Flight Coupling Model Effect Based

Fuzzy Control Logic

Ali Mohamed Elmelhi

Electrical and Electronic Engineering Department

University of Tripoli Tripoli, Libya

ali_elmulhi@yahoo.com

Abstract—The influences of coupling in a launch vehicle between

longitudinal, yaw and roll channels are more serious in the region

of powered or active flight phase. Due to large speed, the effect of

undesired roll and yaw angles induced from coupling dynamics

may cause fluctuations on the performance of a desired

command tracking trajectory in the longitudinal motion. In this

study, the moment of inertia and kinematic couplings are

considered with two objectives in mind. First is to derive their

mathematical models and study their affects on the tracking

accuracy of the desired pitch angle command. Second, by

applying fuzzy proportional derivative controller design, a

significant degradation in the desired command tracking due to

simultaneous effect of these two types of coupling can be

addressed in comparing to the classical design approach. Linear

simulation for a typical launch vehicle is carried out at the end of this paper to demonstrate the objectives of this study.

Keywords-vehicle coupling, fuzzy control, vehicle dynamic

model, proportional derivative controller, cross coupling.

I. INTRODUCTION

Highly maneuvering launch vehicles such as missiles are the most challenging of all guidance and control issues [1] and [2]. Initially Autopilot design is carried out by decoupling the pitch, yaw and roll channels. When this is done, the classical control such as proportional derivative controller [ PD] has proven itself to be a valuable and efficient tool for Autopilot design [3]. Few years later, the coupling problem can be controlled so as to minimize the effects of de-stabilizing roll-yaw couplings [4] and [5]. Recently decoupling controller for high angle of attack of maneuvering missile is designed based on the multivariable control decoupling principle of P and V criterion [6]. Fuzzy logic proposed by Zadeh [7] has much successful application in control theory for system stabilization [8], [9] and [10]. This design technique is applied here as a PD controller in order to stabilize and overcome the fluctuations in pitch angle transient response caused due to exist of the moment of inertia and kinematic couplings. In this paper, the mathematical coupling model have been derived which is the first aim of this work. Next is to study their affects on the accuracy of input desired pitch angle tracking when the Autopilot is designed based on classical and fuzzy PDcontroller design approaches. And the simulation results are only obtained in longitudinal motion where a stability of the launch vehicle is the main purpose of flight control engineer.

In order to accomplish these study objectives, this paper can be arranged as follows. In section II, the moment dynamic equation for the launch vehicle model in three channels is presented. In section III, the derivation of the mathematical models of two types of coupling is obtained. A brief discussion about Autopilot design is included in Section IV. And in section V, linear simulation results are discussed. Finally, the conclusion is presented in section VI.

II. LAUNCH VEHICLE MODEL

The rigid body dynamic equations of motions have been rigorously and systematically derived [11] and [12]. These equations serve as the mathematical model of the Launch vehicle flight dynamics. Which are derived based on the following considerations:

· The earth is ellipsoidal.

· Effect of earth rotation.

In this section, without including of coupling dynamics, the moment dynamic equation of pitch, yaw and roll channels are only presented here. And this model is required for deriving the transfer functions of the launch vehicle in three channels as will be seen in section V.

The dynamic equation of rotations in three axis x , y and zare given as

Iy wy = myβ

QSl β + myr QSl2 v⁄ "r + T(lT − xz) √2⁄ "δψ (1)

Ix wx = mxp

QSl2 v⁄ "p + Tzrδϕ (2)

Izwz = mzαQSl α + mz

qQSl2 v⁄ "q + T(lT − xz) √2⁄ "δθ (3)

And the rate of change of Euler angles can be obtained as follows.

θ = wz (4)

ψ = wy (5)

ϕ = wx (6)

978-1-4673-2679-7/12/$31.00 ©2012 IEEE 385

Where

wx , wy , wz are the roll, yaw and pitch angular rates( deg/

sec ); Ix , Iy , Iz are the moment of inertia in x , y and z

directions ( m2 . sec); myβ is the yaw damping moment due to

side slip angle β; mzα is the pitch damping moment due to angle

of attack α; mz ,q

my r , mx ,

pare the pitch, yaw and roll damping

moments due to angular rates; Q is the dynamic pressure (kg. m−1 . sec−1); S is the reference surface area (m2); l is the length along vehicle axis (m); V is the vehicle velocity (m/sec); α and β are the angle of attack and sideslip (deg); p , q and rare the roll, pitch and yaw angular rates (deg/sec); θ ,ψ , ϕ are the Euler angles (deg); δθ ,δψ , δϕ are the pitch, yaw and roll

swing angles (deg); T is the effective thrust (Newton); lT is the distance from theoretical tip of vehicle to the engine swing axis (m); xz is the distance from theoretical tip of vehicle to center of mass and zr is the distance between the longitudinal axis to the vibration pivot (m) [14].

The basic functional block diagram of the considered launch vehicle Autopilot system in each channel is shown in Fig. 1. In this diagram, the locations where the considered coupling dynamics are inserted to the system are also shown.

autopilotpitch

dynamics

servoelevatordynamicsbody

rigidpitch

autopilotrolldynamics

servoaileron

dynamicsbody

rigidroll

autopilotyawdynamics

servorudder

dynamicsbody

rigidyaw

angle

pitchdesired

angle

rolldesired

angle

yawdesired

dynamics

kinamatic

dynamicsinertia

ofmoment

jθ&

dynamics

kinamatic

jf&

dynamics

kinamatic

dynamicsinertia

ofmoment

jy&

zw

xw

yw

qd

fδ

yd

( )( )yxzyx wwIII -

( )( )zxyxz wwIII -

q

f

y

Figure 1. General block diagram for launch vehicle stabilization

III. COUPLING MODEL

In order to carry out the effect of the considered couplings on the performance of a longitudinal motion, the (1) to (6) are modified by adding the following derived moment of inertia and kinematic models:

A. - Moment of Inertia Model

Under the assumption that the launch vehicle is a rigid body dynamics, the equation of rotation can be derived as follows. The equation of motion called Euler equation is expressed as [12]

dH�� dt⁄ = M��� (7)

Where H�� is a vector of angular momentum of the body and expressed by

H�� = I. w��� b (8)

With I represents the vector of inertia tensor of the body and w��� b is a vector of angular velocity.

From the general point of view, for a given body frame Sb

with axes xb , yb and zb . The vector u� relates to vector v� via a

tensor matrix Ts��� by the following expression:

(u� )b = "Ts��� #

b. (v� )b (9)

Where "Ts��� #

b is given by the following matrix

"Ts��� #

b= $txbxb txbyb tzbxb

txbyb tybyb tybzb

tzbxb tybzb tzbzb

% (10)

Equation (7) and (8) can be expressed in body frame Sb as

"dH�� dt⁄ #b

= "M��� #b (11)

"H�� #b

= "I#b

. (w��� b)b (12)

Where

"M��� #b

= "Mxb Myb Mzb #T (13)

"H�� #b

= "Hxb Hyb Hzb #T(14)

(w��� b)b = �wxb wyb wzb T (15)

Based on (10) and after some manipulations, the expression of

tensor matrix for moment of inertia �I b in (12), can be written

in matrix form as [12]

�I b

= " Ixb −Ixbyb −Izbxb−Ixbyb Iyb −Iybzb−Izbxb −Iybzb Izb

$ (16)

And for completing the derivation procedure, the rule of differentiation in rotating body frame is required which can be obtained as following:

386

Fig 2 shows the vehicle coordinate frame Sb rotating by

angular momentum H%% with angular velocity w%%% b where

w%%% b = wxb ib + wyb jb + wzb k% b (17)

H%% = Hxb ib + Hyb jb + Hzb k% b (18)

by

bz

bwr

H

bx

Figure 2. Vector in rotating frame

The derivative of the vector H%% with respect to time t is

dH%% dt⁄ = (dHxb dt⁄ )ib + �dHyb dt⁄ jb + (dHzb dt⁄ )k% b

+Hxb (dib dt⁄ ) + Hyb (djb dt⁄ ) + Hzb �dk% b dt⁄

It is known that [12]

(dib dt⁄ ) = w%%% b × ib , djb dt⁄ = w%%% b × jb

dk% b dt⁄ = w%%% b × k% b

Then it holds

dH%% dt⁄ = '(dHxb dt⁄ )ib + �dHyb dt⁄ jb + (dHzb dt⁄ )k% b*+ w%%% b × �Hxb ib + Hyb jb + Hzb k% b

The above equation can be written in matrix form as

�dH%% dt⁄ b

= d�H%% b

dt⁄ + (w%%% b)b×�H%%

b(19)

This represents the rule of differentiation in rotating body frame Sb.

Where �dH%% dt⁄ b

is the column matrix of components of

derivative �dH%% dt⁄ , while 'd�H%% b

dt⁄ *b is the derivative of a

column matrix of components of vector H%% . Then from (19) and (11), the equation of rotation motion in body frame is

�dH%% dt⁄ b

= d�H%% b

dt⁄ + (w%%% b)b×�H%%

b= �M%%%

b(20)

Based on (12) and (20), the following alternative expression is obtained:

�I b(d(w%%% b)b dt⁄ ) + (w%%% b)b

�I b(w%%% b)b = �M%%%

b (21)

Although the inertia matrix is a variable, the term 'd�I b

dt⁄ * (w%%% b)b is not appearing in the above equation. This

is because the moments of inertia in three frame axes have been represented by constant numerical values.

Because body axes are taken as principle axes then all products of inertia are equal to zero. As a result, the inertia matrix in (16) becomes

�I b

= "Ix 0 0

0 Iy 0

0 0 Iz

$ (22)

And because the considered launch vehicle is a rigid body with a symmetrical configuration, the moment of inertia in pitch and yaw channels are identical. So let

Iy = Iz = Ic

Then (22) can be rewritten as

�I b

= +Ix 0 0

0 Ic 0

0 0 Ic

, (23)

Similarly the moments and angular velocities given respectively in (13) and (15) can be expressed as

�M%%% b

= �Mx My Mz T (24)

(w%%% b)b = �wx wy wz T (25)

Finally by solving (21), (23), (24) and (25), the moment of inertia coupling can be given by the following dynamic equation of rotations:

wx = Mx Ix⁄ (26)

wy = ((Ic − Ix)wx wz Ic⁄ ) + My Ic⁄ (27)

wz = −'(Ic − Ix)wx wy* Ic. + Mz Ic⁄ (28)

By comparing to the launch vehicle model in the previous section, the total moments in three coordinate axes are substituted by the following expressions:

Mx = �mxp

QSl2 v⁄ p + Tzrδϕ

My = myβ

QSl β + �myr QSl2 v⁄ r + �T(lT − xz) √2⁄ δψ

Mz = mzαQSl α + �mz

qQSl2 v⁄ q + �T(lT − xz) √2⁄ δθ

387

B. Kinematic Dynamic Model

The angular velocities wx , wy and wz given in previous

sections are the components of angular velocity vector w��� b in the body frame Sb of the launch vehicle with respect to inertial frame Si. This case when the earth rotation given by angular

velocity (wE = 7.292115 × 10−5 rad/sec) is not considered. In case of earth rotation, the launch vehicle attitude motion can be described with respect to a special kind of inertial frame

denoted by Sj . This frame with axes Oxj y j zj is different from the

inertial frame Si with axis Oxi y i zi. Where O is the earth center,

axis zi is normal to equatorial plane, axis xi is in equatorial plane and its direction can be specified arbitrarily and yi

complements the right hand system. This is as shown in Fig.3

below. Where Oxe ye ze represents the axes of the earth frame.

iz

ix

iyey

ex

ez

Figure 3. Frames Si and Se

At initial time to , the axes xj , yj and zj coincide with axes

xl , yl and zl of the launch frame Sl [vehicle frame before flight time start]. The inertial frame Sj maintains its direction in the

inertial space and does not rotate with the earth as in case of inertial frame Si [12].

Accordingly, the attitude of the launch vehicle body frame Sb with respect to inertial frame Sj is determined by the Euler

angles θj ,ϕj and ψj. So based in Fig.4, the relation between

the angular velocities wx , wy and wz with respect to change

of angle rates ϕj ,ψj

and θj can be described by the following

matrix form:

bx

by

bz

jx

jz

jf&

jy&

jq&

jf

jfjy

jy

jq

jq

Figure 4. Transition from Sj to Sb

&wx

wy

wz

' = ( dϕj dt⁄*dψj dt⁄ +cosϕj−*dψj dt⁄ +sinϕj

,+ Lbj & 0

0

dθj dt⁄ '

&wx

wy

wz

' = ( *dϕj dt⁄ + − *dθj dt⁄ +sinψj*dψj dt⁄ +cosϕj + *dθj dt⁄ +sinϕjcosψj−*dψj dt⁄ +sinϕj + *dθj dt⁄ +cosϕjcosψj

, (29)

From the basic of flight geometry and because the body frame

Sb is concided with Sj at the starting time of flight, the above

inertial to body transformation matrix Lbj is expressed by

lbj = - cos θj sinθj −sinψj−sinθj cosθj sinϕjcosψjψjcosθj + ϕjsinθj ψjsinθj −ϕjcosθj cosϕjcosψj

.Solving (29) to get

(ϕjψjθj

, = (wx + tanψj*wysinϕj + wzcosϕj+wycosϕj − wzsinϕj*wysinϕj + wzcosϕj cosψj⁄ + , (30)

This represents the rate of change of Euler angles with respect

to inertial frame Sj where the earth rotation is not considered.

And in order to derive the kinematic model, the effect of the earth rotation is taken into account which in turn causes the model in (30) to be modified as follows:

The absolute angular velocity vector w��� b and relative angular velocity w��� r are linked by [12]

(w��� r)b = (w��� b)b − (w��� E)e (31)

Then

&wr,x

wr,y

wr,z

' = &wx

wy

wz

'− lbl lle & 0

0

wE

' (32)

Where

lle = &l11 l12 l13

l21 l22 l23

l31 l32 l33

'Represents the earth to launch frame transformation matrix in which

l11 − cosA sinB cosL− sinA sinL

l12 = −cosA sinB sinL + sinA cosL

l13 = cosA cosB , l21 = cosB cosL ,

l22 = cosB sinL , l23 = sinB

l31 = sinA sinB cosL− cosA sinL

l32 = sinA sinB sinL + cosA cosL , l33 = −sinA cosB

388

With A and L respectively represent the Azimuth and latitude angles. And the transformation matrix lbl is equal to lbj when ϕj = ϕ,ψj = ψ and θ j = θ .

It is known that, the differential equation which describing the variation of attitude angles (ϕ,ψ,θ) with respect to inertial frame Sj is obtained by (30). And by substituting wx , wy and

wz by equivalent relative angular rates and after manipulations,the following kinematic model equation in body frame is obtained:

"ϕk

ψk

θk

$ = %wr,x + tanψ&wr,y sinϕ+ wr,zcosϕ'wr,y cosϕ−wr,zsinϕ&wr,ysinϕ+ wr,zcosϕ' cosψ⁄ , (33)

IV. AUTOPILOT DESIGN

In order to stabilize and addressing the effect of coupling on the considered launch vehicle dynamics in pitch, yaw and roll motions, the following two controller design approaches are designed:

A. PD Controller

This type of controller can be expressed by the following differential equation [13]:

u(t) = kpe(t) + kd e(t) (34)

Where, e(t) is the error difference between the desired objective angle and an actual feedback angle in each channel. kp; is the proportional gain and kd is the detrivative gain. And

u(t) is the control signal which in this case represents directly the actuator deflections δθ ,δψ and δϕ . This is because of a

high bandwidth of the servo actuator dynamics.

B. Fuzzy PD Controller

Due to limited space, a brief description about fuzzy PD controller is given here and for more details, the reader should refers to [8] and [14]. The schematic diagram which describes the structure of this controller in each channel is shown in Fig.5. Where, the triangular membership functions are considered here for fuzziffication process of two dimensional inputs error e(t) and rate of error change e(t) . Where themembership functions indicate the degree of certainty to which a value belongs to the class labeled by the linguistic description. And both of the input linguistic variables can be described by fuzzy set [ NB , NM , NS , ZE , PS , PM , PB].

With NB denotes to Negative Big. NM; Negative Medium. NS; Negative Small. ZE; Zero. PS; Positive Small. PM; Positive Medium and PB represents a Positive Big.

The fuzzy control rules are designed from the expert knowledge which are typically expressed as follows.

if e(t) is Nej and e(t) is Nej

then u(t) is ufj

Where ufj is the jth fuzzy output center rule. Nej

and Nej

represent respectively the linguistic variables for e(t) and e(t). The logic operation in the designed fuzzy rules that combines

the meaning of two linguistic inputs can be represented by minimum operation. This type of process is known as inference mechanism.

The last step for fuzzy PD controller is a deffuzification process. This evaluates the crisp output value via the center of area method given by the following formula [8] and [14]:

u(t) = ∑ /µjufjµj⁄ 1n

j=1 (35)

Where n is the number of rules; µj; is the output of minimum

operation and ufj is the rule center which represents the

conclusion of jth rule.

Inference

mechanism

Rule base

fuzziffica

tio

n

Rigid body

dynamics

de

ffu

zzific

atio

n

)t(e

][Autopilot controllerfuzzy

angle

desireinput

angle

output

+

-dtd

Figure 5. Fuzzy PD structure

V. SIMULATION RESULTS

Based on the launch vehicle model written in section II and III, the block diagram shown in Fig.1 is simulated under the following simplifications:

· The launch vehicle is symmetrical so that the longitudinal and lateral motions are identical.

· The time constant for the elevators, ailerons and rudders is assumed to be 0.01 sec. consequently their dynamic transfer functions are neglected.

· The transfer function dynamics for the free gyro used to detect pitch yaw and roll angles are unity due to its fast response [high bandwidth dynamics]. As a result the outer loop feedback in each channel is unity.

This linear simulation is carried out at one operating point during flight trajectory i.e. at instant of time [ t≈38 sec] where the dynamic pressure is very high as shown in Fig 6.

According to trajectory data obtained from a nonlinear simulation of the equation of motions presented in [12] and [15]. And by solving (2) and (3), the following two transfer functions are obtained:

θ(s)δθ(s)

=48.35

(s2 + 0.1264s + 40.91)

ϕ(s)

δϕ(s)=

91.18

s(s + 0.862)

And because of a symmetrical configuration of the launch

vehicle, the yaw transfer function #ψ(s) δψ(s)⁄ & is ignored

here.

389

Figure 6. Dynamic pressure

In this section, both the classical and fuzzy PD controllers are applied as an Autopilot control dynamics for stabilizing the launch vehicle attitude control motion. And because of mainly concerned to a stabilization of the longitudinal motion, so only the related simulation results are presented here.

In order to test the effectiveness of a controller designed approach, both the classical PD gains and fuzzy PD parameters [membership functions and rule centers] are tuned until an acceptable tracking performance is achieved. In case of fuzzy PD controller, the designed fuzzy rule base centers are shown in TABLE 1 while the triangular membership functions for e(t) and e(t) are shown respectively in Fig.7 and Fig.8.

And for classical PD controller, the design values for

kp and kd are 5.84 and 0.351 respectively.

TABLE I. FUZZY RULE BASE CENTERS FOR PITCH AUTOPILOT

NB NM NS ZERO PS PM PB

NB 21 21 21 21 14 7 0

NM 21 21 21 14 7 0 -7

NS 21 21 14 7 0 -7 -14

ZERO 21 14 7 0 -7 -14 -21

PS 14 7 0 -7 -14 -21 -21

PM 7 0 -7 -14 -21 -21 -21

PB 0 -7 -14 -21 -21 -21 -21

Figure 7. Membership functions for e(t)

Figure 8. Membership functions for e(t)

A. Nominal Condition

In this case, the launch vehicle is decoupled and based on the block diagram shown in Fig.1, the simulation is carried out without including the coupling dynamics. The output response for the pitch angle using classical and fuzzy PD controllers is obtained as shown in Fig.9 and Fig.10 respectively. It is observed that, a better tracking accuracy to an objective desired pitch angle represented by a step input command can be achieved when the fuzzy PD controller is used. Where, a minimum overshoot and steady state error can be obtained as written in Table II.

TABLE II. SIMULATION RESULTS

DesignMethod

Nominal Case Coupling Case

Steady state

error

Overshoot Steady state

error

Overshoot

Classical PD 0.134 0.223 0.134 0.487

Fuzzy PD 0.015 0.0065 0.015 0.025

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8x 10

4

Time

Dynam

ic P

ressure

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Err

or

[rad]

Time (sec)

NB NM NS Zero PS PM PB

-20 -15 -10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [sec]

Change o

f err

or

[rad]

NB NM NS Zero PS PM PB

390

B. Coupling Condition

Based on the mathematical model of the moment of inertia written in (26) to (28) and the kinematic model in (33), both types of couplings are applied. Where, in case of kinematic model, the rate of change of Euler angles will be equal to

θ = θ+ θk , ϕ = ϕ+ϕk , ψ = ψ+ ψk

With θk ,ψk and ϕk are the perturbed angles generated due to effect of kinematic motion.

Accordingly, the simulation is carried out and the results are obtained as shown in Fig.9 and Fig.10. In which the pitch angle tracking under a simultaneous effect of the couplings is shown. And Table II shows the numerical simulation results for overshoot and steady state error under this case. It is concluded that, these two types of coupling have a significant effect on the transient response due to overshoot. This can be observed in case of classical PD control design where fuzzy PD can keep an excellent tracking response.

Figure 9. Step response in case of classical PD. [solid line for nominal case

and dotted line for coupling case]

Figure 10. Step response in case of fuzzy PD. [solid line for nominal case and

dotted line for coupling case]

VI. CONCLUSION

The launch vehicle couplings due to moment of inertia and kinematics have been derived. And its effect on the longitudinal trajectory performance is shown. From the derived

model, it is seen that, the coupling models are caused due to angular rates and earth rotation.

An aerodynamic vehicle has been controlled so as to minimize the effects of these couplings. Where, from simulation results, it is concluded that, they cause degradation on the pitch angle tracking performance due to overshoot in transient response. This can be observed when the longitudinal Autopilot is designed based on classical proportional derivative controller. In contrast, the equivalent fuzzy controller gives a good launch vehicle Autopilot performance where the influence of these undesirable dynamics is minimized and maintains an accurate desired tracking.

Finally, it is recommended that, during Autopilot design, it is preferable to consider the coupling dynamics in order to avoid system un-stabilization and performance degradation.Furthermore, this study emphasis that the fuzzy PD is one of the robust controller design approaches which can be used to overcome of these undesired phenomena’s.

REFERENCES

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[3] Blacke lock, J.H, Automatic Control of Aircraft and Missiles, John

Wiley&Sons, Inc, 1965.

[4] Farrell J. Perdreauville, Roll Control of an Atitude – Controlled

Aerodynamic Vehicel Which has Strong Roll-Yaw Coupling, SAND79-1940, October 1979.

[5] F. William Nesline, Brian H. Wells and Paul Zarchan, A Combined

Optimal/Classical Approach to Robust Missile Autopilot Design, Journal of AIAA, 1979.

[6] Luo Xutao, Jia Xiaohong, Zhang Xiangyang, Design of Decoupling

Control System with High Angle of Attack Based on PV Criterion,Proceedings of the 2009 International Symposium on Web Information

Systems and Applications, Nanchang, P. R. China, May 22-24, pp. 178-181, 2009.

[7] Zadeh. L. A, Fuzzy Sets, Information and Control, vol. 8, pp. 338-353,

1965.

[8] Kevin M. Passino, Stephen Yurkovich, Fuzzy Control, Addison Wesley Longman. Inc, Copyright c_ 1998.

[9] Lee C.C, Fuzzy Logic in Control Systems: Fuzzy Logic Controller Part

I, IEEE Trans. Syst.Man.Cyb, Vol.20, pp.404-418, 1990.

[10] Lee C.C, Fuzzy Logic in Control Systems: Fuzzy Logic Controller Part

II, IEEE Trans. Syst.Man.Cyb, Vol.20, pp.419-435, 1990.

[11] Arthur L. Greensite, Analysis and Design of Space Vehicle Flight Control System, copyright by Spartan books, printed in USA, 1970.

[12] Xiao Ye Lun, Class Lecture Notes, Beijing University of Aeronautics

and Astronautics, Beijing, 2002.

[13] Katsuhiko Ogata, Modern Control Engineering, third edditionj, Pritic Hall, 1996.

[14] Ali Mohamed Elmelhi, Flight Vehicle’s Stabilization Based Fuzzy Logic

Control Design and Optimization, ph. d thesis, Biejing University ofAeronautics and Astronautics, Biejing, china, Nov.2005.

[15] Tahir Mustafa, Flight Program Design and Trajectory Optimization of a

Ballistic Missile, Master thesis, Biejing University of Aeronautics and Astronautics, NOV.2003.

0 0.5 1 1.50

0.5

1

1.5

Time [ sec ]

Pitch a

ngle

[ r

ad ]

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Pitch a

ngle

[ra

d]

391