Adaptive Backstepping Control for Longitudinal Dynamics of...

Research ArticleAdaptive Backstepping Control for LongitudinalDynamics of Hypersonic Vehicle Subject to ActuatorSaturation and Disturbance

Zhiqiang Jia 12 Tianya Li2 and Kunfeng Lu2

1The School of Automation Beijing Institute of Technology Beijing China2Beijing Aerospace Automatic Control Institute Beijing China

Correspondence should be addressed to Zhiqiang Jia chrisjia82163com

Received 8 November 2018 Revised 27 December 2018 Accepted 17 January 2019 Published 3 March 2019

Academic Editor Andrzej Swierniak

Copyright copy 2019 Zhiqiang Jia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper an adaptive backstepping control strategy is presented to solve the longitudinal control problem for a hypersonicvehicle (HSV) subject to actuator saturation and disturbances Small perturbation linearization transforms the dynamics to aseconded-order system at each trimming point with total disturbance including unmodeled dynamics parametric uncertaintiesand external disturbances The disturbance can be estimated and compensated for by an extended state observer (ESO) and thusthe system is decoupled To deal with the actuator saturation and wide flight envelope an adaptive backstepping control strategyis designed A rigorous proof of finite-time convergence is provided applying Lyapunov methodThe effectiveness of the proposedcontrol scheme is verified in simulations

1 Introduction

Attitude control is a typical nonlinear control problem whichis very important for spacecraft missile HSV and so onin engineering practice According to small perturbationassumption the flight dynamics of HSVs can be linearizedaround the trimming pointsThen the classical controlmeth-ods such as PID and feedback linearization are employed todesign the controller [1ndash3] Hypersonic vehicle is requiredto work within a large flight envelope to meet the challengeof highly maneuverable targets in all probable engagements[4] Due to the wide envelope a great number of operatingpoints should be carefully chosen to cover the full envelopand an effective controller should be designed for each pointNote that the designed control algorithms should guaranteestability with superior control performance and robustnessthroughout the flight envelope [5 6] However the equationsof motion that govern the behavior of an HSV are nonlinearand time varying which make flight control system designfor aircraft a complex and difficult problem Consideringthe uncertainties in aerodynamic parameters nonlinearitiesand measurement noises the task of guaranteeing favorable

control performance and robustness throughout the entireflight envelope is a challenging one

Due to the practical importance of HSV the attitudecontrol has attracted extensive attention and a great numberof control methods have been designed to improve con-trol performance In [7ndash10] the backstepping procedure isdesigned for an angle of attack autopilot Furthermore to dealwith the explosion of the complexity associatedwith the back-stepping method improving the dynamic surface controlmethod has been studied in [11] on the longitudinal dynamicsof HSV Although the above literature can obtain somemeaningful results on the HSV control this algorithm haslow robustness under the uncertainties Since sliding modecontrol (SMC) technique provides robustness to internalparameter variations and extraneous disturbance satisfyingthe matching condition addressing the velocity and altitudetracking control of hypersonic vehicle has been introducedin [1 10 12 13] These designed laws guaranteed that bothvelocity and altitude track fast their reference trajectoriesrespectively under both uncertainties and external distur-bances and it has been also confirmed by the simulationresults But the above-mentioned SMC methods inherently

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1019621 10 pageshttpsdoiorg10115520191019621

2 Mathematical Problems in Engineering

suffer from the chattering problem which is an undesiredphenomenon in practice systems In [14] the 119867infin controlwith gain scheduling and dynamic inversion (DI) methodsis designed for the aircraft The time-domain and frequency-domain analysis procedures show that these methods possessstrong robustness and high control performance Based onDI methods the feedback linearization technique is used todesign the controller for the attitude control of HSV [15ndash17] where the nonlinear dynamics of system is convertedinto a chain of integrators in the design of inner-loopnonlinear control law Although DI method can obtain somemeaningful results on the HSV control this algorithm issensitive to the modelling errors [18ndash22]Therefore a model-free framework for the DI strategy is urgently needed

Based on the feedback linearization approach Han pro-poses a novel philosophy active disturbance rejection control(ADRC) which does not rely on a refined dynamic model[23] A linear ADRC controller is then presented for practicalapplications since the parameter tuning process is simplified[24] ESO is the core concept of ADRC and has demonstratedits effectiveness in many fields In [25] an ESO-based controlscheme is presented to handle the initial turning problemfor a vertical launching air-defense missile Based on thework of [25] Tian et al [26] employ ADRC to a genericnonlinear uncertain system with mismatched disturbancesand then a robust output feedback autopilot for an HSV isdevised In [27] a control method combining ADRC andoptimal control is discussed for large aircraft landing attitudecontrol under external disturbancesThe results show that theESO technique can guarantee the unknown disturbance andmodel uncertainties rejection Actually actuator saturationaffects virtually all practical control systems It may resultin performance degradation and even induce instabilityThe hypersonic vehicle dynamics is inherently nonlinearand the number of available results by considering actuatorsaturation in the design and analysis of hypersonic vehicledynamics is still limited [13 28] In this case the flightenvelope of hypersonic vehicle is narrow and the existingactuator saturation deteriorates the control performancethus the effective control method should be investigated forHSV

The paper mainly focuses on the longitudinal controlproblem of HSVs suffering actuator saturation and distur-bances within large envelop The main contributions of thispaper are threefold

(1) A linear ESO is used to estimate and compensatefor the total disturbances of the small-perturbation-linearized longitudinal dynamics of HSV which con-sists of unmodeled dynamics parametric uncertain-ties and external disturbances

(2) An adaptive backstepping control law is designed todeal with the large envelop and actuator saturationproblem and a rigorous proof of stability is providedby employing Lyapunov theory

(3) The control structure considers the complex aerody-namics and ensures the tracking performance withvery little requirement of model information whichis suitable for engineering practices

V

Y

X

mg

Figure 1 Body diagram of an HSV

The remainder of this paper is organized as follows thelongitudinal flight model of HSV and problem formulationare presented in the ldquoPreliminariesrdquo section In the ldquoCon-trol Strategyrdquo section an ESO-based adaptive backsteppingcontroller is designed and the closed-loop convergence isanalyzed The ldquoSimulation Resultsrdquo section gives simulationresults and some discussions Finally conclusions are drawnin the ldquoConclusionrdquo section

2 Preliminaries

21 Longitudinal Dynamic Model This study is concernedwith the longitudinal motion of the vehicle It is assumedthat there is no side slip no lateral motion and no roll forthe hypersonic vehicle As shown in Figure 1 the longitudinaldynamic model for a generic HSV is as follows [6]

119898d119881d119905 = minus119883 minus 119898119892 sin 120579

119898119881d120579d119905 = 119884 minus 119898119892 cos 120579

119869119911 d120596119911d119905 = 119872119911 +119872119892119911d120601d119905 = 120596119911120572 = 120601 minus 120579

(1)

where the drag force lift force and pitchmoment of the HSVare depicted by

119883 = 121205881198812119878119862119883 (120572 119881 120575119911)119884 = 121205881198812119878119862119884 (120572 119881 120575119911)

119872119911 = 121205881198812119878119888119862119872119911 (120572 119881 120575119911)(2)

The parameters 120588 119878 and 119888 represent the air density thereference area and the mean aerodynamic chord and 119862119883119862119884 and119862119872119911 represent the drag lift andmoment coefficientsrespectively

Since 119883 119884 and 119872119911 are all related to 120575119911 the system isstrongly coupled The control objective for this paper is to

Mathematical Problems in Engineering 3

find a feedback control 120575119911 such that the pitch angle can trackthe desired trajectory of pitch angle very well Therefore weshould consider the pitch dynamics and small perturbationmethod is applied

Introduce small perturbation assumption ignore sec-ond order or higher order traces and secondary factors ofaerodynamic forces and moment linearize the equationsand develop the perturbation equation in three-dimensionalspace as follows [6]

119889Δ119881119889119905 = minus119883119881119898 Δ119881 minus 119883120572119898 Δ120572 minus 119892 cos 120579Δ120579119889Δ120579119889119905 = 119884119881119898119881Δ119881 + 119884120572119898119881Δ120572 +

119892 sin 120579119881 Δ120579 + 119884120575119911119898119881Δ120575119911

119889Δ120596119911119889119905 = 119872119881119911119869119911 Δ119881 +119872120572119911119869119911 Δ120572 +

119872120596119911119911119869119911 Δ120596119911 +119872120575119911119911119869119911 Δ120575119911

+ 119872119892119911119869119911119889Δ120601119889119905 = Δ120596119911Δ120572 = Δ120601 minus Δ120579

(3)

where the aerodynamic coefficient 119860119887 stands for 120597119860120597119887 for119860 isin 119883119884 119885 and 119887 isin 119881 120572 120596119911 120575119911 and can be obtained fromprior knowledge

Assuming the velocity to be a constant in a short timethen (3) can be simplified as follows

Δ 120601 minus 1198861Δ 120601 minus 1198862Δ120572 minus 1198863Δ minus 1198864Δ120575119911 = 119872119892119911119869119911 (4)

Δ 120579 minus 1198865Δ120579 minus 1198866Δ120572 = 1198867Δ120575119911 (5)

Δ120572 = Δ120601 minus Δ120579 (6)

where 1198861 = 119872120596119911119911 119869119911 1198862 = 119872120572119911 119869119911 1198863 = 119872119911 119869119911 1198864 = 119872120575119911119911 1198691199111198865 = (119892 sin 120579)119881 1198866 = 119884120572119898119881 and 1198867 = 119884120575119911119898119881Then the second-order dynamics of pitch angle (4) can be

rewritten as follows

120601 (119905) = 119891 (sdot) + 1198864120590 (120575119911) 120575119911 (119905) (7)

with the total disturbance 119891(sdot) defined as

119891 (sdot) = 120601119903 + 1198861Δ 120601 + 1198862Δ120572 + 1198863Δ minus 11988641205751199110 + 119872119892119911119869119911 (8)

with 120601119903 the reference pitch angle and 1205751199110 the deflection anglecalculated by the nominal system with reference 120601119903 which isnot required to be known for subsequent controller design

The function 120590(sdot) is the saturation function of deflectionangle defined as follows

120590 (120575119911) =

1 1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 ⩽ 120575119911120575119911120575119911 (119905) sdot Sign (120575119911 (119905))

1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 gt 120575119911 (9)

where 120575119911 is the maximum allowable value of the deflectionObviously 120590 isin (0 1]

Then an assumption for the disturbance is given asfollows

Assumption 1 The additive disturbances moment 119872119892119911 isdifferentiable and the derivative is bounded

Remark 2 As shown in (8) the total disturbance 119891(sdot) con-tains coupling terms and external disturbances Accordingto Assumption 1 119891(sdot) is physically differentiable which isnecessary for further discussion Applying small perturbationlinearization high-order dynamics and secondary factorsare omitted which brings unmodeled dynamics Note thatthe unmodeled dynamics caused by linearization and theparametric uncertainties can also be concluded in the totaldisturbance which is not presented in the equation forsimplicity

22 Extended State Observer The ESO is a special stateobserver estimating both system states and an extendedstate which consist of the unknown dynamics and externaldisturbance of the system Appropriately designed observerscan provide comparatively accurate estimations that can becompensated in the control inputs

Consider a nonlinear system with uncertainties andexternal disturbances

119909119899 (119905) = 1198911 (119909 (119905) (119905) 119909(119899minus1) (119905)) + 1198891 (119905)+ 119887119906 (119905) (10)

where 1198911(119909(119905) (119905) 119909(119899minus1)(119905)) is an unknown function1198891(119905) is the unknown external disturbance 119906(119905) is the controlinput and 119887 is a known constant

For clarity System (10) can be rewritten as

1 = 11990922 = 1199093

119899 = 119909119899+1 + 119887119906119899+1 = ℎ (119905)119910 = 1199091

(11)

where

119909119899+1 = 1198911 (119909 (119905) (119905) 119909119899minus1 (119905)) + 1198891 (119905) (12)

is the extended stateThen an ESO can be constructed as follows

1199091 = 1199092 minus 1198971 (1199091 minus 119910)

119909119899 = 119909119899+1 minus 119897119899 (1199091 minus 119910) + 119887119906119909119899+1 = minus119897119899+1 (1199091 minus 119910)

(13)


where 119909119894 are the observer outputs and 119897119894 are positive observergains 119894 = 1 2 119899 + 1

Note that the extended state is the total disturbancewhich contains the unknown dynamics and external dis-turbances Appropriately designed observers can providecomparatively accurate estimations that can be compensatedin the control inputs which improves the robustness Moredetailed description of the principle of ESO can be found in[23]

3 Control Strategy

In this section an ESO-based pitch controller for HSV pitchangle controller is devised Adaptive backstepping techniqueis applied with ESO to reject disturbances and unmodeleddynamics In the first stage the estimation of total dis-turbances including external disturbances and unmodeleddynamics is discussedThen in the second stage an adaptivebackstepping controller is designed while the estimationof the disturbances is used as a time-varying parameter toimprove robustness

31 Lumped Disturbance Estimation In order to improve therobustness an extended state observer is used to estimate andcompensate for the disturbance Let 1199091 = 120601 1199092 = 120601 andregard total disturbance 119891(sdot) as an extended state 1199093 then thelongitudinal system in (7) can be expressed as follows

1 = 11990922 = 1199093 + 1198864120590 (120575119911) 1205751199113 = ℎ (sdot)

(14)

where ℎ(sdot) = 119891(sdot) is bounded according to the discussion inRemark 2

Consider the third-order linear ESO as follows1199091 = 1199092 minus 3120596119900 (1199091 minus 1199091)1199092 = 1199093 minus 31205962119900 (1199091 minus 1199091) + 1198864120590 (120575119911) 1205751199111199093 = minus1205963119900 (1199091 minus 1199091)

(15)

where 1199091 1199092 and 1199093 are the observer outputs and 120596119900 gt 0 isthe bandwidth of the ESO Note that few model informationis required for observer design except 1198864

It is obvious that the characteristic polynomial isHurwitzand the observer is bounded-input-bounded-output (BIBO)stable Define the estimation error as 119909119894 = 119909119894 minus 119909119894 119894 = 1 2 3the observer estimation error is as follows

1199091 = 1199092 minus 312059611990011990911199092 = 1199093 minus 3120596211990011990911199093 = ℎ (sdot) minus 12059631199001199091

(16)

Let 120576119894 = 1199091120596119894minus1119900 119894 = 1 2 3 and (16) can be simplified asfollows

120576 = 120596119900119860120576 + 119861ℎ (sdot)1205962119900 (17)

where 120576 = [1205761 1205762 1205763]119879 119861 = [0 0 1]119879 and

119860 = [[[

minus3 1 0minus3 0 1minus1 0 0

]]]

(18)

is Hurwitz

Lemma 3 (see [29]) Assuming ℎ(sdot) is bounded there exist aconstant 984858119894 gt 0 and a finite 1198791 gt 0 so that1003816100381610038161003816119909119894 (119905)1003816100381610038161003816 ⩽ 120588119894

120588119894 = 119874( 11205961198961119900 ) 119894 = 1 2 3 forall119905 ⩾ 1198791 gt 0

(19)

for some positive integer 1198961 where 119874(sdot) represents the infinitelysmall

Remark 4 Since 119860 is Hurwitz error system (17) is BIBOstable Therefore the requirement of ESO is the boundednessof ℎ(sdot) Lemma 3 also indicates a good steady estimationperformance of ESO and the estimation error can be reducedto a sufficient small range within a finite time1198791 by increasingthe bandwidth 120596119900 Since the disturbance 119891(sdot) is partly com-pensated for and the actual disturbance on system becomes119909119894(119905) the steady control performance is likely to be improved

With a well-tuned ESO the total disturbance 119891(sdot) canbe actively estimated by 1199093 For simplicity here we denote119891 = 1199093 Since the observer (15) is BIBO stable the estimationerror of 119891(sdot) is bounded Defining 120578 as the upper bound ofestimation error there is

10038161003816100381610038161003816119891 minus 11989110038161003816100381610038161003816 ⩽ 120578 (20)

32 Adaptive Backstepping Control Thefirst step of backstep-ping design is to define the tracking error as 1198901 = 120601119903 minus 1199091 anda virtual input as

1205941 = 11988811198901 + 120601119903 (21)

where 1198881 is a positive constantThen define the angular velocity tracking error as

1198902 = 1205941 minus 1199092 (22)

and thus

1198901 = 120601119903 minus 1199092 = minus11988811198901 + 1198902 (23)

The derivative of the virtual input is

1205941 = 1198881 1198901 + 120601119903 = minus11988821 1198901 + 11988811198902 + 120601119903 (24)

Design an adaptive controller as

120575119911 = 1198881198864 120574 ( 1205941 minus 119891 + 120578) 1198902 + 11988821198864 1198902 (25)


Nominal HSVLongitudinal Dynamics

Small PerturbationLinearization

Nominal LinearDynamics

(t) = f (middot) + a4(z)z(t)

r r

a4

ESO

f

Adaptive BacksteppingController

z =c

a4 ( 1 minus f + ) e2 +

c2a4

e2

Adaptive Law

= 1ce22

= 2ce22

3( 1 minus f + )

z

HSV LongitudinalDynamics

x1 = x2 minus 3o(x1 minus x1)

x2 = x3 minus 32o (x1 minus x1) + a4(z)z

x3 = minus3o(x1 minus x1)

Figure 2 Block diagram of proposed ESO-based adaptive backstepping control structure for HSV system

with adaption update laws

120578 = 120582111988811989022120574 = 1205822119888119890221205743 (10038161003816100381610038161003816 1205941 minus 11989110038161003816100381610038161003816 + 120578)

(26)

where 119888 1198882 1205821 and 1205822 are designed parametersThe proposed ESO-based adaptive backstepping control

structure for HSV system is shown in Figure 2

33 Stability Analysis The convergence of the tracking errorsis established byTheorems 5 and 6

Theorem 5 For system (1) controlled by (25) and (26) whereinitial estimated values satisfy 141198881198881 lt 120578(0) lt 120578 the error1198902 converges into a small neighborhood of origin |1198902| lt 120598 =1(119888minus 141198881120578(0))within finite time 119905120598 gt 0 and is guaranteed tobe uniformly ultimately bounded (UUB) for 119905 ⩾ 119905120576Proof Consider the Lyapunov function

1198811 = 1211989021 + 1211989022 + 121205821 1205782 + 121205822 120574

2 (27)

where 120578 = 120578 minus 120578 120574 = 120574minus1 minus 120574 and 120574 is a positive constantsatisfying 0 lt 120574 ⩽ 120590(120575119911) ⩽ 1

Considering the dynamics (7) and proposed control law(25) the derivative of 1198811 can be derived as

1 = minus119888111989021 + 11989011198902 + 1198902 1198902 + 11205821 (120578 minus 120578) 120578 minus 11205822 (120574minus1

minus 120574) 120574minus1 120574 = minus119888111989021 + 11989011198902 + 11205821 (120578 minus 120578) 120578 minus 11205822 (120574minus1

minus 120574) 120574minus1 120574 + 1198902 1205941 minus 119891minus 1198864120590 (120575119911) [ 1198881198864 120574 ( 1205941 minus 119891 + 120578) 1198902 + 11988821198864 1198902]= minus119888111989021 + 11989011198902 + 1198902 ( 1205941 minus 119891) minus 11988811989022120574120590 (120575119911) ( 1205941 minus 119891+ 120578) minus 1198882120590 (120575119911) 11989022 + 11205821 (120578 minus 120578) 120578 minus 11205822 (120574

minus1 minus 120574)sdot 120574minus1 120574 = minus119888111989021 + 11989011198902 + 1198902 ( 1205941 minus 119891 + 119891 minus 119891)minus 11988811989022120574120590 (120575119911) (1 minus 119891 + 120578) minus 1198882120590 (120575119911) 11989022 + 11205821 (120578minus 120578) 120578 minus 11205822 (120574

minus1 minus 120574) 120574minus1 120574(28)

Let 120599 = | 1205941 minus 119891| then1 ⩽ minus119888111989021 + 1003816100381610038161003816119890111989021003816100381610038161003816 + 100381610038161003816100381611989021003816100381610038161003816 (120599 + 120578) minus 11988811989022120574120574 (120599 + 120578)

minus 119888212057411989022 + 11205821 (120578 minus 120578) 120578 minus 11205822 (120574minus1 minus 120574) 120574minus1 120574 (29)

Substituting (26) into (29) yields

1 ⩽ minus119888111989021 + 1003816100381610038161003816119890111989021003816100381610038161003816 + 100381610038161003816100381611989021003816100381610038161003816 (120599 + 120578) minus 11988811989022120574120574 (120599 + 120578)minus 119888212057411989022 + 11988811989022 (120578 minus 120578) minus 1198881198902 (120599 + 120578) (1 minus 120574120574)

= minus119888111989021 + 1003816100381610038161003816119890111989021003816100381610038161003816 + 100381610038161003816100381611989021003816100381610038161003816 (120599 + 120578) minus 119888212057411989022minus 11988811989022 (120599 + 120578)


= minus(radic1198881 100381610038161003816100381611989011003816100381610038161003816 minus 12radic1198881100381610038161003816100381611989021003816100381610038161003816)2 + 1198902241198881 minus 1198882120574119890

22

minus 11988811989022 (120599 + 120578) + 100381610038161003816100381611989021003816100381610038161003816 (120599 + 120578)= minus(radic1198881 100381610038161003816100381611989011003816100381610038161003816 minus 12radic1198881

100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)

Provided that 0 lt 120578(0) lt 120578 we obtain 0 lt 120578(0) lt 120599 + 120578that is

1120599 + 120578 lt 1120578 (0) (31)

Therefore

1 ⩽ minus(radic1198881 100381610038161003816100381611989011003816100381610038161003816 minus 12radic1198881100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881120578 (0)) 11989022 minus 100381610038161003816100381611989021003816100381610038161003816]

(32)

Since 141198881198881 lt 120578(0) 120598 = 1(119888 minus 141198881120578(0)) gt 0 If |1198902| gt 1205981 ⩽ minus(radic1198881 100381610038161003816100381611989011003816100381610038161003816 minus 12radic1198881

100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)

whichmeans that1198811 is decreasing and bounded and the error1198902 converges into a small neighborhood of origin ie |1198902| lt 120576within a finite time 119905120576 gt 0 Even though the tracking errorenters the region |1198902| lt 120576 within a finite time it may move inand out since the nonnegativity cannot be guaranteed in therange However when it moves out the Lyapunov function1198811 becomes negative again and the error is driven back to theregion Therefore 1198902 is guaranteed to be UUB for 119905 ⩾ 119905120576119911 Theorem 6 Considering system (1) controlled by (25) and(26) the output tracking can be accomplished with virtualcontrol input (21)

Proof To illustrate the reference state tracking Lyapunovfunction is chosen as follows

1198812 = 1211989021 (34)

The derivative of 1198812 with (23) equals

2 = minus119888111989021 + 11989011198902 (35)

According to Theorem 5 it has been proved that 1198902 isbounded Thus by selecting positive 1198881 large enough weobtain 2 lt 0 when 1198812 is out of a certain boundedregion Therefore 1198901 is also UUB by which 1199091 tracking 120601119903 isguaranteed Note that from (35) it is clear that 1198812 will notconverge to zero due to the existence of 1198902 It also impliesthat the state 1199091 can only converge into a neighborhood ofthe origin and remain within it

Table 1 Control gains at feature points

Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912

Remark 7 Small perturbation linearization is a typical engi-neering method for HSV attitude control and nominal valuesof parameters are usually used in the design process whichbrings the problems of structural and parametric uncertain-ties Applications indicate that ESO can estimate the totaldisturbance well even if 1198864 is not calculated preciselyThus theproposed method needs only a little model information andthe adaptive law guarantees a smooth tracking performancewithin a wide flight envelope which simplifies the designingprocess

4 Simulation Results

In this section simulation results for an HSV are providedto verify the feasibility and efficiency of the proposed controlscheme The reference trajectory used in the simulationsis a typical trajectory of reentry segment The longitudinaldynamics (1) are simulated as the real system while thecontroller design procedure is based on the linearized model(3) The simulations are run for 150 seconds and at 100samples per second The controller gains are 1198881 = 10 1198882 = 20119888 = 3 1205821 = 01 1205822 = 01 and the ESO gains are tuned bybandwidth method with bandwidth 120596119900 = 5 The initial valueof adaptive gains are 120574(0) = 01 120578(0) = 01 and ones ofestimations are 1199091(0) = 1199092(0) = 1199093(0) = 0 The deflectionangle 120575119911 is bounded by [minus30∘ 30∘]

The control performance using an intelligent ADRC con-troller is also given to show the superiority of the proposedmethod In the ADRC controller design the control law is asfollows

120575119911 = 119896119901 (120601119903 minus 120601) + 119896119889 ( 120601119903 minus 120601) minus 119891 (36)

where 119891 is the estimation of total disturbances by an ESOwith the same bandwidth 120596119900 = 5 control gains 119896119901 119896119889 atseveral feature points are optimized by genetic algorithm(GA) according to the nominal linearized model and theninterpolated in order to achieve a smooth and quick trackingThe control gains at feature points are shown in Table 1

To begin with a set of comparative simulations fornominal longitudinal dynamics is studied with no externaldisturbances and parametric uncertainties Figure 3 showsthe pitch angle tracking performance and Figure 4 shows thetracking errors The deflection angles are shown in Figure 5Form the figures it is indicated that both control methodscan track the reference Thanks to the excellent estimationability of ESO to the internal ldquodisturbancerdquo the system can


minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)

ReferenceProposed methodIntelligent ADRC

48 50 523

4

5

35 4011

11512

50 100 1500Time (s)

Figure 3 Pitch angle tracking performance for nominal longitudi-nal control system

Proposed methodIntelligent ADRC

50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)

Figure 4 Pitch angle tracking error for nominal longitudinalcontrol system

be approximately transformed into a second-order integratorwhich is easier to be controlled

It is easily seen from Figure 4 that the proposed controllertracks the reference more precisely The ADRC controllercan be regarded as a PD controller with compensation ofdisturbances When the HSV works in a large flight envelopespecially when the reference pitch angle changes rapidly theset of offline-tuned gains in ADRC may not perform wellAlthough the feature points will be selected more denselyin practical applications it may not reach the performanceof a continuous adaptive method and the computationalload will increase due to interpolation operations Moreoverthe parameter tuning procedure for traditional controllers

5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)

Figure 5 Deflection angle for nominal longitudinal control system

48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal

+20 uncertaintyminus20 uncertainty

minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)

Figure 6 Pitch angle tracking performance using proposed controlmethod subject to disturbance and parametric uncertainties

like PID and ADRC is quite complex while the proposedcontroller can be effective even in large envelop

Then considering the existence of external disturbancesand parametric uncertainties another set of simulation isdone The simulations are done under sustained disturbanceand abrupt reference changes In the simulation a sinusoidalwave disturbance is given as119872119892119911(119905) = 2 times 103 sin(05119905) Nsdotmand uncertainty of plusmn20 in mass119898 and moment of inertia 119869119911is added to show the parameter robustness

The tracking performance and tracking error of systemusing proposed control method are shown in Figures 6 and 7while the ones using intelligent ADRC controller are shownin Figures 8 and 9 It can be inferred that both methods trackthe reference soon and remain stable in general due to the


Nominal+20 uncertaintyminus20 uncertainty

50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)

Figure 7 Pitch angle tracking error using proposed control methodsubject to disturbance and parametric uncertainties

ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)

Figure 8 Pitch angle tracking performance using intelligent ADRCcontroller subject to disturbance and parametric uncertainties

introduction of ESO The time-varying disturbances as wellas parametric uncertainties and unmodeled dynamics can belumped together as the disturbances which can be estimatedby ESO and actively compensated for However the proposedmethod shows certain superiority in tracking error since thecontroller gains are tuned adaptively Also the accuracy ofdisturbance estimation is ensured by setting the ESO gainslarge enough but the existence of measurement noises adds alimit to observer gains in practical applications Hence therewill be a phase lag for estimation and the estimation errorcannot be neglected From this aspect the adaptive method


50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)

Figure 9 Pitch angle tracking error using intelligent ADRC con-troller subject to disturbance and parametric uncertainties

estimates the upper bound 120578 of estimation error and can alsohandle it

5 Conclusion

In this paper the longitudinal control problem for HSVssubject to actuator saturation and disturbance is studiedApplying small perturbation assumption the longitudinaldynamics can be considered as a second-order system atevery trimming point with total disturbance includingunmodeled dynamics and parametric uncertainties Then anESO is constructed to estimate and compensate for the totaldisturbance actively in order to decouple the system andimprove the robustness To deal with the large envelop andactuator saturation an adaptive backstepping control schemeis designed to control the pitch angle The presented methodrequires very little model information and the closed-loopconvergence is proved Finally simulation results indicated aquick and smooth tracking performance and verified that theproposedmethod is effective Further worksmay focus on thetrajectory tracking control of HSV in 6 DoFs

Notation

119881 Velocity of the HSV119898 Mass of the HSV119892 Gravitational constant119869119911 Pitch moment of inertia of the HSV120601 Pitch angle of the HSV120596119911 Pitch angular rate of the HSV119883 Drag force of the HSV119884 Lift force of the HSV119872119911 Pitching moment of the HSV


120579 Flight path angle of the HSV120572 Angle of attack of the HSV120575119911 Deflection angle119872119892119911 External disturbance moment

Data Availability

All the data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Q Wu M Sun Z Chen Z Yang and Z Wang ldquoTuningof active disturbance rejection attitude controller for staticallyunstable launch vehiclerdquo Journal of Spacecraft and Rockets vol54 no 6 pp 1383ndash1389 2017

[2] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[3] J Yang S Li C Sun and L Guo ldquoNonlinear-disturbance-observer-based robust flight control for airbreathing hypersonicvehiclesrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 49 no 2 pp 1263ndash1275 2013

[4] T Cimen ldquoA generic approach tomissile autopilot design usingstate-dependent nonlinear controlrdquo in Proceedings of the 18thIFAC World Congress vol 44 (1) pp 9587ndash9600 Milano ItalySeptember 2011

[5] M Sun Z Wang and Z Chen ldquoPractical solution to atti-tude control within wide enveloperdquo Aircraft Engineering andAerospace Technology vol 86 no 2 pp 117ndash128 2014

[6] Y Xu Z Wang and B Gao ldquoSix-degree-of-freedom digitalsimulations for missile guidance and controlrdquo MathematicalProblems in Engineering vol 2015 Article ID 829473 10 pages2015

[7] D Lin and J Fan ldquoMissile autopilot design using backsteppingapproachrdquo in Proceedings of the 2009 Second InternationalConference on Computer and Electrical Engineering ICCEErsquo09vol 1 pp 238ndash241 IEEE Dubai UAE December 2009

[8] S Lee Y Kim G Moon and B-E Jun ldquoMissile autopilotdesign during boost phase using robust backstepping approchrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference 2015 Kissimme FL USA January 2015

[9] J Sun S Song and G Wu ldquoTracking control of hypersonicvehicle considering input constraintrdquo Journal of AerospaceEngineering vol 30 pp 134ndash142 2017

[10] Z Guo J Guo and J Zhou ldquoAdaptive attitude tracking controlfor hypersonic reentry vehicles via slidingmode-based couplingeffect-triggered approachrdquo Journal of Aerospace Engineeringvol 78 pp 228ndash240 2018

[11] A B Waseem Y Lin and A S Kendrik ldquoAdaptive integraldynamic surface control of a hypersonic flight vehiclerdquo Inter-national Journal of Systems Science vol 46 pp 1ndash13 2015

[12] H Xu M D Mirmirani and P A Ioannou ldquoAdaptive slidingmode control design for a hypersonic flight vehiclerdquo Journal ofGuidance Control and Dynamics vol 27 no 5 pp 829ndash8382004

[13] J-G Sun S-M Song H-T Chen and X-H Li ldquoFinite-timetracking control of hypersonic aircrafts with input saturationrdquoProceedings of the Institution of Mechanical Engineers Part GJournal of Aerospace Engineering vol 232 no 7 pp 1373ndash13892018

[14] C Schumacher andP PKhargonekar ldquoMissile autopilot designsusing Hinfin control with gain scheduling and dynamic inver-sionrdquo Journal of Guidance Control and Dynamics vol 21 no2 pp 234ndash243 1998

[15] R Rysdyk and A J Calise ldquoRobust nonlinear adaptive flightcontrol for consistent handling qualitiesrdquo IEEE Transactions onControl Systems Technology vol 13 no 6 pp 896ndash910 2005

[16] W R van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[17] H An and Q Wu ldquoDisturbance rejection dynamic inversecontrol of air-breathing hypersonic vehiclesrdquoActaAstronauticavol 151 pp 348ndash356 2018

[18] J S Brinker and K A Wise ldquoStability and flying qualitiesrobustness of a dynamic inversion aircraft control lawrdquo Journalof Guidance Control andDynamics vol 19 no 6 pp 1270ndash12771996

[19] J Reiner G J Balas and W L Garrard ldquoRobust dynamicinversion for control of highlymaneuverable aircraftrdquo Journal ofGuidance Control and Dynamics vol 18 no 1 pp 18ndash24 1995

[20] G Wu X Meng and F Wang ldquoImproved nonlinear dynamicinversion control for a flexible air-breathing hypersonic vehi-clerdquo Aerospace Science and Technology vol 78 pp 734ndash7432018

[21] CMu ZNi C Sun andHHe ldquoAir-breathing hypersonic vehi-cle tracking control based on adaptive dynamic programmingrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 28 no 3 pp 584ndash598 2017

[22] J He R Qi B Jiang and J Qian ldquoAdaptive output feedbackfault-tolerant control design for hypersonic flight vehiclesrdquoJournal of The Franklin Institute vol 352 no 5 pp 1811ndash18352015

[23] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[24] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 Denver Colo USA June 2003

[25] Y Huang K Xu J Han and J Lam ldquoFlight control designusing extended state observer and non-smooth feedback inrdquoin Proceedings of the 40th IEEE Conference on Decision andControl vol 1 pp 223ndash228 IEEE 2001

[26] J Tian S Zhang Y Zhang and T Li ldquoActive disturbancerejection control based robust output feedback autopilot designfor airbreathing hypersonic vehiclesrdquo ISA Transactions vol 74pp 45ndash59 2018

[27] G Zhang L Yang J Zhang and C Han ldquoLongitudinal attitudecontroller design for aircraft landing with disturbance usingADRCLQRrdquo in Proceedings of the 2013 IEEE InternationalConference on Automation Science and Engineering CASE 2013pp 330ndash335 August 2013

[28] B Huang A Li and B Xu ldquoAdaptive fault tolerant controlfor hypersonic vehicle with external disturbancerdquo InternationalJournal of Advanced Robotic Systems vol 12 pp 30ndash35 2018


[29] Q Zheng L Q Gao and Z Gao ldquoOn stability analysis ofactive disturbance rejection control for nonlinear time-varyingplants with unknowndynamicsrdquo inProceedings of the 46th IEEEConference on Decision and Control (CDC rsquo07) pp 3501ndash3506New Orleans La USA December 2007

Hindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


suffer from the chattering problem which is an undesiredphenomenon in practice systems In [14] the 119867infin controlwith gain scheduling and dynamic inversion (DI) methodsis designed for the aircraft The time-domain and frequency-domain analysis procedures show that these methods possessstrong robustness and high control performance Based onDI methods the feedback linearization technique is used todesign the controller for the attitude control of HSV [15ndash17] where the nonlinear dynamics of system is convertedinto a chain of integrators in the design of inner-loopnonlinear control law Although DI method can obtain somemeaningful results on the HSV control this algorithm issensitive to the modelling errors [18ndash22]Therefore a model-free framework for the DI strategy is urgently needed

Based on the feedback linearization approach Han pro-poses a novel philosophy active disturbance rejection control(ADRC) which does not rely on a refined dynamic model[23] A linear ADRC controller is then presented for practicalapplications since the parameter tuning process is simplified[24] ESO is the core concept of ADRC and has demonstratedits effectiveness in many fields In [25] an ESO-based controlscheme is presented to handle the initial turning problemfor a vertical launching air-defense missile Based on thework of [25] Tian et al [26] employ ADRC to a genericnonlinear uncertain system with mismatched disturbancesand then a robust output feedback autopilot for an HSV isdevised In [27] a control method combining ADRC andoptimal control is discussed for large aircraft landing attitudecontrol under external disturbancesThe results show that theESO technique can guarantee the unknown disturbance andmodel uncertainties rejection Actually actuator saturationaffects virtually all practical control systems It may resultin performance degradation and even induce instabilityThe hypersonic vehicle dynamics is inherently nonlinearand the number of available results by considering actuatorsaturation in the design and analysis of hypersonic vehicledynamics is still limited [13 28] In this case the flightenvelope of hypersonic vehicle is narrow and the existingactuator saturation deteriorates the control performancethus the effective control method should be investigated forHSV

The paper mainly focuses on the longitudinal controlproblem of HSVs suffering actuator saturation and distur-bances within large envelop The main contributions of thispaper are threefold

(1) A linear ESO is used to estimate and compensatefor the total disturbances of the small-perturbation-linearized longitudinal dynamics of HSV which con-sists of unmodeled dynamics parametric uncertain-ties and external disturbances

(2) An adaptive backstepping control law is designed todeal with the large envelop and actuator saturationproblem and a rigorous proof of stability is providedby employing Lyapunov theory

(3) The control structure considers the complex aerody-namics and ensures the tracking performance withvery little requirement of model information whichis suitable for engineering practices

V

Y

X

mg

Figure 1 Body diagram of an HSV

The remainder of this paper is organized as follows thelongitudinal flight model of HSV and problem formulationare presented in the ldquoPreliminariesrdquo section In the ldquoCon-trol Strategyrdquo section an ESO-based adaptive backsteppingcontroller is designed and the closed-loop convergence isanalyzed The ldquoSimulation Resultsrdquo section gives simulationresults and some discussions Finally conclusions are drawnin the ldquoConclusionrdquo section

2 Preliminaries

21 Longitudinal Dynamic Model This study is concernedwith the longitudinal motion of the vehicle It is assumedthat there is no side slip no lateral motion and no roll forthe hypersonic vehicle As shown in Figure 1 the longitudinaldynamic model for a generic HSV is as follows [6]

119898d119881d119905 = minus119883 minus 119898119892 sin 120579

119898119881d120579d119905 = 119884 minus 119898119892 cos 120579

119869119911 d120596119911d119905 = 119872119911 +119872119892119911d120601d119905 = 120596119911120572 = 120601 minus 120579

(1)

where the drag force lift force and pitchmoment of the HSVare depicted by

119883 = 121205881198812119878119862119883 (120572 119881 120575119911)119884 = 121205881198812119878119862119884 (120572 119881 120575119911)

119872119911 = 121205881198812119878119888119862119872119911 (120572 119881 120575119911)(2)

The parameters 120588 119878 and 119888 represent the air density thereference area and the mean aerodynamic chord and 119862119883119862119884 and119862119872119911 represent the drag lift andmoment coefficientsrespectively

Since 119883 119884 and 119872119911 are all related to 120575119911 the system isstrongly coupled The control objective for this paper is to





119892 sin 120579119881 Δ120579 + 119884120575119911119898119881Δ120575119911

119889Δ120596119911119889119905 = 119872119881119911119869119911 Δ119881 +119872120572119911119869119911 Δ120572 +

119872120596119911119911119869119911 Δ120596119911 +119872120575119911119911119869119911 Δ120575119911

+ 119872119892119911119869119911119889Δ120601119889119905 = Δ120596119911Δ120572 = Δ120601 minus Δ120579

(3)




Δ 120579 minus 1198865Δ120579 minus 1198866Δ120572 = 1198867Δ120575119911 (5)

Δ120572 = Δ120601 minus Δ120579 (6)



120601 (119905) = 119891 (sdot) + 1198864120590 (120575119911) 120575119911 (119905) (7)


119891 (sdot) = 120601119903 + 1198861Δ 120601 + 1198862Δ120572 + 1198863Δ minus 11988641205751199110 + 119872119892119911119869119911 (8)



120590 (120575119911) =

1 1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 ⩽ 120575119911120575119911120575119911 (119905) sdot Sign (120575119911 (119905))

1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 gt 120575119911 (9)







119909119899 (119905) = 1198911 (119909 (119905) (119905) 119909(119899minus1) (119905)) + 1198891 (119905)+ 119887119906 (119905) (10)



1 = 11990922 = 1199093

119899 = 119909119899+1 + 119887119906119899+1 = ℎ (119905)119910 = 1199091

(11)

where

119909119899+1 = 1198911 (119909 (119905) (119905) 119909119899minus1 (119905)) + 1198891 (119905) (12)


1199091 = 1199092 minus 1198971 (1199091 minus 119910)


(13)




3 Control Strategy



1 = 11990922 = 1199093 + 1198864120590 (120575119911) 1205751199113 = ℎ (sdot)

(14)



(15)




(16)


120576 = 120596119900119860120576 + 119861ℎ (sdot)1205962119900 (17)

where 120576 = [1205761 1205762 1205763]119879 119861 = [0 0 1]119879 and

119860 = [[[


]]]

(18)

is Hurwitz


120588119894 = 119874( 11205961198961119900 ) 119894 = 1 2 3 forall119905 ⩾ 1198791 gt 0

(19)




10038161003816100381610038161003816119891 minus 11989110038161003816100381610038161003816 ⩽ 120578 (20)


1205941 = 11988811198901 + 120601119903 (21)


1198902 = 1205941 minus 1199092 (22)

and thus

1198901 = 120601119903 minus 1199092 = minus11988811198901 + 1198902 (23)


1205941 = 1198881 1198901 + 120601119903 = minus11988821 1198901 + 11988811198902 + 120601119903 (24)


120575119911 = 1198881198864 120574 ( 1205941 minus 119891 + 120578) 1198902 + 11988821198864 1198902 (25)






r r

a4

ESO

f


z =c

a4 ( 1 minus f + ) e2 +

c2a4

e2

Adaptive Law

= 1ce22

= 2ce22

3( 1 minus f + )

z







120578 = 120582111988811989022120574 = 1205822119888119890221205743 (10038161003816100381610038161003816 1205941 minus 11989110038161003816100381610038161003816 + 120578)

(26)





1198811 = 1211989021 + 1211989022 + 121205821 1205782 + 121205822 120574

2 (27)















22


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)


1120599 + 120578 lt 1120578 (0) (31)

Therefore



(32)


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)



1198812 = 1211989021 (34)


2 = minus119888111989021 + 11989011198902 (35)



Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018












119892 sin 120579119881 Δ120579 + 119884120575119911119898119881Δ120575119911

119889Δ120596119911119889119905 = 119872119881119911119869119911 Δ119881 +119872120572119911119869119911 Δ120572 +

119872120596119911119911119869119911 Δ120596119911 +119872120575119911119911119869119911 Δ120575119911

+ 119872119892119911119869119911119889Δ120601119889119905 = Δ120596119911Δ120572 = Δ120601 minus Δ120579

(3)




Δ 120579 minus 1198865Δ120579 minus 1198866Δ120572 = 1198867Δ120575119911 (5)

Δ120572 = Δ120601 minus Δ120579 (6)



120601 (119905) = 119891 (sdot) + 1198864120590 (120575119911) 120575119911 (119905) (7)


119891 (sdot) = 120601119903 + 1198861Δ 120601 + 1198862Δ120572 + 1198863Δ minus 11988641205751199110 + 119872119892119911119869119911 (8)



120590 (120575119911) =

1 1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 ⩽ 120575119911120575119911120575119911 (119905) sdot Sign (120575119911 (119905))

1003816100381610038161003816120575119911 (119905)1003816100381610038161003816 gt 120575119911 (9)







119909119899 (119905) = 1198911 (119909 (119905) (119905) 119909(119899minus1) (119905)) + 1198891 (119905)+ 119887119906 (119905) (10)



1 = 11990922 = 1199093

119899 = 119909119899+1 + 119887119906119899+1 = ℎ (119905)119910 = 1199091

(11)

where

119909119899+1 = 1198911 (119909 (119905) (119905) 119909119899minus1 (119905)) + 1198891 (119905) (12)


1199091 = 1199092 minus 1198971 (1199091 minus 119910)


(13)




3 Control Strategy



1 = 11990922 = 1199093 + 1198864120590 (120575119911) 1205751199113 = ℎ (sdot)

(14)



(15)




(16)


120576 = 120596119900119860120576 + 119861ℎ (sdot)1205962119900 (17)

where 120576 = [1205761 1205762 1205763]119879 119861 = [0 0 1]119879 and

119860 = [[[


]]]

(18)

is Hurwitz


120588119894 = 119874( 11205961198961119900 ) 119894 = 1 2 3 forall119905 ⩾ 1198791 gt 0

(19)




10038161003816100381610038161003816119891 minus 11989110038161003816100381610038161003816 ⩽ 120578 (20)


1205941 = 11988811198901 + 120601119903 (21)


1198902 = 1205941 minus 1199092 (22)

and thus

1198901 = 120601119903 minus 1199092 = minus11988811198901 + 1198902 (23)


1205941 = 1198881 1198901 + 120601119903 = minus11988821 1198901 + 11988811198902 + 120601119903 (24)


120575119911 = 1198881198864 120574 ( 1205941 minus 119891 + 120578) 1198902 + 11988821198864 1198902 (25)






r r

a4

ESO

f


z =c

a4 ( 1 minus f + ) e2 +

c2a4

e2

Adaptive Law

= 1ce22

= 2ce22

3( 1 minus f + )

z







120578 = 120582111988811989022120574 = 1205822119888119890221205743 (10038161003816100381610038161003816 1205941 minus 11989110038161003816100381610038161003816 + 120578)

(26)





1198811 = 1211989021 + 1211989022 + 121205821 1205782 + 121205822 120574

2 (27)















22


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)


1120599 + 120578 lt 1120578 (0) (31)

Therefore



(32)


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)



1198812 = 1211989021 (34)


2 = minus119888111989021 + 11989011198902 (35)



Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018











3 Control Strategy



1 = 11990922 = 1199093 + 1198864120590 (120575119911) 1205751199113 = ℎ (sdot)

(14)



(15)




(16)


120576 = 120596119900119860120576 + 119861ℎ (sdot)1205962119900 (17)

where 120576 = [1205761 1205762 1205763]119879 119861 = [0 0 1]119879 and

119860 = [[[


]]]

(18)

is Hurwitz


120588119894 = 119874( 11205961198961119900 ) 119894 = 1 2 3 forall119905 ⩾ 1198791 gt 0

(19)




10038161003816100381610038161003816119891 minus 11989110038161003816100381610038161003816 ⩽ 120578 (20)


1205941 = 11988811198901 + 120601119903 (21)


1198902 = 1205941 minus 1199092 (22)

and thus

1198901 = 120601119903 minus 1199092 = minus11988811198901 + 1198902 (23)


1205941 = 1198881 1198901 + 120601119903 = minus11988821 1198901 + 11988811198902 + 120601119903 (24)


120575119911 = 1198881198864 120574 ( 1205941 minus 119891 + 120578) 1198902 + 11988821198864 1198902 (25)






r r

a4

ESO

f


z =c

a4 ( 1 minus f + ) e2 +

c2a4

e2

Adaptive Law

= 1ce22

= 2ce22

3( 1 minus f + )

z







120578 = 120582111988811989022120574 = 1205822119888119890221205743 (10038161003816100381610038161003816 1205941 minus 11989110038161003816100381610038161003816 + 120578)

(26)





1198811 = 1211989021 + 1211989022 + 121205821 1205782 + 121205822 120574

2 (27)















22


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)


1120599 + 120578 lt 1120578 (0) (31)

Therefore



(32)


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)



1198812 = 1211989021 (34)


2 = minus119888111989021 + 11989011198902 (35)



Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018













r r

a4

ESO

f


z =c

a4 ( 1 minus f + ) e2 +

c2a4

e2

Adaptive Law

= 1ce22

= 2ce22

3( 1 minus f + )

z







120578 = 120582111988811989022120574 = 1205822119888119890221205743 (10038161003816100381610038161003816 1205941 minus 11989110038161003816100381610038161003816 + 120578)

(26)





1198811 = 1211989021 + 1211989022 + 121205821 1205782 + 121205822 120574

2 (27)















22


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)


1120599 + 120578 lt 1120578 (0) (31)

Therefore



(32)


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)



1198812 = 1211989021 (34)


2 = minus119888111989021 + 11989011198902 (35)



Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018










22


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022

minus (120599 + 120578) [(119888 minus 141198881 (120599 + 120578)) 119890

22 minus 100381610038161003816100381611989021003816100381610038161003816]

(30)


1120599 + 120578 lt 1120578 (0) (31)

Therefore



(32)


100381610038161003816100381611989021003816100381610038161003816)2 minus 119888212057411989022 (33)



1198812 = 1211989021 (34)


2 = minus119888111989021 + 11989011198902 (35)



Time(s) 119896119901 1198961198890 7135 1031635 9630 192545 6058 90355 7346 102175 6273 1169100 6780 1591150 1504 1912









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018









minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)


48 50 523

4

5

35 4011

11512

50 100 1500Time (s)



50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

Pitc

h Er

ror (

degr

ee)




5


50 100 1500Time (s)

minus20

minus15

minus10

minus5

0

10

Defl

ectio

n (d

egre

e)


48 50 52 543

35

4

35 40112114116118

12

ReferenceNominal


minus4

minus2

0

2

4

6

8

10

12

14

16Pi

tch

Ang

le (d

egre

e)

50 100 1500Time (s)







50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018










50 100 1500Time (s)

minus002

minus001

0

001

002

003

004

005

Pitc

h Er

ror (

degr

ee)


ReferenceNominal


35 40

11

12

48 50 52 543

35

4

50 100 1500Time (s)

minus4

minus2

0

2

4

6

8

10

12

14

16

Pitc

h A

ngle

(deg

ree)




50 100 1500Time (s)

minus02

minus015

minus01

minus005

0

005

01

015

02

Pitc

h Er

ror (

degr

ee)



5 Conclusion


Notation




Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018










Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018

















Journal of












Journal of







Volume 2018






Volume 2018








Adaptive Backstepping Control for Longitudinal Dynamics of...

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