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


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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






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







Address for Correspondence



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


Technology, West Bengal, India




Electrical Engineering Department, Jadavpur University, West Bengal, India





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





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




1.1 Types of Artificial Satellites and the launching




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




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






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)]




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 ? =





Fig. 1. Attitude control by horizon scanners






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









which is to be minimized, the maximum principle


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


minimizes S, will maximize the scalar





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


i =1





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.




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.




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




Neglecting friction and disturbing forces, the


following equation is obtained:




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?&&





where J = Moment of Inertia of the system. Defining


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


the following equation is obtained:



u (t ) = ?&&






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





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





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)








s 2 + 2??n s + ?n2



The value of ?n is obtained from:


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








u (s)









ts = settling time.





s + 10s + 25








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








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:




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






and the estimator feedback gain matrix is found from:



sI ? ( A ? MC ) = 0,




where K = controller gain; M = feedback gain





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




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










Substituting from eqn. 33, the following equation is





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









Comparing with eqn.33, it is found that



K = [ 25 10]






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






The controller is given as:



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






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






Using eqn. 37, the following matrix equation is





s 0 0 1 m1





+ .[1 0] = 0





0 s 0 0 m2





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



m1 = 30; m2 = 225



So, the characteristic eqn. is:





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





? ( s)



E-ISSN 0976-3945






4.2 The compensator


The equation for the compensator is given as:



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



? K sI ? A + BK + MC












Putting numerical values for the matrices, the


following equation is obtained:





s ?1 0






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




0 s 1







?3000( s + 1.875)


s 2 + 40 s + 550






From equation, the transfer function of the satellite is


found to be:



G ( s) = 1/ s 2





Combining the two the following equation is





Gc ( s)G ( s) =



?3000( s + 1.875)


s 2 ( s 2 + 40s + 550)






International Journal of Advanced Engineering Technology


The resonating frequency,






= ?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




? (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

















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).




























Fig. 6 Root locus of the system




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








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