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Satellite Orbits Control Using 1Adaptive Neural Networks

Predictive Controllers (ANNPC)2

A. F. Aly, M. Naguib Aly, M. A. ZayanFaculty of Engineering Alexandria University, Nilesat Company

Alexandria, Egypt

[email protected]

1 0-7803-7651-X/03/$17.00 © 2003 IEEE 2 IEEEAC paper #1218

Abstract —This paper develops an Adaptive Neural Network Predictive Controller (ANNPC) to predict satellite

thruster force and control osculating orbital elements duringmaneuvers. An adequate mathematical satellite model isimplemented to simulate the satellite orbit trajectory duringthrusting maneuver. When using Adaptive Neural Network

(ANN) for control, two steps, System Identification andControl Design, are used. In The system identificationstage, an ANN model is developed to represent the forwarddynamics of the satellite. The prediction error between the

implemented satellite model output and the ANN output isused as the ANN training signal. In the system controlstage, the ANN model is used to predict future satelliteresponses to potential control signals. Using ANNPC inorbit control will optimize the thrust forces and satellite

parameters due to its inherent characteristic. ANNPC will be efficient in the autonomous satellite generations and canchange the way space segment and missions operate.

TABLE OF CONTENTS

…………………………………………………….

1. INTRODUCTION .................................................1 2. ORBITAL ELEMENTS VARIATION.....................1

3. SATELLITE MANEUVER SIMULATION ..............5 4. SATELLITE ORBIT CONTROL USING NEURAL

NETWORK PREDICTIVE CONTROL.......................9 5. CONCLUSION...................................................13 ACKNOWLEDGEMENT ........................................13 R EFERENCES.......................................................20

1. INTRODUCTION

The satellite motion is affected by different disturbanceforces depending on the satellite altitude, which lead tochange of the desired path and location. Aside from the

natural perturbation forces, the motion of a spacecraft isalso affected by the action of an onboard thruster system[1]. Thrusters are frequently applied for orbit control,attitude control, or combination of both. After a spacecraft

has been placed in an operational orbit about the earth,subsequent maneuvers will be required to correct the

satellite orbit [2]. In this paper, the Model PredictiveControl (MPC) and ANN [3, 4] techniques have been usedsuccessfully to control the satellite orbits during thrustingmaneuvers. MPC was conceived in the 1970s primarily by

industry. Its popularity steadily increased throughout the1980s. Model Predictive Control has developedconsiderably over the last few years. At present, there aremany applications of predictive control successfully in use

not only in the process industry but also applications to thecontrol of a diversity of processes ranging from robotmanipulators [5, 6] to aerospace industry. The reason for this success can be attributed to the fact that the MPC is themost general way of posing the process control problem in

the time domain. MPC formulation integrates optimalcontrol, stochastic control, control of processes with deadtime, multi-variable control, and future references whenavailable. Another advantage of MPC is handling the finitecontrol horizon used constraints and in general non-linear

processes, which are frequently found in industry. Thecapabilities of the multi-layer Neural Network (NN) [7, 3]with non-linear function have been applied very

successfully in the identification and modeling of dynamicsystems. Combing both techniques of the MPC and ANN[8, 9] makes it a popular choice for modeling non-linear systems and for implementing general-purpose non-linear controllers.

2. ORBITAL ELEMENTS VARIATION

The satellite orbit is an ellipse, parabola or hyperbola if it isinfluenced only by the gravitational filed of a point mass or spherical body. The orbit elements can be calculated from position and velocity vector at any time but these elements

will be invariant. Practically, the satellite motion is

perturbated by different forces and the calculation of theorbit elements will yield a different set of values over aninterval of time. This orbit with varying parameters is called

an osculating orbit. The orbital elements can be treated asthe dependent variables of a set of first order differential

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equations. Conversely, the position and velocity vectors can be calculated directly from the set of evolving parameters at

any time. In the following analysis [1] let ∂ indicate a

change in an orbital variable due to the application of a

vector f of acceleration other than due to the sphericallysymmetrical central gravitational field. The change in the

energy per unit mass over a time interval ∂ t is as follows:

2

massunit per 2

1v E kinematic = (1)

)( massunit per r

E potential

µ −= (2)

)2

( massunit per a

E total

µ −= (3)

t a

a E ∂=

∂=∂ f v.

2 2

µ . (4)

Referring to the orbital plane axes (ir, iθ, iz) where:

• ir is along the radius vector, away from

the center of attraction.• iθ is perpendicular to r in the plane of the

motion and in the direction of increasingthe true anomaly ( θ ).

• iz is the normal to the plane motion.

Where the satellite mass is m, velocity vector is v, position

vector is r, the ellipse major-axis of orbit is a, and the Earthgravitational coefficient is µ. In the limit this gives the rate

of change of energy as

).(2 2

f v µ

a

dt

da= . (5)

Since

)).((2 vrvr ××=h . (6)

Where h is the angular momentum per unit mass and the

absolute value h= |h| is known as areal velocity. The rate of

change of angular momentum is

hdt

dh/)).(( f rvr ××= (7)

h/).( vrf r ××= (8)

hr /)].)(.().([ 2f rvrf v −= . (9)

In order to obtain the rate of change of eccentricity (e)

)1( 22 eah −= µ . (10)

By differentiate

).)(.().)([(1 2

f rvrf v +−= r paaedt

de

µ , (11)

where p is the semi-latus rectum

µ /2h p =

Now to calculate out- of -plan elements (i, Ω , ω)

Ω−

Ω

=×=

i

i

i

h

cos

cossin

sinsin

vrh . (12)

Where i is the inclination, Ω is the right ascension of the

ascending node and ω is the argument of perigee.Differentiate this equation and arrange the result in the form

−

Ω

Γ =×

h

ih

ih

sin

2f r , (13)

where

Ω−ΩΩ

ΩΩ−Ω

=Γ

ii

ii

ii

cossin0

sincoscoscossin

sinsincossincos

2 , (14)

which is an orthogonal matrix, transpose equals the inverse.

Therefore

)(

/

/

/sin

2 f r ×Γ =

−

Ω

T

dt dh

dt hdi

dt id h

. (15)

Note that the elements i and Ω refer to the same axes as thevectors on the right hand side of the above equation in the

inertial Geocentric Equatorial Axes (i, j, k) [1, 10]. The

final form depends on the final axes in use. The rate of

change of the true anomaly θ and the rate of change due

only to f is θ ~

∂ are derived by applying the perturbation to

the following equation

θ

µ

cos1

2

e

h

r +

= (16)

)1/( 2e pa −= (17)

µ θ θ θ /2~

sincos hhreer ∂=∂−∂ . (18)

In the limit

dt

dhh

dt

der

dt

d re

µ

θ θ

θ 2

cos

~

sin −= (19)

θ µ sin).( er h =vr . (20)

By differentiating

++−=

dt

dh

hh

p

dt

der

dt

d re

1).().(sin

~

cos vrf rθ θ

θ

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dt

dhh

h

ph p

dt

d re

−+= θ µ

θ θ θ

sin2

).(cos).(cos)/(

~

2vrf r

her er ph

h

p/]cos)cos1([sinsin

2).(cos

2θ θ θ θ

µ θ −++−=

−vr

hr p /)(sin +−= θ ,

and finally

+−=dt dhr p p

rehdt d θ θ θ sin)().(cos1

~

f r . (21)

The variation of the argument of perigee ω is obtained by

)cos(].0sincos[. ω θ +=ΩΩ= r x rri .

Differentiation results in

++−=

ΩΩΩ−

dt

d

dt

d r

dt

d

ω θ ω θ

~

)sin(

].0cossin[ r

. (22)

Therefore

r i

r

)sin(cos

0

0]0cossin[].0cossin[ 12

ω θ +=

Γ Γ ΩΩ−=ΩΩ− r

where

++

+−+

=Γ

100

0)cos()sin(

0)sin()cos(

1 ω θ ω θ

ω θ ω θ

+−=

Ω

dt

d

dt

d

dt

d

i

ω θ ~

cos , (23)

or

dt

d

dt

d i

dt

d θ ω ~

cos −Ω

−= . (24)

We now have the required equations for variations of theorbital elements. The final form of the equations depends onthe axes in use.

Tangential and Normal Components

From the orbital plane axes (ir, iθ, iz) then

r rf =

f r.,

and

γ sin. rv=vr ,

where γ is the angle between the velocity vector v and iθ

measured clockwise from the latter

θ

θ γ

cos1

sintan

e

e

+= .

Also

θ

θ

θ µ

θ θ µ

f r

h f

h

e

f eef p

r

r

+=

++=

sin

])cos1(sin[/.f v

.

The required transformation from i j k axes to r θ z axes

ijk

f r )(

/

/

/sin

2

×Γ =

−

Ω

T

dt dh

dt hdi

dt id h

.

zrf r θ )(122 ×Γ Γ Γ =T

−Γ =

θ rf

rf z

0

1 (25)

Use of Tangential and Normal Component (t, n, z) in orbital

plane

)cos21( 2ee p

v ++= θ µ

θ µ

γ sinsin e p

v =

θ µ

γ cos1(cos e p

v +=

t

nt

nt

vf

f v

h f

vh

re

f f r

=

−=

−=

f v

f r

.

sin

)cossin(.

θ µ

γ γ

by means of transformation from t n z axes to r θ z

−

=Γ

100

0sinsin

0cossin

3 γ γ

γ γ

,

since

+

−

−

=×

γ γ

γ

γ

cossin

sin

cos

)(

t n

z

z

tnz

rf rf

rf

rf

f r ,

then

+

−Γ =

−

Ω

γ γ cossin

0

/

/

/sin

1

t n

z

f f

f r

dt dh

dt hdi

dt id h

. (26)

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Summary Of Equations In Tangential-Normal (t, n, z) Axes

t f va

dt

da

µ

22= (27)

]sin)cos(2[1

nt f a

r f e

vdt

deθ θ −+= (28)

]sin[ nt f e f r

p

pv

rh

dt

dhθ += (29)

ω θ θ +=∗

(30)

dt

d ir h

dt

d

dt

d

dt

d

dt

d Ω−=++= cos/

~2

* ω θ θ θ (31)

])cos2(sin2[1

~

nt f a

r e f

evdt

d θ θ

θ ++−= (32)

z f ih

r

dt

d

sin

)sin( θ ω +=

Ω(33)

z f h

r

dt

di )cos( θ ω += (34)

dt

d

dt

d i

dt

d θ ω ~

cos −Ω

−= (35)

9

THETA+OMEGA

8

omega

7

i

6

OMEGA

5

theta

4

THETA

3

h

2

e

1

a

f(u)

y9

f(u)

y8

f(u)

y7

f(u)

y6

f(u)

y5

f(u)

y4

u(6)

y3

u(5)

y2

u(4)

y1

f(u)

x9

f(u)

x8

f(u)

x7

f(u)

x6

f(u)

x5

f(u)

x4

f(u)

x3

f(u)

x2

f(u)

x1

Mux

SysMux

1/s

Integ9

1/s

Integ8

1/s

Integ7

1/s

Integ6

1/s

Integ5

1/s

Integ4

1/s

Integ3

1/s

Integ2

1/s

Integ1

3

fz

2

ft

1

fn

Figure 1- Satellite Orbits Trajectory Model during Thrusting Maneuver (Satellite Plant Model) Using S IMULINK

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3. SATELLITE MANEUVER SIMULATION

Numerical Integration Methods

The high accuracy, which is required in computation of satellite orbit, can only be achieved by using numericalmethods for the solution of the equation of motion. A

variety of methods have been developed for the numericalintegration of ordinary differential equations. Multi-stepmethods with the availability of variable-order and step-sizeare suited for the satellite orbits from near circular orbits tohigh eccentricity orbits without any precautions. Due to

their flexibility, variable order and step-size multi-stepmethods are ideal candidates for use in general satellite orbit prediction and determination systems.

Satellite Orbits Correction

The three component of a corrective velocity (vn , vt , v z ) maneuver affect the 6 orbital elements, and therefore, it isnot common to require the adjustment of all the orbital

elements. Geostationary satellite orbits [11] are assumed to be equatorial orbits with a period equal to the sidereal day(86164.1 s), i.e. corresponding to the daily rotation of theEarth relative to the stars. A satellite of a circular orbit with

radius of approximately 42164 km will appear stationary toan observer on the earth. Although the perturbations onsatellites in geostationary orbits are very small, they becomeimportant due to the tight tolerance arising from the missionrequirements. Station keeping, therefore has to be

performed, and the spacecraft is maneuvered in order tokeep it within strict latitude and longitude limit defining adead-zone. The magnitude of the dead-zone depends uponthe characteristics of the communication antennas and

transponders. It is common with modern communicationsatellites to require that the satellite remains stationaryrelative to the ground within ± 0.1 degree in both latitudeand longitude due to narrow antenna beam width of theground transmitter. If the inclination of the orbits drifts

away from the Equator then the satellite will appear to havea daily oscillation in latitude equal to the magnitude of thenon-zero inclination. The changes in the inclination of ageostationary orbit arise from the effects of the gravitational

attraction of the Moon and the Sun. The perturbationscaused by the Sun and the Moon are predominantly out-of- plane effects causing a change in the inclination and in theright ascension of the orbit’s ascending node. In-plane

perturbations also occur, but these are second order effectsand need to be considered when extremely tight tolerance,i.e. about ± 0.03 degree, is required.

Thrust Forces

The maneuver may conveniently be treated as instantaneous

velocity increments ∆v occurring at the impulsive maneuver

time t m whenever the thrust duration is small as compared tothe orbital period.

)()()( mmm t t t vvv ∆+=−+

. (36)

A substantial amount of propellant is consumed during asingle maneuver, which results in continuous change of thespacecraft mass along the burn. Despite the variety of the

spacecraft propulsion systems, a simple, constant thrustmodel is sufficient to describe the motion of a spacecraftduring thrust. The propulsion system eject a mass of

propellant per time interval dt at a velocity ve.

dt mdm = .

A space craft mass m experiences a thrust

em vF = .

And the acceleration

em

m

mv

Ff

== .

Integration over the burn time ∆t , the total velocity

increment is given by

0

0

)(

1)(

ln)(0

0

0

0m

t t mdmdt t

t t m

m

em

t t

t

e

∆+−=−==∆ ∫∫

∆+∆+

vvf v

0

1ln(m

t mm

∆−−=∆

Fv .

Assuming that a mass has a constant flow rate and making

use of the total velocity increment ∆v, the acceleration may be expressed [10] as

t

m

t mt m

mt

∆

∆≈

∆−−

=v

f

0

1ln

1

)()(

(37)

Satellite Maneuver Modeling

An adequate mathematical satellite orbit trajectory modelduring thrusting maneuver is implemented by treating theorbital elements as the dependent variables of a set of first

order differential equations in (t, n, z) axes using numerical

integration methods. Due to very small interval time of the propulsion system burning required for the satellitemaneuver the equations of Tangential-Normal axes areapplicable to simulate the satellite maneuver duringthrusting period. A model is implemented, using SIMULINK

[12], to simulate the satellite maneuver. There are threeinputs, which represent the three corrective accelerationvectors samples in Tangential-Normal axes (f n , f t , f z ), andnine outputs, which describe the orbital elements states (a,

e, h, θ *(THETA), θ ~ (theta), Ω (OMEGA), ω (omega), i, and

θ *+Ω (THETA+OMEGA) ) as shown in Figure 1.

Case Study (Satellite Maneuver Simulation)

The following case study contains the information of thetypical satellite data history in geostationary orbit, which isused in the implementing of the North-South maneuvers.The main object of the simulated maneuver is to correct the

non-zero inclination of a geostationary satellite, which is

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called North-South maneuver. North-South maneuver is

defined also as the main Station-Keeping maneuver, whichis accomplished by firing one or several thrusters for the period of time required to achieve the change in orbitalvelocity. The main maneuver is occurring in routine

schedule to compensate the perturbation forces. Pre-maneuver is applied to the satellite in advance of themaneuver with 10% of the full thrust capacity (off-modulation thrust) in order to estimate disturbing torques

variations. After fluids of the propulsion system have been

relocated and the estimator has converged, the open loopcompensation is modified according to estimations done andthen full thrust is used. Orbit bulletin before maneuver inadaptive Keplerian elements are shown in Table 1 where

)cos( ω +Ω= ee x

)sin( ω +Ω= ee y

Ω= cossin ii x

Ω= sinsin ii y

GMST −++Ω≈ θ ω L = True-Longitude.

Where GMST=Greenwich Mean Sidereal Time which

denotes the angles between the mean Vernal equinox of date and the Greenwich meridian. The velocity incrementsvectors in Tangential-Normal axes during pre-maneuver with 50 s pulse duration are shown in Table 2. The velocityincrement vectors in Tangential-Normal axes during

maneuver with 57.094 s pulse duration are shown in Table3.

Results of Maneuver Simulation

Applying the velocity increment at 19/03/2000 9:08:57UTC and propagating the orbital parameters starting fromthe last orbital determination process to get the epoch of theorbital elements at the thrusting time [13]. Using theimplemented model in SIMULINK with sampling time =

0.57094 s and plotting the orbital elements during pre-maneuver and maneuver, Figure 2 shows the accelerationsamples of the pre-maneuver and maneuver respectively.Figures 3, 4, and 5 describe the variation in the orbitalelements due to the thrusting forces. It is shown that

influence of the thrusting forces decrease the inclination andsemi-major axis values. Also, it is clear that the direction of the velocity increment is very important in adjusting theorbital elements. Reviewing the previous differential

equations (27~35), it is seen that the value of the inclinationwill be very sensitive to sign value of the cos(θ * ). Increasingthe value of the velocity increment in -z direction is notsufficient to guarantee the decreasing in the inclinationangle because it is important to supervise and predict all

other parameters related to the satellite orbit duringmaneuver. Tables 4 and 5 contain the initial and final valuesof orbital elements during pre-maneuver and total maneuver (pre-maneuver + maneuver). It is shown that small changes

in inclination leads to large amount changes in the perigeelocation and the right ascension of the ascending node. Thesimulated satellite plant model has succeeded to achieve themain object of the maneuver and produces a typicalscenario of the satellite maneuver.

0 20 40 60 80 100 120-1.5

-1

-0.5

0x 10

-4

f n

0 20 40 60 80 100 120-1.5

-1

-0.5

0x 10

-4

f t

0 20 40 60 80 100 120-0.025

-0.02

-0.015

-0.01

-0.005

f z

time in sec Figure 2- Acceleration Vectors in Tangential-Normal

Axes ( f n, ft , and f z ) in m/s2

TTaabbllee 11 -- OOr r b biitt BBuulllleettiinn EE p poocchh,, BBeef f oor r ee MMaanneeuuvveer r

iinn AAddaa p pttiivvee K K ee p plleer r iiaann EElleemmeennttss aatt 1166//33//22000022 1122::0066::4400 UUTTCC wwiitthh mm==11008833..225555 k k g g

a 42166.35× 103 meter

e _x 0.00036276 e _y -0.000007898

i _x -0.00049069 degree

i _y 0.00026083 degreeL 353.006504 degree

TTaabbllee 22 -- TThhee VVeelloocciittyy IInnccr r eemmeenntt VVeeccttoor r ss iinn TTaannggeennttiiaall N Noor r mmaall AAxxeess dduur r iinngg PPr r ee--MMaanneeuuvveer r

wwiitthh 5500 ss PPuullssee DDuur r aattiioonn

vn pre-maneuver -0.00119 m/s

vt pre-maneuver -0.00146 m/s

v z pre-maneuver -0.25767 m/s

TTaabbllee 33 -- TThhee VVeelloocciittyy IInnccr r eemmeenntt VVeeccttoor r ss iinn TTaannggeennttiiaall N Noor r mmaall AAxxeess dduur r iinngg MMaanneeuuvveer r wwiitthh

5577..009944 ss PPuullssee DDuur r aattiioonn

vn maneuver -0.00644 m/s

vt maneuver -0.00794 m/s

v z maneuver -1.39760 m/s

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0 2 0 4 0 6 0

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6x 1 0

7

a

0 2 0 4 0 6 0

3 . 7 1

3 . 7 1 5

3 . 7 2x 1 0

-4

e

0 2 0 4 0 6 0

1 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4x 1 0

1 1

h

0 2 0 4 0 6 02 9 5

3 0 0

3 0 5

3 1 0

T H E T A

0 2 0 4 0 6 00

0 . 0 5

0 . 1

0 . 1 5

0 . 2

t h e t a

0 2 0 4 0 6 01 5 2

1 5 4

1 5 6

1 5 8

1 6 0

O M E G A

0 2 0 4 0 6 00 . 0 2 9

0 . 0 3

0 . 0 3 1

0 . 0 3 2

i

0 2 0 4 0 6 01 8 8

1 9 0

1 9 2

1 9 4

1 9 6

tim e

o

m e g a

0 2 0 4 0 6 09 8 . 1

9 8 . 2

9 8 . 3

9 8 . 4

T H E T

A + O M

E G A

Figure 3 - Satellite Orbital Variation during Pre-maneuver (a in m, h in m2/s, angles in degree, and time in s)

5 0 1 0 0 1 5 04 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6x 1 0

7

a

5 0 1 0 0 1 5 03 . 7

3 . 7 2

3 . 7 4

3 . 7 6x 1 0

-4

e

5 0 1 0 0 1 5 01 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4

1 . 2 9 6 4x 1 0

1 1

h

5 0 1 0 0 1 5 02 4 0

2 6 0

2 8 0

3 0 0

T H E T A

5 0 1 0 0 1 5 00

0 . 2

0 . 4

0 . 6

0 . 8

t h e t a

5 0 1 0 0 1 5 01 4 0

1 6 0

1 8 0

2 0 0

2 2 0

O M E G A

5 0 1 0 0 1 5 00 . 0 2 4

0 . 0 2 6

0 . 0 2 8

0 . 0 3

i

5 0 1 0 0 1 5 01 2 0

1 4 0

1 6 0

1 8 0

2 0 0

tim e

o m e g a

5 0 1 0 0 1 5 09 8 . 3

9 8 . 4

9 8 . 5

9 8 . 6

T H E T A +

O M E G A

Figure 4 - Satellite Orbital Variation during Maneuver (a in m, h m2/s, angles in degree, and time in s)

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0 5 0 1 0 0 1 5 04 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6

4 . 2 1 6 6x 1 0

7

a

0 5 0 1 0 0 1 5 03 .7

3 . 7 2

3 . 7 4

3 . 7 6x 1 0

-4

e

0 5 0 1 0 0 1 5 01 .29 64

1 .29 64

1 .29 64

1 .29 64x 1 0

1 1

h

0 5 0 1 0 0 1 5 02 4 0

2 6 0

2 8 0

3 0 0

3 2 0

T H E T A

0 5 0 1 0 0 1 5 00

0 .2

0 .4

0 .6

0 .8

t h e t a

0 5 0 1 0 0 1 5 01 4 0

1 6 0

1 8 0

2 0 0

2 2 0

O M

E G A

0 5 0 1 0 0 1 5 00 . 0 2 4

0 . 0 2 6

0 . 0 2 8

0 .0 3

0 . 0 3 2

i

0 5 0 1 0 0 1 5 01 2 0

1 4 0

1 6 0

1 8 0

2 0 0

o m

e g a

0 5 0 1 0 0 1 5 09 8

9 8 . 2

9 8 . 4

9 8 . 6

T H E

T A O

M E G A

t im e

Figure 5 - Satellite Orbital Variation during Total Maneuver Period (Pre-Maneuver + Maneuver) (a in m, h m2/s, angles in

degree, and time in s)

Table 4 – Satellite Orbital Variation during Pre-maneuver

Satellite Orbit Elements Initial Value at t=0 Final Value at t= 50 Amount of Change

a in meter 42166.35×103 42166310.21 -39.78

e 0.00037125 0.00037195 0.00000069

h in m2/s 1.296438395×1011 1.2964377831994×1011 -61180.057

THETA in degree 306.2 298.85 -7.34

theta in degree 0 0.11526 0.11526

OMEGA in degree 152.00 159.55 7.54

i in degree 0.03183960 0.02926856 -0.00257103

Omega in degree 195.71 188.04 -7.66

THETA+OMEGA in degree 98.12 98.33 0.20

Table 5 - Satellite Orbital Variation during Total Maneuver Period (Pre-Maneuver + Maneuver)

Satellite Orbit Elements Initial Value at t=0 Final Value at t= 50 Amount of Changea in meter 42166.35×103 42166.0935837467×103 -256.41

e 0.00037125 0.00037575 0.00000449

h in m2/s 1.29643839500×1011 1.2964344514864×1011 -394351.35

THETA in deg 306.2 245.50 -60.69

theta in deg 0 0.73377979 0.73377979

OMEGA in deg 152.0067823 213.152167139785 61.14

i in deg 0.03183960 0.02816396 -0.00367563

Omega in deg 195.7104787 133.831320930936 -61.87

THETA+OMEGA in deg 98.1295 98.57543008 0.44593008

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9

4. SATELLITE ORBIT CONTROL USING NEURAL

NETWORK PREDICTIVE CONTROL

Model Predictive Control (MPC)

The term MPC does not designate a specific control strategy but a very wide range of control methods, which make an

explicit use of a model of the process to obtain the controlsignal by minimizing an objective function. These designmethods lead to linear controllers, which have practicallythe same structure and present adequate degrees of freedom.The ideas appearing in greater or lesser degree in all the

predictive control family are basically defined in thefollowing points:

• Explicit use of a model to predict the process outputat future time instants (horizon).

• Calculation of a control sequence minimizing anobjective function.

• Receding strategy, so that at each instant the horizonis displaced towards the future, which involves the

application of the first control signal of the sequencecalculated at each step.

While MPC is suitable for almost any kind of problem, it

displays its main strength when applied to problems with alarge number of manipulated and controlled variables. Thevarious MPC algorithms only differ amongst themselves inthe model used to represent the process and the noises and

the cost function to be minimized. This type of control is of an open nature within which many works have beendeveloped, being widely received by the academic worldand by industry. The good performance of theseapplications shows the capacity of the MPC to achieve

highly efficient control systems able to operate during long

periods of time with hardly any intervention

MPC Methodology and Schema

The following strategy represents the methodology of all thecontrollers’ characteristics belonging to the MPC [14]family:

1. The future outputs for a determined horizon N , called the prediction horizon, are predictedat each instant t using the process model.These predicted outputs y(t+k|t) for k = 1...

N , which indicate the value of the variable at

the instant (t+k) calculated at time t, depend

on the known values up to instant t (pastinputs and outputs) and on the future controlsignals u(t + k|t), k = 0... N - 1, which are

those to be sent to the system and to becalculated.

2. To keep the process as close as possible tothe reference trajectory w(t + k), which can be the set point itself or a close

approximation of it, the set of future controlsignals is calculated by optimizing adetermined criterion in order to keep the

process as close as possible to the referencetrajectory. This criterion usually takes theform of a quadratic of the errors between the

predicted output signal and the predictedreference trajectory. An explicit solution can be obtained if the criterion is quadratic, themodel is linear and there are no constraints,

otherwise an iterative optimization method

has to be used.3. The control signal u(t|t) is sent to the process

whilst the next control signals calculated arerejected, because at the next sampling instant (t+1) is already known and step 1 is repeatedwith this new value and all the sequences are brought up to date. Thus, the u(t+1|t+1) iscalculated which in principle will be different

to the u(t+1|t) because of the newinformation available) using the recedinghorizon concept.

In order to implement this strategy, a model is used to predict the future plant outputs, based on past and current

values and on the proposed optimal future control actions.

Object Function

The general aim is that the future output (y) on theconsidered horizon should follow a determined referencesignal (w) and, at the same time, the control effort ( ∆u) necessary for applying. The general expression for such an

object function will be:

∑

∑

=

=

−+∆

++−+

=

u N

j

N

N j

u

jt u j

jt wt jt y j

N N N J

1

2

2

21

)]1()[(

)]()(ˆ)[(

),,(

2

1

λ

δ . (38)

Where N 1 and N 2 are the minimum and maximum cost

horizons and N u is the control horizon. The weightingfactors δ(j) and λ(j) are sequences that consider the future behavior, usually constant values or exponential sequencesare considered.

j N j

−= 2)( α δ .

If the parameter α is a value between 0 and 1, the errorsfarthest from instant t are penalized more than those nearest

to it, giving rise to smoother control with less effort. If α>1

the first errors are more penalized and a tighter control isoccurred. The weighting factor λ acts as a damper on the predicted control. Normally bounds in the amplitude and in

the slew rate of the control signal and limits in the outputwill be considered and adding these constrained to theobjective function to be minimized. The control law isimposed by the use of the control horizon N u, whichconsists of considering that after a certain interval N u<N 2

there is no variation in the proposed control signals. Whenthe condition that the output attains the reference value at adetermined instant, stability results are guaranteed.

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10

MPC Algorithm

The MPC algorithm has the following important steps.1. Generate a reference trajectory.2. Start with the previous calculated control

input vector, and predict the performance of

the plant using the model.

3.

Calculate a new control input that minimizesthe object function,4. Repeat steps 2 and 3 until desired

minimization is achieved,

5. Send the first control input, to the plant,6. Repeat entire process for each time step.

There are several minimization algorithms that have beenimplemented in MPC such as Non-gradient, Simplex, and

Successive Quadratic Programming. The quality of the plant’s model affects the accuracy of a prediction. Areasonable model of the plant is required to implementMPC. With a linear plant there are tools andtechniques available to make modeling easier, but when the

plant is non-linear this task is more difficult. For non-linear plant, the ability of the MPC to make accurate predictionscan be enhanced if a neural network is used to learn thedynamics of the plant instead of standard modeling

techniques. Improved predictions affect rise time, over-shoot, and the energy content of the control signal.

Adaptive Neural Network Predictive Control (ANNPC)

Since a neural network will be used to model the plant, theconfiguration of the network architecture should beconsidered. Figure 6, depicts a multi-layer feed-forwardneural network with a time delayed structure. The inputs to

this network consist of external inputs, u(n) and y(n-l), and

their corresponding delay nodes, u(n-l),…, u(n-nd ), and y(n-2), …, y(n-d d ). The parameters nd and d d represent thenumber of delay nodes associated with their respective inputnode. The second input could instead have been the

estimated output yn(n-l) and it's delayed values. Thenetwork as shown in Figure 6 has one hidden layer containing several hidden nodes that use a general outputfunction. The output node uses a linear output function with

a slope of one for scaling the output.

Prediction Using A Neural Network

The ANNPC algorithm uses the output of the plant’s modelto predict the plant’s dynamics to an arbitrary input from thecurrent time n, to some future time, n+k. Consider anetwork with two hidden nodes, one output node, one inputconsisting of u(n) and two previous inputs, and of three

previous outputs. To produce the output yn(n+2), input

u(n+1) and u(n+2) are needed. The prediction is started attime n, with the initial conditions of [u(n) u(n-1)] and [ y(n)

y(n-1) y(n-2)] and the estimated input u(n+1). The output of

this process is yn(n+1), which is fed back to the network and the process is repeated to produce the predicted plant’soutput yn(n+2). This process is shown in Figure 7. The

network feedback is displayed as one network feedinganother.

Designing Adaptive Neural Predictive Controllers to

Control Satellite OrbitsThe main object of the controllers is to control the satelliteorbits parameters during thrusting maneuver. There aretypically two steps involved when using neural networks for control, system identification and control design. The first

stage of model predictive control is to train a neural network to represent the forward dynamics of the plant. The neuralnetwork plant model is trained off-line, in batch form, usingany of the training algorithms [7, 12]. The prediction error

between the plant output and the neural network output isused as the neural network training signal. The neural

Figure 6 - Multi-Layer Feed-Forward neural Network with a time-delayed structure

Figure 7 – Network Prediction

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11

network plant model uses previous inputs and previous plant outputs to predict future values of the plant output. Inthe controller design phase, the ANN model is used to

predict future plant responses to potential control signals.An optimization algorithm then computes the controlsignals that optimize future plant performance. Theoptimization uses a technique of backtracking trackingalgorithm, which from searches in a given direction to

locate the minimum of the performance function in thatdirection. The backtracking algorithm is a linear searchroutine that begins with a step multiplier of 1 and then backtracks until an acceptable reduction in the performance

is obtained. The structure of the Neural Network PredictiveControl is given in Figure 8. The ANNPC is implementedusing MPC (SIMULINK block), which is contained in the Neural Network Toolbox block-set of the MATLAB V.6[12]. The MPC block-set, as shown in Figure 9, is based on

the receding horizon technique and the optimization is based on Newton’s (sometimes called Newton-Raphson)algorithm [8, 12]. The MATLAB functions PREDOPT [12] andCSRCHBAC [12] have been used.

Three SISO models are designed by using ANNPC to

predict the acceleration vectors ( f n , f t , f z ) increment in (t, n,

z) axes required to control the angular momentum per unitmass (h), the major axis (a), and the inclination (i) respectively. The MPC block-set is used to implement theANNPC controllers and guide the satellite orbits parameters

during maneuvers. The first step in model predictive controlis to determine the ANN model for the satellite orbittrajectory during thrusting maneuver (system identification).

Next, the controller uses this ANN model to predict future performance. Satellite orbit trajectory model during

thrusting maneuver (satellite plant model), which has beensimulated in the previous section, is used to develop theANN model by generating training data. This network istrained off-line in batch mode. In Figure 9 the controller

consists of the ANN model for the satellite trajectory duringthrusting maneuver and the optimization block. Theoptimization block determines the value of the accelerationvectors that minimize the object function of the predictive

control algorithm and then the optimal signal is input to thesatellite plant model as shown in Figure 10. The ANNPC

uses the ANN model to predict the future performance of the satellite. The controller then calculates the control inputsthat will optimize satellite parameters over a specified futuretime horizon. Any poor performance and instability due tothe interactions between processes variables can be avoided

thanks to the setting constraints of the input and the output.

The variation of the perturbation forces depends on the timeof the year and therefore the velocity increment will bevaried according to this change and is limited by the thruster

characteristics. The input and output constraints aredepending on the satellite application and thruster characteristics.

Training data for pre maneuver phase

A set of collective cycles from typical history data of implemented North-South maneuvers for a currentoperating geostationary satellite is used to the set constraints

of the ANNPCs system. The maximum and minimum input

samples of the acceleration vector f n minimum input during pre maneuver are set to value 2.618×10-5 and 2.142×10-5 m/s2 respectively. The maximum and minimum inputs

samples of the acceleration vector f t during pre maneuver are set to value 3.212×10-5 and 2.628×10-5 m/s2 respectively. The maximum and minimum input samples of the acceleration vector f z during pre maneuver are set tovalue 0.0056687 and 0.0046381 m/s2 respectively. The

parameters and characteristic of the controllers are asfollows:

Figure 8 – Neural Network Predictive Control Process

1C o n t ro l S i g n a l

o p t i m i z e ro p t i m i z a te r b l o c k

u

y

y h a t

y h a t 1

N N m o d e l

s i g n a l 1

s i g n a l 2

s i g n a l 3

s i g n a l 4

2 P l a n t o u t p u t

1R e f e re n c e

Figure 9 – Model Predictive Control (MPC) Block-set

Figure 10 – Structure of Neural Network Model

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12

• Two layers feed-forward neural networks with a timedelayed structure as shown in Figure 10. The inputs

to the networks consists of external inputs f n(n), f t (n),

f z (n), and h(n-1), a(n-1), i(n-1) and their

corresponding two delay nodes. The networks haveone hidden layer containing 10 hidden nodes that usea tan-sigmoid transfer function. The output node uses

a linear output function with a slope of one for scaling the output.

• Training function is trainlm, which is a network training function that updates weight and biasvalues according to Levenberg-Marquardt

optimization [12].

• The cost horizon is 7, the control horizon is 2, thecontrol weighting factor. The backtracking algorithm

iteration number is set to two iterations. The search parameter of the, which determines when the linesearch stops is 1.0e-8. The sampling time = 0.57094s.

The simulated satellite plant model, which has been

developed before, is used to generate training data for the NN models. The training data is generated by applying aseries of constrained step inputs to the SIMULINK plantmodel as shown in Figures 12, 13, and 14. The response of

the plant is compared to the response of the trained network and the corresponding error between the plant and thenetwork are shown in Figure 12, 13, and 14.

Neural Network Predictive Control of satell ite orbit trajectory d uring maneuver

i

h

a

satellite model

satellite orbit trajectory model during thrusting maneuver

f z ANNPC

f t ANNPC

f n ANNPC

Plant

Output

Reference

Control

Signal

Optim.

NN

Model

NN Predictive Controller of i (ANNPC 3)

Plant

Output

Reference

Control

Signal

Optim.

NN

Model

NN Predictive Controll er of a (ANNPC 2)

Plant

Output

Reference

Control

Signal

Optim.

NN Model

NN Predicti ve Control ler of h (ANNPC1)

-1

Gain2

-1

Gain1

-1

Gain

3

i ref.

2

a ref.

1

h ref.

Figure 11 – Maneuver Simulation Using ANNPCs to Control satellite Orbit

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13

Training data for maneuver phase:

Repeat the same steps as shown in the previous section butwith the maneuver parameters. The maximum and minimuminputs samples of the acceleration vector f n during premaneuver are set to value 0.00012408 and 0.00010152 m/s2

respectively. The maximum and minimum input samples of

the acceleration vector f t during pre-maneuver are set tovalue 0.00015298 and 0.00012516 m/s2 respectively. Themaximum and minimum input samples of the accelerationvector f z during pre maneuver are set to value 0.0269 and

0.022 m/s2 respectively.The Simulated Satellite Plant model is used to generatetraining data for the NN models as shown in Figures 15, 16,and 17.

Maneuver Simulation using ANNPC

Using SIMULINK to simulate the maneuver, using theimplemented ANNPCs, the simulated output data available

from the previous implemented maneuver is used to drive

(the reference simulate signals of (a, h, i)) the ANNPCs. Numerical integration methods (ode113 function [12]) isused in the simulation with variable step-size and sampling

time = 0.57094 s. The ANNPCs calculate and optimize the

acceleration vectors samples ( f n , f t , f z ) in (t, n, z) axes andinput to the satellite plant model as shown in Figure11. The

simulation is performed over the time of the pre-maneuver and maneuver, which is set to 50, and 57.094 s respectively.

ANNPCs Simulation Results

As shown in Figures 18, 19, 20, 24, 25, and 26, whichrepresent the outputs of the satellite plant model usingANNPCs with the references simulated signals, theANNPCs can follow and trace the reference simulated

signals (or the output constrained) required for controllingthe satellite orbit parameters. Comparing among the finalstates samples of a, h, and i of the reference simulatedmaneuver and the simulated maneuver using the ANNPCs

at t = +107.094 s and calculating the percentage errors asshown in Table 6. It indicates that the high efficiency thatthe ANNPC could achieve during maneuver simulation.Figures 21, 22, 23, 27, 28, and 29 represent the accelerationvectors samples of the reference simulated maneuver in

comparison with the output of the ANNPCs. ANNPC can

minimize the fuel consumption because of its optimizationtechnique. Nonlinear optimizations are computationallyexpensive processes. The use of Newton-Raphson is

intended to produce a computationally efficient process.The Newton-Raphson optimization has been used and it has been found to converge to a good result within twoiterations. Increasing the available training data will refineand enhanced the ANN model. The flexibility and inherent

characteristics of the ANNs in representing non-linear anddynamic models permit realizing high efficiency maneuver and improve the performance of the controllers. Using ANNwith MPC technique is a very good solution due to its

strength in dealing with non-linearity system. The ANNPChas applied the required rehearsal of the satellite maneuver with stable and good performance. The ANNPCs can betuned easily to satisfy satellite application and operationrequirements.

5. CONCLUSION

Using ANNPC in satellite orbits control will optimize the

satellite orbits parameters and the thrust forces and due toits main strength when applied to problems with a large

number of manipulated and controlled multi-variable. Also,the ANNPC have the ability to impose the constraints on both the manipulated and controlled variables. It is veryefficient when future references are known, as in case of the

satellite maneuver. The fuel consumption can be minimized,which is the main issue during satellite lifetime, because of the ANNPC optimization technique. Using Kalman filtersfor on-line parameters estimation [13], the ANNPCs can beused on-board, which will handle the maneuver in

autonomous way, eliminating the need of humancalculations and introduce a robustness controller. It can be

used to control any kind of earth satellite orbits includinggeostationary because it is ability to handle a great variety

of processes, from those with relatively simple dynamics toother more complex ones, including systems with longdelay times or of non-minimum phase or unstable ones.ANNPC permits on-board maneuver planning, calculation,and introduces safety condition when the Earth Control

Stations (ECS) is out of order, which will recover theabsence of ECS and subsequently downgrade the operationcosts. The open methodology of the ANNPC based oncertain basic principles, which allow for future extensions.

ANNPC is a major element in introducing the philosophy of "robustness, better, faster, safety, cheaper" to next

generations of ECS and spacecraft operation.

ACKNOWLEDGEMENT

I would like to thank the Egyptian Satellite company Nilesat(one of the Egyptian Radio and TV Union companies) for providing working and training in the field of aerospace.

TTaabbllee 66 – – AA N N N NPPCCss MMaanneeuuvveer r OOuutt p puutt VVeer r sseess R R eef f eer r eennccee SSiimmuullaatteedd MMaanneeuuvveer r aa iinn mmeetteer r ,, hh iinn mm22//ss,,

ii iinn ddeeggr r eeee

ReferenceSimulatedManeuver Output

ANNPCsManeuver Output

Error %

a

42166.093×103 42166.090×103 6×10-6

h 1.29643445×1011 1.29643440×1011 3×10-6

i 0.02816396 0.02840392 0.85

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14

0 10 20 30 402.35

2.4

2.45

2.5

2.55

2.6x 10

-5 Input

0 10 20 30 40

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

x 1011 Plant Output

0 10 20 30 40-1.5

-1

-0.5

0

0.5

1

Error

0 10 20 30 40

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

x 1011 NN Output

f n

h

h

Figure 12 – Pre-Maneuver Training Data for ANNPC of (h) in m2/s

0 10 20 30 403.004

3.006

3.008

3.01

3.012

3.014x 10

-5

0 10 20 30 40

4.2166

4.2166

4.2166

4.2166

4.2166x 10

7 Plant Output

0 10 20 30 40-2

0

2

4

6

8

10x 10

-6 Error

0 10 20 30 40

4.2166

4.2166

4.2166

4.2166

4.2166x 10

7 NN Output

time s

f t

a

a

Figure 13 – Pre-Maneuver Training Data for ANNPC of (a) in meter

0 10 20 30 405.1

5.15

5.2

5.25

5.3

5.35

5.4x 10

-3 Input

0 10 20 30 40

0.0295

0.03

0.0305

0.031

0.0315

Plant Output

0 10 20 30 40-1

-0.5

0

0.5

1

1.5x 10

-7Error

0 10 20 30 40

0.0295

0.03

0.0305

0.031

0.0315

NN Output

time (s)

f z

i

i

Figure14 – Pre-Maneuver Training Data for ANNPC of (i) in degree

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15

0 10 20 30 40 501.166

1.168

1.17

1.172

1.174

1.176

1.178

1.18x 10

-4 Input

0 10 20 30 40 501.2964

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

x 1011 Plant Output

0 10 20 30 40 50-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Error

0 10 20 30 40 501.2964

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

x 1011 NN Output

time s

f n

h

h

Figure 15 - Maneuver Training Data for ANNPC of (h) in m2/s

0 10 20 30 40 501.365

1.37

1.375

1.38

1.385

1.39

1.395x 10

-4

Input

0 10 20 30 40 50

4.2166

4.2166

4.2166

4.2166

4.2166

x 107 Plant Output

0 10 20 30 40 50-1

-0.5

0

0.5

1x 10

-3 Error

time s

0 10 20 30 40 50

4.2166

4.2166

4.2166

4.2166

4.2166

x 107 NN Output

time s

f t a

a

Figure 16 - Maneuver Training Data for ANNPC of (a) in meter

0 10 20 30 40 500.0254

0.0255

0.0256

0.0257

0.0258

0.0259

0.026

Input

0 10 20 30 40 50

0.026

0.0265

0.027

0.0275

0.028

0.0285

0.029

Plant Output

0 10 20 30 40 50-1

-0.5

0

0.5

1x 10

-7 Error

time (s)

0 10 20 30 40 50

0.026

0.0265

0.027

0.0275

0.028

0.0285

0.029

NN Output

time (s)

f z

i

i

Figure 17 - Maneuver Training Data for ANNPC of (i) in degree

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16

5 10 15 20 25 30 35 40 45 50

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

x 1011

time

refANNPC

h

Figure 18 – Reference Simulated Trajectory of (h) in Comparison With (h) Output of Satellite Plant Model Using ANNPCduring Pre-Maneuver (h) in m/s2

, time in s

5 10 15 20 25 30 35 40 45 50

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

x 107

time

a

refANNPC

Figure 19 – Reference Simulated Trajectory of (a) in Comparison With (a) Output of Satellite Plant Model Using ANNPC

during Pre-Maneuver (a) in m , time in s

0 5 10 15 20 25 30 35 40 45 500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

time

i

refANNPC

Figure20 – Reference Simulated Trajectory of (i) in Comparison With (i) Output of Satellite Plant Model Using ANNPC

during Pre- Maneuver (i) in degree , time in s

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0 5 10 15 20 25 30 35 40 45 50

-2.5

-2

-1.5

-1

-0.5

0x 10

-5

time

f n

refANNPC

Figure 21 – Acceleration Vector Samples in m/s2 (f n ) of Reference Simulated Pre-Maneuver in Comparison with Output of

ANNPC1, time in s

0 5 10 15 20 25 30 35 40 45 50

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-5

time

f t

refANNPC

Figure 22 – Acceleration Vector Samples in m/s2 (f t ) of Reference Simulated Pre-Maneuver in Comparison with Output of

ANNPC2, time in s

0 5 10 15 20 25 30 35 40 45 50

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

time

f z

refANNPC

Figure 23 – Acceleration Vector Samples in m/s2 (f z ) of Reference Simulated Pre-Maneuver in Comparison with Output of

ANNPC3, time in s

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18

10 20 30 40 50 60

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964

1.2964x 10

11

time

refANNPC

h

Figure 24 – Reference Simulated Trajectory of (h) in Comparison With (h) Output of Satellite Plant Model Using ANNPC

during Maneuver (h) in m2

/s , time in s

5 10 15 20 25 30 35 40 45 50 55

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

4.2166

x 107

time

refANNPC

a

Figure 25 – Reference Simulated Trajectory of (a) in Comparison With (a) Output of Satellite Plant Model Using ANNPCduring Maneuver (a) in m , time in s

10 20 30 40 50 60

0.026

0.0265

0.027

0.0275

0.028

0.0285

0.029

time

i

refANNPC

Figure 26 – Reference Simulated Trajectory of (i) in Comparison With (i) Output of Satellite Plant Model Using ANNPC

during Maneuver (i) in degree , time in s

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0 10 20 30 40 50 60

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0x 10

-4

time

f n

refANNPC

Figure 27 – Acceleration Vector Samples in m/s2 (f n ) of Reference Simulated Maneuver in Comparison with Output of

ANNPC1, time in s

0 10 20 30 40 50 60

-1.5

-1

-0.5

0x 10

-4

time

f t

refANNPC

Figure 28 – Acceleration Vector Samples in m/s2 (f t ) of Reference Simulated Maneuver in Comparison with Output of

ANNPC2, time in s

0 10 20 30 40 50 60

-0.025

-0.02

-0.015

-0.01

-0.005

0

time

f z

ref

ANNPC

Figure 29 – Acceleration Vector Samples in m/s2 (f z ) of Reference Simulated Maneuver in Comparison with Output of

ANNPC3, time in s

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R EFERENCES

[1] Maxwell Noton; Space Craft Navigation and Guidance

,”Advances in Industrial Control Series”, Springer-Verlage

London Limited, 1998.

[2] Gwynne Gurevich, Robert Bell, James R. Wertz;

Autonomous On Board Orbit Control, ” Flight Results and

Applications”, American Institute of Aeronautics andAstronautics (AIAA), pp 5226-52644, 2000.

[3] Narendra, K. S. and S. Mukhopadhyay; “Adaptive

Control Using Neural Networks and Approximate Models”,

IEEE Transactions on Neural Networks Vol. 8, pp. 475-485,1997.

[4] Hagan, M.T. and H.B. Demuth, “Neural Networks for

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Prof.Dr. Abd Elsalam F. ALY is a professor of Electrical

Engineering, Faculty of Engineering,

Alexandria University, Egypt. M.Sc. and Ph.D. from

University of Illinois USA, 1966. Alexandria university

since 1966, head of Electrical Department 1992-1995.

Dean of Faculty of Engineering Beirut Arab University

1988-1991. The main interest is Digital control systems and

Artificial intelligence systems.

Prof.Dr .Mohamed Naguib Aly is a professor of Nuclear

Engineering, Faculty of Engineering,

Alexandria University, Egypt. The maininterest are digital simulation and modeling

of nuclear power plants, Artificial

intelligence systems and numerical methods.

Eng. Mohamed Ahmed Zayan was born in 1969, He

received the B.Sc. in 1991, M.Sc. degree in

1998 in digital Communication

Engineering from the Faculty of

Engineering, Alexandria University, Egypt.

Since 1996, he is working as a Satellite

Control Engineer for The Egyptian

Satellite Company (Nilesat) (one of the Egyptian Radio and

TV Union companies). He is currently working toward the PhD at the Department of Electrical Engineering,

Alexandria University. His current research involves

Artificial Intelligence, Fuzzy Logic, Genetic Algorithm,

Satellite Control, and Orbit Determination.

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