nonlinear autopilot for high maneuverability of bank-to ...batti/progetti_fda/40.pdfshown that the...

Nonlinear Autopilot forHigh Maneuverability ofBank-to-Turn Missiles

SANG-YONG LEESamsung Advanced Institute of TechnologyKorea

JU-IL LEESamsung Electronics Co.Korea

IN-JOONG HASeoul National UniversityKorea

This paper presents a novel approach to autopilot design for

highly maneuvering bank-to-turn (BTT) missiles. In the design

and performance analysis of the proposed nonlinear autopilot,

all nonlinearities of missile dynamics including the coupling

between roll, yaw, and pitch channels as well as the asymmetric

structure of missile body are taken into full account. It is shown

that through a kind of feedback linearization technique along

with a singular perturbation-like technique, the input/output (I/O)

dynamic characteristics of pitch, yaw, and roll channels are made

linear, decoupled, and independent of flight conditions such as air

density and missile velocity. In particular, the proposed autopilot

controllers can provide excellent set-point tracking performance

for roll and pitch channels while keeping the side-slip angle

negligible. The generality and practicality of our approach

are demonstrated through mathematical analysis and various

simulation results using an ILAAT missile.

Manuscript received January 23, 2000; revised April 20, 2001;released for publication May 30, 2001.

IEEE Log No. T-AES/37/4/10989.

Refereeing of the contribution was handled by P. K. Willett.

Authors’ addresses: S.-Y. Lee, Samsung Advanced Instituteof Technology, i-Networking Laboratory, 416, Maetan-3Dong,Paldal-Gu, Suwon City, Kyungki-Do 442-742, Korea; J.-I. Lee,Samsung Electronics Co., HDD R&D Group, Storage Division,416, Maetan-3Dong, Paldal-Gu, Suwon City, Kyungki-Do 442-742,Korea; I.-J. Ha, ACRC/ASRI, School of Electrical Engineering,Seoul National University, San 56-1 Shinrim-Dong, Kwanak-Gu,Seoul 151-742, Korea, E-mail: ([email protected]).

0018-9251/01/$17.00 c° 2001 IEEE

I. NOMENCLATURE

(X,Y,Z) Missile body coordinate systemU,V,W X-, Y-, Z-components of the linear

velocity vector of missile,respectively

p,q,r X-, Y-, Z-components of the angularvelocity vector of missile,respectively

Ix,Iy,Iz Moments of inertia about X-, Y-,Z-axes, respectively

Ixy,Iyz ,Izx Products of inertia

m Missile massS,D Aerodynamic reference area, length of

missile, respectivelylb, lf , lg Distances from the nose of the missile

to the center-of-pressure of missilebody, the center-of-pressure of controlfins, and the center-of-gravity ofmissile body, respectively

VM Total velocity of missile(¢=pjUj2 + jVj2 + jWj2)

Vs,½ Sound velocity, air density,respectively

M Mach numberV,W Normalized values of V and W,

respectively (V¢=V=VM ,W

¢=W=VM)

Q Dynamic pressure (¢=½jVM j2=2)

®T Total angle of attack(¢=sin¡1(

pjVj2 + jWj2=VM))

®,¯ Angle of attack, side-slip angle,respectively(®

¢=tan¡1(W=U),¯

¢=tan¡1(V=U))

Á,Ác Roll angular position, roll angularposition command, respectively

±p,±q,±r Deflections of roll, pitch, yaw controlfins, respectively

±cp,±cq,±

cr Roll, pitch, yaw control fin

commands, respectivelyAz ,A

cz Pitch acceleration of missile, pitch

acceleration command,respectively

jaj Absolute value of a 2 <kxk Euclidean norm of x 2 <nxT Transpose of x 2 <nkAk Induced norm of A 2 <n£n¸m(A),¸M(A) Minimum, maximum eigenvalues of

A 2 <n£n, respectivelyDif(x1, : : : ,xn) Partial derivative of f : <n!< at

(x1, : : : ,xn) 2 <n with respect to the ithargument

us(t¡ a) Unit step function (us(t¡ a) = 1, t > a,but us(t¡ a) = 0, t· a)

1236 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 37, NO. 4 OCTOBER 2001

L[f(t)] Laplace transform of a functionf : [0,1)!<

IU Range of X-axis missile linearvelocity U

I½ Range of air density ½I± Range of control fin deflections ±p,

±q, ±r.

II. INTRODUCTION

The bank-to-turn (BTT) missiles can haveseveral advantages over skid-to-turn (STT) missilesand hence the autopilot design for BTT missileshas been widely studied [1]. Since the directionof the aerodynamic normal force is determinedby the roll motion, the BTT missile need not besymmetrical in shape. In fact, an asymmetric structurecan offer the possibility of improved storage andaerodynamic performance [2]. Moreover, the BTTmissile may be designed to use an airbreathingpropulsion system, thereby providing a greaterrange for a given weight of fuel. Proper operationof an airbreathing missile, however, requiresthat the side-slip angle be very small (severaldegrees, at most) and hence precludes an STTautopilot [3].In spite of the aforementioned advantages of BTT

missiles, they pose some severe challenges to thecontrol engineers. First, a high roll rate combined withthe asymmetric structure of the missile body producessevere undesirable coupling between roll, yaw,and pitch channels. Second, the input-output (I/O)dynamic characteristics from the control fin deflectionto the missile acceleration are of nonminimumphase. Third, the associated aerodynamics are highlynonlinear and usually available only in look-up tableform.Recently, a lot of effort has been made in finding

more effective methods of designing autopilots forhighly maneuvering BTT missiles [4—9]. Especiallyin [4—7], the so-called gain-scheduling approachbased on the modern state-space design methodssuch as LQR, LQG/LTR, H1, and ¹-synthesis wastaken to make the autopilot performance uniformlyregardless of equilibrium points. And in [8, 9],robust feedback linearization approaches to BTTautopilot design have been successfully applied.However, such successful applications offeedback linearization technique were made possiblebecause the outputs to be controlled are notacceleration but other variables such as angularposition or angle of attack. In fact, the I/O feedbacklinearization technique, which has been regarded asthe powerful design method for nonlinear systems,cannot be applied directly to acceleration controlof BTT missiles because the pitch channels of

tail-controlled missiles are inherently of nonminimumphase.Adaptive neural-network-based control approaches

were also proposed [10, 11]. In [11], a hybridadaptive control scheme consisting of a radial basisfunction network in parallel with a fixed-parameterlinear feedback loop was proposed to achieveuniform control performance during flight. It wasshown that the hybrid autopilot can compensateadequately for the nonlinearities neglected inlinear approximation of missile dynamics aroundequilibrium points. However, only canard-controlledmissiles, which are of minimum phase, wereconsidered.We propose a new autopilot design method

for high maneuverability of BTT missiles, whichis based on a much more general missile modelthan those previously considered in BTT missilecontrol. This new method does not involve anygain scheduling and is straightforward and practicalthough computationally complex. The proposedroll, yaw, and pitch autopilot controllers can makethe I/O dynamic characteristics for roll, yaw, andpitch channels linear and independent of the flightconditions such as missile velocity and air density.Nonetheless, missile velocity and air density are notassumed to be constant or slow varying. In theserespects, our autopilot design method is believedto significantly improve the previously knownmethods.Basically, a singular perturbation-like technique

and the well-known feedback linearization techniqueare incorporated into the functional inversiontechnique used in [12]. The result given here canbe viewed as the extension of that in [13] to BTTmissiles whose dynamics are much more complexthan those of STT missiles because of asymmetricstructure of missile body and coupling betweenroll, yaw, and pitch channels. We demonstratethe generality and practicality of the proposedcontrol method through mathematical analysisand various simulation results using an ILAATmissile.

III. CONTROLLER DESIGN

In deriving the equations of motion for BTTmissiles, make the following assumptions.

A1. m, Ix, Iy, and Iz are constant.A2. The missile body is symmetrical in the pitch

plane.

Assumption A1 is commonly accepted during theendgame phase of flight [7], while AssumptionA2 is quite natural for BTT missiles. Then thethree-dimensional motion of a BTT missile canbe described by the following nonlinear ordinary

LEE ET AL.: NONLINEAR AUTOPILOT FOR HIGH MANEUVERABILITY OF BANK-TO-TURN MISSILES 1237

differential equations [1, 2]

Roll

Dynamics

8>>>>>>>>>>>><>>>>>>>>>>>>:

_p=Iz

IxIz ¡ I2zxQSDCl

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±p,±r

¶+

IzxIxIz ¡ I2zx

QSDCn

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶

+IyIz ¡ I2z ¡ I2zxIxIz ¡ I2zx

qr+Izx(Iz + Ix¡ Iy)IxIz ¡ I2zx

pq

_Á=p

(1a)

Yaw

Dynamics

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

_V= pW¡Ur+ QSmCy

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶_r=

IxIxIz ¡ I2zx

QSDCn

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶+

IzxIxIz ¡ I2zx

QSDCl

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±p,±r

¶

+Izx(Iy ¡ Iz ¡ Ix)IxIz ¡ I2zx

qr+I2x + I

2zx¡ IxIy

IxIz ¡ I2zxpq

(1b)

Pitch

Dynamics

8>>>>>>>><>>>>>>>>:

_W=¡pV+Uq+ QSmCz

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±q

¶_q=

QSD

IyCm

µtan¡1

µW

U

¶,VMVs,±q

¶+Iz ¡ IxIy

pr+IzxIy(r2¡p2)

Az =QS

mCz

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±q

¶:

(1c)

Here, the functions Cy, Cz, Cm, Cn, and Cl describethe aerodynamic coefficients in terms of V, W, VMand deflections of control fins (±p, ±r, ±q); and areobtained in look-up table form through wind-tunnelexperiments. Note that in (1), the X-axis component Uof the missile velocity is regarded to be a time-varyingexogenous variable. For the derivation of the autopilotcontrollers for BTT missiles, we further assume thefollowings.

A3. The functions Cl, Cy , and Cz are invertiblewith respect to ±p, ±r, and ±q, respectively.A4. The actuator dynamics are fast enough to be

±p = ±cp, ±r = ±

cr , and ±q = ±

cq.

Assumption A3 is well justified in [7] andAssumption A4 is practically reasonable, as shownin Section V.Now, we are ready to design the autopilot

controllers based on the dynamic model for BTTmissiles in (1). Note that our dynamic model is verygeneral compared with those used in previous works.The roll and pitch autopilot controllers are designedto track the desired commands for roll angle and

pitch acceleration, respectively. On the other hand, theside-slip angle, or, equivalently, the yaw velocity V,must be kept as small during flight as possible. Highmaneuverability means not only high aerodynamicaccelerations, but also the ability to change theorientation of the acceleration rapidly. This means thatroll rate may be much larger than it would be in STTmissiles.By Assumption A3, there exists a mapping Kl

satisfying

Cl

Ãtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,

Kl

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,vx,±r

¶,±r

!= vx:

(2)

Using the mapping Kl, we can design the rollautopilot controller as follows

±cp =Kl

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,up,±r

¶(3a)


where the new input up is given by

up =¡IzxIzCn¡

IyIz ¡ I2z ¡ I2zxQSDIz

qr¡ Izx(Iz + Ix¡ Iy)QSDIz

pq

+IxIz ¡ I2zxQSDIz

Ã¡ap¸!np¡ bp¸2!2nÁ

+¸3!3n

Z t

0(Ác(¿ )¡Á(¿ ))d¿

!: (3b)

Here, the positive constants ap, bp, !n, and ¸ aredesign parameters.Then the dynamics of roll channel given by (1a)

and (3) are described as follows

Roll

Dynamics

8<:_p=¡ap¸!np¡ bp¸2!2nÁ+¸2!2n³p_³p= ¸!n(Á

c¡Á)_Á= p:

(4)

Hence, the I/O transfer function of roll channel isgiven by

Gp(s)¢=L[Á(t)]L[Ác(t)] =

1(s=¸!n)

3 + ap(s=¸!n)2 + bp(s=¸!n)+ 1

:

(5)From (5), we see that the roll autopilot controllerin (3) provides exact set-point tracking. As can beseen later, the design parameter !n determines thebandwidth of the pitch channel. Hence, the designparameter ¸ adjusts the bandwidth of the roll channelrelative to that of the pitch channel.Using the roll autopilot controller in (3), we can

also write the dynamics of the yaw channel in thefollowing simpler form

Yaw

Dynamics

8>>>>>>><>>>>>>>:

_V= pW¡Ur+ QSmCy

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶_r=

QSD

IzCn

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶¡ IzxIzqr

+Ix¡ IyIz

pq+IzxIz[¡ap¸!np¡ bp¸2!2nÁ+¸2!2n³p]

(6)

To go further, we define the new functions Ha andHb by

Ha(V,W)¢=

8><>:0, if V = W = 0µlf ¡ lbIz

¶Vq

jVj2 + jWj2fN

µsin¡1

µqjVj2 + jWj2

¶¶, otherwise (7a)

Hb(V,W)¢=

8><>:0, if V = W = 0Ãlf ¡ lbIy

!Wq

jVj2 + jWj2fN

µsin¡1

µqjVj2 + jWj2

¶¶, otherwise: (7b)

Here, the function fN represents the normalcomponent of the aerodynamic force generated by themissile body. And it is well justified in [12] that

fN depends only on ®T, continuouslydifferentiable, and is strictly increasingin the range of 0· ®T < ¼=4such that fN(0) = 0

(8)

and further that the following relationships holdbetween the aerodynamic coefficients:

Cm

µtan¡1

µW

U

¶,VMVs,±q

¶=IySDHb(V,W) +

(lf ¡ lg)D

£Czµtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±q

¶(9a)

Cn

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶

=¡ IzSDHa(V,W)¡

(lf ¡ lg)D

£Cyµtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,±r

¶: (9b)

By Assumption A3, there exist the mappings Ky,Kz satisfying

Cy

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,

Ky

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,vy

¶¶= vy

(10a)


Cz

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,

Kz

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,vz

¶¶= vz

(10b)

respectively. Choose the control fin commands ±cr , ±cq

by

±cr¢=Ky

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,m(uy +Ur)

QS

¶(11a)

±cq¢=Kz

µtan¡1

µW

U

¶, tan¡1

µV

U

¶,VMVs,m(uz ¡Uq)

QS

¶(11b)

respectively, where uy and uz are the new inputs to bedesigned soon.By (9) and (10), the closed-loop system consisting

of the missile dynamics in (6), (1c), and the controllerin (11) then can be described in the following “almostlinear” form

Yaw

Dynamics

8>>>>>>>><>>>>>>>>:

_V= pW+ uy

_r=¡hv1Ur¡ hv1uy ¡½(U2 +V2 +W2)

2

£Ha(V,W)¡IzxIzqr+

Ix¡ IyIz

pq

+IzxIz[¡ap¸!np¡ bp¸2!2nÁ+¸2!2n³p]

(12a)

Pitch

Dynamics

8>>>>>><>>>>>>:

_W=¡pV+ uz_q=¡hv2Uq+ hv2uz +

½(U2 +V2 +W2)2

£Hb(V,W) +Iz ¡ IxIy

pr+IzxIy(r2¡p2)

Az =¡Uq+ uz(12b)

where

hv1¢=(lf ¡ lg)m

Iz, hv2

¢=(lf ¡ lg)m

Iy: (12c)

As mentioned earlier, BTT missiles are desired tohave small side-slip angle during flight. Note that theyaw velocity V goes to zero as the side-slip angle ¯goes to zero since ¯ = tan¡1(V=U). Hence, the yawchannel autopilot is designed so as to keep the yawvelocity V zero. For this purpose, the new input uy in(12a) is chosen as follows.

uy =¡ay!nV¡!2nZ t

0V(¿)d¿ ¡pW (13)

where the constants ay and !n are design parameters.Then the dynamics of the yaw channel given by(12a) and (13) are described by the following

equations

Yaw

Dynamics

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

_V=¡ay!nV+!n³y_³y =¡!nV_r=¡hv1Ur+ hv1ay!nV¡ hv1!n³y + hv1pW

¡½(U2 +V2 +W2)

2Ha

µV

VM,W

VM

¶+Ix¡ IyIz

pq¡ IzxIzqr

+IzxIz[¡ap¸!np¡ bp¸2!2nÁ+¸2!2n³p]:

(14)

From (14), we see that if ay > 0, then V! 0 ast!1.As done for STT missiles in [13], it can be

shown that the direct application of the I/O feedbacklinearization technique [14, 15] to the nonlinearsystem in (12b) produces unstable hidden dynamicsand hence does not give a useful solution foracceleration control of BTT missiles. However, it isfortunately possible to linearize this partial-linearizedsystem in an approximate but stable way by using asingular perturbation-like technique. To show this, weintroduce the following magnitude and time scalingtransformation by

½(t)¢=½µt

!n

¶, U(t)

¢=U

µt

!n

¶V(t)

¢=V

µt

!n

¶, W(t)

¢=W

µt

!n

¶³y(t)

¢=³y

µt

!n

¶, ³p(t)

¢=³p

µt

!n

¶uz(t)

¢=1!nuz

µt

!n

¶, VM(t)

¢=VM

µt

!n

¶Az(t)

¢=Az

µt

!n

¶, Acz(t)

¢=Acz

µt

!n

¶Ác(t)

¢=Ác

µt

!n

¶, Á(t)

¢=Á

µt

!n

¶p(t)

¢=1!np

µt

!n

¶, q(t)

¢=qµt

!n

¶r(t)

¢=rµt

!n

¶:

(15)

It then can be seen that this transformation takesthe closed-loop systems given by (4), (12b), and (14)to a singularly perturbed system as follows

Roll

Dynamics

8>>><>>>:_p=¡ap¸p¡bp¸2Á+¸2³p_³p= ¸(Á

c¡ Á)_Á= p

(16a)


Yaw

Dynamics

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

_V=¡ayV+ ³y_³y =¡V!n_r=¡hv1Ur+ hv1!npW+ hv1!nayV

¡hv1!n³y ¡½(U2 + V2 + W2)

2

£HaµV

VM,W

VM

¶+Ix¡ IyIz

!npq¡IzxIzqr

+IzxIz!2n[¡ap¸p¡ bp¸2Á+¸2³p]

(16b)

Pitch

Dynamics

8>>>>>>>>>>><>>>>>>>>>>>:

_W=¡pV+ uz!n_q=¡hv2Uq+ hv2!nuz

+½(U2 + V2 + W2)

2Hb

µV

VM,W

VM

¶+Iz ¡ IxIy

!npr+IzxIy(r2¡!2np2)

Az =¡Uq+!nuz:(16c)

Letting !n! 0, we obtain the reduced systemgiven by (16a) and

Reduced

Yaw

Dynamics

8>>>>>>>>>><>>>>>>>>>>:

_V=¡ayV+ ³y_³y =¡Vµ

hv1U+IzxIzq

¶r=¡ ½(U

2 + V2 + W2

)2

£Ha

ÃV

VM

,W

VM

!(17a)

Reduced

Pitch

Dynamics

8>>>>>>>>>>><>>>>>>>>>>>:

_W=¡pV+ uz

q=1

hv2U

(½(U2 + V2 + W

2

)2

£HbÃV

VM

,W

VM

!+IzxIyr2

)Az =¡Uq

(17b)

where VM¢=

qU2 + V2 + W

2. Here, we have added

bars on the top of the variables affected by letting!n! 0 to discriminate from those of the originalsystem in (16).Now we attempt to find the feedback law that

can I/O linearize the reduced pitch dynamics. To thisaim, neglect the last term (Izx=Iy)r

2in (17b) and, in

addition to Assumptions A1 through A4, assume

further that

A5. hv2 is constant.

Choose the input uz in (17b) by

uz = pV+F(½,U,V,W,_½,_U)+G(½,U,V,W)vz

(18a)

where vz is the new input and the functions N, F, andG are defined as follows

N(U,V,W)

¢=

WqU2 + V2 + W

2Hb¡

VW

2(U2 + V2 + W2)D1Hb

+U2 + V2

2(U2 + V2 + W2)D2Hb (18b)

F(½,U, V,W, _½,_U)

¢=¡

qU2 + V2 + W

2

2½N¡1(U,V,W)Hb

_½

¡ N¡1(U,V,W)(¡ayV+ ³y)2(U2 + V2 + W

2)

£"2VHb

qU2 + V2 + W

2

+(U2 + W2)D1Hb¡ VWD2Hb

#

+U_UN¡1(U, V,W)

2(U2 + V2 + W2)

£"¡2Hb

qU2 + V2 + W

2+ VD1Hb+ WD2Hb

#(18c)

G(½,U,V,W)

¢=¡ hv2

½

qU2 + V2 + W

2N¡1(U, V,W): (18d)

Here,

Hb

0@ VqU2 + V2 + W

2,

WqU2 + V2 + W

2

1A ,D1Hb

0@ VqU2 + V2 + W

2,

WqU2 + V2 + W

2

1A ,D2Hb

0@ VqU2 + V2 + W

2,

WqU2 + V2 + W

2

1ALEE ET AL.: NONLINEAR AUTOPILOT FOR HIGH MANEUVERABILITY OF BANK-TO-TURN MISSILES 1241

are denoted for notational convenience by Hb, D1Hb,and D2Hb, respectively.Differentiating the output Az directly, we then have

_Az = vz: (19)

Thus, we have seen that the feedback control law in(18) I/O linearizes the reduced pitch dynamics whenthe term (Izx=Iy)r

2is neglected and Assumption A5

holds. Recall that the yaw autopilot has been designedto be V ' 0. As the result of this, along with (17a)and the definition of Ha in (7a), Ha((V=VM), (W=VM))and hence r are always small. Therefore, it is quitenatural to neglect the term (Izx=Iy)r

2in (17b) when

we derive the I/O linearizing controller in (18).We further justify this point through the rigorousperformance analysis in Section IV and the simulationresult in Section V. On the other hand, AssumptionA5 is clearly valid during the endgame phaseflight. Furthermore, even the case where hv2 varieswith time can be handled through use of a kind ofoutput redefinition method but at the cost of someperformance degradation. Finally, as is shown inAppendix A,

N is always positive during flight. (20)

Hence, the feedback control law in (18) is welldefined.Next choose the new input vz in (18) as follows8>>>><>>>>:

_³z1 =¡az³z1¡bzAz + ³ z2_³z2 = A

cz ¡ Az

vz = ³ z1

(21)

where the positive constants az and bz are the design

parameters soon to be discussed and ³ z1(t), ³ z2(t) 2 <.Note that the above dynamic compensator contains anintegral term to eliminate the steady state error. Then,simple calculation shows that the I/O transfer functionof the closed-loop system given by (19) with (21) is

Gz(s)¢=L[Az(t)]L[Acz(t)]

=1

s3 + azs2 + bzs+1: (22)

In what follows, we assume that the constants az, bzare chosen so that

all the roots of (s3 + azs2 + bzs+1)

have negative real parts.(23)

It is clear from (22) that when Acz is a step command,jAz(t)¡ Acz(t)j ! 0 as t!1. Moreover, this desirableproperty remains preserved even when ½ and U aretime varying.Considering the magnitude and time scaling

transformation in (15), we can convert the feedbackcontrol law given by (18) and (21) into the form

required for the new input uz in (11b). Then, we have

uz = pV+!nF(½,U,V,W, _½,_U)+!nG(½,U,V,W)vz

(24a)

_vz =¡az!nvz ¡bz!nAz +!2nZ t

0(Acz(¿)¡Az(¿ ))d¿:

(24b)

In Section IV, we show that the I/O dynamic behaviorof the closed-loop system given by (12b) and (24)tends to be governed by Gz(s=!n) as !n decreases.Furthermore, it is practically possible to choose anappropriate value of !n that can provide not only theI/O dynamic characteristics very close to Gz(s=!n)but also fast transient responses because the physicalvalue of hv2U is generally much larger than thedesired bandwidth of the pitch autopilot.Note that the missile model in (1) does not take

the standard singular perturbation form. However,recall that our partial-linearizing controller in (11)and the magnitude and time scaling transformationin (15) take the missile model in (1) to a singularperturbation-like form. This implies that, as wasdone in [16, 17] for aircraft, a parameter ² may beassociated artificially or naturally with the missileangular rates r and q. Here, we do not convert themissile model into a standard singular perturbed formby introducing such a small parameter ² becausethe design parameter !n in our linear controllerin (16) can play fully the role of such an artificialparameter ².

IV. PERFORMANCE ANALYSIS

In Section III, we already have shown that Á! Ác,V! 0, and p! 0 as t!1. Therefore, it suffices toanalyze the dynamic behaviors of pitch dynamics andyaw rate.To this aim, we consider the time-scaled

closed-loop system consisting of (16a), (16b), and

§q :

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

_W= F(½,U,V,W, _½,

_U) +G(½,U,V,W)³z1

!n_q=¡hv2Uq+ hv2!nfpV+F(½,U,V,W, _½,

_U)

+G(½,U,V,W)³z1g

+½(U2 + V2 + W2)

2Hb

ÃV

VM,W

VM

!

+Iz ¡ IxIy

!npr+IzxIy(r2¡!2np2)

_³z1 =¡az³z1¡ bzAz + ³z2_³z2 = A

cz ¡ Az

Az =¡Uq+!npV+!nF(½,U,V,W, _½,_U)

+!nG(½,U,V,W)³z1:(160)


Then, the reduced system, obtained from thetime-scaled closed-loop system consisting of (16a),(16b), and (16c0) by letting !n! 0 and neglecting(Izx=Iy)r

2, is given by (16a), (17a), and

§q :

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

_W= F(½,U, V,W, _½,

_U) +G(½,U, V,W)³ z1

q=½(U2 + V2 + W

2)

2hv2UHb

ÃV

VM

,W

VM

!_³z1 =¡az³z1¡ bzAz + ³ z2_³z2 = A

cz ¡ Az

Az =¡½(U2 + V2 + W

2)

2hv2Hb

ÃV

VM

,W

VM

!:

(17b0)Furthermore, simple calculation shows that

_Az = ³ z1: (25)

Now, we define some error variables and auxiliaryvariables as follows

e»¢=[ez1 ez2 e´]

T

(26a)ez1

¢= ³z1¡ ³ z1 ez2

¢= ³z2¡ ³ z2

e´¢= ez + Ueq¡!npV¡!nF(½,U,V,W, _½,

_U)

¡!nG(½,U,V,W)³z1 (26b)

ez¢= Az ¡ Az

(26c)

eq¢= q¡ ½(U

2 + V2 + W2)

2hv2UHb

ÃV

VM,W

VM

!:

Through some calculations using (16b), (16c0), (17a),and (17b0), we then can derive the following errorequations

§e :

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

_e» = A11e» + UA12eq¡!nA12g(t)

!n_eq=¡hv2Ueq+!nhv2g(t) +

Iz ¡ IxIy

!npr+IzxIyr2¡ Izx

Iy!2np

2 +!n

U

"³z1 +

_U½(U2 + V2 + W2)

2hv2UHb

µV

VM,W

VM

¶#

!n_r=¡hv1Ur+ hv1!n(pW+ ayV¡ ³y)¡

½(U2 + V2 + W2)2

Ha

µV

VM,W

VM

¶+Ix¡ IyIz

!npeq+Ix¡ IyIz

!np

£ ½(U2 + V2 + W2)

2hv2UHb

µV

VM,W

VM

¶¡ IzxIzreq¡

IzxIzr½(U2 + V2 + W2)

2hv2UHb

µV

VM,W

VM

¶+IzxIz!2n[¡ap¸p¡ bp¸2Á+¸2³p]

ez = e´ ¡ Ueq+!ng(t)(27)

where

A11¢=

264¡az 1 ¡bz0 0 ¡11 0 0

375 , A12¢=

264bz10

375 (28)

g(t)¢= pV+F(½,U,V,W, _½,

_U) +G(½,U,V,W)³z1:

(29)In what follows, we make the following

assumptions for mathematical simplicity.

A6. fN is smooth enough for Ha and Hb to becontinuously differentiable at V = 0, W = 0.A7. During flight, 0· ®T · ®max for a positive

constant ®max < ¼=4.

As seen in Section V, this is practically true. ByAssumptions A6 and A7 along with (7), (8), and (20),there then exist positive constants Jmin, ha, ®M , ¯M1,and ¯M2 such that

N(U, V,W)¸ Jmin (30a)¯¯HaÃ

VpU2 + V2 + W2

,Wp

U2 + V2 + W2

!¯¯· hajVj

(30b)¯¯Hb

ÃVp

U2 + V2 + W2,

WpU2 + V2 + W2

!¯¯· ®M

(30c)¯¯D1Hb

ÃVp

U2 + V2 + W2,

WpU2 + V2 + W2

!¯¯· ¯M1

(30d)¯¯D2Hb

ÃVp

U2 + V2 + W2,

WpU2 + V2 + W2

!¯¯· ¯M2:

(30e)

It is also quite natural to make the followingassumption.


A8. There exist positive constants ½min, ½max, Umin,Umax, ± _½, ± _U such that for all t¸ 0,

½(t) 2 I½¢=[½min,½max],

U(t) 2 IU¢=[Umin,Umax],

j _½(t)j · ± _½, j _U(t)j · ± _U:

By Assumptions A7 and A8 along with (18) and (30),we then can see that

jF(½,U,V,W, _½, _U)j · ·1Umax½min

j _½j+·2j_Uj+·2(ay jVj+ j³yj)

(31)jG(½,U,V,W)j · ·3

½minUmin

where ·i, i= 1,2,3 are the positive constants given by

·1¢=

®Mp2Jmin

,

·2¢=

1Jmin

(®M +12¯M1 +

12¯M2),

·3¢=hv2Jmin

:

(32)

On the other hand, (23) implies that A11 is aHurwitz matrix. Therefore, there exist positive definitesymmetric matrices P11, Q11 2 <3£3 satisfying

P11A11 +AT11P11 =¡Q11: (33)

Define two matrices P0 and Q0 as follows

P0¢=·P11 0

0 I2

¸, ·4

¢=kP11k

q1+ b2z (34a)

Q0¢=

2666664¸m(Q11)¡

2!n·3·4½minUmin

¡µ·4UmaxUmin

+hv2·3½min

+1¶

0

¡µ·4UmaxUmin

+hv2·3½min

+1¶

2hv2Umin!n

0

0 0 q33

3777775 (34b)

where

q33¢=2hv1Umin!n

¡¯IzxIz

¯2½maxUmax®M

!nhv2: (34c)

When Acz and Ác are constant, let

Âp¢=[p (³p¡ bpÁ) (Ác¡ Á)]T

Ây¢=[V ³y]

T

Âz¢=[³ z1 (³ z2¡ bzAz) (Acz ¡ Az)]T:

(35)

Then, (22), (23), and (25) imply the existence ofpositive constants °1, °2, and ¾1 satisfying

j³ z1(t)j · °1kÂz(0)ke¡¾1t

jAz(t)¡ Acz j · °2kÂz(0)ke¡¾1t, 8 t¸ 0:(36)

From (16a) and (16b), we also see that the followinginequalities hold for some positive constants °3, °4, °5,°6, ¾2, and ¾3

jp(t)j · °3kÂp(0)ke¡¾2t,j³p(t)¡ bpÁ(t)j · °4kÂp(0)ke¡¾2t,

jV(t)j · °5kÂy(0)ke¡¾3t,j³y(t)j · °6kÂy(0)ke¡¾3t, 8 t¸ 0:

(37)

We finally introduce the following notations toconcisely describe the performance analysis for theproposed autopilot controllers

¹¢=maxfkÂz(0)k, kÂy(0)kg (38a)

¾0¢=minf¾1, ¾3g, ¾

¢=¸m(Q0)8¸M(P0)

,

(38b)¾min

¢=minf¾0, ¾2, ¾g

d0¢=

s1+

µUmaxUmin

¶2+

!n·3½minUmin

(38c)

d1¢=

sµ!n·3·4½minUmin

¶2+

µhv2·3½min

+1

¶2(38d)

d2¢=·1Umax½min

s·24 +

µhv2Umin!n

¶2(38e)

d3¢=

s(·2·4)

2 +

µhv2·2Umin!n

+½maxUmin®Mhv2!n

¶2(38f)

d4¢=p(!n·4)

2 + (hv2Umin)2 (38g)

d5¢=p(!n·2·4)

2 + (hv2·2Umin)2 + (hv1Umin)

2 (38h)


d6¢=p2hv1UmaxUmin +

¯Ix¡ IyIz

¯½maxUmaxUmin®M

hv2(38i)

d7¢=½maxU

2maxUminha!n

(38j)

d8¢=

¯IzxIy

¯!nUmin (38k)

d9¢=

¯IzxIz

¯!nUmin (38l)

d10¢=d0

r¸M(P0)¸m(P0)

k[eT» (0) Umineq(0) Uminr(0)]k

+d0

¸m(P0)b2

j¾¡¾2j+

d0¸m(P0)

¹b3j¾¡¾0j

+!nf°3°5kÂp(0)k+·2(ay°5 + °6)gkÂy(0)k

+

µ!n·3½minUmin

°1 + °2

¶kÂz(0)k (38m)

d11¢=

8¸M (P0)¸m(P0)¸m(Q0)

d0d2 +·1Umax½min

(38n)

d12¢=

8¸M (P0)¸m(P0)¸m(Q0)

d0d3 +·2 (38o)

b1¢=d2± _½+d3± _U (38p)

b2¢=

¸m(Q0)

4

¯Iz ¡ IxIy

+Ix¡ IyIz

¯

£

8>><>>:d6 +d8¸m(Q0)

4

¯Iz ¡ IxIy

+Ix¡ IyIz

¯ +d9µap¸+¸2 °4°3¶9>>=>>;(38q)

b3¢=d1°1 +d4°5

¸m(Q0)

4

¯Iz ¡ IxIy

+Ix¡ IyIz

¯+d5(ay°5 + °6) + d7°5 (38r)

b4¢=

r¸m(P0)¸M(P0)

¸m(Q0)!n(Umin=Izx)

8

¯1Iz¡ 1Iy

¯ : (38s)

Now, we are ready to state the following theorem.

THEOREM 1 Let the acceleration command Acz tothe closed-loop system consisting of (16a), (16b), and(16c0) be constant. Suppose that Izx is so small suchthat ¯

IzxIz

¯<hv1hv2Umin½maxUmax®M

: (39)

Suppose further that the design parameters !n, ay,az, bz, ap, bp, ¸ and the initial conditions satisfy the

following inequalities

!n <2¸m(Q11)hv2Umin

4·3·4hv2½min

+··4UmaxUmin

+hv2·3½min

+1¸2

(40a)

kÂp(0)k ·¸m(Q0)

4

¯¯ Iz ¡ IxIy

+Ix¡ IyIz

¯¯°3

(40b)

kÂy(0)k,kÂz(0)k

<1b3

264 Ã2s ¸m(P0)¸M(P0)

¡ ¸m(P0)¸M(P0)

!

£ ¸2m(Q0)!n(Umin=Izx)

64

¯¯ 1Iz ¡ 1

Iy

¯¯

¡ b1¡b2

375 (40c)

k[eT» (0) Umineq(0) Uminr(0)]Tk

·s¸m(P0)¸M(P0)

¸m(Q0)!n(Umin=Izx)

8

¯¯ 1Iz ¡ 1

Iy

¯¯

:

(40d)Then,

jAz(t)¡Acz j · d10e¡¾min!nt+ d11± _½+ d12± _U, 8 t ¸ 0:(41)

In particular, when ½ and U are constant, then

jAz(t)¡Acz j · d10e¡¾min!nt, 8 t¸ 0: (42)

The Proof of Theorem 1 is given in AppendixB. Note that Theorem 1 does not necessarily imposestringent limitations on initial conditions and designparameters because Umin=Izx is very large in mostpractical situations. Furthermore, if Izx is assumed tobe zero, as done in most prior works, all assertionsin Theorem 1 hold globally. Nonetheless, the rollrate is required to be sufficiently small in any case.In particular, Theorem 1 shows that when ½ and Uare constant, the proposed pitch controller assuresprecise set-point tracking, while the settling time isdetermined mainly by the design parameters az, bz,and !n. It also suggests that the set-point trackingperformance may be degenerated when ½ and U arefast varying. However, the simulation results presentedin the next section suggest that in practical situations,½ and U cannot vary so fast as to seriously affectmissile acceleration responses. Furthermore, it alsocan be shown that the I/O dynamic characteristics ofthe closed-loop pitch dynamics approach those of thelinear system given by (19) with (21) as !n! 0 orUmin!1 when ½ and U are constant. However, the


Fig. 1. 3D graphic representation of Ha.

details of the assertion are omitted because of limitedspace and mathematical complexity.

V. A PRACTICAL EXAMPLE

In this section, the autopilot design methoddeveloped so far is applied to ILAAT missiles toilluminate further its practicality. We assume that theranges of X-axis missile linear velocity, air density,and control fin deflections are given as

IU¢=[1:0VS ,4:0VS],

I½¢=[0:4 kg/m3,1:2 kg/m3],

I±¢=[¡20±,20±]

(43)

where I½ corresponds to the range of altitude fromsea-level up to about 10 km. The aerodata of theILAAT missile used for our simulation work is thesame as that in [4]. In the aerodata given in [4], Izx =0. To simulate the most general situation, however,we have chosen Izx =¡0:7043 as in [7]. Then, thefunctions Ha and Hb can be determined in look-uptable form by utilizing the relationship in (9), andFig. 1 shows the 3D graphic representation of Ha.Furthermore, both the functions Ha and Hb can beapproximated with fairly high accuracy to

Ha(V,W)'½ks1

qV2 + W2 + kc1

q1¡ V2¡ W2

¾V

(44a)

Hb(0,W)'½ks2

qW2 + kc2

q1¡ W2

¾W (44b)

where ks1 = 3:6842£ 10¡1, kc1 = 9:8686£10¡2,ks2 = 2:8213£10¡1, and kc2 = 1:2788. Here, the firstargument V of the function Hb is set to zero becausethe aerodynamic coefficients Cz and Cm of the ILAATmissile do not depend significantly on V. It canthen be shown through differentiation of the aboveapproximations or numerical differentiation of theactual data for Ha and Hb that the partial derivatives

Fig. 2. Effect of actuator dynamics (U = 2:6Vs,

Acz (t) = 200us(t) (m/s2)).

of Ha and Hb satisfy

0:2564·D1Ha · 0:4565, jD2Haj · 0:08171:1657·D2Hb · 1:4001, jD1Hbj= 0:

(45)Then, it is immediate that N > 0. Hence theapproximate-linearizing controller in (18) is welldefined.Now, suppose that we want to design the autopilot

controllers that can maintain the side-slip angle lessthan 5± and can track the step input commands ofroll angle and pitch acceleration with zero steadystate error, overshoot less than 5%, undershoot lessthan 10%, and have a settling time of roll and pitchchannels shorter than 0.1 s and 0.3 s, respectively,over the whole ranges of U and ½ given in (43).To this aim, we first have chosen az and bz sothat Gz(s) in (22) takes the ITAE (integral timemultiplied absolute error) form [18]. Then, we havechosen ¸= 3 and adjusted !n so as to meet both thesettling time specification and the stability conditionin (40a). Finally, the parameters of roll and yawdynamics have been chosen to satisfy the performancespecifications. In summary, we have determined thedesign parameters:

ap = 2:433, bp = 2:466, ¸= 3,

ay = 1:86, az = 1:75, (46)

bz = 2:15, !n = 18:

We first verify Assumption A4 by investigating theeffect of the following type of actuator dynamics onthe performance of our autopilot controller

¿ _±i =¡±i+ ±ci , i= p,q,r: (47)

The simulation results in Fig. 2 show that the actuatordynamics with reasonable time constant do not causeany noticeable performance degradation.To show that set-point tracking in both pitch and

roll channels is accomplished successfully and thecoupling effect between pitch and roll channels causedby our autopilot controllers is insignificant, we firstapplied a step acceleration command to the pitchchannel at t= 0 s. Then we made step changes inroll angle command Ác from 0± to 45± at t= 1 s andthen to 0± at t= 2 s. Thereby, very high roll rate wasproduced as can be seen from Fig. 3(c). As can be


Fig. 3. Pitch and roll responses to step commands (U = 2:6Vs, Acz (t) = 300us(t) (m/s

2), Ác(t) = 45(us(t¡ 1)¡ us(t¡ 2)) (deg)). (a) Pitchacceleration. (b) Roll angle. (c) Roll rate. (d) Side-slip angle.

Fig. 4. Normalized step responses to various step commands (U = 1:5Vs). (a) Pitch acceleration. (b) Angle of attack.

seen from Fig. 3(a) and (d), however, our autopilotcontroller still provides good set-point trackingperformance. In particular, the pitch accelerationresponse is not affected significantly by high roll rate.Furthermore, Fig. 3(d) shows that the side-slip angleis not affected at all.As shown in Fig. 4(a), the normalized acceleration

responses of the closed-loop missile system to variousstep commands are nearly identical. Hence, thesimulation results in Fig. 4(a) confirm that the I/Odynamic characteristics of the closed-loop missilesystem are almost like those of a linear system.On the other hand, Fig. 4(b) shows that the timeresponses of angle of attack to step pitch accelerationcommands are bounded but not proportional totheir corresponding acceleration commands. Thisimplies that the internal dynamic characteristics ofthe closed-loop missile system from accelerationcommand to angle of attack are still nonlinear.Therefore, these simulation results clearly demonstratethat our autopilot design method can handleeffectively the nonminimum phase characteristics andhigh nonlinearities of BTT missile dynamics.To show through simulation that the performance

of our autopilot controller is quite robust withrespect to uncertainties in aerodynamic data, we haveinvestigated the effect of large scale factor errors inthe aerodynamic coefficients. Our extensive simulation

study has shown that performance degradationby large scale factor errors in Cm, Cn, and Cy areinsignificant, while the effects of those in Cz and Clare noticeable. Nonetheless, our autopilot controllerstill can provide zero steady state error and goodtransient responses, as shown in Fig. 5. We alsohave confirmed through extensive simulation thatthe uncertainties involved in differentiation of theaerodynamic function Hb such as distortion or scalefactor error may cause some degradation in transientperformance but produce no steady state error. Sometypical simulation results are depicted in Fig. 6.Finally, we demonstrate that the performance of

our autopilot is still satisfactory even in ill-conditionedflight situations. To this aim, we have made a scenarioof flight path as shown in Fig. 7(a), where U and½ decreases at constant rate, respectively, from 3Vsto 1:5Vs and from 1:2 kg/m3 to 0:6 kg/m3 in 4 s.This corresponds to a vertical ascent of the missile.Note that along the flight path, Q decreases 1

8times in only 4 s. In this situation, a series of largestep pitch acceleration commands and roll anglecommands were applied as shown in Fig. 7(b) andFig. 7(c). From Fig. 7(b), (c), and (e), we see thatall performance specifications are met even in suchan ill-conditioned flight situation and that the I/Odynamic characteristics of the closed-loop missilesystem are almost independent of flight conditions.


Fig. 5. Step responses in the presence of scale factor errors in Cz and Cl(U = 2:6Vs, Acz (t) = 200us(t) (m/s

2), Ác(t) = 45 (deg)).(a) Effect of uncertainty in Cz . (b) Effect of uncertainty in Cl.

Fig. 6. Step responses in the presence of scale factor errors in[DiHb], i = 1,2. (U = 2:6Vs, A

cz (t) = 200us(t) (m/s

2)).

The simulation results in Fig. 7 also suggest that ourautopilot still can give good performance even for themissile in flight with the thrust burned in.

VI. CONCLUSIONS

In this paper, we have proposed a new approachto autopilot controller design for BTT missiles and

demonstrated its generality and practical use throughmathematical analysis and various simulation results.Our autopilot design method is based on a muchmore general dynamic model for BTT missiles thanthose considered previously and can be appliedto a wide class of BTT missiles. Other interesting

features of the proposed controller are 1) a systematicdesign method which does not require any tediousgain-scheduling procedure; 2) a rigorous stabilityanalysis of our autopilot controller; 3) the excellentset-point tracking performance, which is independentof flight conditions; and 4) insignificant couplingeffect between roll, pitch, and yaw channels.Nonetheless, some important issues still remain

untouched. For instance, fin limitation problem hasnot been addressed. The missile fins, which affect theforces and moments, are deflection and rate limited.Another important issue may be consideration ofsampling effect in the digital implementation of ourautopilot controller.

APPENDIX A. POSITIVENESS OF THE FUNCTION NIN (18b)

Using the definition of the function Hb in (7b), wecan obtain the partial derivatives of Hb as follows

D1Hb(V,W) =

8>><>>:0, if V = W = 0

¡kVWqjVj2 + jWj2

3fN +kVWq

1¡ jVj2¡ jWj2(jVj2 + jWj2)f 0N , otherwise (48a)

D2Hb(V,W) =

8>><>>:kf 0N(0), if V = W = 0

kV2qjVj2 + jWj2

3fN +kW2q

1¡ jVj2¡ jWj2(jVj2 + jWj2)f 0N , otherwise (48b)

where k¢=(lf ¡ lb)=Iy > 0. Here, f 0N denotes the

derivative of fN in (7).By (18b) and (48) along with the definitions of V

and W, the following result can be obtained throughsome tedious calculations

N(U,V,W) =

8>>><>>>:k

2f 0N(0), if V = W = 0

k

2(U2V2 + V4 +3V2W2 +2W4)fN + UW

2pV2 + W2f 0Np

U+ V2 + W2pV2 + W2

3 , otherwise: (49)


Fig. 7. Responses of the closed-loop missile system for a series of step input commands.(Acz (t)

¢=400us(t)¡ 700us(t¡ 1)+600us(t¡ 1:5)¡ 500us(t¡ 2:25)+300us(t¡ 3) (m/s2),

Ác(t)¢=45us(t¡ 0:5)¡ 45us(t¡ 1:5)¡ 45us(t¡ 2:5)+45us(t¡ 3:5) (deg)). (a) Scenario of flight path. (b) Pitch acceleration. (c) Roll angle.

(d) Roll rate. (e) Side-slip angle. (f) Pitch control fin. (g) Yaw control fin. (h) Roll control fin.

Recall from (8) that in the range of 0· ®T < ¼=4,

fN(®T)¸ 0, fN(0) = 0, f 0N(®T)> 0:

(50)

It is obvious that in case of V 6= 0 and W 6= 0 thefunction N is greater than zero. On the other hand,it follows from (49) and (50) that

N(U,0,W) =k

2

"2jWjpU2 + W2

fN +Up

U2 + W2f 0N

#> 0

(51)

N(U,V,0) =k

2

pU2 + V2

jVjfN

µsin¡1

jVjVM

¶> 0 (52)

and

N(U,0,0) =k

2f 0N(0)> 0: (53)

Thus, we see that (20) is true.

APPENDIX B. PROOF OF THEOREM 1

It is the direct consequence of the conditions in(39) and (40a) that Q0 defined in (34b) is positivedefinite and hence that ¸m(Q0)> 0. Choose aLyapunov-like function Vw by

Vw¢= 12 w

TP0w, w¢=[eT» Umineq Uminr]

T:

Take the total derivative of Vw along the trajectoryof the system in (27) and using (31), we then


have

_Vw =

12_eT

» P11e» +12 eT» P11

_e» +U2mineq

_eq+U2min r

_r

·¡ 12¸m(Q11)ke»k2 +·4

UmaxUmin

jUmineqjke»k

¡ hv2!nUminjUmineqj2

+!n·4ke»k(jpj jVj+·1

Umax½min

j _½j

+·2j_Uj+·2(ayjVj+ j³y j) +

·3½minUmin

j³z1j)

+ hv2UminjUmineqj(jpj jVj+·1

Umax½min

j _½j+·2j_Uj

+·2(ay jVj+ j³yj) +·3

½minUminj³z1j)

+

¯IzxIy

¯!nUminjUmineqj jpj2 + jUmineqj j³z1j

+½maxUmin®M

hv2jUmineqj j

_Uj

¡ hv1!nUminjUminrj2 +

½maxU2maxUminha!n

jUmin rj jVj

+p2hv1UmaxUminjUminrj jpj

+ hv1UminjUminrj(ayjVj+ j³y j)

+

¯Ix¡ IyIz

¯½maxUmaxUmin®M

hv2jUminrj jpj

+

¯IzxIz

¯½maxUmax®M!nhv2

jUmin rj2

+

¯IzxIz

¯!nUminjUmin rj(ap¸jpj+¸2j³p¡ bpÁj)

+

¯Iz ¡ IxIy

+Ix¡ IyIz

¯jUmineqj jUmin rj jpj

+

¯IzxIy¡ IzxIz

¯1

!nUminjUmineqj jUminrj2: (54)

Through some calculating using (30) along the factthat j³z1j · jez1j+ j³ z1j · ke»k+ j³ z1j, we can simplifythis inequality as follows

_Vw ·¡ 1

2¸m(Q0)kwk2

+ kwk(d1j³ z1(t)j+!nd2j _½j+!nd3j_Uj+ d4jpj jVj

+ d5(ayjVj+ j³yj)+ d6jpj+ d7jVj

+ d8jpj2 + d9(ap¸jpj+¸2j³p¡ bpÁj))

+

¯¯Iz ¡ IxIy

+Ix¡ IyIz

¯¯ jpjkwk2

+1

!nUmin

¯¯IzxIz ¡ IzxIy

¯¯kwk3: (55)

By (36) and (37), we can obtain the followinginequality

_Vw ·

¡Ã12¸m(Q0)¡

¯¯Iz ¡ IxIy

+Ix¡ IyIz

¯¯°3kÂp(0)ke¡¾2t

!kwk2

+1

!nUmin

¯¯ IzxIz ¡ IzxIy

¯¯kwk3

+ kwk(d1°1kÂz(0)ke¡¾1t+ d2± _½+ d3± _U+ d4°3°5kÂp(0)kkÂy(0)ke¡(¾2+¾3)t

+ d5(ay°5 + °6)kÂy(0)ke¡¾3t

+ d6°3kÂp(0)ke¡¾2t+d7°5kÂy(0)ke¡¾3t

+ d8°23kÂp(0)k2e¡2¾2t

+ d9(ap¸°3 +¸2°4)kÂp(0)ke¡¾2t): (56)

By (40b), this implies that

_Vw · kwkf(kwk) (57)

where

f(kwk) ¢= 1!nUmin

¯¯ IzxIz ¡ IzxIy

¯¯kwk2¡ 1

4¸m(Q0)kwk

+(b1 + b2e¡¾2t+¹b3e

¡¾0t): (58)

Define

Sw¢=fkwk 2 Rjwmin < kwk< wmaxg (59a)

where

¡¢= 116¸

2m(Q0)¡

4!nUmin

¯¯ IzxIz ¡ IzxIy

¯¯ (b1 + b2 +¹b3)

(59b)

wmin¢=!n(Umin=Izx)¯¯ 1Iz ¡ 1

Iy

¯¯( 18¸m(Q0)¡ 1

2

p¡ ) (59c)

wmax¢=!n(Umin=Izx)¯¯ 1Iz ¡ 1

Iy

¯¯( 18¸m(Q0)+

12

p¡ ): (59d)

Then, we see that

_Vw < 0, 8 kwk 2 Sw: (60)


By way of contradiction using (60), we show that

kw(t)k< b4s¸M(P0)¸m(P0)

, 8 t¸ 0: (61)

If (61) is not true, then there exist t1 and t2 such that

kw(t1)k= b4, kw(t2)k= b4s¸M(P0)¸m(P0)

(62a)

and

b4 < kw(t)k< b4s¸M(P0)¸m(P0)

, 8 t 2 (t1, t2)

(62b)since (40c) and (40d) imply that

wmin < b4 < b4

s¸M(P0)¸m(P0)

<wmax: (63a)

andkw(0)k · b4: (63b)

By this along with (60), we have

12¸m(P0)kw(t2)k2 · Vw(w(t2))< Vw(w(t1))

· 12¸M(P0)kw(t1)k2 (64)

which implies that

kw(t2)k< b4s¸M(P0)¸m(P0)

: (65)

This is contradictory, however, to the assumption in(62a), and hence we have shown that the assertion in(61) is true.Now the following inequality is the direct

consequence of the equation (61)

f(kwk)·¡ 18¸m(Q0)kwk+(b1 + b2e¡¾2t+¹b3e¡¾0t):

(66)

Therefore, it follows from (57) and (66) that

_Vw ·¡ 1

8¸m(Q0)kwk2 + (b1 + b2e¡¾2t +¹b3e¡¾0t)kwk

· ¡14¸m(Q0)¸M(P0)

Vw +(b1 + b2e¡¾2t +¹b3e

¡¾0t)

s2Vw¸m(P0)

:

(67)

By the Comparison Principle [19], we then have

vw(t)· vw(0)e¡¾t+1p

2¸m(P0)

Ãb1¾+ b2

Z t

0

e¡¾(t¡¿)e¡¾2¿ d¿

+¹b3

Z t

0

e¡¾(t¡¿)e¡¾0¿ d¿

!(68)

where vw¢=V1=2w . On the other hand, the convolution

integrals in (68) are bounded as followsZ t

0e¡¾(t¡¿ )e¡¾2¿ d¿ · 1

j¾¡¾2je¡¾mint,Z t

0e¡¾(t¡¿ )e¡¾0¿ d¿ · 1

j¾¡¾0je¡¾mint:

(69)

By (68) and (69), we then have

kw(t)k ·s¸M(P0)¸m(P0)

kw(0)ke¡¾t

+8¸M(P0)

¸m(P0)¸m(Q0)(d2± _½+ d3± _U)

+1

¸m(P0)

µb2

j¾¡¾2j+

¹b3j¾¡¾0j

¶e¡¾mint:

(70)

From (26), (31), (36), and (37), we also have thefollowing inequality

jAz(t)¡ Acz j · jez(t)j+ jAz(t)¡ Acz j

·s1+

µUmaxUmin

¶2kw(t)k

+!n

Ã°3°5kÂp(0)kkÂy(0)ke¡(¾2+¾3)t+·1

Umax½min

j _½j

+·2j_Uj+·2(ay°5 + °6)kÂy(0)ke¡¾3t

+·3

½minUmin(°1kÂz(0)ke¡¾1t+ kw(t)k)

!+ °2kÂz(0)ke¡¾1t: (71)

Now, (41) is the immediate consequence of twoinequalities in (70) and (71). Finally, (42) followsdirectly from (41) since ± _½ = ± _U = 0 when ½ and Uare constant.

REFERENCES

[1] Blakelock, J. H. (1991)Automatic Control of Aircraft and Missiles (2nd ed.).New York: Wiley, 1991.

[2] Hemsch, M. J. (1992)Tactical missile aerodynamics: General topics.Progress in Astronautics and Aeronautics, 141 (1992).

[3] Arrow, A. (1985)Status and concerns for bank-to-turn control of tacticalmissiles.Journal of Guidance, Control, and Dynamics, 8, 2 (1985),267—274.

[4] Williams, D. E., and Friedland, B. (1987)Modern control theory for design of autopilots forbank-to-turn missiles.Journal of Guidance, Control, and Dynamics, 10, 4 (1987),378—386.


[5] Sheperd, C. L., and Valavani, L. (1988)Autopilot design for bank-to-turn missiles usingLQG/LTR methodology.In Proceedings of American Control Conference, 1988,579—586.

[6] Nichols, R. A., Reichert, R. T., and Rugh, W. J. (1993)Gain scheduling for H-infinity controllers: A flightcontrol example.IEEE Transactions on Control Systems Technology, 1, 2(June 1993), 69—79.

[7] Carter, L. H., and Shamma, J. S. (1996)Gain-scheduled bank-to-turn autopilot design using linearparameter varing transformation.Journal of Guidance, Control, and Dynamics, 19, 5 (1996),1056—1063.

[8] Lin, C. F. (1994)Advanced Control Systems Design.Englewood Cliffs, NJ: PTR Prentice Hall, 1994.

[9] Lian, K. Y., Fu, L. C., Chuang, D. M., and Kuo, T. S.(1994)Nonlinear autopilot and guidance for a highlymaneuverable missile.In Proceedings of American Control Conference (1994),2293—2297.

[10] Fu, L. C., Chang, W. D., Yang, J. H., and Kuo, T. S. (1997)Adaptive robust bank-to-turn missile autopilot designusing neural networks.Journal of Guidance, Control, and Dynamics, 20, 2 (1997),346—354.

[11] McDowell, D. M., Irwin, G. W., Lightbody, G., andMcConnel, G. (1997)Hybrid neural adaptive control for bank-to-turn missile.IEEE Transactions on Control Systems Technology, 5, 3(1997), 297—308.

[12] Oh, J. H., and Ha, I. J. (1997)Missile autopilot design via functional inversion andtime-scaled transformation.IEEE Transactions on Aerospace and Electronic Systems,33, 1 (Jan. 1997), 64—76.

[13] Lee, J. I., and Ha, I. J. (1999)Autopilot design for highly maneuvering STT missilesvia singular preturbation-like technique.IEEE Transactions on Control Systems Technology, 7, 5(Sept. 1999), 527—541.

[14] Isidori, A. (1995)Nonlinear Control System (3rd ed.).New York: Springer-Verlag, 1995.

[15] Ha, I. J. (1988)The standard decomposed system and noninteractingfeedback control of nonlinear systems.SIAM Journal Control and Optimization, 26, 5 (Sept.1988), 1235—1249.

[16] Naidu, A. S., and Calise, A. J. (1995)Singular perturbations and time scales in guidance,navigation and control of aerospace systems: Survey.In Proceedings of AIAA Conference, 1995, 1338—1363.

[17] Calise, A. J., and Markopoulos, N. (1994)Nondimensional forms for singular perturbation analysesof aircraft energy climbs.Journal of Guidance, Control, and Dynamics, 17, 3 (1994),584—590.

[18] Franklin, G. F., Powell, J. D., and Emami-Naeini, A. (1994)Feedback Control of Dynamic Systems (3rd ed.).Reading, MA: Addison-Wesley, 1994.

[19] Khalil, H. (1996)Nonlinear Systems (2nd ed.).Englewood Cliffs, NJ: Prentice-Hall, 1996.

[20] Rugh, W. J. (1996)Linear System Theory (2nd ed.).Englewood Cliffs, NJ: Prentice-Hall, 1996.

[21] Apostol, T. M. (1974)Mathematical Analysis.Reading, MA: Addison-Wesley, 1974.


Sang-Yong Lee received the B.S. and M.S. degrees in control and instrumentationengineering in 1993 and 1996, respectively, and the Ph.D. degree in electricalengineering in 2000, all from Seoul National University, Seoul, Korea.He is presently a member of the Research Staff at the i-Networking

Laboratory, Samsung Advanced Institute of Technology, Suwon, Korea. Hiscurrent research interests include control-theoretic approaches to the design offlow control in high-speed networks and their applications to home networksolutions.

Ju-Il Lee received the B.S., M.S., and Ph.D. degrees in control andinstrumentation engineering from Seoul National University, Seoul, Korea, in1992, 1994, and 1999, respectively.He is presently a Senior Engineer at HDD R&D Group, Storage Division,

Samsung Electronics Co., Suwon, Korea. His current research interests includenonlinear control theory and its application to flight control and high precisionservo systems for data storage devices.

In-Joong Ha received the B.S. and M.S. degrees in electronics engineering fromSeoul National University, Seoul, Korea, in 1973 and 1980, respectively, and thePh.D. degree in computer, information, and control engineering (CICE) from theUniversity of Michigan, Ann Arbor, in 1985.From 1973 to 1981, he worked as a Senior Research Engineer in the area

of missile guidance and control at the Agency for Defense Development inTaejon, Korea. From 1982 to 1985, he was a Research Assistant at the Centerfor Research on Integrated Manufacturing, University of Michigan. From 1985to 1986, he worked as a Senior Research Engineer at General Motors ResearchLaboratories, Troy, MI. Since 1986 he has been with Seoul National University,where he is currently a Professor in the School of Electrical Engineering. Hisfields of interest are nonlinear control theory and its application to missiles,robots, electric machines, and other high precision servo systems for factoryautomation and multimedia.Dr. Ha was the recipient of the 1985 Outstanding Achievement Award in the

CICE program.


nonlinear autopilot for high maneuverability of bank-to ...batti/progetti_fda/40.pdfshown that the...

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