Nonlinear fuzzy robust adaptive control of a longitudinal hypersonic aircraft model

Yan-bin LIU College of Astronautics

Nanjing University of Aeronautics and Astronautics Nanjing, China

E-mail: nuaa_liuyanbin@163.com

Yu-ping LU College of Astronautics

Nanjing University of Aeronautics and Astronautics Nanjing, China

E-mail: yplac@nuaa.edu.cn

Abstract—A Multi Input Multi Output (MIMO) robust adaptive fuzzy control is presented for the longitudinal dynamics of a hypersonic aircraft. Because of various sources such as modeling errors, in-flight failure and external disturbances, the vehicle dynamics are partially or completely unknown. Furthermore, the vehicle state variables are unavailable for measure in the complicated flight conditions. As a result, the traditional control methods cannot be used for a hypersonic aircraft. In this article, the unknown dynamics are approximated with fuzzy logic systems, and the state observers are constructed for estimating the state variables. In addition, the “dominant input” concept is applied, and ∞H control is designed to improve the system performance by attenuating the effects of both the external disturbances and the approximation errors to expected level. Simulation studies are conducted for the trimmed cruise conditions at an altitude of 110,000 ft and at a velocity of 15 Mach to evaluate the response of the hypersonic aircraft to a step function with magnitude of 80 ft/s in airspeed and a sinusoidal function 100sin(0.3t) ft in altitude. Simulation results demonstrate that the robust adaptive fuzzy control can guarantee the flight stability and also maintain the tracking performance.

Keywords-hypersonic vehicle; Nonlinear control; flight control; robust adaptive control；fuzzy logic.

I. INTRODUCTION The dynamic characteristics of the hypersonic aircrafts

will vary considerably over the flight envelop than other aircrafts due to their extremely wide range of operating conditions and rapid change of mass distributions. Moreover many aerodynamic and propulsion characteristics still remain uncertain and are hard to predict due to the effect of the structural dynamics, propulsion heat, aerodynamics and coupling between them. As a result, the controlled system in the guidance analysis is multivariable, unstable, non-minimum phase and possesses significant input-output cross-coupling. Therefore, the control laws must deliver an extremely robust guidance and control system. In addition, a significant amount of guidance and control system integration will be necessary in order to achieve the requisite system performance and stability robustness [1,2,3].

In order to satisfy the above demands, a number of studies on the control of the hypersonic aircrafts have been done. In ref.4, the robust nonlinear controller of a hypersonic aircraft is proposed when the model parameter variations and uncertainties are small. In ref.5 and 13, the adaptive sliding mode controller of a hypersonic flight vehicle is presented in the presence of the small uncertain model parameter variations and the unknown state variables. In ref.6 and 7, the works have considered the model dynamics to be unknown but the state variables to be available for measurement. Nevertheless, relatively few sources address that the vehicle dynamics is completely unknown and the state variables cannot be available for measurement. Therefore, this paper will design the controller in the presence of the completely unknown vehicle dynamics and the unavailable state variables for measurement.�

Fuzzy control methods have emerged in recent years as prosing ways to handle nonlinear control problems. Based on the universal approximation theorem, several stable adaptive fuzzy control schemes are developed for multi-input-multi-output (MIMO) nonlinear systems [10,11]. However, these adaptive control techniques are only limited to the MIMO nonlinear systems whose states are assumed to be available for measurement. The adaptive fuzzy logic system together with ∞H control theory is proposed for controlling nonlinear MIMO systems with unknown dynamics in ref.8 and 9. In this case, the fuzzy logic system is used to approximate the unknown vector functions and ∞H control is designed to improve the system performance by suppressing influence of the external disturbances and removing the fuzzy approximation errors. Furthermore, the state observer is constructed in order to predict the actual system states. The control design is based on the fuzzy approximation model and the state observer outputs rather than the actual model and the system outputs, so the robustness may greatly improve.

In this paper, the robust adaptive fuzzy controllers are presented for the longitudinal motion of a hypersonic aircraft. The vehicle dynamics are nonlinear partially or completely unknown and the state variables are unavailable for measurement. In order to guarantee closed loop system stability and convergence of the tracking error, the robust adaptive fuzzy control methods have been introduced. Simulation studies are conducted for the trimmed cruise conditions at an altitude of 110,000 ft and at a velocity of 15

2009 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-3816-7/09 $26.00 © 2009 IEEE

DOI 10.1109/AICI.2009.13

31

Mach to evaluate the response of the hypersonic aircraft to a step function with magnitude of 80ft/s in airspeed and a sinusoidal function 100sin(0.3t) ft in altitude. The simulation results demonstrate the adaptive fuzzy robust control can improve robustness and provide good performances.

II. MATHEMATICAL MODEL OF A HYPERSONIC AIRCRAFT A model for the longitudinal dynamics of a hypersonic

aircraft was presented in Refs.5 and 6. The equations of motion developed include an inverse-square-law gravitational model, and the centripetal acceleration. The nominal flight at trimmed cruise condition

)0deg,0,000,110,/060,15( ==== qfthsftV γ is considered. The equations of motion describing the model [4,5,6] are:

2sincos

rmDTV γμα −−= (1)

2cos)2(sin

Vr

rVmVTL γμαγ −−+= (2)

γsinVh = (3) γα −= q (4)

yyIyyMq /= (5)

Where [4,5,6]

LsCVL 25.0 ρ= (6)

DsCVD 25.0 ρ= (7)

TsCVT 25.0 ρ= (8)

[ ])()()(25.0 qMCeMCMCcsVyyM ++= δαρ (9)

Rhr += (10)

α6203.0=LC (11)

003772.00043378.026450.0 ++= ααDC (12)

⎩⎨⎧

+=

ββ

00336.00224.002576.0

TCifif

11

><

ββ

(13)

6103261.5036617.02035.0)( −×++−= αααMC (14)

)2289.03015.02796.6()2/()( −+−= ααqVcqMC (15)

)()( αδδ −= eecEMC (16)

The engine dynamics are modeled by a second order system[4,5,6]:

cnnn βωβωβξωβ 222 +−−= (17)

The control inputs are the throttle setting, cβ , and the elevator deflection, eδ ; the outputs are the velocity,V , and the altitude, h .

III. INPUT-OUTPUT LINEARIZATION The nominal flight at trimmed cruise condition

( mhsmV 33528,/3.4590 00 == ) is considered, and the equations of motion describing the model are: According to the methods in Refs.5 and 6, when the outputs V and h are differentiated continuously by 3 and 4 times respectively, a control input component appears in the resulting equation. Furthermore, the relative degree of the system equals to the order of the system. Therefore, the nonlinear longitudinal model can be linearized completely and the closed loop system will have no zero dynamics [5,6]. We have:

⎪⎪⎩

⎪⎪⎨

⎧

Ω+=

=

=

mxTxxV

mxVxVfV

/)21(

/1

)(

ω

ω (18)

and

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−−+−+=

+−+=+=

=

γγγγγγγγγγγγγγ

γγγγγγγγγγ

coscos3sin3cos3sin23cos3sin)4(

cossin2cos2sincossin

)(

VVVVVVVh

VVVVhVVh

xhfh

(19)

where

⎪⎪⎩

⎪⎪⎨

⎧

Π+=

=

=

xTxx

x

xf

21

1

)(

πγ

πγγγ

(20)

The symbols )( xfV , )( xf γ and )( xf h are the short hand expressions of the right-hand-side of (1), (2), and (3) respectively, and ,/,/)(,/ 12112 xxxfx ∂∂=Π∂∂=∂∂=Ω ππω γ

,/)(1 xxf V ∂∂=ω [ ]ThVx βαγ= . By separating the second differentiation of α and

β into control-irrelevant and control relevant parts [6], we have:

⎪⎩

⎪⎨

⎧

+=

+=

cn

eyyIcsvec

βωββ

δραα2

0

)2/2(0 (21)

where

32

[ ] γααρα −−+= yyIecqMCMCcsV /)()(25.00 (22)

βωβξωβ 220 nn −−= (23)

Defining [ ]ThVx 000 βαγ= , the output dynamics V and h can be written as:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

ec

bbbb

hFVF

hV

δβ

22211211)4( (24)

where

mxTxxFV /)201( Ω+= ω (25)

γγγγγγγγγγγ cos3sin3cos3sin23cos3 VVVVVFh −−+−=

)201(cos/sin)201( xTxxVmxTxx Π++Ω++ πγγω (26)

αωβρ cos)2/22(11 mnscVb = (27)

)cossin)(2/2(12 ααααρ TDTyymIcsVecb −+−= (28)

)sin()2/22(21 γαωβρ += mnscVb (29)

γαγαρ cos)cos()[2/2(22 LTyymIcsVecb ++=

]sin)sin( γαγαα DT −++ (30)

ββαααααα ∂∂=∂∂=∂∂=∂∂= /,/,/,/ TccTTLLDD (31)

IV. NONLINEAR ADAPTIVE ROBUST FUZZY CONTROL The control objective is to design a controller so that the

output V and h can track the reference input dV and dh . Generally speaking, inputs cβ and eδ all affect output V and h in (24). But according to the ideas of “dominant input”

[8], we can select inputs cβ and eδ work the dominant actions on output V and h respectively, the other inputs are taken as the external disturbances, then we have:

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

c

e

h

V

h

Vc

h

V

bb

dd

dd

ebb

FF

hV

βδδβ

21

12

22

11)4(

(32)

where hV dd , are regarded as the external disturbances.

Define Th

TV hhhhxVVVx ],,,[,],,[ == and

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000100010

VA ,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

001

VB ,

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100

TVC (33)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0000100001000010

hA ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0001

VB ,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000

TVC (34)

Then (32) is equivalent to the following system

⎩⎨⎧

=+++=

VV

VcVVVVVVV

xCVdxbxFBxAx ])()([ 11 β

(35)

⎩⎨⎧

=+++=

hh

hehhhhhhh

xChdxbxFBxAx ])()([ 22 δ

(36)

Where the nonlinear functions )(),(),(),( 2211 hVhhVV xbxbxFxF are partially or completely unknown, so we apply following fuzzy logic systems )(ˆ),(ˆ),(ˆ),(ˆ

22221111 bhbVFhhhFVVV xbxbxFxF θθθθ to approximate them respectively.

The fuzzy logic systems can be constructed by the )1( >MM fuzzy rules as follows:

lR : IF V is lVN and h is l

hN then VF is lFVN and hF

is lFhN and 11b is l

bN 11 and 22b is lbN 12 , Ml ,,2,1=

After being defuzzified by center average defuzzifier, we have

⎩⎨⎧

====

,)(,)(,)(,)(

222212111111 ζθθζθθζθθζθθ

Tbbh

TbbV

TFhFhhh

TFVFVVV

xbxbxFxF

(37)

Where TMFhFhFhFh

TMFVFVFVFV ],,[,],,[ 2121 θθθθθθθθ ==

and TMbbbb

TMbbbb ],,,[,],,,[ 12

212

1121211

211

11111 θθθθθθθθ == with

each variable lb

lb

lFh

lFV 2211,,, θθθθ as the point at which the fuzzy

membership of lb

lb

lFh

lFV NNNN 2211,,, achieve the maximum value

respectively [10]. As a practical matter, 2211 ,,, bbFhFV θθθθ are the adjustable parameter vectors on line, and

TM],,[ 21 ζζζζ = with each variable lζ as the fuzzy basis

function defined as

∑=

= M

l

lNh

lNV

lNh

lNV

l

hV

hV

1)()(

)()(

μμ

μμζ Ml ,,2,1= (38)

Where lNh

lNV μμ , are the given membership function of

lh

lV NN , respectively, so T

M ],,[ 21 ζζζζ = is completely known.

To (35), TV VVVx ],,[= cannot be available for

33

measurement due to various sources, so we design the velocity state observer for (35) as

⎪⎩

⎪⎨⎧

=−+−−++=

VV

VVDVsVaVcbVFVVVVVVV

xCVxCVKuuxbxFBxAx

ˆˆ)ˆ())ˆ(ˆ)ˆ(ˆ(ˆˆ 1111 βθθ (39)

where TV VVVx ]ˆ,ˆ,ˆ[ˆ = , T

DVDVDVDV KKKK ],,[ 321= is observer gain vector to make sure that the characteristic polynominal of CKA DVV − is strict Hurwitz, and aVu is a

∞H robust control to attenuate the disturbance effect on system outputs, and sVu is the feedback control for Vx and need to be designed later.

Design the observation error as VVV xxe ˆ−= and VVV ˆ~ −= , then subtracting (39) from (35), we obtain

⎪⎩

⎪⎨

⎧

=+++−

+−+−=

VV

VsVaVcbVV

FVVVVVVVVDVVV

eCVduuxbxb

xFxFBeCKAe

~]))ˆ(ˆ)((

))ˆ(ˆ)([()(

111111 βθθ

(40)

Design optimal parameter vectors ∗FVθ and ∗

11bθ

⎪⎪⎩

⎪⎪⎨

⎧

⎭⎬⎫

⎩⎨⎧

−=

⎭⎬⎫

⎩⎨⎧

−=

∈∈Ω∈

∗

∈∈Ω∈

∗

)ˆ(ˆ)(supminarg

)ˆ(ˆ)(supminarg

111111ˆ,

11

ˆ,

ˆ1111

ˆ

bVVUxUx

b

FVVVVVUxUx

FV

xbxb

xFxF

VxVxbb

VxVxFVFV

θθ

θθ

θ

θ

θ

θ (41)

and fuzzy approximation errorϖ

cbVVFVVVVV xbxbxFxF βθθϖ ])ˆ(ˆ)([)ˆ(ˆ)( 111111∗∗ −+−= (42)

Then (40) can be formulated as

⎩⎨⎧

=+++++−=

VV

VsVaVcT

bTFVVVVDVVV

eCVuuBeCKAe

~]~~[)( 11 ϖζβθζθ (43)

where VV d+=ϖϖ , FVFVFV θθθ −= ∗~ and 111111

~bbb θθθ −= ∗ .

Let velocity command vector TcccVc VVVy ),,(= and

VVcV xye ˆˆ −= , the controller will be chosen as

)ˆ)ˆ(ˆ()ˆ(ˆ

1

1111sVaVV

TCVVcFVVV

bVc uueKVxF

xb++++−= θ

θβ (44)

where TCVCVCVCV KKKK ],,[ 321= .

Substituting (44) into (39), we have

VVDVVTCVVVV eCKeKBAe −−= ˆ)(ˆ (45)

Control Objectives: For the (35), we design the robust fuzzy controller and an adaptive law for adjusting the parameter vectors such that the output V can track the reference input dV and the following conditions are met:

1). All the signals involved are uniformly bounded. 2). For the prescribed attenuation level 0>Vρ , the

following ∞H tracking performance is achieved

++≤∫ ))0(~)0(~(1)0()0(1

0 FVT

FVV

VVTVV

T

VTV EPEdtEQE θθ

γ

∫+T

VVbT

bV

dt0

21111

2

2))0(~)0(~(1 ϖρθθγ

(46)

where ],,ˆ[ VVV eeE = 0,0 ≥=≥= VT

VVT

V PPQQ are

weighting matrices , 0,0 21 >> VV γγ are adaptation gains. Theorem: For the (35), the robust adaptive fuzzy

controller is designed as (47)-(59) and the parameter adaptation laws in (50), (51), the proposed fuzzy control scheme can guarantee the ∞H tracking performance (46) is achieved for a prescribed attenuation level Vρ .

)ˆˆ()ˆ(

ˆ2

11

11sVaVV

TCVcV

Vc uueKVF

bb ++++−

+=

δβ (47)

VVTV

VaV ePB

ru 2

1−= (48)

VVDVsV ePKu ˆ1= (49) with

ζγθ VVTVVFV BPe 21= (50)

cVVTVVb BPe ζβγθ 2211 = (51)

where 0>Vδ , a small design parameter may free control laws of becoming singular, 0>Vr is weight factor

The block diagram of the overall closed loop system is shown in Fig.1.

Robust Adaptive

Fuzzy Controller

Hypersonic

Aircraft Model

V

h

cβ

eδ

Observer

cV

chRobust

compensator

On-line Parameter

Adaptive Laws

he

sVu

aVu

shu

ahu

Ve

he

Fuzzy Approximation

Model

Ve

Vh

11ˆb

FVF

hF

22ˆbF

FVθ

11bθ

22bθFhθ

cβ eδ

cV ch

h V

he

Ve

Ve

he

V h

Fig.1. The block diagram of the overall control closed loop system

V. SIMULATION AND ANALYSIS To illustrate the effectiveness of the adaptive fuzzy

robust control for a hypersonic aircraft longitudinal model, the simulation is conducted at trimmed condition of )0deg,0,000,110,/060,15( ==== qfthsftV γ .

34

The altitude h and the velocity V are required to track the sinusoidal function ftthc )3.0sin(100= and the step function with magnitude of sft /80 respectively. In the simulation, we select fuzzy rules 6=M . Following the robust adaptive fuzzy control scheme, we select

],4,6,4,1[,]3,1,1[],2.0,3.0,2.0,05.0[,]15.0,15.0,05.0[ ==== ChT

CVDhT

DV KKKK01.0;1.0,02.0,2.0,1.0 2211 ========== hVhVhVhVhV rrρργγγγδδ .Du

ring the 50 seconds, the aircraft responses to the velocity and altitude commands from the trimmed conditions, and relevant simulation results are shown as follows.

Figure 2 Response to 80 ft/s step-velocity command

Fig.3 Response to the sinusoidal function 100sin(0.3t) ft altitude command

Figure 4 The angle of attack, thrust and elevator deflection at the first 50

seconds Fig. 2, 3 show the flight velocity, flight altitude and the

observation output converge to the desired value, and Fig. 4 shows the observation errors approach zeros in a short of time. That demonstrates the system has good tracking performance. The following figures illustrate the changes in the angle of attack, thrust and elevator deflection during the first 50 seconds.

From Fig.4, we observe that the angle of attack, thrust and elevator deflection converge to the steady states shortly. All simulation results demonstrate that the adaptive fuzzy robust controller has good tracking performance and high

robustness for a hypersonic aircraft. Furthermore, the system stability is guaranteed in the overall simulation process.

VI. CONCLUSION In this paper, the robust adaptive fuzzy controller for a

MIMO nonlinear longitudinal model of a hypersonic aircraft is presented. It is assumed that hypersonic aircraft dynamics are partially or completely unknown and the system outputs cannot be available for measurement. The main feature of the control method adopts the fuzzy logic system to approximate the unknown part, and ∞H control is used to suppress influence of the external disturbances and remove the fuzzy approximation errors. Moreover, the fuzzy weights for estimating the nonlinear model can adjust on-line and the state observer introduced can approach the system real state outputs. As a result, the system stability, the tracking performance and robustness are guaranteed even in the case where the hypersonic vehicle dynamics are uncertain or unknown.

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