbound with a satisfactory prescribed performance in spite ofmultiple disturbances including unmodeled dynamics,uncertainties, and external disturbances, i.e., the objectivecan be expressed as follows:

(1) The state x1 can accurately track the desired com-mand x1c with the unknown multiple disturbances

(2) The output tracking error e1 is stabilized at the originwith a prescribed maximum overshoot, the steady-state error, and convergence rate

(3) The closed-loop system states are both stable androbust to the uncertainties and disturbances

3. Control System Design and Stability Analysis

In this part, a novel ASV attitude controller is proposed basedon the LADRC method by introducing the PPC technique,which can improve the steady-state and transient perfor-mances of the ASV attitude control system. Then, the

e (0)

0

e (t)

t (s)

𝜌 (t)

–𝜌 (t)

–𝜌0

𝜌0

–𝜌∞

𝜌∞

Figure 1: The prescribed performance illustration.

3International Journal of Aerospace Engineering

convergence property and stability analysis for the controlsystem are provided on the basic of the Lyapunov method.

3.1. Attitude Control System Design. The block diagram of thepresented control system is depicted in Figure 2, whichmainly consists of three parts: prescribed performance con-trol (PPC), linear extended state observer (LESO), and linearstate error feedback (LSEF).

In view of the previous description, the design process ofthe ASV attitude control system is illustrated as follows:

Step 1.The LESO is the core part of the LADRC; it can generate

the estimation of the states and the disturbances in real time;the estimated value can be used to compensate the distur-bances to the controller which can enhance the robustnessof the system. According to system (3), a three-order LESOis constructed as follows:

_z1 = z2 + β1 x1 − z1ð Þ,_z2 = z3 + β2 x1 − z1ð Þ + bu,

_z2 = β3 x1 − z1ð Þ,

8>><>>: ð10Þ

where z1, z2, and z3 are the observer states, and β1, β2, and β3are the designed gains which can be chosen with the pole-placement method [24] as follows:

β1 = 3ω0,

β2 = 3ω20,

β3 = ω30,

ð11Þ

where ω0 is the observer bandwidth.With the properly selected gains, z1, z2 will converge to

the system states x1 and x2, respectively, and z3 will accu-rately track the total disturbance term f , i.e., z1 ⟶ x1,z2 ⟶ x2, z3 ⟶ f .

Note that the LESO can estimate the states and totaldisturbance exactly without the mathematical precisemodel, it is only dependent of the system input and outputinformation.

Step 2.The PPC can transform the tracking error into an equiv-

alent unconstrained one by incorporating the transformation

function and the prescribed performance function such thatthe tracking error can be limited in the envelope of the pre-scribed performance bounds (PPB).

Here, we define the error ~e1 as

~e1 = x1c − z1, ð12Þ

where x1c is the desired command and z1 is the estimationvalue of x1 obtained by LESO.

Subsequently, the normalized error λ1 is given by

λ1 =~e1

ρ1 tð Þ , ð13Þ

where ρ1ðtÞ is a similar PPF defined in (6), and it is expressedas

ρ1 tð Þ = ρ10 − ρ1∞ð Þ exp −κ1tð Þ + ρ1∞, ð14Þ

whereρ10,ρ1∞, and κ1 are all positive constants.Then, the transformed error is obtained by

ε1 = T−1 λ1ð Þ = 12ln 1 + λ1ð Þ − ln 1 − λ1ð Þ½ �: ð15Þ

Step 3.Based on the accurate estimation and the designed trans-

formed error, the LSEF can approximately simplify the sys-tem to a disturbance-free form meanwhile reconciling theperformance of the system. The final control law is designas follows:

u = u0 + uf , ð16Þ

where term u0 denotes a linear state error feedback controlterm that guarantees the system is asymptotically stable,and term uf represents the dynamic compensation controlterm to suppress the unfavourable consequence of the totaldisturbance such that it can enhance the robustness of theASV attitude control system.

Here, the control subitem u0 adopted the linear pro-portional and derivative (PD) control framework by

Errornormalized

Errortransformed

PPC

LSEF

Totaldisturbances

PPF

PPC-based LADRC scheme

LESO

ASVattitudedynamic

e1 𝜆1 𝜀1 𝛿zx1c

z1z2 z3

𝜌1 (t)

Figure 2: The block diagram of the presented controller.

4 International Journal of Aerospace Engineering

introducing the transformed error ε1 of PPC which isdesigned as follows:

u0 =kpε1 − kdz2

b, ð17Þ

where kp, kd are the controller gains and z2 is the estima-tion value of x2 obtained by LESO, and expression of b isgiven in (3).

Meanwhile, the control subitem uf compensates the dis-turbances with the estimated value z3, which is given by

uf = −z3b: ð18Þ

Thus, the controller u is obtained in terms of (16), (17),and (18) as follows:

u =kpε1 − kdz2 − z3

b: ð19Þ

Note that the controller u0 introduces the transformed errorε1 instead of ~e1 which can enhance the transient and steady-state performances with the PPC technique.

3.2. Stability Analysis. In this subsection, the observer con-vergence and the closed-loop system stability will be analysedwith the listed theorem. Prior to investigating, the followinglemmas are introduced.

Lemma 1. For system _ηðtÞ =NηðtÞ + gðtÞ, where N is a n × nmatrix, and ηðtÞ = ½η1ðtÞ, η2ðtÞ,⋯, ηnðtÞ�, gðtÞ = ½g1ðtÞ,g2ðtÞ,⋯, gnðtÞ�, if lim

t→∞kgðtÞk = 0 and N is Hurwitz, then

limt→∞

kηðtÞk = 0 holds.

This lemma has been proofed in detail which can be seenin [48].

Lemma 2. If the transformed error εðtÞ can be controlled to bebounded, i.e., the condition jεðtÞj ≤ εM holds where εM > 0 isconstant; then, the tracking error eðtÞ can be controlled withina prescribed boundary, i.e., −ρðtÞ < eðtÞ < ρðtÞ, ∀t > 0 holds.

This lemma has been proofed in detail which can be seenin [38].

Theorem 1. For system (3) and LESO (10), if _f = hðx1, x2, ΔÞsatisfies the Lipschitz condition in the definition domain, thereexist a constant ω0 > 0 such that the estimated states z1, z2,and z3 can converge to the state x1, x2, and f , respectively,i.e., the observer errors satisfy that lim

t→∞ez iðtÞ = 0, i = 1, 2, 3,

where the observer errors are defined as ez1 = x1 − z1, ez2 = x2− z2, and ez3 = f − z3.

Proof. Define the observer errors as

ez1 = x1 − z1,

ez2 = x2 − z2,

ez3 = f − z3:

ð20Þ

From (3) and (10), the observer error dynamic can begiven by

_ez1 = −3ω0 ez1 + ez2,

_ez2 = −3ω20 ez1 + ez3,

_ez3 = −ω30 ez1 + _f :

8>><>>: ð21Þ

Let ~e = ½~e1,~e2,~e3� = ½ez1, ez 2/ω0, ez 3/ω02� and _f = hðx1,

x2, ΔÞ, then (21) can be rewritten as follows:

_~e = ω0~A~e + ~B h x1, x2, Δð Þ

ω03 , ð22Þ

where ~A =

−3 1 0

−3 0 0

−1 0 0

2664

3775, ~B =

0

0

1

2664

3775.

As ~Asatisfies the Hurwitz stability, the condition P~A +~ATP = −I holds, where the matrix I is 3-order identity matrixand P is a positive definite Hermitian matrix and isexpressed as

P =

1 −12

−1

−12

1 −12

−1 −12

4

26666664

37777775: ð23Þ

Design the following Lyapunov function candidate

V1 = ~eTP~e: ð24Þ

Substituting (23) into (24) yields

V1 = ~e21 +~e22 + 4~e23 −~e1 ~e2 − 2~e1 ~e3 − ~e2 ~e3: ð25Þ

The time derivative of V1 along (25) is

_V1 = 2~e1 _~e1 + 2~e2 _~e2 + 8~e3 _~e3 − _~e1~e2 −~e1 _~e2− 2_~e1~e3 − 2~e1 _~e3 − _~e2~e3 − ~e2 _~e3,

= −ω0 ~e21 +~e22 +~e

23

� �+ −2~e1 −~e2 + 8~e3ð Þ h x1, x2, Δð Þ

ω02

≤ −ω0 ~ek k2 + −2~e1 −~e2 + 8~e3ð Þ h x1, x2, Δð Þj jω0

2 :

ð26Þ

5International Journal of Aerospace Engineering

The term hðx1, x2, ΔÞ satisfies [49].

h x1, x2, Δð Þj j ≤ k ~ek k, ð27Þ

where k > 0 is constant.Meanwhile, as the equation −2~e1 −~e2 + 8~e3 = 2~eT P~B

holds, we can obtain that

−2~e1 −~e2 + 8~e3ð Þ h x1, x2, Δð Þj jω0

2 ≤ 2~eTP~Bk ~ek kω0

2 : ð28Þ

In addition, the following inequality holds

P~Bk�� ��2 − 2P~Bk + 1 ≥ 0: ð29Þ

Thus, we obtain

2~eTP~Bk ~ek kω0

2 ≤ K ~ek k2, ð30Þ

where K = ðkPBZkk2 + 1Þ/ω02.

Based on (26), (28), and (30), we can obtain that

_V1 ≤ − ω0 − Kð Þ ~eZk k2: ð31Þ

That is, if ω0 > K such that _V1 < 0. Therefore, limt→∞

ez iðtÞ =0, i = 1, 2, 3 can be achieved, i.e., the estimation errors of theLESO (10) are asymptotically stable. The proof of Theorem1 is completed.

Theorem 2. For system (3), LESO (10), and the proposedcontroller (19) with the prescribed performance function(14), if _f = hðx1, x2, ΔÞ satisfies the Lipschitz condition indefinition domain, there exists constants ω0 > 0 and ωc > 0that can guarantee the system (3) to be asymptotically stable.Furthermore, the tracking error e1 can be maintained in a pre-defined set, i.e., the inequality −ρ1ðtÞ < e1ðtÞ < ρ1ðtÞ holds.

Proof. Define the tracking error of system (3) as

e1 = x1c − x1,

e2 = x2c − x2:ð32Þ

As the control command x1c is assumed as a constant, wehave x2c = _x1c = 0. The time derivative of e1 and e2 along (3) is

_e1 = _x1c − _x1 = −x2 = e2, ð33Þ

_e2 = _x2c − _x2 = − _x2 = −f − bu: ð34ÞBased on the expression of the transformed error ε1, it

yields

ε1 =12ln 1 + λ1ð Þ − ln 1 − λ1ð Þ½ �: ð35Þ

Then, the function ln ð1 + λ1Þ and ln ð1 + λ1Þ can bederived using Taylor’s expansion as follows:

ln 1 + λ1ð Þ = λ1 −12λ21 + R2 λ1ð Þ, ð36Þ

ln 1 − λ1ð Þ = −λ1 +12λ21 + R2 λ1ð Þ, ð37Þ

where R2ðλ1Þ is the 2-order Taylor remainder. As the error λ1is relatively small near zero, thus the R2ðλ1Þ can be neglected.

Substituting (36) into (35), the transformed error ε1 canbe approximately expressed as

ε1 = λ1 =x1c − z1ρ1 tð Þ : ð38Þ

Substituting (38) into (19), the proposed controller (18)can be rewritten as

u =kp x1c − z1/ρ1 tð Þð Þ − kdz2 − z3

b: ð39Þ

Substituting (39) into (34) yields

_e2 = −f − bu = −kpx1c − z1ρ tð Þ + kd z2 + z3 − f : ð40Þ

Combining the expressions of the observer error and sys-tem error, we can obtain that

x1c − z1 = e1 + ez1, ð41Þ

z2 = − e2 + ez2ð Þ, ð42Þz3 − f = −ez3: ð43Þ

Substituting (41) into (40) yields

_e2 = −kpρ tð Þ e1 + ez1ð Þ − kd e2 + ez2ð Þ − ez3: ð44Þ

From (33) and (44), the tracking error model is given by

_e =Ae e + Be ez , ð45Þ

where e = ½e1, e2�T and ez = ½ez1, ez2�T are the system trackingerror vector and the observer error vector, respectively. Thematrix Ae and Be are described as

Ae =0 1

−kp

ρ1 tð Þ −kd

264

375,

Be =0 0 0

−kp

ρ1 tð Þ −kd −1

264

375:

ð46Þ

6 International Journal of Aerospace Engineering

Meanwhile, the prescribed performance function ρ1ðtÞsatisfies

limt→∞

ρ1 tð Þ = limt→∞

ρ10 − ρ1∞ð Þ exp −κ1tð Þ + ρ1∞½ � = ρ1∞:

ð47Þ

Thus, under the condition t⟶∞, the matrix Ae and Becan be given as

Ae =0 1

−kpρ1∞

−kd

264

375,

Be =0 0 0

−kpρ1∞

−kd −1

264

375:

ð48Þ

According to Theorem 1, we can obtain that limt→∞

kezk = 0.Furthermore, if the characteristic polynomial s2 + kds + kp/ρ1∞ satisfies the Routh criterion, here, we select s2 + kds +kp/ρ1∞ = ðs + ωcÞ2, where ωc > 0 is the controller bandwidth,i.e., kp = ρ∞ω2

c , kd = 2ωc such that the matrix Ae is Hurwitz.Based on Lemma 1, we can obtain that the lim

t→∞eiðtÞ = 0, i =

1, 2 holds, i.e., the system (3) is asymptotically stable.Besides that, we can obtain the following equation by

conducting the inverse operation based on the transformederror (15) as

exp 2ε1ð Þ = 1 + λ11 − λ1

: ð49Þ

Based on Lemma 2, if the transformed errors ε1 = T−1ðλ1Þ can be controlled to be bounded, i.e., jε1j ≤ εM1 holdsfor positive constants εM1 > 0. This further implies

−1 <exp −εM1ð Þ − 1exp −εM1ð Þ + 1

≤ λ1 ≤exp εM1ð Þ − 1exp εM1ð Þ + 1

< 1: ð50Þ

From the fact λ1 = e1ðtÞ/ρ1ðtÞ and ρ1ðtÞ > 0, the error e1can be maintained within a predefined set, −ρ1ðtÞ < e1ðtÞ <ρ1ðtÞ holds. This completes the proof.

4. Simulation Results and Analysis

In order to evaluate the performance of the proposed ASVattitude controller, three numerical simulation cases are con-ducted in different configurations and scenarios.

4.1. Contrast Scheme. In order to demonstrate effectivenessand superiority of the proposed controller with the pre-scribed performance control-based LADRC (LADRC-PPC)method, the LADRC method [30] is introduced to the simu-

lation scenarios for comparison study, and the controller isdevised as follows:

uLADC =kp x1c − z1ð Þ − kdz2 − z3

b, ð51Þ

where z1, z2, and z3 is obtained by LESO.

4.2. Flight Conditions. The vehicle is assumed to maintainaltitude H = 12 km cruise at constant velocity V = 1000m/ssuch that the flight path angle is θ = 0 deg, i.e., the variationof the angle of attack α is the same as that of the pitch angleϑ based on the system (1). The aerodynamic coefficients arem0 = 0:0005, mα

z = −0:018 deg−1, mα2z = −0:003 deg−2 , and

mδzz = −0:035 deg−1. The initial conditions for the ASV are

set as ϑð0Þ = 0 deg, ωzð0Þ = 0 deg/s.

4.3. Controller Parameters. The relevant parameters of PPFis designed as ρ10 = 1:5, ρ1∞ = 0:05, and κ1 = 5, i.e., thePPF is selected as ρ1ðtÞ = ð1:5 − 0:05Þ exp ð−5tÞ + 0:05. Theobserver bandwidth ω0 is set as ω0 = 40, and controllerbandwidth ωc is designed as ωc = 10. Then, the observergains β1, β2, and β3 and controller gains kp, kd can beobtained according to the correspondence with the band-width ω0 and ωc.

4.4. Evaluation Index. For quantitatively contrasting the per-formance, the following performance indexes are introducedto evaluate the above control schemes.

(1) Convergence time (CT) tc of the tracking error (TE)

The convergence time tc is assumed as the corre-sponding time that the tracking error e1 is e1 ≤ 10−6and continues at least 10 sampling periods.

(2) Average value (AV) μe of the TE

μe =1n〠n

i=1e1 ið Þj j, ð52Þ

where n is the sample point number

(3) Standard deviation (SD) σe of the TE

σe =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n〠n

i=1e1 ið Þj j − μe½ �2

sð53Þ

(4) Amount of control consumption (ACC) Q

Q =ðt ft0

u2dt, ð54Þ

where t0 and t f are the start time and end time,respectively, u is the control input.

7International Journal of Aerospace Engineering

Case 1. This simulation is conducted in a standard condition.The aerodynamic coefficients are set to be the nominal valueswithout external disturbance, i.e., the term Δ in system (3) isΔ = 0.

The comparative results are summarized in Table 1 andFigures 3–10.

As depicted in Figures 3 and 4, the ASV can track thedesired attitude command with the listed control approaches

Table 1: The comparative simulation results of Case 1.

Index LADRC LADRC-PPC Ratio

CT tc 2.65 s 0.99 s 2.57 : 1

AV μe 0.07 deg 0.03 deg 2.33 : 1

SD σe 0.18 deg 0.14 deg 1.28 : 1

ACC Q 1.26 1.30 1 : 1.03

1.2

1

0.8

0.6

0.4

0.2

𝜗 (d

eg)

00 1 2 3

t (s)4

LADRCLADRC-PPC

5

Figure 3: Pitch angle.

Trac

king

erro

r of 𝜗

(deg

)

t (s)

1.5

1

0.5

0

–0.5

–1

–1.50 1 2 3 4 5

LADRCLADRC-PPCPPF

Figure 4: Tracking error of pitch angle.

𝜔z (

deg/

s)

t (s)

5

4

3

2

1

0

–10 1 2 3 4 5

LADRCLADRC-PPC

Figure 5: Pitch angular rate.

𝛿z (

deg)

t (s)

0

–0.2

–0.4

–0.6

–0.8

–1

–1.2

–1.40 1 2 3 4 5

LADRCLADRC-PPC

Figure 6: Control input signal-elevator deflection.

0

–0.2

–0.4

–0.6

–0.8

–10 1 2 3

t (s)4

Actual - LADRC

5

Estimated - LADRC

f (d

eg/s

2 )

Figure 7: Estimation of f with LADRC.

8 International Journal of Aerospace Engineering

accurately. It is worth noting that the tracking error is limitedin a prescribed set with a fairly satisfactory tracking errorresponse by utilizing the presented LADRC-PPC controller.As listed in Table 1, the tracking error remains remarkablysmall with a faster rate and a much smaller average errorwhich the average error is 0.03 deg, and converges to theneighbourhood of zero in approximately 0.99 s with theLADRC-PPC controller. However, the corresponding valueswith the LADRC scheme are 0.07 deg and 2.55 s, respectively.That is, the steady-state and transient performances of theASV attitude system under the designed LADRC-PPC con-troller is obviously superior than the contrast LADRC whichis owing to the help of introducing the PPC technology. Wecan see that initial value with the LADRC-PPC controller isslightly larger than the one with LADRC which is due tothe faster convergence in the initial phase with the LADRC-PPC approach as shown in Figures 5 and 6, and the amountof control consumption under the two schemes is virtuallyidentical, i.e., the novel proposed LADRC-PPC scheme canachieve an excellent control performance with approximatelythe same control consumption in comparison with theLADRC method. Figures 7 and 8 illustrate the estimationsof disturbances; it can be seen that LESO can precisely andrapidly estimate the total disturbances of the system, andthe convergence rate of LESO under the LADRC-PPC is fas-ter than that of LADRC as show in Figures 9 and 10.

Case 2. This case is performed in a perturbed condition toverify the robustness of the proposed method. The uncer-tainty term Δ is mainly considered two parts: (1) the modeluncertainties Δ1, and the aerodynamic coefficients are pre-sumed to decrease by 10% on the basis of standard values,the density of air ρ is perturbed to be +5% and (2) externaldisturbance Δ2, and it is considered an abrupt one Δ2 = 0:1deg/s2ðt ≥ 3Þ, i.e., the uncertainty term Δ is set as follows

Δ =Δ1 t < 3ð ÞΔ1 + 0:1 t ≥ 3ð Þ

( deg/s2: ð55Þ

The comparative results are summarized in Table 2 andFigures 11–18.

From Figure 11, the ASV attitude system can track thedesired command successfully by using the mentioned twocontrol approaches in the presence of unknown dynamics,uncertainties, and disturbances. The tracking error responseof the system with the proposed LADRC-PPC scheme is alsobetter than the classic LADRC method in Figure 12. Espe-cially, when the abrupt external disturbance occurs, thetracking error under the LADRC-PPC controller can con-verge to a compact set less than 0.5 s, and the correspondingconvergence time under the LADRC approach is nearly 2.0 s.That is, the transient performance under the LADRC-PPCcontroller is significantly improved than the contrast LADRCin spite of unknown uncertainties and disturbances. FromFigures 13 and 14, it can be seen that the variations of thestates under the perturbed condition are generally accordantwith the standard condition. Meanwhile, the amount of con-trol consumption under the two controller is similar. In

0.2

0

–0.2

–0.4

–0.6

–0.8

–10 1 2 3

t (s)4 5

Actual - LADRC-PPCEstimated - LADRC-PPC

f (d

eg/s

2 )

Figure 8: Estimation of f with LADRC-PPC.

0.05

0

–0.05

–0.1

–0.15

–0.20 1 2 3

t (s)4 5

LADRC

Estim

atio

n er

ror (

deg/

s2 )

Figure 9: Estimation error of f with LADRC.

0.05

0

–0.05

–0.1

–0.15

–0.2

–0.25

–0.30 1 2 3

t (s)4 5

LADRC-PPC

Estim

atio

n er

ror (

deg/

s2 )

Figure 10: Estimation error of f with LADRC-PPC.

9International Journal of Aerospace Engineering

addition, the two controllers can achieve an excellent track-ing performance owing to the total disturbances can be pre-cisely estimated with the LESO as shown in Figures 15–18.

Case 3. For further evaluating the robust performance of theproposed control scheme, a Monte Carlo analysis consisting

Table 2: The comparative simulation results of Case 2.

Index LADRC LADRC-PPC Ratio

CT tc 2.74 s 1.10s 2.49 : 1

AV μe 0.07 deg 0.03 deg 2.33 : 1

SD σe 0.18 deg 0.14 deg 1.28 : 1

ACC Q 1.06 1.10 1 : 1.04

1.2

1

0.8

0.6

0.4

0.2

𝜗 (d

eg)

00 1

1.03

1.02

1.01

1

3 3.5 4 4.5 5

2 3t (s)

4

LADRCLADRC-PPC

5

Figure 11: Pitch angle.

1.5

1

0.5

0

–0.5 –0.01

–0.02

–0.03

0

3 3.5 4.54 5

–1

–1.50 1 2 3

t (s)4

LADRCLADRC-PPCPPF

5

Trac

king

erro

r of 𝜗

(deg

)

Figure 12: Tracking error of pitch angle.

5

4

3

2

1

0

–10 1 2 3

t (s)4

LADRCLADRC-PPC

5

𝜔z (

deg/

s)

Figure 13: Pitch angular rate.

0

–0.5

–1

–1.5

0 1 2 3t (s)

4

LADRCLADRC-PPC

5

𝛿z (

deg)

Figure 14: Control input signal-elevator deflection.

0 1 2 3t (s)

4

Actual - LADRCEstimated - LADRC

5

0

–0.2

–0.4

–0.6

–0.8

f (d

eg/s

2 )

Figure 15: Estimation of f with LADRC.

10 International Journal of Aerospace Engineering

of N = 1000 sample runs is developed, and the parameterperturbations are considered

(1) There is a random variation between -20% and +20%obeying normal distribution for the aerodynamicscoefficients

(2) The perturbation of parameter Jz and ρ is -10%~+10% of the standard value, which follows the ran-dom variation with normal distribution

Figures 19–24 depicts the 1000-run Monte Carlo simula-tion results, and Table 3 illustrates the corresponding statisti-cal results including the expectation value μ and averagevariance σ of the above indexes CT tc, AV μe, and ACC Q.

The Monte Carlo simulation results are depicted inFigures 19–24, and Table 3 lists the statistical results of1000 Monte Carlo simulations which included the expecta-tion μ and average variance σ of the concerned indexes CTtc, AV μe, and ACC Q.

0

–0.2

–0.4

–0.6

–0.80 1 2 3

t (s)4 5

Actual - LADRC-PPCEstimated - LADRC-PPC

f (d

eg/s

2 )

Figure 16: Estimation of f with LADRC-PPC.

0.1

0

0.05

–0.05

–0.150 1 2 3 4 5

54

0

0.01

0.02

0.03

4.53.5

–0.1

t (s)

Estim

atio

n er

ror (

deg/

s2 )

LADRC

Figure 17: Estimation error of f with LADRC.

0.1

0

0.05

–0.05

–0.15

–0.1

–0.2

0.08

0.06

0.04

0.02

3.2 3.4 3.6 3.80

0 1 2 3 4 5t (s)

Estim

atio

n er

ror (

deg/

s2 )

LADRC-PPC

Figure 18: Estimation error of f with LADRC-PPC.

3.1

2.9

2.8

2.7t c (s

)

2.6

2.5

2.40 200 400 600

N (times)800 1000

3

LADRC

Figure 19: CT of Monte Carlo.

1.4

1.3

1.2

1.1

0.9

0.8

0.70 200 400 600 800 1000

1t c (s

)

N (times)

LADRC-PPC

Figure 20: CT of Monte Carlo.

11International Journal of Aerospace Engineering

Figures 19 and 20 depict the convergence time of thetracking error history for 1000 times with the mentionedtwo controllers, we can obtain that the expectation of CT tcwith the proposed LADRC-PPC 1.07 s that is shorter thanthe one of the classic LADRC approach, and it can achievea smaller dispersion with LADRC-PPC. The statistical resultsμe exhibit the similar regularity as shown in Figures 21 and22. The distributions of the amount of control consumptionQ with the above controllers are illustrated in Figures 23and 24, and the expectation of Q under LADRC-PPC is1.228, which is very close to the one of the LADRC method.The above results further verify that the proposed LADRC-PPC approach exhibits enhanced robustness despite multipledisturbances, and it can achieve an obviously superiorsteady-state and transient performances than the classicLADRC-PPC method.

5. Conclusion

In this paper, a novel LADRC scheme is proposed for thrASV attitude control system with multiple disturbances andprescribed performance constraint. The chief feature is thatit introduces the PPC technique into the LADRC design

𝜇e (

deg)

0.07

0.068

0.066

0.064

0.062

0.060 200 400 600

N (times)800 1000

LADRC

Figure 21: AV of Monte Carlo.

𝜇e (

deg)

0.041

0.04

0.039

0.038

0.037

0.036

0.0350 200 400 600 800 1000

N (times)

LADRC-PPC

Figure 22: AV of Monte Carlo.

Q

2.5

1.5

0.5

2

1

0 200 400 600N (times)

800 1000

LADRC

Figure 23: ACC of Monte Carlo.

Q

2.5

1.5

0.5

2

1

0 200 400 600 800 1000N (times)

LADRC-PPC

Figure 24: ACC of Monte Carlo.

Table 3: The statistical results of the Monte Carlo simulations.

Statistical data LADRC LADRC-PPC

tcμ 2.7257 1.0710

σ 0.1022 0.0750

μe

μ 0.0641 0.0374

σ 0.0015 0.0007

Q

μ 1.2270 1.2880

σ 0.2253 0.2353

12 International Journal of Aerospace Engineering

process so that the tracking error can be strictly confined toan adjustable residual set with the prescribed steady-stateand transient performances. Meanwhile, the LESO isemployed to estimate the total disturbances such that it caneffectively solve the unknown model dynamic problem. The-oretical analysis and contrast simulation results validate thata superior robust tracking performance of the ASV attitudesystem with the proposed control strategy design. Further-more, both the transient and steady-state performances ofthe classical LADRC can be significantly improved by intro-ducing the PPC method. Future work will consider thecompletely unknown dynamic control system design withactuator saturation via the presented control approach.

Nomenclature

ϑ: Pitch angleωz : Pitch angular rateθ: Flight path angleα: Angle of attackJz : Pitch moment of inertiaMz : Pitch momentS: Reference areaL: Reference length�q = 0:5ρV2: Dynamic pressureρ: Density of airV : Velocityδz : Elevator deflectionexp ðÞ: Exponential function with natural constant.

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 declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Acknowledgments

The authors are grateful for the projects supported by theNational Natural Science Foundation of China (Grant No.11176012).

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