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Nonlinear-Disturbance-Observer-Based Robust Flight Control for Airbreathing Hypersonic Vehicles

JUN YANG

SHIHUA LI

CHANGYIN SUN

Southeast University

China

LEI GUO

Beihang University

The work presented here is concerned with the robust flight control problem for the longitudinal dynamics of a generic airbreathing hypersonic vehicles(AHVs)under mismatched disturbances via a nonlinear-disturbance-observer-based control (NDOBC)https://www.wendangku.net/doc/2211232432.html,pared with other robust flight control method for AHV,the proposed method obtains not only promising robustness and disturbance rejection performance but also the property of nominal performance recovery.The merits of the proposed method are validated by simulation studies.

Manuscript received January30,2012;revised May22,2012; released for publication October8,2012.

IEEE Log No.T-AES/49/2/944534.

Refereeing of this contribution was handled by I.Hwang.

This work was supported in part by National Basic Research Program of China(973Program)under Grant2012CB720003, National Natural Science Foundation of China under Grants 91016004and61203011,Program for New Century Excellent Talents in University under Grant NCET-10-0328,Ph.D.Program Foundation of Ministry of Education of China under Grant 20120092120031,and Natural Science Foundation of Jiangsu Province under Grant BK2012327.

Authors’addresses:J.Yang,S.Li,and C.Sun,School of Automation,Southeast University,Si Pai Lou2,Nanjing210096, P.R.China,E-mail:(lsh@https://www.wendangku.net/doc/2211232432.html,);L.Guo,Institute of Instrument Science and Opto-Electronics Engineering,Beihang University,Beijing100083,P.R.China.

0018-9251/13/$26.00c°2013IEEE I.INTRODUCTION

Due to the growing interest in development

of new technologies in cost efficient space access

and speedy global reach,airbreathing hypersonic vehicle(AHV)technique has become more and more crucial[1—3].The so-called scramjet propulsion

is usually employed by the AHV to execute its advantages,e.g.,high speed,cost efficient travel

and large payload[3].However,such propulsion mechanism always brings peculiar characteristics

of the vehicle dynamics,such as strong nonlinear cross couplings between propulsive and aerodynamic forces results from the integration of the scramjet engine[2—4].In addition,the flight of the AHV is also significantly affected by uncertainties as a result of the variability of the vehicle characteristics with flight conditions,fuel consumption,and thermal effects on the structure[1—8].Another great challenge for the flight control design of AHV lies in that its longitudinal dynamics are always severely affected by various external disturbances due to the complex flight environment[8].

Facing the above-mentioned attractive features and control challenges,the design of guidance and control systems for the longitudinal dynamics of AHVs has attracted a great deal of attention by research institutes all over the world in the past few years.Based on the models obtained by approximate linearization around specified trim condition or input/output linearization technique,many advanced control methods,such

as adaptive control[2],robust control[3—5,7], sliding mode control[8],back-stepping control

[9,18],guaranteed cost control[10],gain-scheduling, and linear parameter-varying control[11,12],

have been employed for the control design of the longitudinal dynamics of the AHVs.These methods are conceptually intuitive and improve the robust tracking performance from different aspects.It is

also reported that thus designed controllers may exhibit undesired performances especially if the AHV dynamics are highly nonlinear and experience great parameter perturbations due to tough flight task and severe flight environment.

Recently,to deal with the possible problems encountered by the above control approaches,the artificial-intelligence-based method has become

an active research topic and is widely investigated

in literature,e.g.,neural-network-based control

[9,13],and fuzzy-logic-based control[14—17].It

is claimed that these methods obtain fine control performances as they take into account the full nonlinear dynamics and parameter uncertainties

of the AHVs.Generally speaking,these control designs employ an artificial-intelligence-based approach to approximate nonlinear dynamics or

the model uncertainties of AHVs first,and then additional advanced feedback controllers,such

as adaptive control[9,13],sliding mode control [14],robust control[15],and fault tolerant control [16,17]are integrated to achieve desired control specifications,such as robust command tracking, specified disturbance attenuation,and satisfactory fault handling.

Although these control methods have obtained extensive applications and proved to be efficient

for the AHVs,they mainly focus on the stability

or robust stability of the uncertain AHVs,and in general,the robustness and disturbance rejection performance of these controllers are achieved at the price of sacrificing the nominal control performance. In addition,most of the above-mentioned methods do not consider active and direct disturbance rejection

in the controller design.It is difficult for these controllers to react promptly in the presence of strong disturbances,although they can finally counteract the disturbances through a passive feedback regulation

in a relatively slow way.It is well known that disturbance-observer-based control(DOBC)provides an active approach to handle system disturbances

and improve robustness against uncertainties [19—39].

A novel nonlinear-disturbance-observer-based control(NDOBC)approach is proposed in this paper to enhance the disturbance rejection performance

of the longitudinal dynamics of a generic of AHVs developed at NASA Langley Research Center[7,8]. The AHVs under consideration are essentially

multi-input multi-output(MIMO)nonlinear systems subject to mismatched disturbances and uncertainties, which brings big challenges for the NDOBC design since most of the existing NDOBC methods are

only able to cope with matched disturbances except for[31],[37]—[39].In[31],[37],[38]mismatched disturbance attenuation approaches are proposed for linear systems.An NDOBC method is proposed for

a special class of single-input single-output nonlinear systems in[39].Of course,these method cannot be employed directly for the control design of the AHVs here.

In the proposed NDOBC framework,a finite-time disturbance observer is employed for fast disturbance estimation firstly.Secondly,a baseline nonlinear dynamic inversion control law is designed to stabilize the nominal nonlinear dynamics.Finally,by integrating the disturbance observer and the baseline controller via a disturbance compensation gain,a composite NDOBC method is constructed to reject the mismatched disturbances.It is proved that the proposed composite controller is able to eliminate the mismatched disturbances and uncertainties from the output channels of the AHVs by properly designing the disturbance compensation gain matrix.

In the context of the aforementioned literature,

the major contribution of this work is that an active antidisturbance controller with a novel disturbance compensation gain design method is developed

to counteract the mismatched disturbances and uncertainties in the AHVs.As compared with the existing flight controllers for the AHVs,the major merits of the proposed method lies in that the fine robustness against mismatched disturbances and model uncertainties is achieved without sacrificing

the nominal control performance.

II.LONGITUDINAL DYNAMICS OF A GENERIC AHVS

The longitudinal dynamic models for a generic AHV under consideration is taken from[8],

which were developed at NASA Langley Research Center.The equations of motion consist of an inverse-square-law gravitational model and the centripetal acceleration that results from a curved flight path.The longitudinal dynamics of the vehicle model can be described by a set of five-order differential equations in terms of forward velocity V, flight-path angle°,altitude h,angle of attack?,and pitch rate q,respectively,

_V=T cos??D

1sin°

+d1(1) _°=

L+T sin?

mV?

(1?V2r)cos°

+d2(2)

_h=V sin°+d

(3)

_?=q?_°+d

(4)

_q=

M yy

+d5(5)

where m,I yy,and1represent the vehicle mass, moment of inertia,and gravity constant,respectively. Variables d i(i=1,:::,5)denote the unknown external disturbances.L,D,T,M yy,and r represent lift,drag, thrust forces,pitching moment,and the radial distance from Earth’s center,respectively,depicted by

L=1

?V2SC L

D=1

?V2SC D

T=1?V2SC T

M yy=1

?V2Sˉc[C M(?)+C M(±e)+C M(q)]

r=h+R E

where C D,C L,and C T represent the drag,lift,and thrust coefficients,respectively.C M(q),C M(?),

and C M(±e)denote the moment coefficients due to pitch rate,angle of attack,and elevator deflection, respectively.The parameters?,S,ˉc,and R E

represent the air density,the reference area,the mean aerodynamic chord and the radius of the Earth, respectively.

The engine dynamics are modeled by a

second-order system

¨ˉ=?2&!

n _ˉ?!2

ˉ+!2nˉc+d6:(6)

whereˉis the throttle setting,ˉc is the throttle setting command,and d6is the external disturbance to the engine dynamics.

The physical and aerodynamic coefficients under this study are simplified around the nominal cruising flight[8],which is M=15(the Mach number),V= 15,060ft/s,h=110,000ft,°=0rad,and q=0rad/s, respectively.The parametric uncertainty is modeled as an additive perturbation￠to its nominal values,

C L=0:6203?

C D=0:6450?2+0:0043378?+0:003772

C T=?0:02576ˉ,ifˉ<1

0:0224+0:00336ˉ,ifˉ>1

C M(?)=?0:035?2+0:036617(1+￠C M?)?

+5:3261￡10?6

C M(q)=ˉc

q(?6:796?2+0:3015??0:2289)

C M(±e)=c e(±e??)

m=m0(1+￠m)

I yy=I0(1+￠I)

S=S0(1+￠S)

ˉc=ˉc

(1+￠ˉc)

?=?0(1+￠?)

c e=0:0292(1+￠c e)

where the nominal values of the parameters m,1,I yy,?,S,ˉc,and R E are listed as follows.

m0=9,375slugs

1=1:39￡1016ft3=s2

I0=7￡106slugs￠ft2

?0=0:24325￡10?4slugs/ft3

S0=3,603ft2

ˉc

=80ft

R E=20,903,500ft:

The maximum values of the additive parameter perturbations under consideration are taken as

j￠m j·0:25

j￠I j·0:25

j￠S j·0:25

j￠ˉc j·0:25

j￠?j·0:25

j￠c e j·0:25

j￠C M?j·0:25:

(7)

R EMARK1To exhibit the superiority of the proposed method,severe parameter uncertainties are considered in later simulation studies.The maximum values

of the additive perturbation terms￠are up to25%

as shown in(7),which are much larger than those considered in Xu,et al.[8].

The control inputs of the AHV are the throttle setting commandˉc and the elevator deflection±e, while the controlled outputs are the velocity V and the altitude h.

III.PRELIMINARIES AND PROBLEM FORMULATION A.Preliminaries

Since one major contribution of this paper is concerned with mismatched disturbance compensation for the AHVs,the concept of matched and mismatched disturbances for a MIMO nonlinear system are presented in this subsection for the convenience of the readers.Consider a MIMO nonlinear system depicted by

_x=f(x)+g(x)u+p(x)w

y=h(x)

(8)

where x=[x1,:::,x n]T2R n,u=[u1,:::,u m]T2R m,

w=[w1,:::,w n]T2R n,and y=[y1,:::,y m]T2R m denote the state,input,disturbance and output vectors, respectively.f(x),g(x)=[g1(x),:::,g m(x)],p(x)=

[p1(x),:::,p n(x)],and h(x)=[h1(x),:::,h m(x)]T are smooth vector or matrix fields on R n.Without loss

of generality,it is supposed that the equilibrium of system(8)in the absence of disturbances is x0=0. The standard Lie derivative notation is used in this paper for the sake of simplified notation,stated as follows[41].

D EFINITION1[41]The vector relative degree from the control inputs to the outputs of system(8)is

(?1,:::,?m)at the equilibrium x0if L g

L k

h i(x)=0 (1·j·m,1·i·m)for all k

x in a neighborhood of x0,and the m￡m matrix

A(x)=2

666

664

L g

L?1?1

h1L g

L?1?1

h1￠￠￠L g m L?1?1f h1

L g

L?2?1

h2L g

L?2?1

h2￠￠￠L g m L?2?1f h2

L g

L?m?1

h m L g

L?m?1

h m￠￠￠L g m L?m?1f h m

777

775

(9)

nonsingular at x=x0.For simplicity,it is referred

to as the input relative degree(IRD).Similarly,the disturbance relative degree(DRD)at x0is defined as (o1,:::,om).

D EFINITION2The disturbances in(8)is called mismatched if some DRD of nonlinear system(8)is strictly lower than some IRD,i.e.,oi

i,j2f1,:::,m g.

The cases of mismatched and matched disturbances can be intuitively illustrated by the following typical examples[40],

=arctan x2+w

=u(10a)

y=x1

=arctan x2

=u+w(10b)

y=x1:

The disturbance in system(10a)is clearly a mismatched one as it appears in a different channel from that of the control input,or equivalently,the DRDo=1is lower than its IRD?=2.The existing DOBC law u=K(x)??w(where?w is the disturbance estimation)is able to solve the disturbance attenuation problem of system(10b)where the disturbance

is a matched one,but clearly cannot effectively compensate the disturbance in system(10a).

B.Problem Formulation

The AHV dynamic models(1)—(6)can be rewritten as the following compact form

_x=[f(x)+±f]+[g(x)+±g]u+pd

y=h(x)

(11)

where the states x=[x1,:::,x7]T=[V,°,h,?,q,ˉ,_ˉ]T, the controls u=[u1,u2]T=[±e,ˉc]T,and the

outputs y=[y1,y2]T=[V,h]T,respectively.f(x)=

[f1(x),:::,f7(x)]T,g(x)=[g1(x),:::g7(x)]T,and h(x)= [h1(x),h2(x)]T are vector or matrix fields in terms of state x,and p=[p1,:::,p6].±f=[±f1,:::,±f7]T and

±g=[±g1,:::,±g7]T denote uncertainties caused by physical and aerodynamic parameter perturbations. Considering±f6=0,±g6=0here for the AHV system and lumping the external disturbances and uncertainties together,the AHV dynamics can be expressed as

_x=f(x)+g(x)u+pw

y=h(x)

(12)

where w=[w1,:::,w6]T=p T[±f+±gu]+d denotes the lumped disturbances.According to the definition of relative degrees for nonlinear systems[41],the IRDs and DRDs of the AHV system are calculated

as(?1,?2)=(3,4),and(o1,o2)=(1,1),respectively.In the absence of disturbances and uncertainties,a new group of coordinate transformation which linearizes

the AHV system(12)is given by z=?(x)=[z T

,z T

]T, where

z i=

666

z i

z i?

777

75=

666

664

h i(x)

L f h i(x)

L?i?1

h i(x)

777

775

for i=1,2.The description of system(12)under

the new coordinates is expressed as two subsystems, where the velocity and altitude subsystems are depicted by

S1:

8>>

>><

>>>>:

_z1

=z1

+L p h1w

_z1

=z1

+L p L f h1w

_z1

=L3

h1+L g L2f h1u+L p L2f h1w

y1=h1=z1

S2:

8>>

>>>>>

>>:

_z2

=z2

+L p h2w

_z2

=z2

+L p L f h2w

_z2

=z2

+L p L2f h2w

_z2

=L4f h2+L g L3

h2u+L p L3

h2w

y2=h2=z2

(13)

R EMARK2It can be observed from the definition

of relative degrees of the AHV systems and also (13)that the disturbances under consideration are mismatched ones.The existing NDOBC method

is unavailable for MIMO nonlinear systems with mismatched disturbances[39],which motivates further research on NDOBC for the AHV system.

IV.NONLINEAR-DISTURBANCE-OBSERVER-BASED ROBUST FLIGHT CONTROL

A.Nonlinear Disturbance Observer Design

Assumption1The disturbances w in the AHV system(12)are twice differentiable,and¨w(t)has a group of Lipshitz constants L i>0(i=1,:::,6).

Note that the sixth state equation of the AHV system(12)is not affected by the disturbances;

we only consider the state equations that contain disturbances when designing the disturbance observer.

To this end letˉx=p T x=[x1,:::,x5,x7]T,which gives

_ˉx=ˉf(x)+ˉg(x)u+w(14) whereˉf(x)=[f1(x),:::,f5(x),f7(x)]T,ˉg(x)=

[g1(x),:::,g5(x),g7(x)]T.

Inspired from[21]—[23],a nonlinear disturbance observer(NDO)with finite-time convergence property,which provides an adequate way to estimate the disturbances for the AHV system(12),is constructed first

=v0+ˉf(x)+ˉg(x)u

=v1

=v2

?ˉx=z

?w=z

?_w=z

(15)

where z0=[z01,:::,z06]T,z1=[z11,:::,z16]T,z2=

[z21,:::,z26]T,v0=[v01,:::,v06]T,v1=[v11,:::,v16]T,

v2=[v21,:::,v26]T,?ˉx is the estimate ofˉx,?w is the estimate of w,?_w is the estimate of_w,and

v0i=??0L1=3

i j z0i?ˉx i j2=3sign(z0i?ˉx i)+z1i

v1i=??1L1=2

i j z1i?v0i j1=2sign(z1i?v0i)+z2i

v2i=??2L i sign(z2i?v1i)

for i=1,:::,6,and?0,?1,?2>0are observer coefficients to be designed.

Combining(14)and(15),the i th subobserver estimation error is given as

_e 0i =??0L1=3

i j e0i j2=3sign(e0i)+e1i

_e 1i =??1L1=2

i j e1i?_e0i j1=2sign(e1i?_e0i)+e2i

_e 2i =??2L i sign(e2i?_e1i)?¨w i

(16)

where the estimation errors are defined as e0i=

z0i?ˉx i,e1i=z1i?w i,and e2i=z2i?_w i,respectively. Considering the condition given in Assumption1,it follows from[21]—[23]that system(16)is finite-time stable,that is,there is a time constant t f such that

e0i(t)=e1i(t)=e2i(t)=0(or equivalently?ˉx i(t)=ˉx i(t),

?w i (t)=w i(t),?_w i(t)=_w i(t))for t?t f.The readers can

refer to[21]for detailed guidance of parameter tuning of?i(i=0,1,2).The parameter L i(i=0,1,2)is usually tuned to regulate the convergence rate of the observer estimation error system(16).

Let e=[e T

0,e T

,e T

]T,the full observer error

dynamics is governed by

_e=?(e,¨w,?

,?1,?2)(17) which is finite-time https://www.wendangku.net/doc/2211232432.html,posite Control Law Design

A novel NDO-based robust flight control for the AHV system is presented by Theorem1.

T HEOREM1Consider the AHV system(12)with mismatched disturbances satisfying Assumption1,and also suppose that the observer coefficients?0,?1,?2of the NDO(15)are selected such that the observer error system(16)is finite-time stable.A novel NDO-based robust flight control law which compensates the mismatched disturbances from the output channels of the AHV system(12)is given by

u=A?1(x)[?B(x)+V(x,y r)+?(x)?w](18) with

A(x)=

"L g L2f h1

L g L3

V(x,y r)=·v1(x,y r)

v2(x,y r)

"?c10(h1?y r1)?c11L f h1?c12L2f h1

?c20(h2?y r2)?c21L f h2?c22L2f h2?c23L3f h2

# B(x)=

"L3f h1

?(x)=·°1(x)°2(x)

"?c11L p h1?c12L p L f h1?L p L2f h1

?c21L p h2?c22L p L f h2?c23L p L2f h2?L p L3f h2

where y r=[y r1,y r2]T,y r1and y r2denote the reference signals of the velocity and altitude,respectively.

Parameters c i

(i=1,2;k=0,1,:::,?i?1)have to be designed such that the polynomials

p i0(s)=c i0+c i1s+￠￠￠+c i?

i?1

s?i?1+s?i(19) are Hurwitz stable.?w is the disturbance estimation by the NDO(15).

P ROOF Collecting the last states of the velocity subsystem S1and altitude subsystem S2in(13) together,gives

·_z13

_z2

?=B(x)+A(x)u+"L p L2f h1w

L p L3

h2w

:(20)

Substituting the proposed control law(18)into (20),yields

·_z13

_z2

?="v1(x,y r)+°1(x)?w+L p L2f h1w

v2(x,y r)+°2(x)?w+L p L3

h2w

:(21)

Combining(21)with(13),the closed-loop system is obtained

_z=A z+?(x)e

+D(x)w+R(y r)

y=Cz

(22)

where e1=?w?w,and A=

?(x)=666 666 666 664

°1(x)

°2(x)

777

775

D(x)=2

666

L p h1

L p L f h1

°1(x)+L p L2

L p h2

L p L f h2

L p L2

°2(x)+L p L3

777

R(y r)=2 666 666 666 664

y r1

y r2

777

775

C=:

It can be verified by simple calculation that

C A?1D(x)=0,C A?1R(y r)=?y r:(23) It follows from(22)and(23)that

y=C A?1[_z??(x)e1?D(x)w?R(y

)]

=C A?1[_z??(x)e1]+y r:(24) It can be concluded from(24)that the disturbances are decoupled from the output channels in an asymptotical way.Suppose that the closed-loop system reaches a bounded steady state and let x s,

e1s,and y s denote the steady-state values of x,e1,

and y,respectively.The steady-state value y s can be represented as

y s=?C A?1?(x s)e1s+y r:(25)

Fig.1.Block diagram of proposed NDO-based robust flight

control structure for AHV system. Considering the fact that e1s=0(according to the result given in Section IV-A),it follows from(25)that y s=y r.

R EMARK3In the absence of disturbances,the disturbance estimation by(15)satisfies?w(t)′0if the initial state of the NDO(15)is selected as z0(0)=

ˉx(0),and z

(0)=z2(0)=0.In this case the control performance under the proposed control law(18) recovers to that under the baseline feedback control law,which implies that the property of nominal performance recovery is obtained by the proposed NDOBC.

The proposed NDO-based robust flight control structure for the AHV system is shown in Fig.1.

C.Stability Analysis

The stability of the resultant closed-loop system is established by Theorem2.

T HEOREM2The AHV system(12)under the proposed NDO-based robust flight control law(18)is locally input-to-state stable(ISS)if the following conditions are satisfied:a)Assumption1is satisfied for the mismatched disturbances;b)the control parameters

c i

in the control law(18)are selected such that the

polynomials p i

(s)in(19)are Hurwitz stable;c)the observer coefficients?0,?1,?2are selected such that the system(17)is finite-time stable;d)the disturbance compensation gain is selected such that?(x)and D(x) are continuously differentiable around the operating point.

P ROOF Combining(17),(18)with(22),the augmented closed-loop AHV system is described by

_z=A z+?(x)e

+D(x)w+R(y r)

_e=?(e,¨w,?

,?1,?2):

(26)

The asymptotical stability of the following system

_z=A z+?(x)e

_e=?(e,¨w,?

,?1,?2)

(27)

is proved first.The asymptotical stability of system _z=A z has been guaranteed by condition b). Considering this result as well as condition d)and taking e1as an input of system_z=A z+?(x)e1,it can be concluded from[42,Lemma5.4]that_z=

A z+?(x)e1is locally https://www.wendangku.net/doc/2211232432.html,bining this result with condition c),it follows from[42,Lemma5.6]that system(27)is asymptotically stable.

Let X=[z T,e T]T,the closed-loop system(26)is rewritten as

_X=F(X)+G(X)w+R(y

r )(28)

where

F(X)=·A z+?(x)e1

?(e,¨w,?0,?1,?2)

G(X)=·D(x)0

R(y r)=·R(y r)0

Since system_X=F(X)is asymptotically stable

and G(X)is continuously differentiable(which

can be derived from condition d)),it follows from [42,Lemma5.4]that the augmented closed-loop system(28)is locally ISS.

V.SIMULATION STUDIES

To evaluate the effectiveness of the proposed NDOBC method,the simulation studies are carried out in the presence of unknown external disturbances and parameter uncertainties,respectively.The tracking commands of the velocity and altitude

are15,180ft/s(a step change of120ft/s from its nominal flight value15,060ft/s)and112,000ft

(a step change of2,000ft from its nominal flight value110,000ft),respectively.To show the efficiency of the proposed method in improving

the robustness of the baseline controller,the baseline controller without compensation is also employed in the simulation studies for the purpose

of comparison.

The control parameters in these simulation studies are selected as

c10=0:16

c11=0:92

c12=1:7

c20=0:096

c21=0:712

c22=1:94

c23=2:3:The parameters of the NDO(15)is selected as

?0=3

?1=1:5

?2=1:1

L1=10

L2=0:01

L3=100

L4=0:5

L5=1

L6=5:

A.External Disturbance Rejection

In this simulation scenario the unknown

external disturbances are considered and taken as

!1=?5,!3=30,!6=0:5at t=50s,and!2=

0:001sin(0:2?t),!4=0:05,!5=0:2+0:1sin(0:3?t+?) at t=100s for the AHV system.Response curves

of the outputs,inputs,and states of the AHV system under both the proposed NDOBC and the baseline controller are shown by Fig.2.

It is observed from Fig.2that the baseline controller results in undesirable control performance, while the resultant NDOBC obtains fine external disturbance rejection properties.A brief observation

of Fig.2shows that response curves of the variables under the proposed NDOBC method are overlapped with those under the baseline control method during the first50s when there are no disturbances imposed on the system,which is a convincing evidence that the nominal performance under the proposed method is retained.

B.Robustness Against Parameter Uncertainties

To make the work more challenging,two cases

of severe parameter uncertainties are considered when doing robustness tests,actually the uncertain parameters(7)are taken as their maximum values.

Case I All the uncertain parameters(7)are taken as their positive maximum values,i.e.,￠=0:25.

The response curves of all the variables of the AHV system under the model uncertainties in case I are shown in Fig.3.

As shown by Fig.3,the baseline controller method leads to a poor tracking performance,while the outputs under the NDOBC are able to track their commands timely and without any steady state errors.

Case II All the uncertain parameters(7)are taken as their negative maximum values,i.e.,￠=?0:25.

The response curves of all the variables of the AHV system under the model uncertainties in case II are shown in Fig.4.

It is observed from Fig.4that the baseline controller results in a very poor tracking performance.

Fig.2.Variable response curves in presence of unknown external disturbances under proposed NDOBC(solid line)and baseline control(dashed line)methods.Tracking commands denoted by dash-dotted line.

All the variables under the baseline controller tend

to be unstable,which implies that the parameter uncertainties in this case may be very large.However, as shown in Fig.4,the proposed NDOBC method achieves excellent robust tracking performance in such case of severe model uncertainties.VI.CONCLUSION

A novel robust flight control design method has been proposed for the longitudinal dynamic models of a generic AHV system via a nonlinear disturbance observer.It has been proved that the mismatched uncertainties in the AHV system can be eliminated

Fig.3.Variable response curves in presence of parameter uncertainties(case I)under proposed NDOBC(solid line)and baseline control(dashed line)methods.Tracking commands denoted by dash-dotted line.

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Jun Yang was born in Anlu,Hubei Province,China in 1984.He received his B.S.degree in the Department of Automatic Control from Northeastern University,Shenyang,China in 2006.In 2011,he received the Ph.D.degree in control theory and control engineering from the School of Automation,Southeast University,Nanjing,China.

He is currently a lecturer in the School of Automation,Southeast University.His research interests include disturbance estimation and compensation,advanced control theory and its application to flight control systems,motion control systems and process control

systems.

Shihua Li was born in Pingxiang,Jiangxi Province,China in 1975.He received his B.S.,M.S.,and Ph.D.degrees all in automatic control from Southeast University,Nanjing,China in 1995,1998,and 2001,respectively.

Since 2001he has been with the School of Automation,Southeast University,where he is currently a professor.His main research interests include nonlinear control theory with applications to robots,spacecraft,AC motors and other mechanical

systems.

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He is a professor in the School of Automation at Southeast University,China. His research interests include intelligent control,neural networks,SVM,pattern recognition,optimal theory,etc.

Dr.Sun has received the First Prize of Nature Science of Ministry of Education,China,has published more than70papers,and is an awardee of the national science fund for distinguished young scholars of China.He was an Associate Editor of IEEE Transactions on Neural Networks,Neural Processing Letters,International Journal of Swarm Intelligence Research,and Recent Patents on Computer Science,et al.

Lei Guo was born in Qufu,China,in1966.He received his B.S.and M.S. degrees from Qufu Normal University(QFNU),China in1988and1991, respectively and the Ph.D.degree in control engineering from Southeast University(SEU)in1997.

From1991to1994,he was with Qingdao University(QDU)as a lecturer.

From1997to1999,he was a post-doctoral fellow at Southeast University.From

2000to2003,he was a research fellow at Glasgow University,Loughborough University and UMIST,UK.In2004,he joined the Institute of Automation in Southeast University as a professor.In2006he became a professor at the School of Instrumentation and Opto-Electronics Engineering,and now at the School of Automation and Electronic Engineering,Beihang University,Beijing,China.His research interests include robust control,stochastic systems,fault detection,filter design and nonlinear control with their applications to aerospace systems.

Dr.Guo is an Awardee of the National Science Fund for Distinguished Young Scholars of China,and a Changjiang Distinguished Professor of the Ministry of Education of China.He has published more than120papers and one monograph

and served as an editor for5journals.