Research ArticleRobust Adaptive Neural Control of MorphingAircraft with Prescribed Performance

ZhonghuaWu1 Jingchao Lu1 Jingping Shi1 Yang Liu1 and Qing Zhou2

1School of Automation Northwestern Polytechnical University Xirsquoan 710072 China2Xirsquoan Aeronautics Computing Technique Research Institute AVIC Xirsquoan 710068 China

Correspondence should be addressed to Zhonghua Wu 463897575qqcom

Received 14 October 2016 Revised 28 March 2017 Accepted 11 April 2017 Published 17 May 2017

Academic Editor Asier Ibeas

Copyright copy 2017 Zhonghua Wu 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

This study proposes a low-computational composite adaptive neural control scheme for the longitudinal dynamics of a swept-back wing aircraft subject to parameter uncertainties To efficiently release the constraint often existing in conventional neuraldesigns whose closed-loop stability analysis always necessitates that neural networks (NNs) be confined in the active regionsa smooth switching function is presented to conquer this issue By integrating minimal learning parameter (MLP) techniqueprescribed performance control and a kind of smooth switching strategy into back-stepping design a new composite switchingadaptive neural prescribed performance control scheme is proposed and a new type of adaptive laws is constructed for the altitudesubsystem Compared with previous neural control scheme for flight vehicle the remarkable feature is that the proposed controllernot only achieves the prescribed performance including transient and steady property but also addresses the constraint on NNTwo comparative simulations are presented to verify the effectiveness of the proposed controller

1 Introduction

Morphing aircraft has received considerable interest sinceit possesses distinct advantages which is capable of alter-ing autonomously its aerodynamic configuration to obtainoptimal flight performance adapting different flight envi-ronments and high efficiency executing multiple types ofmissions [1] As regards the control issue ofmorphing vehiclethe key point is to design a flight control system that hascapability to guarantee the stability of the aircraft [2 3] Thedifficulty associated with the control design of such systemarises from the fact that morphing aircraft manifests time-varying characteristic of aerodynamic forces moments andmass distribution as well as strong nonlinear nature [4 5] Inthe literature several effective methods have been presentedto tackle the control problem of folding-wing and swept-backwing aircraft A multiloop control structure comprised oflinear inner-loop and outer-loop controller is proposed for akind of folding-wing aircraft [4] similarly the idea can also befound in [1] depending on gain self-scheduled119867infin techniqueSubsequently a finite-time boundedness control approach [2]

and a switching linear parameter varying method [6] areinvestigated for swept-back wing aircraft respectively Thecommon feature in [1 2 4 6] is that the controllers are capableof ensuring the aircraft flight steady subject to the wingshape changes but those designs are highly dependent onthe precise prior knowledge of the dynamic model Howeverthe aerodynamic forces and moments are also quite difficultto model accurately Moreover general aircraft dynamicspossess strong nonlinearities and uncertainties which havenecessitated the use of nonlinear control methodsThereforedesigning a nonlinear control method independent of priorknowledge of the aerodynamic model for the morphingaircraft is still an interesting yet challenging problem

Adaptive back-stepping method has been widely studiedin tracking control designs for nonlinear systems in strict-feedback or pure-feedback form because it owns capabil-ity of systematically manipulating mismatched uncertain-ties [7ndash10] Lately some significant works regarding adap-tive neuralfuzzy control for nonlinear systems with totallyunknown or nonlinearly parameterized nonlinearities are

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1401427 16 pageshttpsdoiorg10115520171401427

2 Mathematical Problems in Engineering

investigated [11ndash13] and the noticeable problem of ldquoexplosionof itemsrdquo is elegantly avoided by virtue of dynamic surfacecontrol or nonlinear differentiator technique [7 14ndash16]More specifically adaptive neuralfuzzy approaches are alsoextensively utilized to the control problem of flight vehiclesaccompanied with aerodynamic parameters uncertainty in[17ndash20] Although the previous adaptive neural controlmeth-ods have the ability to guarantee the steady-state performanceconverging to a small residual set few results consider thetransient performance related to overshoot and convergencerate [21] More recently the prescribed performance control(PPC) scheme which denotes that the tracking error shouldconverge to a predefined boundwith convergence rate no lessthan a certain value has been an active research area [21ndash30]Based on an error transformation technique that incorpo-rates the desirable performance Bechlioulis and Rovithakisfirstly presented an adaptive PPC method for strict-feedbacknonlinear systems [22] Afterwards an observer based fuzzyadaptive prescribed performance tracking approach is inves-tigated for nonlinear stochastic systems subject to inputsaturation [26] Subsequently the PPC approach has beenextended to deal with MIMO state and output feedbacknonlinear control problems [27ndash30] Moreover this tool hasalso been successfully applied to flight control area [24 31]

Despite the recent progress in the neural networkscontrol of unknown nonlinear systems certain issues stillremain open In practice even though the stability anal-ysis of aforementioned adaptive neural control schemes isproven it relies on the condition that the approximationability of NN must be effective all the time (ie the NNshould be permanently working in the neural active region)Therefore the deterioration of the tracking performanceor even instability may happen provided that the transientstates overstep the neural active region Additionally sucha condition is also difficult to verify beforehand in realapplications [32]This difficulty can be naturally eliminated incase the invariance of the neural active region is guaranteedIn [33] by introducing 119899 error transformation functionsinto the back-stepping design a priori guaranteed evolutionwithin the NN approximation set methodology is proposedto conquer this issue However the method developed in[33] constitutes an overparameterized solution which is hardto implement in practical application Another alternativeapproach is designing a smooth switching function whichhas the capability of switching the normal neural controller toa robust controller for pulling back the escaped transient intothe neural effective regions Several smooth switching func-tion based neural adaptive back-stepping control schemesare proposed to solve aforementioned problem where eachvirtual controller contains a neural controller and a robustcontroller effectively working inside or outside the activeregion respectively [32 34ndash37] However numerous adaptiveparameters still need to be updated online and the transientperformance problem is also omitted in those methods

Motivated by the aforementioned discussion a switchingstrategy based composite adaptive neural control schemeis proposed for a swept-wing morphing aircraft NNs areemployed to approximate unknown functions thus a prioriknowledge of the aerodynamic parameters is not necessary

It is proven that all the signals in the closed-loop systems arebounded The main contributions of this work are shown asfollows

(1) Different from the works [32 34ndash37] which com-pletely neglect the transient performance related toovershoot and convergence rate by introducing anerror transformation the proposed controller is capa-ble of allowing attributes such as a lower bound onthe convergence rate and steady error to be specifiedThe MLP and FOSD are incorporated into the neuralback-stepping design to reduce the online updat-ing parameters of NN and to conquer the problemof ldquoexplosion of the itemsrdquo thus deriving a low-computational scheme

(2) In contrast to traditional neural control schemes[21ndash24 26ndash30] whose stability analysis relies onthe condition that the NNs should always be keptwithin the neural active region a smooth switchingfunction based composite control scheme presentedto manipulate the exchange of control authoritiesbetween normal neural controller working in theactive region and a robust controller out of this scopeis constructed to relax this constraint and a new kindof adaptive laws is proposed Note that it is also thefirst low-computational neural control scheme whichcan efficiently address the prescribed performanceissue and relax the constraint on NN simultaneously

2 Model Dynamics and Problem Formulation

21 Morphing AircraftModel The longitudinal dynamics of amorphing aircraft considered in this study are derived from[5 38] This model includes state variables (119881 ℎ 120572 120574 119902) andcontrol inputs (120575119890 119879) where 119881 denotes the velocity ℎ is thealtitude 120572 denotes angle of attack 120574 represents the flight pathangle (FPA) and 119902 is the pitch rate120575119890 and119879 represent elevatordeflection and thrust respectively

= minus119863 + 119879 cos120572 minus 119898119892 sin 120574 + 119865119868119909119898 (1)

ℎ = 119881 sin 120574 (2)

120574 = 119871 + 119879 sin120572 minus 119898119892 cos 120574 minus 119865119868119870119911119898119881 (3)

= minus119871 minus 119879 sin120572 + 119898119892 cos 120574 + 119865119868119911119898119881 + 119902 (4)

119902 = minus 119868119910119902 minus 119878119909119892 cos 120579 + 119872119860 + 119879119885119879 + 119872119868119910119868119910 (5)

119865119868119909 = 119878119909 ( 119902 sin120572 + 1199022 cos120572) + 2 119878119909119902 sin120572 minus 119878119909 cos120572119865119868119911 = 119865119868119896119911

= 119878119909 ( 119902 cos120572 minus 1199022 sin120572) + 2 119878119909119902 cos120572 + 119878119909 sin120572119872119868119910 = 119878119909 ( sin120572 + 119881 cos120572 minus 119881119902 cos120572)

(6)

Mathematical Problems in Engineering 3

where 119863 119871 and 119872119860 denote drag force lift force and pitchmoment respectively119898 119868119910 and119892 denote themass of aircraftmoment of inertia about pitch axis and gravity constant 119865119868119909119865119868119911 119865119868119896119911 and 119872119868119910 are the inertial force and moment causedby morphing process 119885119879 is the position of engine in thebody axis 119878119909 is the static moment caused by wing sweepTherelated definitions are given as follows

119862119863 (120577) = 1198621198630 (120577) + 119862119863120572 (120577) 120572 + 1198621198631205722 (120577) 1205722119878119909 (120577) asymp [211989811199031119909 + 11989831199033119909]

119871 = 119862119871 (120577) 119876119878119908 (120577) 119863 = 119862119863 (120577) 119876119878119908 (120577)

119862119898 (120577) = 1198621198980 (120577) + 119862119898120572 (120577) 120572 + 119862119898120575119890 (120577) 120575119890+ 119862119898119902119902119888119860 (120577)2119881

119876 = 12120588ℎ1198812119862119871 (120577) asymp 1198621198710 (120577) + 119862119871120572 (120577) 120572

119872119860 = 119862119898 (120577) 119876119878119908 (120577) 119888119860 (120577)

(7)

where 120577 is the sweep angle the detailed explanation of theother parameters can be found in [5]

22 Model Transformation and Control Objective Thrust 119879mainly affects velocity 119881 and elevator deflection 120575119890 has adominant contribution to altitude ℎ change thus the dynam-ics model is reasonably decomposed into two subsystemsincluding altitude and velocity subsystems

221 Altitude Subsystem Define 1199091 = ℎ 1199092 = 120574 1199093 = 120579 and1199094 = 119902 where 120579 = 120572 + 120574 and 119909 = (1199092 1199093 1199094) Thereforethe altitude subsystem can be converted into the followingformulation

1 = 11988111990922 = 1198912 (1199092 1199093) + 11990933 = 11990944 = 1198914 (119909 120575119890) + 119906119910 = 1199091119906 = minus120575119890

(8)

where 119910 is the output signal of altitude subsystem (8)1198912(1199092 1199093) = [119871 + 119879 sin(120579 minus 120574) minus 119898119892 cos 120574](119898119881) minus 120579 and1198914(119909 119906) = [minus 119868119910119902 + (minus119878119909119892 cos 120579 + 119872119860 + 119879119885119879 + 119872119868119910)]119868119910 + 120575119890are unknown functions

222 Velocity Subsystem Velocity subsystem is transformedinto the following formulation

= 119891119881 (119909119881 119879) + 119879 (9)

where 119891119881(119909119881 119879) = (minus119863 + 119879 cos120572 minus 119898119892 sin 120574 + 119865119868119909)119898 minus 119879 isan unknown function

Remark 1 In order to transform the altitude system intopure-feedback system119865119868119896119911 in (3) is regarded as an unmodeledterm Since we only consider the cruise phase in this paper 120574is quite small and we can take sin 120574 asymp 120574 in (2) to simplify thesystem

Control Objective The control objective in this study is todesign adaptive controllers 120575119890 and 119879 such that

(1) the altitude and velocity can track the desired trajec-tory ℎ119889 and119881119889 while guaranteeing that all the signalsin the closed loop are bounded

(2) the corresponding altitude and velocity trackingerrors achieve prescribed transient and steady-stateperformance

23 Some Preliminaries

231 Prescribed Performance To achieve the control objec-tive the tracking error 119911119894(119905) 119894 = 1 119881 should satisfy thefollowing prescribed performance bounds [22 39]

minus120582119894 (119905) lt 119911119894 (119905) lt 120582119894 (119905) (10)

where 120582119894(119905) gt 0 named performance function is defined as

120582119894 (119905) = (1205821198940 minus 120582119894infin) exp (minus119897119894119905) + 120582119894infin (11)

where 1205821198940 120582119894infin and 119897119894 are positive constants 1205821198940 = 120582119894(0)and 120582119894infin = lim119905rarrinfin120582119894(119905) 119897119894 denotes the minimum speed ofconvergence and 120582119894infin is the maximum steady-state error

To transform the constrained tracking error condition(10) into an equivalent unconstrained one the following statetransformation is employed So we have

120583119894 (119905) = 119877119894 ( 119911119894 (119905)120582119894 (119905)) (12)

where 120583119894(119905) is the transformed error and 119877119894(sdot) is an increasingtransformation function shown as follows

119877119894 (119911 (119905)120582 (119905)) = ln((1 + 119911119894 (119905) 120582119894 (119905))1 minus 119911119894 (119905) 120582119894 (119905) ) (13)

The derivative of (12) is shown as

120583119894 (119905) = 119903119894 (119894 (119905) minus 119894 (119905)120582119894 (119905)119911119894 (119905)) (14)

where 119903119894 = (120597119877119894120597(119911119894(119905)120582119894(119905)))(1120582119894(119905))Using (13) the following inequalities can be obtained

119903119894 gt 119903119894min gt 0 (15)

where 119903min = 2(120582119894(0))

4 Mathematical Problems in Engineering

232 Useful Function and Key Lemmas

Definition 2 (see [35]) The boundaries of the compactsubsets Ω119894 are defined by several prescribed constants 0 lt1199031198941 lt 1199031198942 119894 = 2 4meanwhile someuseful switching functionsare described as

119898119894 (119909119894) ≜ 119894prod119896=1

119861119896 (119909119896) 119861119896 (119909119896)

≜

1 if 10038161003816100381610038161199091198961003816100381610038161003816 lt 119903119896111990321198962 minus 1199092

11989611990321198962

minus 11990321198961

119890minus((11990321198962minus1199092119896)((11990321198962minus11990321198961)120596119896))21198871 if 1199031198961 le 10038161003816100381610038161199091198961003816100381610038161003816 le 1199031198962

0 if 10038161003816100381610038161199091198961003816100381610038161003816 gt 1199031198962

(16)

where 120596119896 gt 0 and 1198871 gt 0 are positive constantsLemma 3 (see [40]) The following inequality holds for any1205960 gt 0 and 120578 isin R

0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh( 1205781205960

) le 12058101205960 (17)

where 1205810 is a constant satisfying 1205810 = 119890minus(1205810+1) that is 1205810 =02785Lemma 4 (see [41]) The ldquofirst-order sliding mode differentia-tor (FOSD)rdquo is designed as

1205890 = minus120583010038161003816100381610038161205890 minus 119897 (119905)100381610038161003816100381605 sign (1205890 minus 119897 (119905)) + 1205891

1205891 = minus1205831 sign (1205891 minus 1205890) (18)

where 1205890 and 1205891 are the states of system (18) 1205830 and 1205831 are thedesigned parameters of FOSD and 119897(119905) is an input function1205890 can estimate 119897(119905) to an arbitrary precision in case the initialvalues 1205890 minus 119897(1199050) and 1205890 minus 119897(1199050) are bounded3 Controller Design

In order to process the derivation motived by [8 12] filteredsignals are used to circumvent algebraic loop problemsencountered in the following design thus we define

Δ1198912 = 1198912 (1199092 1199093) minus 1198912 (1199092 1199093119891) Δ1198914 = 1198914 (119909 119906) minus 1198914 (119909 119906119891) Δ119891119881 = 119891119881 (119909119881 119879) minus 119891119881 (119909119881 119879119891)

(19)

where 1199093119891 119906119891 and 119879119891 are the filtered signals defined by [8]

1199093119891 = 119867119871 (119904) 1199093 asymp 1199093119906119891 = 119867119871 (119904) 119906 asymp 119906119879119891 = 119867119871 (119904) 119879 asymp 119879

(20)

where119867119871(119904) is a Butterworth low-pass filterThe correspond-ing filter parameters of Butterworth filters can be obtained in[8]

Assumption 5 In this paper we assume that all of the systemstates are measurable

Assumption 6 The functions 119891119894(sdot) 119894 = 2 4 are unknownand are bounded by |119891119894(sdot)| le 119891119906

119894 (sdot) where 119891119906119894 (sdot) are known

nonnegative smooth functionsMeanwhile it is also assumedthat Δ119891119894 are bounded

Obviously there exist ideal weight vectors 119882lowast2 119882lowast

4 and119882lowast119881 such that

1198912 (1199092 1199093119891) = 119882lowast1198792 Φ2 (1199092 1199093119891) + 1205762 100381610038161003816100381612057621003816100381610038161003816 le 1205762119872

1198914 (119909 119906119891) = 119882lowast1198794 Φ4 (119909 119906119891) + 1205764 100381610038161003816100381612057641003816100381610038161003816 le 1205764119872

119891119881 (119909119881 119879119891) = 119882lowast119879119881 Φ119881 (119909119881 119879119891) + 120576119881 10038161003816100381610038161205761198811003816100381610038161003816 le 120576119881119872

(21)

where 120576119894 and 120576119894119872 denote the approximation errors and theirupper bounds respectively 119882lowast

119894 is the weight of NN Φ119894(sdot) =[1206011(sdot) 120601119899(sdot)] is the basis function vector with 120601119894(sdot) =exp(minus(sdot minus 119888119894)119879(sdot minus 119888119894)1198872119894 ) wherein 119888119894 and 119887119894 are the centersand widths of 120601119894(sdot) Obviously the ideal weights 119882lowast

2 119882lowast4

and119882lowast119881 are completely unknownThus theMLP technique is

employed to estimate the norm of119882lowast2 119882lowast

4 and119882lowast119881 to reduce

the computation burden Those parameters are defined as120593119894 = 119882lowast119894 2 (119894 = 2 4 119881) In the following we replace 120593119894(sdot)

with 120593119894 to simplify the expression

31 Velocity Controller Design Define velocity tracking erroras

119911119881 = 119881 minus 119881119889 (22)

The time derivative of 119911119881 can be described as

119881 = 119891119881 + 119879 minus 119889 (23)

According to (14) and (23) the time derivation of thetransformed error 120583119881(119905) is shown as

120583119881 (119905) = 119903119881 (119881 minus 119881120582119881

119911119881)= 119903119881 (119891119881 + 119879 minus 119889 minus 119881120582119881

119911119881) (24)

where 119903119881 = (120597119877119881120597(119911119881120582119881))(1120582119881) gt 119903119881min gt 0 and 120582119881(119905) =(1205821198810 minus 120582119881infin) exp(minus119897119881119905) + 120582119881infinBy employing MLP technique the controller 119879119889 is

designed as

119879119889 = minus(1198961198811 minus 11990311988121199032119881)120583119881 minus 12120583119881120593119881Φ119879119881Φ119881

minus 119889119881 tanh( 1205831198811205961198811

) + 119889 + 119881120582119881

119911119881(25)

where 1198961198811 and 1205961198811 are positive design parameters 120593119881 and119889119881 denote the estimation of 120593119881 and 119889119881119872 respectively 119889119881 =Δ119891119881 + 120576119881 is the lump approximation error with |119889119881| le 119889119881119872

Mathematical Problems in Engineering 5

and 119903119881 denotes the estimation of 119903119881 bymeans of slidingmodedifferentiator According to Lemma 4 we can easily obtain| 119903119881 minus 119903119881| le 119897 with 119897 gt 0

Consider the following adaptive laws for 120593119881 and 119889119881

120593119881 = 12058811988112 (1205832119881Φ119879

119881Φ119881 minus 21205901198811120593119881) 119889119881 = 1205881198812 [120583119881 tanh( 1205831198811205961198811

) minus 1205901198812119889119881] (26)

where 1205881198811 1205881198812 1205901198811 and 1205901198812 denote positive design parame-ters

Theorem 7 Suppose that the velocity subsystem (9) satisfiesAssumption 5 if the adaptive controller is selected as (25) andupdating laws are selected as (26) the signals including 120583119881 120593119881and 119889119881 are ensured to be bounded

Remark 8 The velocity design is partially derived from [24]Note that the FOSD is used to estimate unknown item 119903119881 Byintroducing 05 1199031198811205831198811199032119881 in (25) the stability analysis problemin [24] is overcome

32 Altitude Controller Design The following coordinatechange is constructed to facilitate the control design

1199111 = 1199091 minus 1199101198891199112 = 1199092 minus 12057211199113 = 1199093 minus 12057221199114 = 1199094 minus 1205723(27)

where 1205721 1205722 and 1205723 are the virtual controllers to be designedat Steps 1 2 and 3 respectively119910119889 = ℎ119889 is the reference signalThe control scheme for the altitude subsystem is developed inthe framework of back-stepping technique which contains 4-step recursive design procedure

Step 1 The time derivative of 1199111 = 1199091 minus 119910119889 is expressed as

1 = 1 minus 119910119889 = 1198811199092 minus 119910119889 = 119881 (1199112 + 1205721) minus 119910119889 (28)

By using (15) and (28) the time derivative of the trans-formed altitude error 1205831(119905) is shown as follows

1205831 (119905) = 1199031 (1 minus 11205821

1199111) = 1199031 (1198811199092 minus 119910119889 minus 11205821

1199111) (29)

where 1199031 = (1205971198771120597(11991111205821))(11205821) gt 1199031min gt 0 and 1205821(119905) =(12058210 minus 1205821infin) exp(minus1198971119905) + 1205821infinThe virtual controller 1205721 is designed as

1205721 = (minus (1198961 minus 1199031211990321) 1205831 + 119910119889 + (11205821) 1199111)119881 (30)

where 1198961 is positive parameter It is worth noticing that 1199031 canbe easily obtained via system states

Invoking (28) and (30) one has

1 = 1199031 (1198811199112 minus (1198961 minus 1199031211990321 )1205831) (31)

In order to avoid the tedious computation of 1 thefollowing FOSD is adopted to estimate it

12058910 = minus12058310 100381610038161003816100381612058910 minus 1205721100381610038161003816100381605 sign (12058910 minus 1205721) + 1205891112058911 = minus12058311 sign (12058911 minus 12058910) (32)

where 12058910 and 12058911 are the states of FOSD (32) and 12058310 and 12058311are the positive design constants

Then we have

1 = 12058910 + 1205911 (33)

where 1205911 is the estimation error of the FOSD with |1205911| le 1205911Step 2 The differentiation of 1199112 is obtained as follows

2 = 2 minus 1 = 1198912 (1199092 1199093) + 1199113 + 1205722 minus 1 (34)

The virtual controller 1205722 is designed as

1205722 = minus11989621199112 + 12058910 minus 11989821199061198732 minus (1 minus 1198982) 1199061199032 (35)

with

1199061198732 = 1211991121205932Φ1198792Φ2 + 1198892 tanh( 119911212059621

) 1199061199032 = 119891119906

2 tanh(1199112119891119906212059622

) (36)

where 1198962 12059621 and 12059622 are positive design parameters 1198892 =Δ1198912 + 1205762 is bounded with |1198892| le 1198892119872 1205932 and 1198892 denote theestimations of 1205932 and 1198892119872 respectively

The structure of adaptive control laws is expressed asfollows

1205932 = 120588212 (119898211991122Φ1198792Φ2 minus 2120590211205932)

1198892 = 12058822 [11989821199112 tanh( 119911212059621

) minus 120590221198892] (37)

Substituting (35) into (34) (34) can be rewritten as

2 = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (1198912 minus 1199061198732 ) + (1 minus 1198982)sdot (1198912 minus 1199061199032) = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (119882lowast119879

2 Φ2

minus 1211991121205932Φ1198792Φ2 + 1198892 minus 1198892 tanh( 119911212059621

)) + (1minus 1198982) (1198912 minus 119891119906

2 tanh(1199112119891119906212059622

))

(38)

6 Mathematical Problems in Engineering

The following FOSD is adopted to estimate 212058920 = minus12058320 100381610038161003816100381612058920 minus 1205722100381610038161003816100381605 sign (12058920 minus 1205722) + 1205892112058921 = minus12058321 sign (12058921 minus 12058920) (39)

where 12058920 and 12058921 are the states of system (32) and 12058320 and 12058321are positive design constants

From (39) and Lemma 3 we have

2 = 12058920 + 1205912 (40)

where 1205912 is the estimation error with |1205912| le 1205912Step 3 The differentiation of 1199113 is obtained as follows

3 = 3 minus 2 = 1199114 + 1205723 minus 2 (41)

The virtual control law 1205723 is designed as

1205723 = minus11989631199113 + 12058920 minus 1199112 (42)

where 1198963 is a positive design parameterSubstituting (42) into (41) yields

3 = 1199114 minus 1199112 minus 11989631199113 + 12058920 minus 2 = 1199114 minus 1199112 minus 11989631199113 minus 1205912 (43)

As done previously the following FOSD is employed toestimate 3

12058930 = minus12058330 100381610038161003816100381612058930 minus 1205723100381610038161003816100381605 sign (12058930 minus 1205723) + 1205893112058931 = minus12058331 sign (12058931 minus 12058930) (44)

where 12058930 and 12058931 are the states of the system and 12058330 and 12058331are the positive design constants

Thus we have

3 = 12058930 + 1205913 (45)

where 1205913 is an estimation error with |1205913| le 1205913Step 4 In this step the actual controller 119906 will be developedThe differentiation of 1199114 can be obtained as follows

4 = 1198914 (119909 119906) + 119906 minus 3= 11989841198914 (119909 119906) + (1 minus 1198984) 1198914 (119909 119906) + 119906 minus 3 (46)

The controller 119906 is designed as

119906 = minus11989641199114 + 12058930 minus 1199113 minus 11989841199061198734 minus (1 minus 1198984) 1199061199034 (47)

with

1199061198734 = 1211991141205934Φ1198794Φ4 + 1198894 tanh( 119911412059641

) 1199061199034 = 119891119906

4 tanh(1199114119891119906412059642

) (48)

where 1198964 12059641 and 12059642 are the positive design constants 1198894 =Δ1198914 + 1205764 is the lump approximation error with |1198894| le 1198894119872 1205934

and 1198894 denote the estimations of 1205934 and 1198894119872 respectively 1205934

and 1198894 are updated as

1205934 = 120588412 (119898411991124Φ1198794Φ4 minus 2120590411205934)

1198894 = 12058842 (11989841199114 tanh( 119911412059641

) minus 120590421198894) (49)

Thus (46) can be rewritten as

4 = minus11989641199114 minus 1199113 minus 1205913 + 1198984 (119882lowast1198794 Φ4 minus 1211991141205934Φ119879

4Φ4

+ 1198894 minus 1198894 tanh( 119911412059641

)) + (1 minus 1198984) (1198914minus 119891119906

4 tanh(1199114119891119906412059642

)) (50)

Theorem9 Consider the altitude subsystem (8) with Assump-tions 5 and 6 if the switching adaptive neural prescribedperformance control scheme is selected as (30) (35) (42) and(47) adaptive laws are selected as (37) and (49) and FOSD isselected as (32) (39) and (44) the signals 1205831 119911119894=234 120593119894=24and 119889119894=24 in the closed-loop system are bounded

Remark 10 The altitude controller composed of a normaladaptive neural controller working in the neural activeregion a robust controller being in charge outside the neuralapproximation region and a switching strategy supervisingthe exchange of the former two controllers is constructed

Remark 11 In this paper in order to estimate the derivativeof virtual controllers 1205721 1205722 and 1205723 the FOSD (first-ordersliding mode differentiator) is employed Using (33) as anexample 12058910 is the estimation of 1198861 and 1205911 is the estimationerror between actual 1198861 and 12058910 It must be noted that 1205911 is notused in the controller design but is just employed for stabilityanalysis (please see (B6))

4 Simulations

In this section two comparative cases are presented to illus-trate the effectiveness of the switching functions based adap-tive neural control for longitudinal model of the morphingaircraft The aerodynamic coefficients and model parametersare the same as [5] The initial conditions are set as 1198830 =[1205740 1205720 1199020 ℎ0 1198810] = [0∘ 099512∘ 0∘s 1000m 30ms] Thecontrol parameters are selected as 1198961 = 03 1198962 = 03 1198963 =005 1198964 = 375 and 1198961198811 = 3 Gains for the adaptive laws areset as 12058821 = 40 12059021 = 001 12058822 = 10 12059022 = 01 12058841 = 25012059041 = 001 12058842 = 100 12059042 = 001 1205881198811 = 10 1205901198811 = 011205881198812 = 100 1205901198812 = 01 12059621 = 1 12059622 = 1 12059641 = 1 and12059642 = 5 The aforementioned transient and steady outputerror bounds are prescribed by the performance functions120582119894(119905) = (1205821198940 minus 120582119894infin) exp(minus119897119894119905) + 120582119894infin 119894 = 1 119881 where 12058210 =05 1205821infin = 02 1205821198810 = 035 1205821infin = 01 and 119897119894 = 005The corresponding neural active regions are defined as 119881 isin[30ms 40ms] 1199092 isin [minus4 deg 4 deg] 1199093 isin [minus6 deg 6 deg]

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where 119863 119871 and 119872119860 denote drag force lift force and pitchmoment respectively119898 119868119910 and119892 denote themass of aircraftmoment of inertia about pitch axis and gravity constant 119865119868119909119865119868119911 119865119868119896119911 and 119872119868119910 are the inertial force and moment causedby morphing process 119885119879 is the position of engine in thebody axis 119878119909 is the static moment caused by wing sweepTherelated definitions are given as follows

119862119863 (120577) = 1198621198630 (120577) + 119862119863120572 (120577) 120572 + 1198621198631205722 (120577) 1205722119878119909 (120577) asymp [211989811199031119909 + 11989831199033119909]

119871 = 119862119871 (120577) 119876119878119908 (120577) 119863 = 119862119863 (120577) 119876119878119908 (120577)

119862119898 (120577) = 1198621198980 (120577) + 119862119898120572 (120577) 120572 + 119862119898120575119890 (120577) 120575119890+ 119862119898119902119902119888119860 (120577)2119881

119876 = 12120588ℎ1198812119862119871 (120577) asymp 1198621198710 (120577) + 119862119871120572 (120577) 120572

119872119860 = 119862119898 (120577) 119876119878119908 (120577) 119888119860 (120577)

(7)

where 120577 is the sweep angle the detailed explanation of theother parameters can be found in [5]

22 Model Transformation and Control Objective Thrust 119879mainly affects velocity 119881 and elevator deflection 120575119890 has adominant contribution to altitude ℎ change thus the dynam-ics model is reasonably decomposed into two subsystemsincluding altitude and velocity subsystems

221 Altitude Subsystem Define 1199091 = ℎ 1199092 = 120574 1199093 = 120579 and1199094 = 119902 where 120579 = 120572 + 120574 and 119909 = (1199092 1199093 1199094) Thereforethe altitude subsystem can be converted into the followingformulation

1 = 11988111990922 = 1198912 (1199092 1199093) + 11990933 = 11990944 = 1198914 (119909 120575119890) + 119906119910 = 1199091119906 = minus120575119890

(8)

where 119910 is the output signal of altitude subsystem (8)1198912(1199092 1199093) = [119871 + 119879 sin(120579 minus 120574) minus 119898119892 cos 120574](119898119881) minus 120579 and1198914(119909 119906) = [minus 119868119910119902 + (minus119878119909119892 cos 120579 + 119872119860 + 119879119885119879 + 119872119868119910)]119868119910 + 120575119890are unknown functions

222 Velocity Subsystem Velocity subsystem is transformedinto the following formulation

= 119891119881 (119909119881 119879) + 119879 (9)

where 119891119881(119909119881 119879) = (minus119863 + 119879 cos120572 minus 119898119892 sin 120574 + 119865119868119909)119898 minus 119879 isan unknown function

Remark 1 In order to transform the altitude system intopure-feedback system119865119868119896119911 in (3) is regarded as an unmodeledterm Since we only consider the cruise phase in this paper 120574is quite small and we can take sin 120574 asymp 120574 in (2) to simplify thesystem

Control Objective The control objective in this study is todesign adaptive controllers 120575119890 and 119879 such that

(1) the altitude and velocity can track the desired trajec-tory ℎ119889 and119881119889 while guaranteeing that all the signalsin the closed loop are bounded

(2) the corresponding altitude and velocity trackingerrors achieve prescribed transient and steady-stateperformance

23 Some Preliminaries

231 Prescribed Performance To achieve the control objec-tive the tracking error 119911119894(119905) 119894 = 1 119881 should satisfy thefollowing prescribed performance bounds [22 39]

minus120582119894 (119905) lt 119911119894 (119905) lt 120582119894 (119905) (10)

where 120582119894(119905) gt 0 named performance function is defined as

120582119894 (119905) = (1205821198940 minus 120582119894infin) exp (minus119897119894119905) + 120582119894infin (11)

where 1205821198940 120582119894infin and 119897119894 are positive constants 1205821198940 = 120582119894(0)and 120582119894infin = lim119905rarrinfin120582119894(119905) 119897119894 denotes the minimum speed ofconvergence and 120582119894infin is the maximum steady-state error

To transform the constrained tracking error condition(10) into an equivalent unconstrained one the following statetransformation is employed So we have

120583119894 (119905) = 119877119894 ( 119911119894 (119905)120582119894 (119905)) (12)

where 120583119894(119905) is the transformed error and 119877119894(sdot) is an increasingtransformation function shown as follows

119877119894 (119911 (119905)120582 (119905)) = ln((1 + 119911119894 (119905) 120582119894 (119905))1 minus 119911119894 (119905) 120582119894 (119905) ) (13)

The derivative of (12) is shown as

120583119894 (119905) = 119903119894 (119894 (119905) minus 119894 (119905)120582119894 (119905)119911119894 (119905)) (14)

where 119903119894 = (120597119877119894120597(119911119894(119905)120582119894(119905)))(1120582119894(119905))Using (13) the following inequalities can be obtained

119903119894 gt 119903119894min gt 0 (15)

where 119903min = 2(120582119894(0))

4 Mathematical Problems in Engineering

232 Useful Function and Key Lemmas

Definition 2 (see [35]) The boundaries of the compactsubsets Ω119894 are defined by several prescribed constants 0 lt1199031198941 lt 1199031198942 119894 = 2 4meanwhile someuseful switching functionsare described as

119898119894 (119909119894) ≜ 119894prod119896=1

119861119896 (119909119896) 119861119896 (119909119896)

≜

1 if 10038161003816100381610038161199091198961003816100381610038161003816 lt 119903119896111990321198962 minus 1199092

11989611990321198962

minus 11990321198961

119890minus((11990321198962minus1199092119896)((11990321198962minus11990321198961)120596119896))21198871 if 1199031198961 le 10038161003816100381610038161199091198961003816100381610038161003816 le 1199031198962

0 if 10038161003816100381610038161199091198961003816100381610038161003816 gt 1199031198962

(16)

where 120596119896 gt 0 and 1198871 gt 0 are positive constantsLemma 3 (see [40]) The following inequality holds for any1205960 gt 0 and 120578 isin R

0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh( 1205781205960

) le 12058101205960 (17)

where 1205810 is a constant satisfying 1205810 = 119890minus(1205810+1) that is 1205810 =02785Lemma 4 (see [41]) The ldquofirst-order sliding mode differentia-tor (FOSD)rdquo is designed as

1205890 = minus120583010038161003816100381610038161205890 minus 119897 (119905)100381610038161003816100381605 sign (1205890 minus 119897 (119905)) + 1205891

1205891 = minus1205831 sign (1205891 minus 1205890) (18)

where 1205890 and 1205891 are the states of system (18) 1205830 and 1205831 are thedesigned parameters of FOSD and 119897(119905) is an input function1205890 can estimate 119897(119905) to an arbitrary precision in case the initialvalues 1205890 minus 119897(1199050) and 1205890 minus 119897(1199050) are bounded3 Controller Design

In order to process the derivation motived by [8 12] filteredsignals are used to circumvent algebraic loop problemsencountered in the following design thus we define

Δ1198912 = 1198912 (1199092 1199093) minus 1198912 (1199092 1199093119891) Δ1198914 = 1198914 (119909 119906) minus 1198914 (119909 119906119891) Δ119891119881 = 119891119881 (119909119881 119879) minus 119891119881 (119909119881 119879119891)

(19)

where 1199093119891 119906119891 and 119879119891 are the filtered signals defined by [8]

1199093119891 = 119867119871 (119904) 1199093 asymp 1199093119906119891 = 119867119871 (119904) 119906 asymp 119906119879119891 = 119867119871 (119904) 119879 asymp 119879

(20)

where119867119871(119904) is a Butterworth low-pass filterThe correspond-ing filter parameters of Butterworth filters can be obtained in[8]

Assumption 5 In this paper we assume that all of the systemstates are measurable

Assumption 6 The functions 119891119894(sdot) 119894 = 2 4 are unknownand are bounded by |119891119894(sdot)| le 119891119906

119894 (sdot) where 119891119906119894 (sdot) are known

nonnegative smooth functionsMeanwhile it is also assumedthat Δ119891119894 are bounded

Obviously there exist ideal weight vectors 119882lowast2 119882lowast

4 and119882lowast119881 such that

1198912 (1199092 1199093119891) = 119882lowast1198792 Φ2 (1199092 1199093119891) + 1205762 100381610038161003816100381612057621003816100381610038161003816 le 1205762119872

1198914 (119909 119906119891) = 119882lowast1198794 Φ4 (119909 119906119891) + 1205764 100381610038161003816100381612057641003816100381610038161003816 le 1205764119872

119891119881 (119909119881 119879119891) = 119882lowast119879119881 Φ119881 (119909119881 119879119891) + 120576119881 10038161003816100381610038161205761198811003816100381610038161003816 le 120576119881119872

(21)

where 120576119894 and 120576119894119872 denote the approximation errors and theirupper bounds respectively 119882lowast

119894 is the weight of NN Φ119894(sdot) =[1206011(sdot) 120601119899(sdot)] is the basis function vector with 120601119894(sdot) =exp(minus(sdot minus 119888119894)119879(sdot minus 119888119894)1198872119894 ) wherein 119888119894 and 119887119894 are the centersand widths of 120601119894(sdot) Obviously the ideal weights 119882lowast

2 119882lowast4

and119882lowast119881 are completely unknownThus theMLP technique is

employed to estimate the norm of119882lowast2 119882lowast

4 and119882lowast119881 to reduce

the computation burden Those parameters are defined as120593119894 = 119882lowast119894 2 (119894 = 2 4 119881) In the following we replace 120593119894(sdot)

with 120593119894 to simplify the expression

31 Velocity Controller Design Define velocity tracking erroras

119911119881 = 119881 minus 119881119889 (22)

The time derivative of 119911119881 can be described as

119881 = 119891119881 + 119879 minus 119889 (23)

According to (14) and (23) the time derivation of thetransformed error 120583119881(119905) is shown as

120583119881 (119905) = 119903119881 (119881 minus 119881120582119881

119911119881)= 119903119881 (119891119881 + 119879 minus 119889 minus 119881120582119881

119911119881) (24)

where 119903119881 = (120597119877119881120597(119911119881120582119881))(1120582119881) gt 119903119881min gt 0 and 120582119881(119905) =(1205821198810 minus 120582119881infin) exp(minus119897119881119905) + 120582119881infinBy employing MLP technique the controller 119879119889 is

designed as

119879119889 = minus(1198961198811 minus 11990311988121199032119881)120583119881 minus 12120583119881120593119881Φ119879119881Φ119881

minus 119889119881 tanh( 1205831198811205961198811

) + 119889 + 119881120582119881

119911119881(25)

where 1198961198811 and 1205961198811 are positive design parameters 120593119881 and119889119881 denote the estimation of 120593119881 and 119889119881119872 respectively 119889119881 =Δ119891119881 + 120576119881 is the lump approximation error with |119889119881| le 119889119881119872

Mathematical Problems in Engineering 5

and 119903119881 denotes the estimation of 119903119881 bymeans of slidingmodedifferentiator According to Lemma 4 we can easily obtain| 119903119881 minus 119903119881| le 119897 with 119897 gt 0

Consider the following adaptive laws for 120593119881 and 119889119881

120593119881 = 12058811988112 (1205832119881Φ119879

119881Φ119881 minus 21205901198811120593119881) 119889119881 = 1205881198812 [120583119881 tanh( 1205831198811205961198811

) minus 1205901198812119889119881] (26)

where 1205881198811 1205881198812 1205901198811 and 1205901198812 denote positive design parame-ters

Theorem 7 Suppose that the velocity subsystem (9) satisfiesAssumption 5 if the adaptive controller is selected as (25) andupdating laws are selected as (26) the signals including 120583119881 120593119881and 119889119881 are ensured to be bounded

Remark 8 The velocity design is partially derived from [24]Note that the FOSD is used to estimate unknown item 119903119881 Byintroducing 05 1199031198811205831198811199032119881 in (25) the stability analysis problemin [24] is overcome

32 Altitude Controller Design The following coordinatechange is constructed to facilitate the control design

1199111 = 1199091 minus 1199101198891199112 = 1199092 minus 12057211199113 = 1199093 minus 12057221199114 = 1199094 minus 1205723(27)

where 1205721 1205722 and 1205723 are the virtual controllers to be designedat Steps 1 2 and 3 respectively119910119889 = ℎ119889 is the reference signalThe control scheme for the altitude subsystem is developed inthe framework of back-stepping technique which contains 4-step recursive design procedure

Step 1 The time derivative of 1199111 = 1199091 minus 119910119889 is expressed as

1 = 1 minus 119910119889 = 1198811199092 minus 119910119889 = 119881 (1199112 + 1205721) minus 119910119889 (28)

By using (15) and (28) the time derivative of the trans-formed altitude error 1205831(119905) is shown as follows

1205831 (119905) = 1199031 (1 minus 11205821

1199111) = 1199031 (1198811199092 minus 119910119889 minus 11205821

1199111) (29)

where 1199031 = (1205971198771120597(11991111205821))(11205821) gt 1199031min gt 0 and 1205821(119905) =(12058210 minus 1205821infin) exp(minus1198971119905) + 1205821infinThe virtual controller 1205721 is designed as

1205721 = (minus (1198961 minus 1199031211990321) 1205831 + 119910119889 + (11205821) 1199111)119881 (30)

where 1198961 is positive parameter It is worth noticing that 1199031 canbe easily obtained via system states

Invoking (28) and (30) one has

1 = 1199031 (1198811199112 minus (1198961 minus 1199031211990321 )1205831) (31)

In order to avoid the tedious computation of 1 thefollowing FOSD is adopted to estimate it

12058910 = minus12058310 100381610038161003816100381612058910 minus 1205721100381610038161003816100381605 sign (12058910 minus 1205721) + 1205891112058911 = minus12058311 sign (12058911 minus 12058910) (32)

where 12058910 and 12058911 are the states of FOSD (32) and 12058310 and 12058311are the positive design constants

Then we have

1 = 12058910 + 1205911 (33)

where 1205911 is the estimation error of the FOSD with |1205911| le 1205911Step 2 The differentiation of 1199112 is obtained as follows

2 = 2 minus 1 = 1198912 (1199092 1199093) + 1199113 + 1205722 minus 1 (34)

The virtual controller 1205722 is designed as

1205722 = minus11989621199112 + 12058910 minus 11989821199061198732 minus (1 minus 1198982) 1199061199032 (35)

with

1199061198732 = 1211991121205932Φ1198792Φ2 + 1198892 tanh( 119911212059621

) 1199061199032 = 119891119906

2 tanh(1199112119891119906212059622

) (36)

where 1198962 12059621 and 12059622 are positive design parameters 1198892 =Δ1198912 + 1205762 is bounded with |1198892| le 1198892119872 1205932 and 1198892 denote theestimations of 1205932 and 1198892119872 respectively

The structure of adaptive control laws is expressed asfollows

1205932 = 120588212 (119898211991122Φ1198792Φ2 minus 2120590211205932)

1198892 = 12058822 [11989821199112 tanh( 119911212059621

) minus 120590221198892] (37)

Substituting (35) into (34) (34) can be rewritten as

2 = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (1198912 minus 1199061198732 ) + (1 minus 1198982)sdot (1198912 minus 1199061199032) = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (119882lowast119879

2 Φ2

minus 1211991121205932Φ1198792Φ2 + 1198892 minus 1198892 tanh( 119911212059621

)) + (1minus 1198982) (1198912 minus 119891119906

2 tanh(1199112119891119906212059622

))

(38)

6 Mathematical Problems in Engineering

The following FOSD is adopted to estimate 212058920 = minus12058320 100381610038161003816100381612058920 minus 1205722100381610038161003816100381605 sign (12058920 minus 1205722) + 1205892112058921 = minus12058321 sign (12058921 minus 12058920) (39)

where 12058920 and 12058921 are the states of system (32) and 12058320 and 12058321are positive design constants

From (39) and Lemma 3 we have

2 = 12058920 + 1205912 (40)

where 1205912 is the estimation error with |1205912| le 1205912Step 3 The differentiation of 1199113 is obtained as follows

3 = 3 minus 2 = 1199114 + 1205723 minus 2 (41)

The virtual control law 1205723 is designed as

1205723 = minus11989631199113 + 12058920 minus 1199112 (42)

where 1198963 is a positive design parameterSubstituting (42) into (41) yields

3 = 1199114 minus 1199112 minus 11989631199113 + 12058920 minus 2 = 1199114 minus 1199112 minus 11989631199113 minus 1205912 (43)

As done previously the following FOSD is employed toestimate 3

12058930 = minus12058330 100381610038161003816100381612058930 minus 1205723100381610038161003816100381605 sign (12058930 minus 1205723) + 1205893112058931 = minus12058331 sign (12058931 minus 12058930) (44)

where 12058930 and 12058931 are the states of the system and 12058330 and 12058331are the positive design constants

Thus we have

3 = 12058930 + 1205913 (45)

where 1205913 is an estimation error with |1205913| le 1205913Step 4 In this step the actual controller 119906 will be developedThe differentiation of 1199114 can be obtained as follows

4 = 1198914 (119909 119906) + 119906 minus 3= 11989841198914 (119909 119906) + (1 minus 1198984) 1198914 (119909 119906) + 119906 minus 3 (46)

The controller 119906 is designed as

119906 = minus11989641199114 + 12058930 minus 1199113 minus 11989841199061198734 minus (1 minus 1198984) 1199061199034 (47)

with

1199061198734 = 1211991141205934Φ1198794Φ4 + 1198894 tanh( 119911412059641

) 1199061199034 = 119891119906

4 tanh(1199114119891119906412059642

) (48)

where 1198964 12059641 and 12059642 are the positive design constants 1198894 =Δ1198914 + 1205764 is the lump approximation error with |1198894| le 1198894119872 1205934

and 1198894 denote the estimations of 1205934 and 1198894119872 respectively 1205934

and 1198894 are updated as

1205934 = 120588412 (119898411991124Φ1198794Φ4 minus 2120590411205934)

1198894 = 12058842 (11989841199114 tanh( 119911412059641

) minus 120590421198894) (49)

Thus (46) can be rewritten as

4 = minus11989641199114 minus 1199113 minus 1205913 + 1198984 (119882lowast1198794 Φ4 minus 1211991141205934Φ119879

4Φ4

+ 1198894 minus 1198894 tanh( 119911412059641

)) + (1 minus 1198984) (1198914minus 119891119906

4 tanh(1199114119891119906412059642

)) (50)

Theorem9 Consider the altitude subsystem (8) with Assump-tions 5 and 6 if the switching adaptive neural prescribedperformance control scheme is selected as (30) (35) (42) and(47) adaptive laws are selected as (37) and (49) and FOSD isselected as (32) (39) and (44) the signals 1205831 119911119894=234 120593119894=24and 119889119894=24 in the closed-loop system are bounded

Remark 10 The altitude controller composed of a normaladaptive neural controller working in the neural activeregion a robust controller being in charge outside the neuralapproximation region and a switching strategy supervisingthe exchange of the former two controllers is constructed

Remark 11 In this paper in order to estimate the derivativeof virtual controllers 1205721 1205722 and 1205723 the FOSD (first-ordersliding mode differentiator) is employed Using (33) as anexample 12058910 is the estimation of 1198861 and 1205911 is the estimationerror between actual 1198861 and 12058910 It must be noted that 1205911 is notused in the controller design but is just employed for stabilityanalysis (please see (B6))

4 Simulations

In this section two comparative cases are presented to illus-trate the effectiveness of the switching functions based adap-tive neural control for longitudinal model of the morphingaircraft The aerodynamic coefficients and model parametersare the same as [5] The initial conditions are set as 1198830 =[1205740 1205720 1199020 ℎ0 1198810] = [0∘ 099512∘ 0∘s 1000m 30ms] Thecontrol parameters are selected as 1198961 = 03 1198962 = 03 1198963 =005 1198964 = 375 and 1198961198811 = 3 Gains for the adaptive laws areset as 12058821 = 40 12059021 = 001 12058822 = 10 12059022 = 01 12058841 = 25012059041 = 001 12058842 = 100 12059042 = 001 1205881198811 = 10 1205901198811 = 011205881198812 = 100 1205901198812 = 01 12059621 = 1 12059622 = 1 12059641 = 1 and12059642 = 5 The aforementioned transient and steady outputerror bounds are prescribed by the performance functions120582119894(119905) = (1205821198940 minus 120582119894infin) exp(minus119897119894119905) + 120582119894infin 119894 = 1 119881 where 12058210 =05 1205821infin = 02 1205821198810 = 035 1205821infin = 01 and 119897119894 = 005The corresponding neural active regions are defined as 119881 isin[30ms 40ms] 1199092 isin [minus4 deg 4 deg] 1199093 isin [minus6 deg 6 deg]

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

232 Useful Function and Key Lemmas

Definition 2 (see [35]) The boundaries of the compactsubsets Ω119894 are defined by several prescribed constants 0 lt1199031198941 lt 1199031198942 119894 = 2 4meanwhile someuseful switching functionsare described as

119898119894 (119909119894) ≜ 119894prod119896=1

119861119896 (119909119896) 119861119896 (119909119896)

≜

1 if 10038161003816100381610038161199091198961003816100381610038161003816 lt 119903119896111990321198962 minus 1199092

11989611990321198962

minus 11990321198961

119890minus((11990321198962minus1199092119896)((11990321198962minus11990321198961)120596119896))21198871 if 1199031198961 le 10038161003816100381610038161199091198961003816100381610038161003816 le 1199031198962

0 if 10038161003816100381610038161199091198961003816100381610038161003816 gt 1199031198962

(16)

where 120596119896 gt 0 and 1198871 gt 0 are positive constantsLemma 3 (see [40]) The following inequality holds for any1205960 gt 0 and 120578 isin R

0 le 10038161003816100381610038161205781003816100381610038161003816 minus 120578 tanh( 1205781205960

) le 12058101205960 (17)

where 1205810 is a constant satisfying 1205810 = 119890minus(1205810+1) that is 1205810 =02785Lemma 4 (see [41]) The ldquofirst-order sliding mode differentia-tor (FOSD)rdquo is designed as

1205890 = minus120583010038161003816100381610038161205890 minus 119897 (119905)100381610038161003816100381605 sign (1205890 minus 119897 (119905)) + 1205891

1205891 = minus1205831 sign (1205891 minus 1205890) (18)

where 1205890 and 1205891 are the states of system (18) 1205830 and 1205831 are thedesigned parameters of FOSD and 119897(119905) is an input function1205890 can estimate 119897(119905) to an arbitrary precision in case the initialvalues 1205890 minus 119897(1199050) and 1205890 minus 119897(1199050) are bounded3 Controller Design

In order to process the derivation motived by [8 12] filteredsignals are used to circumvent algebraic loop problemsencountered in the following design thus we define

Δ1198912 = 1198912 (1199092 1199093) minus 1198912 (1199092 1199093119891) Δ1198914 = 1198914 (119909 119906) minus 1198914 (119909 119906119891) Δ119891119881 = 119891119881 (119909119881 119879) minus 119891119881 (119909119881 119879119891)

(19)

where 1199093119891 119906119891 and 119879119891 are the filtered signals defined by [8]

1199093119891 = 119867119871 (119904) 1199093 asymp 1199093119906119891 = 119867119871 (119904) 119906 asymp 119906119879119891 = 119867119871 (119904) 119879 asymp 119879

(20)

where119867119871(119904) is a Butterworth low-pass filterThe correspond-ing filter parameters of Butterworth filters can be obtained in[8]

Assumption 5 In this paper we assume that all of the systemstates are measurable

Assumption 6 The functions 119891119894(sdot) 119894 = 2 4 are unknownand are bounded by |119891119894(sdot)| le 119891119906

119894 (sdot) where 119891119906119894 (sdot) are known

nonnegative smooth functionsMeanwhile it is also assumedthat Δ119891119894 are bounded

Obviously there exist ideal weight vectors 119882lowast2 119882lowast

4 and119882lowast119881 such that

1198912 (1199092 1199093119891) = 119882lowast1198792 Φ2 (1199092 1199093119891) + 1205762 100381610038161003816100381612057621003816100381610038161003816 le 1205762119872

1198914 (119909 119906119891) = 119882lowast1198794 Φ4 (119909 119906119891) + 1205764 100381610038161003816100381612057641003816100381610038161003816 le 1205764119872

119891119881 (119909119881 119879119891) = 119882lowast119879119881 Φ119881 (119909119881 119879119891) + 120576119881 10038161003816100381610038161205761198811003816100381610038161003816 le 120576119881119872

(21)

where 120576119894 and 120576119894119872 denote the approximation errors and theirupper bounds respectively 119882lowast

119894 is the weight of NN Φ119894(sdot) =[1206011(sdot) 120601119899(sdot)] is the basis function vector with 120601119894(sdot) =exp(minus(sdot minus 119888119894)119879(sdot minus 119888119894)1198872119894 ) wherein 119888119894 and 119887119894 are the centersand widths of 120601119894(sdot) Obviously the ideal weights 119882lowast

2 119882lowast4

and119882lowast119881 are completely unknownThus theMLP technique is

employed to estimate the norm of119882lowast2 119882lowast

4 and119882lowast119881 to reduce

the computation burden Those parameters are defined as120593119894 = 119882lowast119894 2 (119894 = 2 4 119881) In the following we replace 120593119894(sdot)

with 120593119894 to simplify the expression

31 Velocity Controller Design Define velocity tracking erroras

119911119881 = 119881 minus 119881119889 (22)

The time derivative of 119911119881 can be described as

119881 = 119891119881 + 119879 minus 119889 (23)

According to (14) and (23) the time derivation of thetransformed error 120583119881(119905) is shown as

120583119881 (119905) = 119903119881 (119881 minus 119881120582119881

119911119881)= 119903119881 (119891119881 + 119879 minus 119889 minus 119881120582119881

119911119881) (24)

where 119903119881 = (120597119877119881120597(119911119881120582119881))(1120582119881) gt 119903119881min gt 0 and 120582119881(119905) =(1205821198810 minus 120582119881infin) exp(minus119897119881119905) + 120582119881infinBy employing MLP technique the controller 119879119889 is

designed as

119879119889 = minus(1198961198811 minus 11990311988121199032119881)120583119881 minus 12120583119881120593119881Φ119879119881Φ119881

minus 119889119881 tanh( 1205831198811205961198811

) + 119889 + 119881120582119881

119911119881(25)

where 1198961198811 and 1205961198811 are positive design parameters 120593119881 and119889119881 denote the estimation of 120593119881 and 119889119881119872 respectively 119889119881 =Δ119891119881 + 120576119881 is the lump approximation error with |119889119881| le 119889119881119872

Mathematical Problems in Engineering 5

and 119903119881 denotes the estimation of 119903119881 bymeans of slidingmodedifferentiator According to Lemma 4 we can easily obtain| 119903119881 minus 119903119881| le 119897 with 119897 gt 0

Consider the following adaptive laws for 120593119881 and 119889119881

120593119881 = 12058811988112 (1205832119881Φ119879

119881Φ119881 minus 21205901198811120593119881) 119889119881 = 1205881198812 [120583119881 tanh( 1205831198811205961198811

) minus 1205901198812119889119881] (26)

where 1205881198811 1205881198812 1205901198811 and 1205901198812 denote positive design parame-ters

Theorem 7 Suppose that the velocity subsystem (9) satisfiesAssumption 5 if the adaptive controller is selected as (25) andupdating laws are selected as (26) the signals including 120583119881 120593119881and 119889119881 are ensured to be bounded

Remark 8 The velocity design is partially derived from [24]Note that the FOSD is used to estimate unknown item 119903119881 Byintroducing 05 1199031198811205831198811199032119881 in (25) the stability analysis problemin [24] is overcome

32 Altitude Controller Design The following coordinatechange is constructed to facilitate the control design

1199111 = 1199091 minus 1199101198891199112 = 1199092 minus 12057211199113 = 1199093 minus 12057221199114 = 1199094 minus 1205723(27)

where 1205721 1205722 and 1205723 are the virtual controllers to be designedat Steps 1 2 and 3 respectively119910119889 = ℎ119889 is the reference signalThe control scheme for the altitude subsystem is developed inthe framework of back-stepping technique which contains 4-step recursive design procedure

Step 1 The time derivative of 1199111 = 1199091 minus 119910119889 is expressed as

1 = 1 minus 119910119889 = 1198811199092 minus 119910119889 = 119881 (1199112 + 1205721) minus 119910119889 (28)

By using (15) and (28) the time derivative of the trans-formed altitude error 1205831(119905) is shown as follows

1205831 (119905) = 1199031 (1 minus 11205821

1199111) = 1199031 (1198811199092 minus 119910119889 minus 11205821

1199111) (29)

where 1199031 = (1205971198771120597(11991111205821))(11205821) gt 1199031min gt 0 and 1205821(119905) =(12058210 minus 1205821infin) exp(minus1198971119905) + 1205821infinThe virtual controller 1205721 is designed as

1205721 = (minus (1198961 minus 1199031211990321) 1205831 + 119910119889 + (11205821) 1199111)119881 (30)

where 1198961 is positive parameter It is worth noticing that 1199031 canbe easily obtained via system states

Invoking (28) and (30) one has

1 = 1199031 (1198811199112 minus (1198961 minus 1199031211990321 )1205831) (31)

In order to avoid the tedious computation of 1 thefollowing FOSD is adopted to estimate it

12058910 = minus12058310 100381610038161003816100381612058910 minus 1205721100381610038161003816100381605 sign (12058910 minus 1205721) + 1205891112058911 = minus12058311 sign (12058911 minus 12058910) (32)

where 12058910 and 12058911 are the states of FOSD (32) and 12058310 and 12058311are the positive design constants

Then we have

1 = 12058910 + 1205911 (33)

where 1205911 is the estimation error of the FOSD with |1205911| le 1205911Step 2 The differentiation of 1199112 is obtained as follows

2 = 2 minus 1 = 1198912 (1199092 1199093) + 1199113 + 1205722 minus 1 (34)

The virtual controller 1205722 is designed as

1205722 = minus11989621199112 + 12058910 minus 11989821199061198732 minus (1 minus 1198982) 1199061199032 (35)

with

1199061198732 = 1211991121205932Φ1198792Φ2 + 1198892 tanh( 119911212059621

) 1199061199032 = 119891119906

2 tanh(1199112119891119906212059622

) (36)

where 1198962 12059621 and 12059622 are positive design parameters 1198892 =Δ1198912 + 1205762 is bounded with |1198892| le 1198892119872 1205932 and 1198892 denote theestimations of 1205932 and 1198892119872 respectively

The structure of adaptive control laws is expressed asfollows

1205932 = 120588212 (119898211991122Φ1198792Φ2 minus 2120590211205932)

1198892 = 12058822 [11989821199112 tanh( 119911212059621

) minus 120590221198892] (37)

Substituting (35) into (34) (34) can be rewritten as

2 = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (1198912 minus 1199061198732 ) + (1 minus 1198982)sdot (1198912 minus 1199061199032) = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (119882lowast119879

2 Φ2

minus 1211991121205932Φ1198792Φ2 + 1198892 minus 1198892 tanh( 119911212059621

)) + (1minus 1198982) (1198912 minus 119891119906

2 tanh(1199112119891119906212059622

))

(38)

6 Mathematical Problems in Engineering

The following FOSD is adopted to estimate 212058920 = minus12058320 100381610038161003816100381612058920 minus 1205722100381610038161003816100381605 sign (12058920 minus 1205722) + 1205892112058921 = minus12058321 sign (12058921 minus 12058920) (39)

where 12058920 and 12058921 are the states of system (32) and 12058320 and 12058321are positive design constants

From (39) and Lemma 3 we have

2 = 12058920 + 1205912 (40)

where 1205912 is the estimation error with |1205912| le 1205912Step 3 The differentiation of 1199113 is obtained as follows

3 = 3 minus 2 = 1199114 + 1205723 minus 2 (41)

The virtual control law 1205723 is designed as

1205723 = minus11989631199113 + 12058920 minus 1199112 (42)

where 1198963 is a positive design parameterSubstituting (42) into (41) yields

3 = 1199114 minus 1199112 minus 11989631199113 + 12058920 minus 2 = 1199114 minus 1199112 minus 11989631199113 minus 1205912 (43)

As done previously the following FOSD is employed toestimate 3

12058930 = minus12058330 100381610038161003816100381612058930 minus 1205723100381610038161003816100381605 sign (12058930 minus 1205723) + 1205893112058931 = minus12058331 sign (12058931 minus 12058930) (44)

where 12058930 and 12058931 are the states of the system and 12058330 and 12058331are the positive design constants

Thus we have

3 = 12058930 + 1205913 (45)

where 1205913 is an estimation error with |1205913| le 1205913Step 4 In this step the actual controller 119906 will be developedThe differentiation of 1199114 can be obtained as follows

4 = 1198914 (119909 119906) + 119906 minus 3= 11989841198914 (119909 119906) + (1 minus 1198984) 1198914 (119909 119906) + 119906 minus 3 (46)

The controller 119906 is designed as

119906 = minus11989641199114 + 12058930 minus 1199113 minus 11989841199061198734 minus (1 minus 1198984) 1199061199034 (47)

with

1199061198734 = 1211991141205934Φ1198794Φ4 + 1198894 tanh( 119911412059641

) 1199061199034 = 119891119906

4 tanh(1199114119891119906412059642

) (48)

where 1198964 12059641 and 12059642 are the positive design constants 1198894 =Δ1198914 + 1205764 is the lump approximation error with |1198894| le 1198894119872 1205934

and 1198894 denote the estimations of 1205934 and 1198894119872 respectively 1205934

and 1198894 are updated as

1205934 = 120588412 (119898411991124Φ1198794Φ4 minus 2120590411205934)

1198894 = 12058842 (11989841199114 tanh( 119911412059641

) minus 120590421198894) (49)

Thus (46) can be rewritten as

4 = minus11989641199114 minus 1199113 minus 1205913 + 1198984 (119882lowast1198794 Φ4 minus 1211991141205934Φ119879

4Φ4

+ 1198894 minus 1198894 tanh( 119911412059641

)) + (1 minus 1198984) (1198914minus 119891119906

4 tanh(1199114119891119906412059642

)) (50)

Theorem9 Consider the altitude subsystem (8) with Assump-tions 5 and 6 if the switching adaptive neural prescribedperformance control scheme is selected as (30) (35) (42) and(47) adaptive laws are selected as (37) and (49) and FOSD isselected as (32) (39) and (44) the signals 1205831 119911119894=234 120593119894=24and 119889119894=24 in the closed-loop system are bounded

Remark 10 The altitude controller composed of a normaladaptive neural controller working in the neural activeregion a robust controller being in charge outside the neuralapproximation region and a switching strategy supervisingthe exchange of the former two controllers is constructed

Remark 11 In this paper in order to estimate the derivativeof virtual controllers 1205721 1205722 and 1205723 the FOSD (first-ordersliding mode differentiator) is employed Using (33) as anexample 12058910 is the estimation of 1198861 and 1205911 is the estimationerror between actual 1198861 and 12058910 It must be noted that 1205911 is notused in the controller design but is just employed for stabilityanalysis (please see (B6))

4 Simulations

In this section two comparative cases are presented to illus-trate the effectiveness of the switching functions based adap-tive neural control for longitudinal model of the morphingaircraft The aerodynamic coefficients and model parametersare the same as [5] The initial conditions are set as 1198830 =[1205740 1205720 1199020 ℎ0 1198810] = [0∘ 099512∘ 0∘s 1000m 30ms] Thecontrol parameters are selected as 1198961 = 03 1198962 = 03 1198963 =005 1198964 = 375 and 1198961198811 = 3 Gains for the adaptive laws areset as 12058821 = 40 12059021 = 001 12058822 = 10 12059022 = 01 12058841 = 25012059041 = 001 12058842 = 100 12059042 = 001 1205881198811 = 10 1205901198811 = 011205881198812 = 100 1205901198812 = 01 12059621 = 1 12059622 = 1 12059641 = 1 and12059642 = 5 The aforementioned transient and steady outputerror bounds are prescribed by the performance functions120582119894(119905) = (1205821198940 minus 120582119894infin) exp(minus119897119894119905) + 120582119894infin 119894 = 1 119881 where 12058210 =05 1205821infin = 02 1205821198810 = 035 1205821infin = 01 and 119897119894 = 005The corresponding neural active regions are defined as 119881 isin[30ms 40ms] 1199092 isin [minus4 deg 4 deg] 1199093 isin [minus6 deg 6 deg]

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

and 119903119881 denotes the estimation of 119903119881 bymeans of slidingmodedifferentiator According to Lemma 4 we can easily obtain| 119903119881 minus 119903119881| le 119897 with 119897 gt 0

Consider the following adaptive laws for 120593119881 and 119889119881

120593119881 = 12058811988112 (1205832119881Φ119879

119881Φ119881 minus 21205901198811120593119881) 119889119881 = 1205881198812 [120583119881 tanh( 1205831198811205961198811

) minus 1205901198812119889119881] (26)

where 1205881198811 1205881198812 1205901198811 and 1205901198812 denote positive design parame-ters

Theorem 7 Suppose that the velocity subsystem (9) satisfiesAssumption 5 if the adaptive controller is selected as (25) andupdating laws are selected as (26) the signals including 120583119881 120593119881and 119889119881 are ensured to be bounded

Remark 8 The velocity design is partially derived from [24]Note that the FOSD is used to estimate unknown item 119903119881 Byintroducing 05 1199031198811205831198811199032119881 in (25) the stability analysis problemin [24] is overcome

32 Altitude Controller Design The following coordinatechange is constructed to facilitate the control design

1199111 = 1199091 minus 1199101198891199112 = 1199092 minus 12057211199113 = 1199093 minus 12057221199114 = 1199094 minus 1205723(27)

where 1205721 1205722 and 1205723 are the virtual controllers to be designedat Steps 1 2 and 3 respectively119910119889 = ℎ119889 is the reference signalThe control scheme for the altitude subsystem is developed inthe framework of back-stepping technique which contains 4-step recursive design procedure

Step 1 The time derivative of 1199111 = 1199091 minus 119910119889 is expressed as

1 = 1 minus 119910119889 = 1198811199092 minus 119910119889 = 119881 (1199112 + 1205721) minus 119910119889 (28)

By using (15) and (28) the time derivative of the trans-formed altitude error 1205831(119905) is shown as follows

1205831 (119905) = 1199031 (1 minus 11205821

1199111) = 1199031 (1198811199092 minus 119910119889 minus 11205821

1199111) (29)

where 1199031 = (1205971198771120597(11991111205821))(11205821) gt 1199031min gt 0 and 1205821(119905) =(12058210 minus 1205821infin) exp(minus1198971119905) + 1205821infinThe virtual controller 1205721 is designed as

1205721 = (minus (1198961 minus 1199031211990321) 1205831 + 119910119889 + (11205821) 1199111)119881 (30)

where 1198961 is positive parameter It is worth noticing that 1199031 canbe easily obtained via system states

Invoking (28) and (30) one has

1 = 1199031 (1198811199112 minus (1198961 minus 1199031211990321 )1205831) (31)

In order to avoid the tedious computation of 1 thefollowing FOSD is adopted to estimate it

12058910 = minus12058310 100381610038161003816100381612058910 minus 1205721100381610038161003816100381605 sign (12058910 minus 1205721) + 1205891112058911 = minus12058311 sign (12058911 minus 12058910) (32)

where 12058910 and 12058911 are the states of FOSD (32) and 12058310 and 12058311are the positive design constants

Then we have

1 = 12058910 + 1205911 (33)

where 1205911 is the estimation error of the FOSD with |1205911| le 1205911Step 2 The differentiation of 1199112 is obtained as follows

2 = 2 minus 1 = 1198912 (1199092 1199093) + 1199113 + 1205722 minus 1 (34)

The virtual controller 1205722 is designed as

1205722 = minus11989621199112 + 12058910 minus 11989821199061198732 minus (1 minus 1198982) 1199061199032 (35)

with

1199061198732 = 1211991121205932Φ1198792Φ2 + 1198892 tanh( 119911212059621

) 1199061199032 = 119891119906

2 tanh(1199112119891119906212059622

) (36)

where 1198962 12059621 and 12059622 are positive design parameters 1198892 =Δ1198912 + 1205762 is bounded with |1198892| le 1198892119872 1205932 and 1198892 denote theestimations of 1205932 and 1198892119872 respectively

The structure of adaptive control laws is expressed asfollows

1205932 = 120588212 (119898211991122Φ1198792Φ2 minus 2120590211205932)

1198892 = 12058822 [11989821199112 tanh( 119911212059621

) minus 120590221198892] (37)

Substituting (35) into (34) (34) can be rewritten as

2 = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (1198912 minus 1199061198732 ) + (1 minus 1198982)sdot (1198912 minus 1199061199032) = 1199113 minus 11989621199112 + 12058910 minus 1 + 1198982 (119882lowast119879

2 Φ2

minus 1211991121205932Φ1198792Φ2 + 1198892 minus 1198892 tanh( 119911212059621

)) + (1minus 1198982) (1198912 minus 119891119906

2 tanh(1199112119891119906212059622

))

(38)

6 Mathematical Problems in Engineering

The following FOSD is adopted to estimate 212058920 = minus12058320 100381610038161003816100381612058920 minus 1205722100381610038161003816100381605 sign (12058920 minus 1205722) + 1205892112058921 = minus12058321 sign (12058921 minus 12058920) (39)

where 12058920 and 12058921 are the states of system (32) and 12058320 and 12058321are positive design constants

From (39) and Lemma 3 we have

2 = 12058920 + 1205912 (40)

where 1205912 is the estimation error with |1205912| le 1205912Step 3 The differentiation of 1199113 is obtained as follows

3 = 3 minus 2 = 1199114 + 1205723 minus 2 (41)

The virtual control law 1205723 is designed as

1205723 = minus11989631199113 + 12058920 minus 1199112 (42)

where 1198963 is a positive design parameterSubstituting (42) into (41) yields

3 = 1199114 minus 1199112 minus 11989631199113 + 12058920 minus 2 = 1199114 minus 1199112 minus 11989631199113 minus 1205912 (43)

As done previously the following FOSD is employed toestimate 3

12058930 = minus12058330 100381610038161003816100381612058930 minus 1205723100381610038161003816100381605 sign (12058930 minus 1205723) + 1205893112058931 = minus12058331 sign (12058931 minus 12058930) (44)

where 12058930 and 12058931 are the states of the system and 12058330 and 12058331are the positive design constants

Thus we have

3 = 12058930 + 1205913 (45)

where 1205913 is an estimation error with |1205913| le 1205913Step 4 In this step the actual controller 119906 will be developedThe differentiation of 1199114 can be obtained as follows

4 = 1198914 (119909 119906) + 119906 minus 3= 11989841198914 (119909 119906) + (1 minus 1198984) 1198914 (119909 119906) + 119906 minus 3 (46)

The controller 119906 is designed as

119906 = minus11989641199114 + 12058930 minus 1199113 minus 11989841199061198734 minus (1 minus 1198984) 1199061199034 (47)

with

1199061198734 = 1211991141205934Φ1198794Φ4 + 1198894 tanh( 119911412059641

) 1199061199034 = 119891119906

4 tanh(1199114119891119906412059642

) (48)

where 1198964 12059641 and 12059642 are the positive design constants 1198894 =Δ1198914 + 1205764 is the lump approximation error with |1198894| le 1198894119872 1205934

and 1198894 denote the estimations of 1205934 and 1198894119872 respectively 1205934

and 1198894 are updated as

1205934 = 120588412 (119898411991124Φ1198794Φ4 minus 2120590411205934)

1198894 = 12058842 (11989841199114 tanh( 119911412059641

) minus 120590421198894) (49)

Thus (46) can be rewritten as

4 = minus11989641199114 minus 1199113 minus 1205913 + 1198984 (119882lowast1198794 Φ4 minus 1211991141205934Φ119879

4Φ4

+ 1198894 minus 1198894 tanh( 119911412059641

)) + (1 minus 1198984) (1198914minus 119891119906

4 tanh(1199114119891119906412059642

)) (50)

Theorem9 Consider the altitude subsystem (8) with Assump-tions 5 and 6 if the switching adaptive neural prescribedperformance control scheme is selected as (30) (35) (42) and(47) adaptive laws are selected as (37) and (49) and FOSD isselected as (32) (39) and (44) the signals 1205831 119911119894=234 120593119894=24and 119889119894=24 in the closed-loop system are bounded

Remark 10 The altitude controller composed of a normaladaptive neural controller working in the neural activeregion a robust controller being in charge outside the neuralapproximation region and a switching strategy supervisingthe exchange of the former two controllers is constructed

Remark 11 In this paper in order to estimate the derivativeof virtual controllers 1205721 1205722 and 1205723 the FOSD (first-ordersliding mode differentiator) is employed Using (33) as anexample 12058910 is the estimation of 1198861 and 1205911 is the estimationerror between actual 1198861 and 12058910 It must be noted that 1205911 is notused in the controller design but is just employed for stabilityanalysis (please see (B6))

4 Simulations

In this section two comparative cases are presented to illus-trate the effectiveness of the switching functions based adap-tive neural control for longitudinal model of the morphingaircraft The aerodynamic coefficients and model parametersare the same as [5] The initial conditions are set as 1198830 =[1205740 1205720 1199020 ℎ0 1198810] = [0∘ 099512∘ 0∘s 1000m 30ms] Thecontrol parameters are selected as 1198961 = 03 1198962 = 03 1198963 =005 1198964 = 375 and 1198961198811 = 3 Gains for the adaptive laws areset as 12058821 = 40 12059021 = 001 12058822 = 10 12059022 = 01 12058841 = 25012059041 = 001 12058842 = 100 12059042 = 001 1205881198811 = 10 1205901198811 = 011205881198812 = 100 1205901198812 = 01 12059621 = 1 12059622 = 1 12059641 = 1 and12059642 = 5 The aforementioned transient and steady outputerror bounds are prescribed by the performance functions120582119894(119905) = (1205821198940 minus 120582119894infin) exp(minus119897119894119905) + 120582119894infin 119894 = 1 119881 where 12058210 =05 1205821infin = 02 1205821198810 = 035 1205821infin = 01 and 119897119894 = 005The corresponding neural active regions are defined as 119881 isin[30ms 40ms] 1199092 isin [minus4 deg 4 deg] 1199093 isin [minus6 deg 6 deg]

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The following FOSD is adopted to estimate 212058920 = minus12058320 100381610038161003816100381612058920 minus 1205722100381610038161003816100381605 sign (12058920 minus 1205722) + 1205892112058921 = minus12058321 sign (12058921 minus 12058920) (39)

where 12058920 and 12058921 are the states of system (32) and 12058320 and 12058321are positive design constants

From (39) and Lemma 3 we have

2 = 12058920 + 1205912 (40)

where 1205912 is the estimation error with |1205912| le 1205912Step 3 The differentiation of 1199113 is obtained as follows

3 = 3 minus 2 = 1199114 + 1205723 minus 2 (41)

The virtual control law 1205723 is designed as

1205723 = minus11989631199113 + 12058920 minus 1199112 (42)

where 1198963 is a positive design parameterSubstituting (42) into (41) yields

3 = 1199114 minus 1199112 minus 11989631199113 + 12058920 minus 2 = 1199114 minus 1199112 minus 11989631199113 minus 1205912 (43)

As done previously the following FOSD is employed toestimate 3

12058930 = minus12058330 100381610038161003816100381612058930 minus 1205723100381610038161003816100381605 sign (12058930 minus 1205723) + 1205893112058931 = minus12058331 sign (12058931 minus 12058930) (44)

where 12058930 and 12058931 are the states of the system and 12058330 and 12058331are the positive design constants

Thus we have

3 = 12058930 + 1205913 (45)

where 1205913 is an estimation error with |1205913| le 1205913Step 4 In this step the actual controller 119906 will be developedThe differentiation of 1199114 can be obtained as follows

4 = 1198914 (119909 119906) + 119906 minus 3= 11989841198914 (119909 119906) + (1 minus 1198984) 1198914 (119909 119906) + 119906 minus 3 (46)

The controller 119906 is designed as

119906 = minus11989641199114 + 12058930 minus 1199113 minus 11989841199061198734 minus (1 minus 1198984) 1199061199034 (47)

with

1199061198734 = 1211991141205934Φ1198794Φ4 + 1198894 tanh( 119911412059641

) 1199061199034 = 119891119906

4 tanh(1199114119891119906412059642

) (48)

where 1198964 12059641 and 12059642 are the positive design constants 1198894 =Δ1198914 + 1205764 is the lump approximation error with |1198894| le 1198894119872 1205934

and 1198894 denote the estimations of 1205934 and 1198894119872 respectively 1205934

and 1198894 are updated as

1205934 = 120588412 (119898411991124Φ1198794Φ4 minus 2120590411205934)

1198894 = 12058842 (11989841199114 tanh( 119911412059641

) minus 120590421198894) (49)

Thus (46) can be rewritten as

4 = minus11989641199114 minus 1199113 minus 1205913 + 1198984 (119882lowast1198794 Φ4 minus 1211991141205934Φ119879

4Φ4

+ 1198894 minus 1198894 tanh( 119911412059641

)) + (1 minus 1198984) (1198914minus 119891119906

4 tanh(1199114119891119906412059642

)) (50)

Theorem9 Consider the altitude subsystem (8) with Assump-tions 5 and 6 if the switching adaptive neural prescribedperformance control scheme is selected as (30) (35) (42) and(47) adaptive laws are selected as (37) and (49) and FOSD isselected as (32) (39) and (44) the signals 1205831 119911119894=234 120593119894=24and 119889119894=24 in the closed-loop system are bounded

Remark 10 The altitude controller composed of a normaladaptive neural controller working in the neural activeregion a robust controller being in charge outside the neuralapproximation region and a switching strategy supervisingthe exchange of the former two controllers is constructed

Remark 11 In this paper in order to estimate the derivativeof virtual controllers 1205721 1205722 and 1205723 the FOSD (first-ordersliding mode differentiator) is employed Using (33) as anexample 12058910 is the estimation of 1198861 and 1205911 is the estimationerror between actual 1198861 and 12058910 It must be noted that 1205911 is notused in the controller design but is just employed for stabilityanalysis (please see (B6))

4 Simulations

In this section two comparative cases are presented to illus-trate the effectiveness of the switching functions based adap-tive neural control for longitudinal model of the morphingaircraft The aerodynamic coefficients and model parametersare the same as [5] The initial conditions are set as 1198830 =[1205740 1205720 1199020 ℎ0 1198810] = [0∘ 099512∘ 0∘s 1000m 30ms] Thecontrol parameters are selected as 1198961 = 03 1198962 = 03 1198963 =005 1198964 = 375 and 1198961198811 = 3 Gains for the adaptive laws areset as 12058821 = 40 12059021 = 001 12058822 = 10 12059022 = 01 12058841 = 25012059041 = 001 12058842 = 100 12059042 = 001 1205881198811 = 10 1205901198811 = 011205881198812 = 100 1205901198812 = 01 12059621 = 1 12059622 = 1 12059641 = 1 and12059642 = 5 The aforementioned transient and steady outputerror bounds are prescribed by the performance functions120582119894(119905) = (1205821198940 minus 120582119894infin) exp(minus119897119894119905) + 120582119894infin 119894 = 1 119881 where 12058210 =05 1205821infin = 02 1205821198810 = 035 1205821infin = 01 and 119897119894 = 005The corresponding neural active regions are defined as 119881 isin[30ms 40ms] 1199092 isin [minus4 deg 4 deg] 1199093 isin [minus6 deg 6 deg]

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Sweep signal

0

10

20

30

40

50

10 20 300Time (s)

(d

eg)

(a)

m2

m4

08

09

1

11

mi

10 20 300Time (s)

(b)

Figure 1 Sweep reference and switch signal

SPPCPPC

SAN1

minus15

minus1

minus05

0

05

her

ror

(m)

10 20 300Time (s)

(a)

SPPCPPC

SAN

minus04

minus02

0

02

04Z

V(m

s)

10 20 300Time (s)

V

(b)

Figure 2 Altitude and velocity tracking errors

and 1199094 isin [minus6 degs 6 degs] The centers such as 1198882 1198884 and119888119881 including 50 nodes are evenly spaced in their bounds Thewidths of Gaussian functions are chosen as 1198871198942 = 2 1198871198944 = 15and 119887119894119881 = 4Theparameters of switching function are selectedas 11990321 = 4 deg 11990322 = 7 deg 11990331 = 6 deg 11990332 = 10 deg 11990341 =5 degs 11990342 = 10 degs 120596119894=123 = 10 and 1198871 = 1 Referencecommands are smoothened via several second-order filterswhich are given in (51)

ℎ119889ℎ1198890 = 0041199042 + 04119904 + 004 1198811198891198811198890

= 0041199042 + 04119904 + 004 1205771198891205771198890 = 11199042 + 2119904 + 1 (51)

Case 1 In this simulation we assume that the aircraft iscruising at trim states and only the morphing process isconsidered The initial tracking errors are assumed to be1199111(0) = 01m and 119911119881(0) = 03ms For comparison pur-poses the switching function based adaptive neural control(SAN which means the PPC technique is not employed inthe control design) and adaptive neural prescribed perfor-mance control (PPC described in Remark 10) are utilizedmeanwhile the control gains are kept fixed to the valuesused in the proposed control scheme (SPPC) Simulationresults are presented in Figures 1ndash4 Specifically the outputtracking errors are presented in Figure 2 Moreover therequired control input 120575119890 119879 system states and the evolutionof NNsrsquo weight are provided in Figures 3 and 4 Notice thatduring the morphing process the input states (1199092 1199093 1199094)

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

of NNs always stay in the active region which can beexplained in Figure 1(b) thus the proposed SPPC schemeis equivalent to PPC scheme As expected output trackingwith prescribed performance as well as states boundedness isachieved with reasonable control effort In contrast to PPCor SPPC approach although the SAN control scheme cankeep the stability of the aircraft it owns relatively poor systemperformance as shown in Figure 2 Moreover the superiorityof the proposed SPPC schemewill be further revealed in Case2 simulation

Case 2 In this simulation the control gains are kept thesame as Case 1 meanwhile the SAN and PPC approachesare also used as comparison The upper bounds of 119891119906

2 and1198911199064 are set as 119891119906

2 = 05 and 1198911199064 = 1 Simulation results

are presented in Figures 5ndash9 The output tracking for SANand SPPC is presented in Figures 5(a) and 5(c) and thecorresponding tracking errors are presented in Figures 5(b)and 5(d) along with their performance bounds Particularlythe control inputs and system states as well as the value ofthe switch functions are provided in Figures 6 7 and 8As expected output tracking with prescribed performanceand states boundedness are both achieved by using theproposed SPPC scheme Unfortunately the altitude andvelocity tracking errors of SAN transcend the prescribedbounds 1205821 and 120582119881 It is worth noting that different fromCase 1 the switch values are not always equal to one asdepicted in Figure 8 which means that the NNs are outof the neural approximation regions The curve of 120575119890 isobtained as shown in Figure 7(a) After about 11 secondsthe robust controller pulls back the escaped transient to theneural effective regions Some corresponding responses forPPC scheme are pictured in Figure 9 From this figure wecan conclude that the altitude error oversteps the prescribedbounds without the utilization of switching strategy thusleading to the instability of the system In all comparedwith simulation results the superiority of SPPC scheme isobvious

5 Conclusion

A composite switching neural prescribed performance con-trol scheme has been proposed for the longitudinal dynamicmodel of the morphing aircraft In the control design byusing neural networks to approximate the unknown func-tions the prior information of the aerodynamic parametersis unnecessary By introducing the performance functionthe proposed controller is able to permit attributes suchas a lower bound on the convergence rate and maxi-mum allowable steady error to be specified A switchingmechanism supervising the exchange of control authoritiesbetween the normal neural controller and a robust con-troller is used to relax the constraint that NN should bekept in the active regions all the time Two comparativesimulations have revealed the superiority of this controlscheme

Appendix

A Proof of Theorem 7

Proof Invoking (24) and (25) yields

120583119881 (119905) = 119903119881 [minus(1198961198811 minus 11990311988121199032119881)120583119881 + 119882lowast119879119881 Φ119881

minus 12120583119881120593119881Φ119879119881Φ119881 + 119889119881 minus 119889119881 tanh( 1205831198811205961198811

)] (A1)

Consider the following candidate Lyapunov function

119871119881 = 121199031198811205832119881 + 1212058811988111205932

119881 + 121205881198812 1198892119881 (A2)

where 120593119881 = 120593119881 minus 120593119881 and 119889119881 = 119889119881119872 minus 119889119881Based on (26) and (A1) the time derivative of 119871119881 is given

by

119881 = 1119903119881120583119881 120583119881 minus 119903119881211990321198811205832119881 minus 11205881198811120593119881

120593119881 minus 11205881198812 119889119881 119889119881

= minus(1198961198811 minus 119903119881 minus 11990311988121199032119881 )1205832119881 + 120583119881119882lowast119879

119881 Φ119881

minus 121205832119881120593119881Φ119879

119881Φ119881 + 119889119881120583119881minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) + 121205901198811120593119881120593119881

+ 1205901198812119889119881119889119881

(A3)

Note that the following inequalities hold

120583119881119882lowast119879119881 Φ119881 le 121205832

119881120593119881Φ119879119881Φ119881 + 12

1205901198812119889119881119889119881 le 12059011988122 (1198892119881119872 minus 1198892

119881) 119889119881120583119881 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

)le 119889119881119872

10038161003816100381610038161205831198811003816100381610038161003816 minus 119889119881119872120583119881 tanh( 1205831198811205961198811

) le 120581012059611988111198891198811198721205901198811120593119881120593119881 = 12059011988112 (1205932

119881 minus 1205932119881 minus 1205932

119881) le 12059011988112 (1205932119881 minus 1205932

119881)

(A4)

By considering (A4) 119881 can be reformulated as

119881 le minus(1198961198811 minus 11989711988121199032119881min)1205832

119881 minus 12059011988112 1205932119881 minus 12059011988122 1198892

119881

+ 119862119881(A5)

where 119862119881 = 05 + 12058101205961198811119889119881119872 + (12059011988112)1205932119881 + (12059011988122)1198892

119881119872

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

SPPCPPCSAN

(d

eg)

minus1

minus05

0

05

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

2

4

6

8

(d

eg)

10 20 300Time (s)

(b)

SPPCPPCSAN

minus2

0

2

4

6

q(d

egs

)

10 20 300Time (s)

(c)

SPPCPPCSAN

0

002

004

006 2

10 20 300Time (s)

(d)

SPPCPPCSAN

4

0

05

1

15

2

10 20 300Time (s)

(e)

SPPCPPCSAN

0

2

4

6

8

10

10 20 300Time (s)

v

(f)

Figure 3 System states and NN weights

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

SPPCPPCSAN

e

(deg

)

minus3

minus2

minus1

0

1

10 20 300Time (s)

(a)

SPPCPPCSAN

0

50

100

150

T(N

)

10 20 300Time (s)

(b)

Figure 4 Control inputs

If 1198961198811 minus 051198971198811199032119881min gt 0 define the following compactsets

Ω120583119881= 120583119881 | 10038161003816100381610038161205831198811003816100381610038161003816 le radic 119862119881(1198961198811 minus 051198971198811199032119881min)

Ω120593119881

= 120593119881 | 10038161003816100381610038161205931198811003816100381610038161003816 le radic 119862119881(051205901198811)

Ω119889119881= 119889119881 | 1003816100381610038161003816100381611988911988110038161003816100381610038161003816 le radic 119862119881(051205901198812)

(A6)

It is obvious that 119881 is negative if 120583119881 notin Ω120583119881 120593119881 notin Ω120593119881

and 119889119881 notin Ω119889119881

Therefore the signals in 120583119881 120593119881 and 119889119881 in theclosed-loop system are bounded

B Proof of Theorem 9

Proof Select the candidate Lyapunov function as follows

119871 = 1198711 + 1198712 + 1198713 + 1198714 (B1)

where 1198711 = (1211988121199031)12058321 1198712 = (12)11991122 + 1205932

2212058821 + 119889222120588221198713 = (12)11991123 1198714 = (12)11991124 +1205932

4212058841 +11988924212058842 1205932 = 1205932 minus12059321198892 = 1198892119872minus1198892 1205934 = 1205934minus1205934 and 1198894 = 1198894119872minus1198894 and119881 denotes

the upper bound of 119881On the basis of (31) the time derivative of 1198711 is given by

1 = 111988121199031 1205831 1205831 minus 11990312119881211990321 12058321 = minus11989611198812

12058321 + 1198811198812

12058311199112 (B2)

Differentiating 1198712 with respect to time invoking (37) and(38) we have

2 = 11991122 minus 11205882112059321205932 minus 112058822 1198892 1198892 le minus119896211991122 + 11991121199113

minus 11991121205911 + 1198982 (1199112119882lowast1198792 Φ2 minus 12119911221205932Φ119879

2Φ2

+ 1198892119872 100381610038161003816100381611991121003816100381610038161003816 minus 11988921198721199112 tanh( 119911212059621

)) + (1 minus 1198982)sdot (10038161003816100381610038161199112119891119906

21003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

)) + 1205902112059321205932

+ 1205902211988921198892

(B3)

Employing (43) the time derivative of 1198713 is obtained as

3 = 1199113 (1199114 minus 1199112 minus 11989631199113 minus 1205912)le minus(1198963 minus 1211989612)11991123 minus 11991121199113 + 11991131199114 + 119896122 12059122 (B4)

Using (49) and (50) results in the time derivative of 1198714

4 = 11991144 minus 11205884112059341205934 minus 112058842 1198894 1198894 le minus119896411991124 minus 11991131199114

minus 11991141205913 + 1198984 (1199114119882lowast1198794 Φ4 minus 12119911241205934Φ119879

4Φ4

+ 100381610038161003816100381611991141003816100381610038161003816 1198894119872 minus 11991141198894119872 tanh( 119911412059641

)) + (1 minus 1198984)sdot (1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

)) + 1205904112059341205934

+ 1205904211988941198894

(B5)

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

SPPCSAN

h(m

)

h

1000

1010

1020

1030

1040

1050

10 5030 40200Time (s)

(a)

SPPCSAN1

minus1

minus05

0

05

1

her

ror

(m)

5020 30 40100Time (s)

(b)

SPPCSAN

V(m

s)

V

30

32

34

36

38

40

5020100 30 40Time (s)

(c)

SPPCSAN

minus04

minus02

0

02

04

5020 30 40100Time (s)

ZV

(ms

)

V

(d)

Figure 5 Altitude and velocity tracking

Consider the following facts

11988112058311199112 le 1211989611 12058321 + 119896112 119881211991122

1205902112059321205932 = 120590212 (12059322 minus 1205932

2 minus 12059322) le 120590212 (1205932

2 minus 12059322)

1199112119882lowast1198792 Φ2 le 12119911221205932Φ119879

2Φ2 + 12 1205902211988921198892 le 12 (120590221198892

2119872 minus 1205902211988922)

minus 11991121205911 le 1211991122 + 1212059121100381610038161003816100381611991121003816100381610038161003816 1198892 minus 11991131198892119872 tanh( 11991131199082

) le 1205810119889211987212059621

1003816100381610038161003816119911211989111990621003816100381610038161003816 minus 1199112119891119906

2 tanh(1199112119891119906212059622

) le 120581012059622minus 12059121199113 le 1211989612 11991123 + 119896122 120591221205904112059341205934 = 120590412 (1205932

4 minus 12059324 minus 1205932

4) le 120590412 (12059324 minus 1205932

4) 1205904211988941198894 le 1212059042 (1198892

4119872 minus 11988924)

1003816100381610038161003816119911411988941003816100381610038161003816 minus 11991141198894119872 tanh( 119911412059641

) le 1205810119889411987212059641minus 11991141205913 le 12 (11991124 + 12059123)

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

SPPCSAN

(d

eg)

minus2

0

2

4

6

8

30 40 5010 200Time (s)

(a)

SPPCSAN

minus2

0

2

4

6

8

(d

eg)

4030 5010 200Time (s)

(b)

SPPCSAN

minus2

0

2

4

6

q(d

egs

)

30 40 5010 200Time (s)

(c)

SPPCSAN

0

0005

001

0015 2

10 20 30 40 500Time (s)

(d)

SPPCSAN

4

0

05

1

15

10 20 30 40 500Time (s)

(e)

SPPCSAN

0

5

10

15

20

10 20 30 40 500Time (s)

v

(f)

Figure 6 System states and NN weights

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

SPPCSAN

e

(deg

)

minus2

minus1

0

1

2

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

20

40

60

80

100

T(N

)

10 20 30 40 500Time (s)

(b)

Figure 7 Control inputs

SPPCSAN

m2

0

02

04

06

08

1

10 20 30 40 500Time (s)

(a)

SPPCSAN

0

02

04

06

08

1

m4

10 20 30 40 500Time (s)

(b)

Figure 8 Switching functions

1199114119882lowast1198794 Φ le 12119911241205934Φ119879

4Φ4 + 12 1003816100381610038161003816119911411989141003816100381610038161003816 minus 1199114119891119906

4 tanh(1199114119891119906412059642

) le 120581012059642(B6)

We have

le minus 11198812(1198961 minus 1211989611)1205832

1 minus (1198962 minus 05 minus 0511989611) 11991122minus 12120590211205932

2 minus 121205902211988922 minus (1198963 minus 1211989612)11991123

minus (1198964 minus 05) 11991124 minus 121205904112059324 minus 12120590421198892

4 + 1198622(B7)

where the corresponding design parameters should be chosensuch that 1198961minus0511989611 gt 0 1198962minus05minus0511989611 gt 0 1198963minus0511989612 gt 0(1198964 minus 05) gt 0 and 120590119894119895 gt 0 119894 = 2 4 119895 = 1 2

1198622 = 05 (1205902112059322 + 120590221198892

2119872 + 1198982 + 12059121 + 1198961212059122+ 120590421198892

4119872 + 1205904112059324 + 12059123 + 1198984) + 11989821205810119889211987212059621+ (1 minus 1198982) 120581012059622 + 11989841205810119889411987212059641

(B8)

Define the following compact sets

Ω1205831= 1205831 | 100381610038161003816100381612058311003816100381610038161003816 le radic 1198622(1198961 minus 0511989611) 1198812

Ω1199112

= 1199112 | 100381610038161003816100381611991121003816100381610038161003816 le radic 1198622(1198962 minus 05 minus 0511989611)

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

PPC1

minus05

0

05h

erro

r(m

)

5 10 150Time (s)

(a)

PPC

e

(deg

)

minus20

minus10

0

10

20

2 4 6 8 100Time (s)

(b)

PPC

minus5

0

5

10

q(d

egs

)

5 10 150Time (s)

(c)PPC

4

0

10

20

30

5 10 150Time (s)

(d)

Figure 9 Partial response of PPC scheme

Ω1199113= 1199113 | 100381610038161003816100381611991131003816100381610038161003816 le radic 1198622(1198963 minus 0511989612)

Ω1199114= 1199114 | 100381610038161003816100381611991141003816100381610038161003816 le radic 1198622(1198964 minus 05)

Ω1205932= 1205932 | 10038161003816100381610038161205932

1003816100381610038161003816 le radic 2119862212059021 Ω1205934

= 1205934 | 100381610038161003816100381612059341003816100381610038161003816 le radic 2119862212059041

Ω1198892= 1198892 | 10038161003816100381610038161003816119889210038161003816100381610038161003816 le radic 2119862212059022

Ω1198894= 1198894 | 10038161003816100381610038161003816119889410038161003816100381610038161003816 le radic 2119862212059042

(B9)

If 1205831 notin Ω1205831 1199112 notin Ω1199112

1199113 notin Ω1199113 1199114 notin Ω1199114 1205932 notin Ω1205932

1205934 notin Ω1205934 1198892 notin Ω1198892

and 1198894 notin Ω1198894 we know that will be

negative Therefore the signals 1205831 119911119894=234 120593119894=24 and 119889119894=24 inthe closed-loop system are bounded

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant nos 61374032 and61573286)

References

[1] T Yue L Wang and J Ai ldquoGain self-scheduled119867infin control formorphing aircraft in the wing transition process based on an

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

LPV modelrdquo Chinese Journal of Aeronautics vol 26 no 4 pp909ndash917 2013

[2] T Wang C Dong and Q Wang ldquoFinite-time bounded-ness control of morphing aircraft based on switched systemsapproachrdquo Optik vol 126 no 23 pp 4436ndash4445 2015

[3] Z Wu J Lu Q Zhou and J Shi ldquoModified adaptive neuraldynamic surface control for morphing aircraft with input andoutput constraintsrdquo Nonlinear Dynamics vol 87 no 4 pp2367ndash2383 2017

[4] D H Baldelli D-H Lee R S Sanchez Penal and B CannonldquoModeling and control of an aeroelastic morphing vehiclerdquoJournal of Guidance Control and Dynamics vol 31 no 6 pp1687ndash1699 2008

[5] Z Wu J Lu J Rajput J Shi and W Ma ldquoAdaptive neuralcontrol based on high order integral chained differentiator formorphing aircraftrdquoMathematical Problems in Engineering vol2015 pp 1ndash12 2015

[6] W Jiang C Dong and Q Wang ldquoA systematic method ofsmooth switching LPV controllers design for a morphingaircraftrdquo Chinese Journal of Aeronautics vol 28 no 6 pp 1640ndash1649 2015

[7] D Wang and J Huang ldquoNeural network-based adaptivedynamic surface control for a class of uncertain nonlinearsystems in strict-feedback formrdquo IEEE Transactions on NeuralNetworks vol 16 no 1 pp 195ndash202 2005

[8] A-M Zou Z-G Hou and M Tan ldquoAdaptive control of a classof nonlinear pure-feedback systems using fuzzy backsteppingapproachrdquo IEEE Transactions on Fuzzy Systems vol 16 no 4pp 886ndash897 2008

[9] B S Kim and S J Yoo ldquoApproximation-based adaptive controlof uncertain non-linear pure-feedback systems with full stateconstraintsrdquo IET Control Theory amp Applications vol 8 no 17pp 2070ndash2081 2014

[10] Q Shen B Jiang and V Cocquempot ldquoFuzzy logic system-based adaptive fault-tolerant control for near-space vehicleattitude dynamics with actuator faultsrdquo IEEE Transactions onFuzzy Systems vol 21 no 2 pp 289ndash300 2013

[11] Y Li S Tong andT Li ldquoAdaptive fuzzy output-feedback controlfor output constrained nonlinear systems in the presence ofinput saturationrdquo Fuzzy Sets and Systems vol 248 pp 138ndash1552014

[12] YM Li and S C Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead-zonerdquo IEEE Transactions on Fuzzy Systems vol 22 pp1341ndash1347 2014

[13] K P Tee S S Ge and E H Tay ldquoBarrier Lyapunov functionsfor the control of output-constrained nonlinear systemsrdquo Auto-matica vol 45 no 4 pp 918ndash927 2009

[14] M Chen G Tao and B Jiang ldquoDynamic surface control usingneural networks for a class of uncertain nonlinear systems withinput saturationrdquo IEEE Transactions on Neural Networks andLearning Systems vol 26 no 9 pp 2086ndash2097 2015

[15] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44pp 2626ndash2634 2014

[16] Q Shen B Jiang and V Cocquempot ldquoAdaptive fuzzyobserver-based active fault-tolerant dynamic surface controlfor a class of nonlinear systems with actuator faultsrdquo IEEETransactions on Fuzzy Systems vol 22 no 2 pp 338ndash349 2014

[17] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[18] B Xu Z Shi C Yang and S Wang ldquoNeural control ofhypersonic flight vehicle model via time-scale decompositionwith throttle setting constraintrdquo Nonlinear Dynamics vol 73no 3 pp 1849ndash1861 2013

[19] X Bu X Wu D Wei and J Huang ldquoNeural-approximation-based robust adaptive control of flexible air-breathing hyper-sonic vehicles with parametric uncertainties and control inputconstraintsrdquo Information Sciences vol 346-347 pp 29ndash43 2016

[20] Q Zong F Wang B Tian and R Su ldquoRobust adaptivedynamic surface control design for a flexible air-breathinghypersonic vehicle with input constraints and uncertaintyrdquoNonlinear Dynamics vol 78 no 1 pp 289ndash315 2014

[21] L Zhang S Sui Y Li and S Tong ldquoAdaptive fuzzy output feed-back tracking controlwith prescribed performance for chemicalreactor of MIMO nonlinear systemsrdquo Nonlinear Dynamics vol80 no 1-2 pp 945ndash957 2015

[22] C P Bechlioulis and G A Rovithakis ldquoAdaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systemsrdquoAutomatica vol 45 no 2 pp 532ndash5382009

[23] J Na Q Chen X Ren and Y Guo ldquoAdaptive prescribed perfor-mance motion control of servo mechanisms with friction com-pensationrdquo IEEE Transactions on Industrial Electronics vol 61no 1 pp 486ndash494 2014

[24] X Bu X Wu F Zhu J Huang Z Ma and R ZhangldquoNovel prescribed performance neural control of a flexible air-breathing hypersonic vehicle with unknown initial errorsrdquo ISATransactions vol 59 pp 149ndash159 2015

[25] Y Huang J Na X Wu X Liu and Y Guo ldquoAdaptive control ofnonlinear uncertain active suspension systems with prescribedperformancerdquo ISA Transactions vol 54 pp 145ndash155 2015

[26] S Sui S Tong and Y Li ldquoObserver-based fuzzy adaptive pre-scribed performance tracking control for nonlinear stochasticsystems with input saturationrdquo Neurocomputing vol 158 pp100ndash108 2015

[27] Y Li and S Tong ldquoPrescribed performance adaptive fuzzyoutput-feedback dynamic surface control for nonlinear large-scale systems with time delaysrdquo Information Sciences vol 292pp 125ndash142 2015

[28] S I Han and J M Lee ldquoPartial tracking error constrained fuzzydynamic surface control for a strict feedback nonlinear dynamicsystemrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5 pp1049ndash1061 2014

[29] C P Bechlioulis andG A Rovithakis ldquoPrescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systemsrdquo IEEE Transactions on AutomaticControl vol 55 no 5 pp 1220ndash1226 2010

[30] S Tong S Sui and Y Li ldquoFuzzy adaptive output feedbackcontrol ofMIMOnonlinear systemswith partial tracking errorsconstrainedrdquo IEEE Transactions on Fuzzy Systems vol 23 pp729ndash742 2015

[31] Z Wu J Lu J Shi Q Zhou and X Qu ldquoTracking errorconstrained robust adaptive neural prescribed performancecontrol for flexible hypersonic flight vehiclerdquo InternationalJournal of Advanced Robotic Systems vol 14 no 1 2017

[32] J-T Huang ldquoGlobal tracking control of strict-feedback systemsusing neural networksrdquo IEEE Transactions on Neural Networksand Learning Systems vol 23 no 11 pp 1714ndash1725 2012

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Mathematical Problems in Engineering

[33] C P Bechlioulis and G A Rovithakis ldquoA priori guaranteedevolution within the neural network approximation set androbustness expansion via prescribed performance controlrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 23 no 4 pp 669ndash675 2012

[34] J Wu W Chen D Zhao and J Li ldquoGlobally stable directadaptive backstepping NN control for uncertain nonlinearstrict-feedback systemsrdquo Neurocomputing vol 122 pp 134ndash1472013

[35] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015

[36] J-T Huang ldquoGlobal adaptive neural dynamic surface control ofstrict-feedback systemsrdquoNeurocomputing vol 165 pp 403ndash4132015

[37] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems withpredefined output accuracyrdquo Information Sciences vol 301 pp27ndash43 2015

[38] W Chen J Lu X Wang andW Zhang ldquoDesign of a controllerfor morphing aircraft based on backsteppingRHOrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 40 no 8pp 1060ndash1065 2014

[39] Y Yang C Ge H Wang X Li and C Hua ldquoAdaptive neuralnetwork based prescribed performance control for teleoper-ation system under input saturationrdquo Journal of the FranklinInstitute Engineering and Applied Mathematics vol 352 no 5pp 1850ndash1866 2015

[40] B Ren S S Ge K P Tee andTH Lee ldquoAdaptive neural controlfor output feedback nonlinear systems using a barrier lyapunovfunctionrdquo IEEE Transactions on Neural Networks vol 21 no 8pp 1339ndash1345 2010

[41] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 pp 7706ndash7716 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of