Neural Back-Stepping Control of Hypersonic Flight Vehicle ...
6
Research Article Neural Back-Stepping Control of Hypersonic Flight Vehicle with Actuator Fault Qi Wu 1 and Yuyan Guo 2 1 Department of Computer Science, Guangdong Police College, Guangzhou 510230, China 2 School of Automation, Northwestern Polytechnical University, Xi’an 710072, China Correspondence should be addressed to Qi Wu; [email protected] Received 14 October 2017; Accepted 3 January 2018; Published 1 March 2018 Academic Editor: Youqing Wang Copyright © 2018 Qi Wu and Yuyan Guo. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper addresses the fault-tolerant control of hypersonic flight vehicle. To estimate the unknown function in flight dynamics, neural networks are employed in controller design. Moreover, in order to compensate the actuator fault, an adaptive signal is introduced in the controller design to estimate the unknown fault parameters. Simulation results demonstrate that the proposed approach could obtain satisfying performance. 1. Introduction Fault diagnosis and fault-tolerant control have been a hot issue since its significance on actual systems with actuator faults. By identifying or estimating unknown faults, the influence of faults could be considered and eliminated in controller design. Representative study on this issue could be found in [1, 2], while in this paper, fault diagnosis have been studied and applied on hypersonic flight vehicle (HFV) whose control methods have attracted a lot of attention. Hypersonic flight vehicle has a much higher speed than tradi- tional aircraſt, however due to its special flight environment as well as structure, the controller design of HFV faces more challenges [3]. Currently, most studies on HFV control did not take the actuator fault into consideration. However, unknown fault such as actuator dead-zone is quite common in nonlinear systems and may cause serious consequences [4]. In order to remove the influence caused by actuator fault, several approaches have been studied such as fuzzy control [5], robust control [6, 7], and adaptive compensation [8, 9]. e model of the HFV also contains uncertain nonlinearity. As a result, studies on HFV have concentrated on the estimation of the uncertainty. Among all the methods, neural network has been widely used due to its good performance in approximating unknown nonlinear functions [10]. e model of HFV is highly nonlinear; meanwhile the altitude subsystem could be considered as a strict-feedback system [11]. erefore, back-stepping method [12–14] in HFV control has been widely studied. To eliminate continuous derivatives of each virtual control in back-stepping design, multiple methods such as dynamic surface control [15] and differentiator [14] are combined with back-stepping which makes the approach practicable. It is noticed that in [16] the multiple actuator fault is considered and the robust design is presented to guarantee the system stability. Moreover, in [17], the input dead-zone is considered where the Nussbaum design is included to make the adaptive design available. In this paper, we try to estimate the parameters of system fault in the hypersonic flight dynamics and then the “dynamic inversion” of the actuator fault can be included in the control signal. e whole design is using a back-stepping control law with neural networks and adaptive method. e paper is organized in 6 parts. In Section 2, the control-oriented model (COM) of HFV considered in this paper is simply introduced. In Sections 3 and 4, the back- stepping controller based on neural networks and adaptive fault estimation is designed and system stability is analysed. Hindawi Journal of Control Science and Engineering Volume 2018, Article ID 2198423, 5 pages https://doi.org/10.1155/2018/2198423
Transcript of Neural Back-Stepping Control of Hypersonic Flight Vehicle ...
Research ArticleNeural Back-Stepping Control of Hypersonic Flight Vehicle withActuator Fault
QiWu 1 and Yuyan Guo 2
1Department of Computer Science Guangdong Police College Guangzhou 510230 China2School of Automation Northwestern Polytechnical University Xirsquoan 710072 China
Correspondence should be addressed to Qi Wu wuqi888126com
Received 14 October 2017 Accepted 3 January 2018 Published 1 March 2018
Academic Editor Youqing Wang
Copyright copy 2018 Qi Wu and Yuyan Guo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper addresses the fault-tolerant control of hypersonic flight vehicle To estimate the unknown function in flight dynamicsneural networks are employed in controller design Moreover in order to compensate the actuator fault an adaptive signal isintroduced in the controller design to estimate the unknown fault parameters Simulation results demonstrate that the proposedapproach could obtain satisfying performance
1 Introduction
Fault diagnosis and fault-tolerant control have been a hotissue since its significance on actual systems with actuatorfaults By identifying or estimating unknown faults theinfluence of faults could be considered and eliminated incontroller design Representative study on this issue couldbe found in [1 2] while in this paper fault diagnosis havebeen studied and applied on hypersonic flight vehicle (HFV)whose control methods have attracted a lot of attentionHypersonic flight vehicle has amuch higher speed than tradi-tional aircraft however due to its special flight environmentas well as structure the controller design of HFV faces morechallenges [3]
Currently most studies on HFV control did not take theactuator fault into consideration However unknown faultsuch as actuator dead-zone is quite common in nonlinearsystems and may cause serious consequences [4] In orderto remove the influence caused by actuator fault severalapproaches have been studied such as fuzzy control [5]robust control [6 7] and adaptive compensation [8 9]The model of the HFV also contains uncertain nonlinearityAs a result studies on HFV have concentrated on theestimation of the uncertainty Among all the methods neural
network has been widely used due to its good performance inapproximating unknown nonlinear functions [10]
The model of HFV is highly nonlinear meanwhile thealtitude subsystem could be considered as a strict-feedbacksystem [11]Therefore back-steppingmethod [12ndash14] in HFVcontrol has been widely studied To eliminate continuousderivatives of each virtual control in back-stepping designmultiple methods such as dynamic surface control [15] anddifferentiator [14] are combined with back-stepping whichmakes the approach practicable It is noticed that in [16] themultiple actuator fault is considered and the robust designis presented to guarantee the system stability Moreover in[17] the input dead-zone is considered where the Nussbaumdesign is included to make the adaptive design available Inthis paper we try to estimate the parameters of system faultin the hypersonic flight dynamics and then the ldquodynamicinversionrdquo of the actuator fault can be included in the controlsignal The whole design is using a back-stepping control lawwith neural networks and adaptive method
The paper is organized in 6 parts In Section 2 thecontrol-oriented model (COM) of HFV considered in thispaper is simply introduced In Sections 3 and 4 the back-stepping controller based on neural networks and adaptivefault estimation is designed and system stability is analysed
HindawiJournal of Control Science and EngineeringVolume 2018 Article ID 2198423 5 pageshttpsdoiorg10115520182198423
2 Journal of Control Science and Engineering
The simulation results of the designed controller are shownin Section 5 Finally the summary is given in Section 6
2 Longitudinal Dynamics of HFV withActuator Fault
The COM in [18] is employed for study
= 119879 cos120572 minus 119863119898 minus 119892 sin 120574ℎ = 119881 sin 120574120574 = 119871 + 119879 sin120572119898119881 minus 119892 cos 120574119881 = 119902 minus 120574119902 = 119872119910119910119868119910119910
(1)
The detail of the dynamics can be found in [18]Consider the following actuator fault model
120575119890 =
119906 minus 119887 if 119906 ge 1198870 if minus 119887 lt 119906 lt 119887119906 + 119887 if 119906 le minus119887
(2)
where 119906 is the designed control input and 119887 lt 0 is anunknown fault parameter to be estimated
Remark 1 In HFV systems the fault will cause error betweendesigned and actual control input and it will result in trackingerror or even flight instability
3 Adaptive Back-Stepping Controller
The strict-feedback form altitude subsystem based on thehypersonic flight dynamics is considered
1 = 1198911 + 119892111990922 = 11990933 = 1198913 + 1198923120575119890
(3)
where 1199091 = 120574 1199092 = 120574 + 120572 and 1199093 = 119902 119891119894 119892119894 are nonlinearfunctions of the HFV model
Assumption 2 In altitude subsystem (3) 1198923 is a boundedfunction
Flight path angle (FPA) tracking error 1199091 is defined as
1199091 = 1199091 minus 1199091119889 (4)
where 1199091119889 is the command signal of FPA which is designedvia altitude reference signal
Step 1 Choose virtual control of 1199092 as1199092119889 = 11198921 (minus1198911 minus 12058911199091 + 1119889) (5)
where 1205891 is the control gainTo avoid the continuous derivative of virtual control the
following first-order differentiator is designed
2119888 = minus11989720radic10038161003816100381610038161206032119888 minus 11990921198891003816100381610038161003816sgn (1206032119888 minus 1199092119889) + 12060321198892119889 = minus11989721sgn (1206032119889 minus 2119888)
(6)
where 11989720 and 11989721 are positive designed parameters
Step 2 Define the tracking error 11990921199092 = 1199092 minus 1199092119889 (7)
Choose virtual control of 1199093 as1199093119889 = minus12058921199092 + 2119888 minus 11989211199091 (8)
where 1205892 is the control gainThe following first-order differentiator is designed
3119888 = minus11989730radic10038161003816100381610038161206033119888 minus 11990931198891003816100381610038161003816sgn (1206033119888 minus 1199093119889) + 12060331198893119889 = minus11989731sgn (1206033119889 minus 3119888)
(9)
where 11989730 and 11989731 are positive designed parameters
Step 3 Define the tracking error 11990931199093 = 1199093 minus 1199093119889 (10)
Due to the uncertainty caused by imprecise model thenonlinear function 1198913 may be unknown Therefore its NN-based estimation value is employed in the control law
1199060 = 11198923 (minus1198793Θ3 minus 12058931199093 minus 1199092 + 3119888) (11)
where 1205893 is the control gain and 3 is the estimation of theoptimal NN weights 120596lowast3 which is obtained via adaptive law
1205963 = Γ31199093Θ3 (1199093) minus Γ312057533 (12)
where Γ3 and 1205753 are positive parameters Θ3 is obtained viaradial basis function Define 3 = 120596lowast3 minus 3
In order to eliminate the influence of unknown constant119887 signal is introduced in controller design to compensatethe actuator
119906 = 1199060 + sgn (1199060) (13)
The adaptive law is designed as
119887 = minusΓ11988711989231199093sgn (1199060) minus Γ119887120590119887 (14)
where Γ119887 and 120590119887 are positive designed parameters Define theestimation error = 119887 minus
Journal of Control Science and Engineering 3
Remark 3 In previous work on HFV dead-zone fault control[17] the paper regarded the dead-zone as a part of thecompound disturbance where robust technique is employedIn our paper we proposed an adaptive law to estimate theunknown fault parameter and added a compensating signalin the controller so that the influence of dead-zone fault couldbe directly eliminated which is shown by Figures 3 and 5 inour manuscript
Define the velocity tracking error as
= 119881 minus 119881119903 (15)
where 119881119903 is the reference signal Then the following PIDcontroller is designed
Φ = 119896119901V + 119896119894V int (119905) d119905 + 119896119889V 119881 (16)
4 Stability Analysis
Theorem4 Consider theHFV altitude system (3) with controllaws (5) (8) (11) and (13) and adaptive laws (12) and (14) allof the error signals are uniformly ultimately bounded
Proof The Lyapunov function candidate is chosen as
119881119871 = 3sum119894=1
119881119894 (17)
where
1198811 = 12119909211198812 = 12119909221198813 = 1211990923 + 121198793 Γminus13 3 + 12Γminus1119887 2
(18)
The derivative of 119881119871 is obtained as
119871 = 3sum119894=1
119909119894 (119894 minus 119894119889) minus 1198793 Γminus13 1205963 minus Γminus1119887 119887 (19)
According to the conclusion in [19] 2119888 3119888 in differ-entiators (6) (9) could estimate 2119889 and 3119889 to arbitraryaccuracy Therefore there exists
119894119889 = 119894119888 + 120594119894 119894 = 2 3 (20)
where 120594119894 le |120594119894119898| 120594119894119898 is a positive constant Substitute thecontrol laws and adaptive laws into (19) there exists
119871 = minus 3sum119894=1
1205891198941199092119894 + 1205753 (1198793120596lowast3 minus 1198793 3) + 12057631199093+ 11988711990931198923 [sgn (1199060) minus sgn (119906)] + 120590119887 (119887 minus 2)minus 3sum119894=2
119909119894120594119894(21)
where 1205763 is the bounded inevitable NN reconstruction errorsatisfying |1205763| le 1205763119898 where 1205763119898 is a positive constant Define120578 = 1198871198923[sgn(1199060) minus sgn(119906)] consider Assumption 1 120578 is abounded signal satisfying |120578| le 120578119898 where 120578119898 is a positiveconstant
The following inequalities hold
1205753 (1198793120596lowast3 minus 1198793 3) le 121205753 (1003817100381710038171003817120596lowast3 10038171003817100381710038172 minus 1198793 3)120590119887 (119887 minus 2) le 12120590119887 (1198872 minus 2)
minus 3sum119894=2
119909119894120594119894 le 3sum119894=2
(121199092119894 + 121205942119894119898)12057631199093 le 1212057623119898 + 12119909231205781199093 le 121205782119898 + 1211990923
(22)
The derivative of 119881119871 satisfies119871 le minus120589111990921 minus (1205892 minus 12) 11990922 minus (1205893 minus 32) 11990923 minus 1212057531198793 3
minus 121205901198872 + 121205753 1003817100381710038171003817120596lowast3 10038171003817100381710038172 + 1212057623119898 + 121205782119898+ 121205901198871198872 +
3sum119894=2
121205942119894119898le minus120590119881119871 + 119875
(23)
where 120590 = min21205891 (21205892 minus 1) (21205892 minus 3) 1205753Γ3 120590119887Γ119887 and119875 = (12)1205753120596lowast3 2 + (12)12057623119898 + (12)1205782119898 + (12)1205901198871198872 +sum3119894=2(12)1205942119894119898 Then all of the signals in 119881119871 are uniformlyultimately bounded This completes the proof
5 Simulation
Let altitude increase 1524m from the initial value noticethat the signal will pass the following filter to make sure thereference signal is smooth enough for tracking
Δ 119903Δ = 120596112059622(119904 + 1205961) (1199042 + 21205851205962119904 + 12059622) (24)
where 1205961 = 08 1205962 = 05 and 120585 = 01 The FPA commandsignal is obtained as
1199091119889 = arcsin(minus119896ℎℎ minus 119896119894 int ℎ d119905 + ℎ119903119881 ) (25)
where 119896ℎ = 05 119896119894 = 005 ℎ = ℎ minus ℎ119903 is the altitude trackingerror and ℎ119903 is the altitude reference signal
The parameters in the controller are set as 1205891 = 2 1205892 = 21205893 = 3 119896119901V = 5 119896119889V = 001 and 119896119894V = 001 The parametersof first-order differentiator and adaptive laws are chosen as11989720 = 120 11989730 = 100 11989721 = 09 11989731 = 005 Γ3 = 05 1205753 = 0002
4 Journal of Control Science and Engineering
AltitudeReference signal
262
2625
263
2635
264
Alti
tude
(m)
10 20 30 40 50 60 70 800Time (sec)
minus2
minus1
0
1
2
erro
r (m
)A
ltitu
de tr
acki
ng
10 20 30 40 50 60 70 800Time (sec)
times104
Figure 1 Altitude
Flig
ht p
ath
Pitc
hing
angl
ePi
tchi
ng ra
te
10 20 30 40 50 60 70 800Time (sec)
minus05
0
05
angl
e (de
g)
0
5
10
(deg
)
minus20
0
20
(deg
s)
10 20 30 40 50 60 70 800Time (sec)
10 20 30 40 50 60 70 800Time (sec)
p
q
Figure 2 FPA pitching angle and pitching rate
Γ119887 = 0113 and 120590119887 = 0005 Take the initial states as ℎ0 =26212m V0 = 2743ms 1205740 = 0 deg 1205720 = 0 deg and 1199020 =0 degs The fault parameter is set as 119887 = 005 The results areas follows
Figure 1 indicates that the designed controller couldobtain good performance on trajectory tracking Figure 2shows that the all state variables are boundedThe response ofdesigned adaptive signal and NN weights norm are shown
002
0025
003
0035
004
0045
005
0055
006
Am
plitu
de
10 20 30 40 50 60 70 800Time (sec)
b
Figure 3 Response of
0
002
004
006
008
01
012
014A
mpl
itude
10 20 30 40 50 60 70 800Time (sec)
3
Figure 4 NN weights norm
in Figures 3 and 4 The designed control input and actualcontrol input are shown in Figure 5 respectively The resultsverify that the adaptive compensation design could eliminatethe influence of actuator fault
6 Conclusion
This paper proposes an adaptive back-stepping control lawwith NN learning for HFV control The influence of actuatorfault is eliminated by constructing an adaptive compensationsignal Meanwhile the unknown nonlinearity is estimated byneural networks The simulation results clearly present theconsequence of the above design and verify that the approachcould reach the desired tracking performance when actuatorfault and model uncertainty exist
Journal of Control Science and Engineering 5
Designed inputActual input
10 20 30 40 50 60 70 800Time (sec)
minus30
minus20
minus10
0
10
20
30
Elev
ator
(deg
)
Figure 5 Control input
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyt-ical and knowledge-based redundancy a survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] D Xu B Jiang and P Shi ldquoRobust NSV fault-tolerant controlsystem design against actuator faults and control surface dam-age under actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 62 no 9 pp 5919ndash5928 2015
[3] 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
[4] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[5] Q Shen B Jiang and V Cocquempot ldquoFault diagnosis andestimation for near-space hypersonic vehicle with sensor faultsrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 226 no 3 pp302ndash313 2012
[6] Y Li S Tong Y Liu and T Li ldquoAdaptive fuzzy robust outputfeedback control of nonlinear systems with unknown deadzones based on a small-gain approachrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 164ndash176 2014
[7] D Zhao YWang Y Li and S XDing ldquoHinfin fault estimation for2-D linear discrete time-varying systems based on krein spacemethodrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol PP no 99 Article ID 2723623 pp 1ndash10 2017
[8] D Zhao D Shen and Y Wang ldquoFault diagnosis and compen-sation for two-dimensional discrete time systems with sensorfaults and time-varying delaysrdquo International Journal of Robustand Nonlinear Control vol 27 no 16 pp 3296ndash3320 2017
[9] Y Wang D Zhao Y Li and S Ding ldquoUnbiased minimumvariance fault and state estimation for linear discrete time-varying two-dimensional systemsrdquo Institute of Electrical andElectronics Engineers Transactions onAutomatic Control vol 62no 10 pp 5463ndash5469 2017
[10] S Zhang C Li and J Zhu ldquoComposite dynamic surface controlof hypersonic flight dynamics using neural networksrdquo ScienceChina Information Sciences vol 58 no 7 2015
[11] J-L Wu ldquoStabilizing controllers design for switched nonlinearsystems in strict-feedback formrdquo Automatica vol 45 no 4 pp1092ndash1096 2009
[12] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014
[13] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64 no11 pp 8676ndash8685 2017
[14] B Xu D Yang Z Shi Y Pan B Chen and F SunldquoOnline recorded data-based composite neural control ofstrict-feedback systems with application to hypersonic flightdynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 Article ID 2743784 pp 1ndash11 2017
[15] D Swaroop J K Hedrick P P Yip and J Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 45 no 10 pp 1893ndash1899 2000
[16] B Xu Y Guo Y Yuan Y Fan and D Wang ldquoFault-tolerantcontrol using command-filtered adaptive back-stepping tech-nique application to hypersonic longitudinal flight dynamicsrdquoInternational Journal of Adaptive Control and Signal Processingvol 30 no 4 pp 553ndash577 2016
[17] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015
[18] J T Parker A Serrani S Yurkovich M A Bolender and D BDoman ldquoControl-orientedmodeling of an air-breathing hyper-sonic vehiclerdquo Journal of Guidance Control and Dynamics vol30 no 3 pp 856ndash869 2007
[19] A Levant ldquoRobust Exact Differentiation via Sliding ModeTechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998
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2 Journal of Control Science and Engineering
The simulation results of the designed controller are shownin Section 5 Finally the summary is given in Section 6
2 Longitudinal Dynamics of HFV withActuator Fault
The COM in [18] is employed for study
= 119879 cos120572 minus 119863119898 minus 119892 sin 120574ℎ = 119881 sin 120574120574 = 119871 + 119879 sin120572119898119881 minus 119892 cos 120574119881 = 119902 minus 120574119902 = 119872119910119910119868119910119910
(1)
The detail of the dynamics can be found in [18]Consider the following actuator fault model
120575119890 =
119906 minus 119887 if 119906 ge 1198870 if minus 119887 lt 119906 lt 119887119906 + 119887 if 119906 le minus119887
(2)
where 119906 is the designed control input and 119887 lt 0 is anunknown fault parameter to be estimated
Remark 1 In HFV systems the fault will cause error betweendesigned and actual control input and it will result in trackingerror or even flight instability
3 Adaptive Back-Stepping Controller
The strict-feedback form altitude subsystem based on thehypersonic flight dynamics is considered
1 = 1198911 + 119892111990922 = 11990933 = 1198913 + 1198923120575119890
(3)
where 1199091 = 120574 1199092 = 120574 + 120572 and 1199093 = 119902 119891119894 119892119894 are nonlinearfunctions of the HFV model
Assumption 2 In altitude subsystem (3) 1198923 is a boundedfunction
Flight path angle (FPA) tracking error 1199091 is defined as
1199091 = 1199091 minus 1199091119889 (4)
where 1199091119889 is the command signal of FPA which is designedvia altitude reference signal
Step 1 Choose virtual control of 1199092 as1199092119889 = 11198921 (minus1198911 minus 12058911199091 + 1119889) (5)
where 1205891 is the control gainTo avoid the continuous derivative of virtual control the
following first-order differentiator is designed
2119888 = minus11989720radic10038161003816100381610038161206032119888 minus 11990921198891003816100381610038161003816sgn (1206032119888 minus 1199092119889) + 12060321198892119889 = minus11989721sgn (1206032119889 minus 2119888)
(6)
where 11989720 and 11989721 are positive designed parameters
Step 2 Define the tracking error 11990921199092 = 1199092 minus 1199092119889 (7)
Choose virtual control of 1199093 as1199093119889 = minus12058921199092 + 2119888 minus 11989211199091 (8)
where 1205892 is the control gainThe following first-order differentiator is designed
3119888 = minus11989730radic10038161003816100381610038161206033119888 minus 11990931198891003816100381610038161003816sgn (1206033119888 minus 1199093119889) + 12060331198893119889 = minus11989731sgn (1206033119889 minus 3119888)
(9)
where 11989730 and 11989731 are positive designed parameters
Step 3 Define the tracking error 11990931199093 = 1199093 minus 1199093119889 (10)
Due to the uncertainty caused by imprecise model thenonlinear function 1198913 may be unknown Therefore its NN-based estimation value is employed in the control law
1199060 = 11198923 (minus1198793Θ3 minus 12058931199093 minus 1199092 + 3119888) (11)
where 1205893 is the control gain and 3 is the estimation of theoptimal NN weights 120596lowast3 which is obtained via adaptive law
1205963 = Γ31199093Θ3 (1199093) minus Γ312057533 (12)
where Γ3 and 1205753 are positive parameters Θ3 is obtained viaradial basis function Define 3 = 120596lowast3 minus 3
In order to eliminate the influence of unknown constant119887 signal is introduced in controller design to compensatethe actuator
119906 = 1199060 + sgn (1199060) (13)
The adaptive law is designed as
119887 = minusΓ11988711989231199093sgn (1199060) minus Γ119887120590119887 (14)
where Γ119887 and 120590119887 are positive designed parameters Define theestimation error = 119887 minus
Journal of Control Science and Engineering 3
Remark 3 In previous work on HFV dead-zone fault control[17] the paper regarded the dead-zone as a part of thecompound disturbance where robust technique is employedIn our paper we proposed an adaptive law to estimate theunknown fault parameter and added a compensating signalin the controller so that the influence of dead-zone fault couldbe directly eliminated which is shown by Figures 3 and 5 inour manuscript
Define the velocity tracking error as
= 119881 minus 119881119903 (15)
where 119881119903 is the reference signal Then the following PIDcontroller is designed
Φ = 119896119901V + 119896119894V int (119905) d119905 + 119896119889V 119881 (16)
4 Stability Analysis
Theorem4 Consider theHFV altitude system (3) with controllaws (5) (8) (11) and (13) and adaptive laws (12) and (14) allof the error signals are uniformly ultimately bounded
Proof The Lyapunov function candidate is chosen as
119881119871 = 3sum119894=1
119881119894 (17)
where
1198811 = 12119909211198812 = 12119909221198813 = 1211990923 + 121198793 Γminus13 3 + 12Γminus1119887 2
(18)
The derivative of 119881119871 is obtained as
119871 = 3sum119894=1
119909119894 (119894 minus 119894119889) minus 1198793 Γminus13 1205963 minus Γminus1119887 119887 (19)
According to the conclusion in [19] 2119888 3119888 in differ-entiators (6) (9) could estimate 2119889 and 3119889 to arbitraryaccuracy Therefore there exists
119894119889 = 119894119888 + 120594119894 119894 = 2 3 (20)
where 120594119894 le |120594119894119898| 120594119894119898 is a positive constant Substitute thecontrol laws and adaptive laws into (19) there exists
119871 = minus 3sum119894=1
1205891198941199092119894 + 1205753 (1198793120596lowast3 minus 1198793 3) + 12057631199093+ 11988711990931198923 [sgn (1199060) minus sgn (119906)] + 120590119887 (119887 minus 2)minus 3sum119894=2
119909119894120594119894(21)
where 1205763 is the bounded inevitable NN reconstruction errorsatisfying |1205763| le 1205763119898 where 1205763119898 is a positive constant Define120578 = 1198871198923[sgn(1199060) minus sgn(119906)] consider Assumption 1 120578 is abounded signal satisfying |120578| le 120578119898 where 120578119898 is a positiveconstant
The following inequalities hold
1205753 (1198793120596lowast3 minus 1198793 3) le 121205753 (1003817100381710038171003817120596lowast3 10038171003817100381710038172 minus 1198793 3)120590119887 (119887 minus 2) le 12120590119887 (1198872 minus 2)
minus 3sum119894=2
119909119894120594119894 le 3sum119894=2
(121199092119894 + 121205942119894119898)12057631199093 le 1212057623119898 + 12119909231205781199093 le 121205782119898 + 1211990923
(22)
The derivative of 119881119871 satisfies119871 le minus120589111990921 minus (1205892 minus 12) 11990922 minus (1205893 minus 32) 11990923 minus 1212057531198793 3
minus 121205901198872 + 121205753 1003817100381710038171003817120596lowast3 10038171003817100381710038172 + 1212057623119898 + 121205782119898+ 121205901198871198872 +
3sum119894=2
121205942119894119898le minus120590119881119871 + 119875
(23)
where 120590 = min21205891 (21205892 minus 1) (21205892 minus 3) 1205753Γ3 120590119887Γ119887 and119875 = (12)1205753120596lowast3 2 + (12)12057623119898 + (12)1205782119898 + (12)1205901198871198872 +sum3119894=2(12)1205942119894119898 Then all of the signals in 119881119871 are uniformlyultimately bounded This completes the proof
5 Simulation
Let altitude increase 1524m from the initial value noticethat the signal will pass the following filter to make sure thereference signal is smooth enough for tracking
Δ 119903Δ = 120596112059622(119904 + 1205961) (1199042 + 21205851205962119904 + 12059622) (24)
where 1205961 = 08 1205962 = 05 and 120585 = 01 The FPA commandsignal is obtained as
1199091119889 = arcsin(minus119896ℎℎ minus 119896119894 int ℎ d119905 + ℎ119903119881 ) (25)
where 119896ℎ = 05 119896119894 = 005 ℎ = ℎ minus ℎ119903 is the altitude trackingerror and ℎ119903 is the altitude reference signal
The parameters in the controller are set as 1205891 = 2 1205892 = 21205893 = 3 119896119901V = 5 119896119889V = 001 and 119896119894V = 001 The parametersof first-order differentiator and adaptive laws are chosen as11989720 = 120 11989730 = 100 11989721 = 09 11989731 = 005 Γ3 = 05 1205753 = 0002
4 Journal of Control Science and Engineering
AltitudeReference signal
262
2625
263
2635
264
Alti
tude
(m)
10 20 30 40 50 60 70 800Time (sec)
minus2
minus1
0
1
2
erro
r (m
)A
ltitu
de tr
acki
ng
10 20 30 40 50 60 70 800Time (sec)
times104
Figure 1 Altitude
Flig
ht p
ath
Pitc
hing
angl
ePi
tchi
ng ra
te
10 20 30 40 50 60 70 800Time (sec)
minus05
0
05
angl
e (de
g)
0
5
10
(deg
)
minus20
0
20
(deg
s)
10 20 30 40 50 60 70 800Time (sec)
10 20 30 40 50 60 70 800Time (sec)
p
q
Figure 2 FPA pitching angle and pitching rate
Γ119887 = 0113 and 120590119887 = 0005 Take the initial states as ℎ0 =26212m V0 = 2743ms 1205740 = 0 deg 1205720 = 0 deg and 1199020 =0 degs The fault parameter is set as 119887 = 005 The results areas follows
Figure 1 indicates that the designed controller couldobtain good performance on trajectory tracking Figure 2shows that the all state variables are boundedThe response ofdesigned adaptive signal and NN weights norm are shown
002
0025
003
0035
004
0045
005
0055
006
Am
plitu
de
10 20 30 40 50 60 70 800Time (sec)
b
Figure 3 Response of
0
002
004
006
008
01
012
014A
mpl
itude
10 20 30 40 50 60 70 800Time (sec)
3
Figure 4 NN weights norm
in Figures 3 and 4 The designed control input and actualcontrol input are shown in Figure 5 respectively The resultsverify that the adaptive compensation design could eliminatethe influence of actuator fault
6 Conclusion
This paper proposes an adaptive back-stepping control lawwith NN learning for HFV control The influence of actuatorfault is eliminated by constructing an adaptive compensationsignal Meanwhile the unknown nonlinearity is estimated byneural networks The simulation results clearly present theconsequence of the above design and verify that the approachcould reach the desired tracking performance when actuatorfault and model uncertainty exist
Journal of Control Science and Engineering 5
Designed inputActual input
10 20 30 40 50 60 70 800Time (sec)
minus30
minus20
minus10
0
10
20
30
Elev
ator
(deg
)
Figure 5 Control input
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyt-ical and knowledge-based redundancy a survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] D Xu B Jiang and P Shi ldquoRobust NSV fault-tolerant controlsystem design against actuator faults and control surface dam-age under actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 62 no 9 pp 5919ndash5928 2015
[3] 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
[4] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[5] Q Shen B Jiang and V Cocquempot ldquoFault diagnosis andestimation for near-space hypersonic vehicle with sensor faultsrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 226 no 3 pp302ndash313 2012
[6] Y Li S Tong Y Liu and T Li ldquoAdaptive fuzzy robust outputfeedback control of nonlinear systems with unknown deadzones based on a small-gain approachrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 164ndash176 2014
[7] D Zhao YWang Y Li and S XDing ldquoHinfin fault estimation for2-D linear discrete time-varying systems based on krein spacemethodrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol PP no 99 Article ID 2723623 pp 1ndash10 2017
[8] D Zhao D Shen and Y Wang ldquoFault diagnosis and compen-sation for two-dimensional discrete time systems with sensorfaults and time-varying delaysrdquo International Journal of Robustand Nonlinear Control vol 27 no 16 pp 3296ndash3320 2017
[9] Y Wang D Zhao Y Li and S Ding ldquoUnbiased minimumvariance fault and state estimation for linear discrete time-varying two-dimensional systemsrdquo Institute of Electrical andElectronics Engineers Transactions onAutomatic Control vol 62no 10 pp 5463ndash5469 2017
[10] S Zhang C Li and J Zhu ldquoComposite dynamic surface controlof hypersonic flight dynamics using neural networksrdquo ScienceChina Information Sciences vol 58 no 7 2015
[11] J-L Wu ldquoStabilizing controllers design for switched nonlinearsystems in strict-feedback formrdquo Automatica vol 45 no 4 pp1092ndash1096 2009
[12] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014
[13] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64 no11 pp 8676ndash8685 2017
[14] B Xu D Yang Z Shi Y Pan B Chen and F SunldquoOnline recorded data-based composite neural control ofstrict-feedback systems with application to hypersonic flightdynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 Article ID 2743784 pp 1ndash11 2017
[15] D Swaroop J K Hedrick P P Yip and J Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 45 no 10 pp 1893ndash1899 2000
[16] B Xu Y Guo Y Yuan Y Fan and D Wang ldquoFault-tolerantcontrol using command-filtered adaptive back-stepping tech-nique application to hypersonic longitudinal flight dynamicsrdquoInternational Journal of Adaptive Control and Signal Processingvol 30 no 4 pp 553ndash577 2016
[17] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015
[18] J T Parker A Serrani S Yurkovich M A Bolender and D BDoman ldquoControl-orientedmodeling of an air-breathing hyper-sonic vehiclerdquo Journal of Guidance Control and Dynamics vol30 no 3 pp 856ndash869 2007
[19] A Levant ldquoRobust Exact Differentiation via Sliding ModeTechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Journal of Control Science and Engineering 3
Remark 3 In previous work on HFV dead-zone fault control[17] the paper regarded the dead-zone as a part of thecompound disturbance where robust technique is employedIn our paper we proposed an adaptive law to estimate theunknown fault parameter and added a compensating signalin the controller so that the influence of dead-zone fault couldbe directly eliminated which is shown by Figures 3 and 5 inour manuscript
Define the velocity tracking error as
= 119881 minus 119881119903 (15)
where 119881119903 is the reference signal Then the following PIDcontroller is designed
Φ = 119896119901V + 119896119894V int (119905) d119905 + 119896119889V 119881 (16)
4 Stability Analysis
Theorem4 Consider theHFV altitude system (3) with controllaws (5) (8) (11) and (13) and adaptive laws (12) and (14) allof the error signals are uniformly ultimately bounded
Proof The Lyapunov function candidate is chosen as
119881119871 = 3sum119894=1
119881119894 (17)
where
1198811 = 12119909211198812 = 12119909221198813 = 1211990923 + 121198793 Γminus13 3 + 12Γminus1119887 2
(18)
The derivative of 119881119871 is obtained as
119871 = 3sum119894=1
119909119894 (119894 minus 119894119889) minus 1198793 Γminus13 1205963 minus Γminus1119887 119887 (19)
According to the conclusion in [19] 2119888 3119888 in differ-entiators (6) (9) could estimate 2119889 and 3119889 to arbitraryaccuracy Therefore there exists
119894119889 = 119894119888 + 120594119894 119894 = 2 3 (20)
where 120594119894 le |120594119894119898| 120594119894119898 is a positive constant Substitute thecontrol laws and adaptive laws into (19) there exists
119871 = minus 3sum119894=1
1205891198941199092119894 + 1205753 (1198793120596lowast3 minus 1198793 3) + 12057631199093+ 11988711990931198923 [sgn (1199060) minus sgn (119906)] + 120590119887 (119887 minus 2)minus 3sum119894=2
119909119894120594119894(21)
where 1205763 is the bounded inevitable NN reconstruction errorsatisfying |1205763| le 1205763119898 where 1205763119898 is a positive constant Define120578 = 1198871198923[sgn(1199060) minus sgn(119906)] consider Assumption 1 120578 is abounded signal satisfying |120578| le 120578119898 where 120578119898 is a positiveconstant
The following inequalities hold
1205753 (1198793120596lowast3 minus 1198793 3) le 121205753 (1003817100381710038171003817120596lowast3 10038171003817100381710038172 minus 1198793 3)120590119887 (119887 minus 2) le 12120590119887 (1198872 minus 2)
minus 3sum119894=2
119909119894120594119894 le 3sum119894=2
(121199092119894 + 121205942119894119898)12057631199093 le 1212057623119898 + 12119909231205781199093 le 121205782119898 + 1211990923
(22)
The derivative of 119881119871 satisfies119871 le minus120589111990921 minus (1205892 minus 12) 11990922 minus (1205893 minus 32) 11990923 minus 1212057531198793 3
minus 121205901198872 + 121205753 1003817100381710038171003817120596lowast3 10038171003817100381710038172 + 1212057623119898 + 121205782119898+ 121205901198871198872 +
3sum119894=2
121205942119894119898le minus120590119881119871 + 119875
(23)
where 120590 = min21205891 (21205892 minus 1) (21205892 minus 3) 1205753Γ3 120590119887Γ119887 and119875 = (12)1205753120596lowast3 2 + (12)12057623119898 + (12)1205782119898 + (12)1205901198871198872 +sum3119894=2(12)1205942119894119898 Then all of the signals in 119881119871 are uniformlyultimately bounded This completes the proof
5 Simulation
Let altitude increase 1524m from the initial value noticethat the signal will pass the following filter to make sure thereference signal is smooth enough for tracking
Δ 119903Δ = 120596112059622(119904 + 1205961) (1199042 + 21205851205962119904 + 12059622) (24)
where 1205961 = 08 1205962 = 05 and 120585 = 01 The FPA commandsignal is obtained as
1199091119889 = arcsin(minus119896ℎℎ minus 119896119894 int ℎ d119905 + ℎ119903119881 ) (25)
where 119896ℎ = 05 119896119894 = 005 ℎ = ℎ minus ℎ119903 is the altitude trackingerror and ℎ119903 is the altitude reference signal
The parameters in the controller are set as 1205891 = 2 1205892 = 21205893 = 3 119896119901V = 5 119896119889V = 001 and 119896119894V = 001 The parametersof first-order differentiator and adaptive laws are chosen as11989720 = 120 11989730 = 100 11989721 = 09 11989731 = 005 Γ3 = 05 1205753 = 0002
4 Journal of Control Science and Engineering
AltitudeReference signal
262
2625
263
2635
264
Alti
tude
(m)
10 20 30 40 50 60 70 800Time (sec)
minus2
minus1
0
1
2
erro
r (m
)A
ltitu
de tr
acki
ng
10 20 30 40 50 60 70 800Time (sec)
times104
Figure 1 Altitude
Flig
ht p
ath
Pitc
hing
angl
ePi
tchi
ng ra
te
10 20 30 40 50 60 70 800Time (sec)
minus05
0
05
angl
e (de
g)
0
5
10
(deg
)
minus20
0
20
(deg
s)
10 20 30 40 50 60 70 800Time (sec)
10 20 30 40 50 60 70 800Time (sec)
p
q
Figure 2 FPA pitching angle and pitching rate
Γ119887 = 0113 and 120590119887 = 0005 Take the initial states as ℎ0 =26212m V0 = 2743ms 1205740 = 0 deg 1205720 = 0 deg and 1199020 =0 degs The fault parameter is set as 119887 = 005 The results areas follows
Figure 1 indicates that the designed controller couldobtain good performance on trajectory tracking Figure 2shows that the all state variables are boundedThe response ofdesigned adaptive signal and NN weights norm are shown
002
0025
003
0035
004
0045
005
0055
006
Am
plitu
de
10 20 30 40 50 60 70 800Time (sec)
b
Figure 3 Response of
0
002
004
006
008
01
012
014A
mpl
itude
10 20 30 40 50 60 70 800Time (sec)
3
Figure 4 NN weights norm
in Figures 3 and 4 The designed control input and actualcontrol input are shown in Figure 5 respectively The resultsverify that the adaptive compensation design could eliminatethe influence of actuator fault
6 Conclusion
This paper proposes an adaptive back-stepping control lawwith NN learning for HFV control The influence of actuatorfault is eliminated by constructing an adaptive compensationsignal Meanwhile the unknown nonlinearity is estimated byneural networks The simulation results clearly present theconsequence of the above design and verify that the approachcould reach the desired tracking performance when actuatorfault and model uncertainty exist
Journal of Control Science and Engineering 5
Designed inputActual input
10 20 30 40 50 60 70 800Time (sec)
minus30
minus20
minus10
0
10
20
30
Elev
ator
(deg
)
Figure 5 Control input
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyt-ical and knowledge-based redundancy a survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] D Xu B Jiang and P Shi ldquoRobust NSV fault-tolerant controlsystem design against actuator faults and control surface dam-age under actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 62 no 9 pp 5919ndash5928 2015
[3] 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
[4] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[5] Q Shen B Jiang and V Cocquempot ldquoFault diagnosis andestimation for near-space hypersonic vehicle with sensor faultsrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 226 no 3 pp302ndash313 2012
[6] Y Li S Tong Y Liu and T Li ldquoAdaptive fuzzy robust outputfeedback control of nonlinear systems with unknown deadzones based on a small-gain approachrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 164ndash176 2014
[7] D Zhao YWang Y Li and S XDing ldquoHinfin fault estimation for2-D linear discrete time-varying systems based on krein spacemethodrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol PP no 99 Article ID 2723623 pp 1ndash10 2017
[8] D Zhao D Shen and Y Wang ldquoFault diagnosis and compen-sation for two-dimensional discrete time systems with sensorfaults and time-varying delaysrdquo International Journal of Robustand Nonlinear Control vol 27 no 16 pp 3296ndash3320 2017
[9] Y Wang D Zhao Y Li and S Ding ldquoUnbiased minimumvariance fault and state estimation for linear discrete time-varying two-dimensional systemsrdquo Institute of Electrical andElectronics Engineers Transactions onAutomatic Control vol 62no 10 pp 5463ndash5469 2017
[10] S Zhang C Li and J Zhu ldquoComposite dynamic surface controlof hypersonic flight dynamics using neural networksrdquo ScienceChina Information Sciences vol 58 no 7 2015
[11] J-L Wu ldquoStabilizing controllers design for switched nonlinearsystems in strict-feedback formrdquo Automatica vol 45 no 4 pp1092ndash1096 2009
[12] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014
[13] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64 no11 pp 8676ndash8685 2017
[14] B Xu D Yang Z Shi Y Pan B Chen and F SunldquoOnline recorded data-based composite neural control ofstrict-feedback systems with application to hypersonic flightdynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 Article ID 2743784 pp 1ndash11 2017
[15] D Swaroop J K Hedrick P P Yip and J Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 45 no 10 pp 1893ndash1899 2000
[16] B Xu Y Guo Y Yuan Y Fan and D Wang ldquoFault-tolerantcontrol using command-filtered adaptive back-stepping tech-nique application to hypersonic longitudinal flight dynamicsrdquoInternational Journal of Adaptive Control and Signal Processingvol 30 no 4 pp 553ndash577 2016
[17] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015
[18] J T Parker A Serrani S Yurkovich M A Bolender and D BDoman ldquoControl-orientedmodeling of an air-breathing hyper-sonic vehiclerdquo Journal of Guidance Control and Dynamics vol30 no 3 pp 856ndash869 2007
[19] A Levant ldquoRobust Exact Differentiation via Sliding ModeTechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
4 Journal of Control Science and Engineering
AltitudeReference signal
262
2625
263
2635
264
Alti
tude
(m)
10 20 30 40 50 60 70 800Time (sec)
minus2
minus1
0
1
2
erro
r (m
)A
ltitu
de tr
acki
ng
10 20 30 40 50 60 70 800Time (sec)
times104
Figure 1 Altitude
Flig
ht p
ath
Pitc
hing
angl
ePi
tchi
ng ra
te
10 20 30 40 50 60 70 800Time (sec)
minus05
0
05
angl
e (de
g)
0
5
10
(deg
)
minus20
0
20
(deg
s)
10 20 30 40 50 60 70 800Time (sec)
10 20 30 40 50 60 70 800Time (sec)
p
q
Figure 2 FPA pitching angle and pitching rate
Γ119887 = 0113 and 120590119887 = 0005 Take the initial states as ℎ0 =26212m V0 = 2743ms 1205740 = 0 deg 1205720 = 0 deg and 1199020 =0 degs The fault parameter is set as 119887 = 005 The results areas follows
Figure 1 indicates that the designed controller couldobtain good performance on trajectory tracking Figure 2shows that the all state variables are boundedThe response ofdesigned adaptive signal and NN weights norm are shown
002
0025
003
0035
004
0045
005
0055
006
Am
plitu
de
10 20 30 40 50 60 70 800Time (sec)
b
Figure 3 Response of
0
002
004
006
008
01
012
014A
mpl
itude
10 20 30 40 50 60 70 800Time (sec)
3
Figure 4 NN weights norm
in Figures 3 and 4 The designed control input and actualcontrol input are shown in Figure 5 respectively The resultsverify that the adaptive compensation design could eliminatethe influence of actuator fault
6 Conclusion
This paper proposes an adaptive back-stepping control lawwith NN learning for HFV control The influence of actuatorfault is eliminated by constructing an adaptive compensationsignal Meanwhile the unknown nonlinearity is estimated byneural networks The simulation results clearly present theconsequence of the above design and verify that the approachcould reach the desired tracking performance when actuatorfault and model uncertainty exist
Journal of Control Science and Engineering 5
Designed inputActual input
10 20 30 40 50 60 70 800Time (sec)
minus30
minus20
minus10
0
10
20
30
Elev
ator
(deg
)
Figure 5 Control input
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyt-ical and knowledge-based redundancy a survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] D Xu B Jiang and P Shi ldquoRobust NSV fault-tolerant controlsystem design against actuator faults and control surface dam-age under actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 62 no 9 pp 5919ndash5928 2015
[3] 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
[4] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[5] Q Shen B Jiang and V Cocquempot ldquoFault diagnosis andestimation for near-space hypersonic vehicle with sensor faultsrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 226 no 3 pp302ndash313 2012
[6] Y Li S Tong Y Liu and T Li ldquoAdaptive fuzzy robust outputfeedback control of nonlinear systems with unknown deadzones based on a small-gain approachrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 164ndash176 2014
[7] D Zhao YWang Y Li and S XDing ldquoHinfin fault estimation for2-D linear discrete time-varying systems based on krein spacemethodrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol PP no 99 Article ID 2723623 pp 1ndash10 2017
[8] D Zhao D Shen and Y Wang ldquoFault diagnosis and compen-sation for two-dimensional discrete time systems with sensorfaults and time-varying delaysrdquo International Journal of Robustand Nonlinear Control vol 27 no 16 pp 3296ndash3320 2017
[9] Y Wang D Zhao Y Li and S Ding ldquoUnbiased minimumvariance fault and state estimation for linear discrete time-varying two-dimensional systemsrdquo Institute of Electrical andElectronics Engineers Transactions onAutomatic Control vol 62no 10 pp 5463ndash5469 2017
[10] S Zhang C Li and J Zhu ldquoComposite dynamic surface controlof hypersonic flight dynamics using neural networksrdquo ScienceChina Information Sciences vol 58 no 7 2015
[11] J-L Wu ldquoStabilizing controllers design for switched nonlinearsystems in strict-feedback formrdquo Automatica vol 45 no 4 pp1092ndash1096 2009
[12] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014
[13] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64 no11 pp 8676ndash8685 2017
[14] B Xu D Yang Z Shi Y Pan B Chen and F SunldquoOnline recorded data-based composite neural control ofstrict-feedback systems with application to hypersonic flightdynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 Article ID 2743784 pp 1ndash11 2017
[15] D Swaroop J K Hedrick P P Yip and J Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 45 no 10 pp 1893ndash1899 2000
[16] B Xu Y Guo Y Yuan Y Fan and D Wang ldquoFault-tolerantcontrol using command-filtered adaptive back-stepping tech-nique application to hypersonic longitudinal flight dynamicsrdquoInternational Journal of Adaptive Control and Signal Processingvol 30 no 4 pp 553ndash577 2016
[17] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015
[18] J T Parker A Serrani S Yurkovich M A Bolender and D BDoman ldquoControl-orientedmodeling of an air-breathing hyper-sonic vehiclerdquo Journal of Guidance Control and Dynamics vol30 no 3 pp 856ndash869 2007
[19] A Levant ldquoRobust Exact Differentiation via Sliding ModeTechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Journal of Control Science and Engineering 5
Designed inputActual input
10 20 30 40 50 60 70 800Time (sec)
minus30
minus20
minus10
0
10
20
30
Elev
ator
(deg
)
Figure 5 Control input
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] P M Frank ldquoFault diagnosis in dynamic systems using analyt-ical and knowledge-based redundancy a survey and some newresultsrdquo Automatica vol 26 no 3 pp 459ndash474 1990
[2] D Xu B Jiang and P Shi ldquoRobust NSV fault-tolerant controlsystem design against actuator faults and control surface dam-age under actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 62 no 9 pp 5919ndash5928 2015
[3] 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
[4] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007
[5] Q Shen B Jiang and V Cocquempot ldquoFault diagnosis andestimation for near-space hypersonic vehicle with sensor faultsrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 226 no 3 pp302ndash313 2012
[6] Y Li S Tong Y Liu and T Li ldquoAdaptive fuzzy robust outputfeedback control of nonlinear systems with unknown deadzones based on a small-gain approachrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 164ndash176 2014
[7] D Zhao YWang Y Li and S XDing ldquoHinfin fault estimation for2-D linear discrete time-varying systems based on krein spacemethodrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol PP no 99 Article ID 2723623 pp 1ndash10 2017
[8] D Zhao D Shen and Y Wang ldquoFault diagnosis and compen-sation for two-dimensional discrete time systems with sensorfaults and time-varying delaysrdquo International Journal of Robustand Nonlinear Control vol 27 no 16 pp 3296ndash3320 2017
[9] Y Wang D Zhao Y Li and S Ding ldquoUnbiased minimumvariance fault and state estimation for linear discrete time-varying two-dimensional systemsrdquo Institute of Electrical andElectronics Engineers Transactions onAutomatic Control vol 62no 10 pp 5463ndash5469 2017
[10] S Zhang C Li and J Zhu ldquoComposite dynamic surface controlof hypersonic flight dynamics using neural networksrdquo ScienceChina Information Sciences vol 58 no 7 2015
[11] J-L Wu ldquoStabilizing controllers design for switched nonlinearsystems in strict-feedback formrdquo Automatica vol 45 no 4 pp1092ndash1096 2009
[12] B Xu XHuangDWang and F Sun ldquoDynamic surface controlof constrained hypersonic flightmodels with parameter estima-tion and actuator compensationrdquo Asian Journal of Control vol16 no 1 pp 162ndash174 2014
[13] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol 64 no11 pp 8676ndash8685 2017
[14] B Xu D Yang Z Shi Y Pan B Chen and F SunldquoOnline recorded data-based composite neural control ofstrict-feedback systems with application to hypersonic flightdynamicsrdquo IEEE Transactions on Neural Networks and LearningSystems vol PP no 99 Article ID 2743784 pp 1ndash11 2017
[15] D Swaroop J K Hedrick P P Yip and J Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 45 no 10 pp 1893ndash1899 2000
[16] B Xu Y Guo Y Yuan Y Fan and D Wang ldquoFault-tolerantcontrol using command-filtered adaptive back-stepping tech-nique application to hypersonic longitudinal flight dynamicsrdquoInternational Journal of Adaptive Control and Signal Processingvol 30 no 4 pp 553ndash577 2016
[17] B Xu ldquoRobust adaptive neural control of flexible hypersonicflight vehicle with dead-zone input nonlinearityrdquo NonlinearDynamics vol 80 no 3 pp 1509ndash1520 2015
[18] J T Parker A Serrani S Yurkovich M A Bolender and D BDoman ldquoControl-orientedmodeling of an air-breathing hyper-sonic vehiclerdquo Journal of Guidance Control and Dynamics vol30 no 3 pp 856ndash869 2007
[19] A Levant ldquoRobust Exact Differentiation via Sliding ModeTechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
