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Hi, 1.1/a)I plot the bode plot for question already just need to-(Answered)

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Hi,

1.1/a)I plot the bode plot for question already just need to comment on?the gain and phase curves of each. I need to double check if my plot is correct

1.1/b) I have simulate k=10 and k=25?

1.2/a) I did convert those transfer function into state space via matlab (need confirm if it is correct or not)

I will send you my work once we have agreed with the price. Also, I found?Docking to Refuel the Satellite paper so it might help you finish even faster.


International Journal of Advanced Engineering Technology

 


 

E-ISSN 0976-3945

 


 

Research Article

 

DESIGN OF ATTITUDE CONTROL SYSTEM OF A SPACE

 

SATELLITE

 

Sourish Sanyala, Ranjit Kumar Baraib, Pranab Kumar Chattopadhyayb, Rupendranath

 

Chakrabortyb

 

a

 


 

Address for Correspondence

 


 

Electronics & Communication Department, College of Engineering & Management, West Bengal University of

 

Technology, West Bengal, India

 

b

 

Electrical Engineering Department, Jadavpur University, West Bengal, India

 

ABSTRACT

 


 

The orientation of space satellites in stationary orbits gradually change with time. Their attitudes have to be controlled so as

 

to make the deviation from the reference zero by firing on-board jets. It is an optimal control problem which has been

 

addressed in this paper by the maximum principle of Pontryagrin. It has revealed that a relay element is required to

 

minimize the response time, backed up by a dead zone to minimize the fuel consumption, and a limiter to minimize the

 

energy. A controller has been designed along with a full order estimator using linear state variable feedback. This type of

 

compensated linear controllers gives satisfactory performance for limited dynamic range and limited input. The design has

 

been made by specially constructed programs and the results have been checked up using MATLAB tools.

 

KEYWORDS Satellite, Attitude control, dynamic programming, State variable feedback

 


 

1.0 INTRODUCTION:

 

Artificial satellites are purposely placed into orbit

 

around Earth, other planets, or the Sun. They have

 

come into use very recently. Since the launching of

 

the first artificial satellite in 1957, thousands of these

 

?man-made moons? have been rocketed into Earth?s

 

orbit. Today, artificial satellites play key roles in the

 

communications industry, in military intelligence,

 

and in the scientific study of both Earth and outer

 

space.

 

1.1 Types of Artificial Satellites and the launching

 

process

 

There are many kinds of satellites, each designed to

 

serve a specific purpose or mission. These are: a)

 

Communications Satellites, b) Navigation Satellites,

 

c) Weather Satellites, d) Military Satellites, and e)

 

Scientific Satellites

 

Satellite Launchers are used to place a satellite into

 

orbit. It requires a tremendous amount of energy,

 

which must come from the launch vehicle, or device

 

that launches the satellite. Satellites receive this

 

combination of potential energy (altitude) and kinetic

 

energy (speed) from multistage rockets burning

 

chemical fuels. The rockets are multistage. The first

 

stage gives the thrust and lifts the launch vehicle into

 

space, the second places the satellite into orbit and

 

the rest depends on the specific mission. All rocketengine burns occur at a precise moment and last for a

 

precise amount of time so that the satellite achieves

 

its proper position in space.

 

1.2 Operations in Space

 

The satellites must be robust enough to survive the

 

launch. They must be designed to operate in spatial

 

environment of direct rays from the Sun and in the

 

cold blackness of space, high levels of radiation and

 

collisions with micrometeoroids. They have to carry

 

their own power source and must remain pointed in a

 

specific direction, or orientation, to accomplish their

 

mission. They generally have onboard computers to

 

guide them. The most common source of power for

 

Earth-orbiting satellites is a combination of solar

 

cells

 

1.3 Orientation

 


 

IJAET/Vol.III/ Issue II/April-June, 2012/13-16

 


 

A satellite has to keep the solar panels pointed toward

 

the Sun. It has to keep its antennas and sensors point

 

toward Earth or toward the object the satellite is

 

observing. Methods of maintaining orientation

 

include small rocket engines, known as attitude

 

thrusters; large spinning wheels that turn the satellite;

 

and magnets that interact with Earth?s magnetic field

 

to correctly orient the satellite. Attitude thrusters can

 

make large changes to orientation quickly, but they

 

are not good under critical conditions of stability of

 

turn. They also require fuel which is limited.

 

A spinning wheel on a satellite acts as a gyroscope.

 

The rotational motion of the wheel keeps the satellite

 

in one orientation, and any change in it causes the

 

satellite to turn. Spinning wheels and magnets are

 

slower than thrusters but are better for attitude

 

stability. They require only electric power.

 

1.4 Propulsion

 

Once in orbit, a spacecraft relies on its own rocket

 

engines to change its orientation (or attitude) in

 

space, the shape or orientation of its orbit, and its

 

altitude. Of these three tasks, changes in orientation

 

require the least energy. The spacecraft turns on one

 

or more of its three principal axes: roll, pitch, and

 

yaw. Roll is a spacecraft?s rotation around its

 

longitudinal axis, the horizontal axis that runs from

 

front to rear. Pitch is rotation around the craft?s

 

lateral axis, the horizontal axis that runs from side to

 

side. Yaw is a spacecraft?s rotation around a vertical

 

axis. A change in attitude might be required to point a

 

scientific instrument at a particular target, to prepare

 

a spacecraft for an upcoming maneuver in space, or

 

to line the craft up for docking with another

 

spacecraft.

 

2.0 THE CONTROL PROBLEM

 

The satellite is assumed to be a rigid body, operating

 

in frictionless space having no disturbance (fig.1).

 

?(t) is the deviation of attitude angle from the

 

reference (fig. 2). The desired value for ?(t) is zero.

 

The satellite will drift with time. Jets aboard will

 

have to be fired in a manner so as to keep the value of

 

?(t) at zero. This is a problem of optimal control

 

which can be solved made by the application of

 


 

International Journal of Advanced Engineering Technology

 

Pontryagin?s maximum principle. The principle

 

comes out of the concept of dynamic programming.

 

2.1 Dynamic programming

 

The concept of dynamic programming was originally

 

developed by Bellman. It is based on the principle of

 

optimality and imbedding approach.

 

Statement of the principle: Regardless of initial state

 

and initial decision, the remaining decision must

 

form an optimal control policy with respect to the

 

state resulting from the first decision. The imbedding

 

approach is a multistage process in which an optimal

 

decision-making problem is broken up into a series of

 

smaller problems which are easier to solve

 

2.2 The maximum principle

 

The maximum principle was developed by Russian

 

mathematician Pontryagrin and his associates in

 

1956. It has its footing on calculus of variation and is

 

closely related to dynamic programming. If the

 

process dynamics is given as:

 

Ci(t) = fi[c(t), u(t)]

 

1

 

and the performance index as:

 


 

E-ISSN 0976-3945

 


 

3. Let the reference input to the horizon tracker be

 

R(s), the resultant output position be C(s), the rate of

 

change of local vertical with respect to earth be ?1

 

and the rate of vehicle with respect to the local

 

vertical be ?2.The vehicle inertial rate is the sum ? =

 

?1+?2.

 


 

Fig. 1. Attitude control by horizon scanners

 


 

T

 


 

S = ? G[c(t ), u (t ), r (t ), t ]dt

 


 

2

 


 

0

 


 

which is to be minimized, the maximum principle

 

requires that that the optimal control input u0(t) that

 

minimizes S, will maximize the scalar

 

n

 


 

H (t ) = ? pi (t ). fi [c(t ), u (t )] ? G[c(t ), u (t ), r (t ), to ]

 

i =1

 


 

3

 

where the process pi(t) is defined as:

 


 

? H (t )

 

pi (t ) =

 

pi(t) = ?H(t)/?ci(t),

 

? ci (t )

 


 

Fig 2

 


 

Single axis attitude control problem

 


 

i = 1,2,??.n.

 

4

 

The scalar H is called the Hamiltonian function, the

 

vector p is known as the co-state. From equation 1 &

 

2, ci(t) can also be written as:

 


 

ci (t ) =

 


 

? H (t )

 

,

 

? pi (t )

 


 

i = 1,2,??.n.

 

5

 

The maximum principle can be obtained directly

 

from dynamic programming equations by a simple

 

change of variables.

 

2.3 Application to space attitude control problem

 

The optimal control theory can be applied to solve a

 

variety of space vehicle problems e.g. attitude

 

control, lunar soft landing, transfer of orbit etc.

 

Attitude control faces variety of problems. During

 

flight, it receives commands from the guidance

 

system and exercises control in accordance to it. This

 

causes the vehicle to pitch or yaw and results in

 

change of attitude or direction of the flight. After the

 

vehicle is placed in the desired orbit, its attitude is

 

stabilized with respect to a reference. During reentry

 

to earth, the vehicle is pitched over to proper angle

 

before retro-rockets are fired to place the vehicle on a

 

transfer orbit in the earth?s atmosphere. The attitude

 

control problem is more sophisticated for the manned

 

space vehicle.

 

The present problem deals with attitude stabilization.

 

It is to be stabilized perpendicular to the earth?s local

 

vertical as shown in fig. 1. Its physical model is given

 

in fig. 2 and the block diagram representation in fig.

 

IJAET/Vol.III/ Issue II/April-June, 2012/13-16

 


 

Fig. 3 Space attitude control: block diagram

 

3.0 MATHEMATICAL DESCRIPTION

 

Neglecting friction and disturbing forces, the

 

following equation is obtained:

 

6

 

c&&(t ) = T (t ) / J = u (t ) (say) , u (t ) ?1

 

where, T (t ) = control torque generated by the space

 

vehicle for alignment with the ref. axes.

 

Expressions using state variable notation are given as

 

c1 (t ) = c(t ) ; c&1 (t ) = c2 (t ) ; c&2 (t ) = u (t ) As

 

par state variable notation the equation may be

 

written as:

 

c&(t ) = A.c (t ) + B.u (t ) ,

 

where A= state matrix and B = Control matrix

 

The following problems are associated with optimal

 

attitude control of the vehicle:

 

a. Minimum time problem

 

b. Minimum fuel-consumption problem

 

c. Minimum energy problem

 

For every case, it is assumed that: ?1 ? u (t ) ?1

 

3.1 The design problem

 

The satellite is assumed to be rigid and operating in a

 

frictionless environment; the disturbance is

 


 

International Journal of Advanced Engineering Technology

 

negligible. ?(t), the angle of mismatch has to be

 

reduced to zero. But the satellite will drift with time.

 

Jets will be fired so that ?(t) becomes zero. T(t), the

 

torque produced by jet firings is input to the system.

 

It is related to the attitude angle by the following eqn.

 


 

T (t ) = J?&&

 


 

7

 

where J = Moment of Inertia of the system. Defining

 

u(t) as: u(t) = T(t)/J,

 

the following equation is obtained:

 


 

u (t ) = ?&&

 


 

8

 


 

3.2 The design specification

 

The specifications of the attitude control system are

 

given below:

 

? The system controller is to be critically damped.

 

? The settling time is to be ? 0.8 sec

 

? The estimator is to be critically damped and it

 

must be twice faster than the controller.

 

The specs can be met if the attitude control is of

 

bang-bang type with a dead zone. Such a system can

 

be implemented very easily.

 

3.3 The characteristic equation

 

The controller is assumed to have the following

 

form:

 


 

u (t ) = ? k1 x1 (t ) ? k2 x2 (t )

 


 

9

 

where u(t) is the input torque, .x1(t) is the attitude

 

angle and .x2(t) is the attitude velocity.

 

The controller?s closed loop transfer function is

 

of 2nd. order given as:

 


 

? ( s)

 

u (s)

 


 

=

 


 

?n2

 

s 2 + 2??n s + ?n2

 


 

The value of ?n is obtained from:

 

?n = 4 / (? ts ) = 5 r/s.,

 

where

 


 

10

 


 

u (s)

 


 

=

 


 

11

 


 

ts = settling time.

 


 

25

 

s + 10s + 25

 

2

 


 

12

 


 

? c ( s) = s + 10 s + 25 = 0

 


 

13

 


 

4.0 LINEAR STATE VARIABLE FEEDBACK

 

It is advantageous to use linear state variable

 

technique for control system design in lieu of the

 

classical methods. In this case the input is given as:

 

14

 

u (t ) = Kr (t ) ? hx (t )

 

It is the difference between the ref. command and the

 

weighted sum of the state variables multiplied by the

 

gain K. The states must be controllable and

 

observable. This is used where the closed loop

 

transfer function is given and the open loop transfer

 

function is to be determined.

 

In many practical systems all the states may not be

 

controllable, observable and measurable. An

 

estimator is to be added in such cases. The closed

 

loop operation aims at finding the difference between

 

measured output c(t ) and the estimated output c(t )

 

from the process model. The gain factor M is selected

 

suitably so that the error gradually reduces to zero.

 

The goal is to make the estimator much faster

 

compared to the controller and the roots are placed

 

IJAET/Vol.III/ Issue II/April-June, 2012/13-16

 


 

sI ? A + BK = 0

 


 

15

 


 

and the estimator feedback gain matrix is found from:

 


 

sI ? ( A ? MC ) = 0,

 

16

 

where K = controller gain; M = feedback gain

 


 

factor.

 

4.1 The controller and the estimator

 

Defining states as: x1 (t ) = ? (t ) ; x2 (t ) =

 


 

x&1 (t ) ,

 


 

the following equation is obtained:

 


 

x&1 0 1 x1 0

 

x1

 

x& = 0 0 x + 1 .u ; ? (t) = [1 0] x

 

2

 

2

 

2

 

17

 

Substituting from eqn. 33, the following equation is

 

obtained:

 


 

? c ( s ) = s 2 + K 2 s + K1

 


 

=0

 


 

18

 


 

Comparing with eqn.33, it is found that

 


 

K = [ 25 10]

 


 

Therefore,

 


 

? c ( s) = ( s + 5) 2 = 0

 


 

19

 


 

The controller is given as:

 


 

u (t ) = ? Kx (t ) = ?25 x1 (t ) ? 10 x2 (t )

 


 

20

 


 

The full order estimator is to be 3 times faster than

 

the controller

 

i.e. ?ne = 3?n = 3 x 5 = 15 r/s.

 


 

? e (s)

 


 

=

 


 

( s + 15) 2 = s 2 + 30 s + 225 = 0

 


 

21

 


 

Using eqn. 37, the following matrix equation is

 

obtained:

 


 

s 0 0 1 m1

 


 

?

 

+ .[1 0] = 0

 


 


 


 

0 s 0 0 m2

 


 

2

 

Which yeilds: s + m1s + m2 = 0 ; resulting in:

 


 

m1 = 30; m2 = 225

 


 

So, the characteristic eqn. is:

 

2

 


 

accordingly. While a compensator is combined with

 

the controller and the estimator for best possible

 

performance, the controller gain is obtained from:

 


 

Therefore eqn. of the estimator is given as:

 


 

Therefore, the closed loop transfer function reduces

 

to:

 


 

? ( s)

 


 

E-ISSN 0976-3945

 


 

22

 


 

4.2 The compensator

 

The equation for the compensator is given as:

 


 

Gc ( s) = U ( s ) / ? ( s ) =

 


 

? K sI ? A + BK + MC

 


 

?1

 


 

23

 


 

M

 


 

Putting numerical values for the matrices, the

 

following equation is obtained:

 

?1

 


 

s ?1 0

 

30

 

30

 

[ 25 10] + [ 25 10] + [1 0]

 

225

 

0 s 1

 

225

 

=

 


 

?3000( s + 1.875)

 

s 2 + 40 s + 550

 


 

24

 


 

From equation, the transfer function of the satellite is

 

found to be:

 


 

G ( s) = 1/ s 2

 


 

25

 

Combining the two the following equation is

 

obtained

 


 

Gc ( s)G ( s) =

 


 

?3000( s + 1.875)

 

s 2 ( s 2 + 40s + 550)

 


 

26

 


 

International Journal of Advanced Engineering Technology

 

The resonating frequency,

 


 

?n

 


 

= ?550= 23.45 r /s and

 


 

the damping factor, ? = (40/2/23.45) = 0.8528. The

 

pole-zero configuration of the closed loop system

 

obtained from MATLAB is given in fig. 4

 


 

E-ISSN 0976-3945

 


 

This type of compensated linear controllers gives

 

satisfactory performance unless the dynamic range is

 

too large or the input is too high. MATLAB tools are

 

very often used by the control engineers to find out a

 

feasible solution and also for optimizing the same, if

 

that be the goal.

 

It is found from the case study that the phase margin

 

has increased from 0o to 46.78o (at gain cross-over

 

frequency of 4.174 r/s) and the gain margin has

 

increased from ? ? to 15.41 db (at phase crossover

 

frequency of 15.36 r/s) by the use of the compensator

 

and the estimator. The gain crossover frequency of

 

4.174 r/s is consistent with the controller?s closed

 

loop roots of ?n= 4 and ? = 1

 

Symbols

 

? (t ) :Angular position of the satellite

 


 

t1 :

 

Fig. 4 Pole-zero configuration of the system

 

The gain and phase margins of the system found out

 

by using Bode plot, are given in fig. 5. It is observed

 

that the phase crossover occurs at 21.8 r/s and the

 

gain margin is 16 db. It has improved to this value

 

from -?. The gain crossover occurs at 5.6 r/s and the

 

phase margin is 48.1o. It has improved to this value

 

from zero.

 


 

?n :Natural frequency of oscillation

 

J : Moment of Inertia of the satellite

 

? : Damping factor

 

REFERENCES:

 

1.

 


 

2.

 

3.

 

4.

 


 

5.

 


 

Fig. 5 Bode plot of the open loop system

 

Now the system will be examined by root locus.

 

Therefore, the numerical value of the gain is being

 

replaced by K. The observations made analytically

 

are confirmed by the root locus (fig. 6).

 


 

6.

 

7.

 

8.

 

9.

 

10.

 

11.

 

12.

 

13.

 

14.

 

15.

 

16.

 

17.

 


 

Fig. 6 Root locus of the system

 

5.0 CONCLUSION

 

Pontryagrin?s maximum principle reveals that a relay

 

element is required to minimize the response time, a

 

relay with a dead zone to minimize the fuel

 

consumption over a bounded time, and a limiter to

 

minimize the energy.

 

IJAET/Vol.III/ Issue II/April-June, 2012/13-16

 


 

Settling time, s

 


 

? (t ) :Angular velocity, r/s

 

K : System Gain

 

T (t ) : Input torque due to jet-firing

 


 

18.

 

19.

 


 

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