### Satelite telescope orientation - Part 1

This post is the first in a series of post going through the design for the control system which controls the orientation of a telescope on a satelite which must stay fixed on the star that is being observed using another, brighter star as it's reference.  The telescope must remain fixed on the observed star regardless of disturbances like collisions with meteorites, gravity forces or others.

This post will deal with the information and parameters of the problem statement and the work that needs to be done to complete the task at hand.

Work with me on this one and let's compare notes in the end.  Post questions and comments.  We have a month to complete all the calculations so take your time.

The first figure (denoted 'a') shows the satelite and the axis on which it operates and the second figure (denoted 'b') shows the schematic diagram for the closed-loop control system.

The control system works as follows:  the telescope's orientation is achieved by using a easy identifiable star (much brighter than the star being observed) as reference.  To start off, assume the telescope is in line with the star that is being observed.

At this position the axis (X, Y, Z as indicated in figure a) forms an angle of   Î˜r with the line connecting the telescoped with the reference star.  This angle Î˜r is know in advance and is the system's excitation.  The angle between the telescope (Z-axis) and the reference star is Î˜y.  If this angle is different to the desired angle (Î˜r) then the control system must align the z-axis so that
Î˜r = Î˜y.

The block designated "thruster" in figure b, is excited by the controllers output and it produces torque forces around the X-axis and these forces affect the correction in the angle Î˜y so that Î˜y = Î˜r.

The various blocks in figure b is described by the following equations:

•          Kt is the conversion constant of the position angle  to an electrical signal
•          Gc(s) is the transfer function of the controller
•          Kb is the constant of the thruster in which the voltage V(s) is converted to torque T(s)
•          The satelite's transfer function is Gs(s)
•          J is the moment of inertia about the X-axis

Tasks that are expected to be done while working on this design:
1 -  Construct the block diagram model of this system
2 -  Derive the transfer function of the closed-loop system
3 -  If the controller is just an amplifier, derive the differential equation of the output (Î˜y) and then say if this controller is feasible for this purpose.

4 -  Design a PID controller for this system, making the maximum overshoot less than 10%
5 -  Use the values for KI and KD obtained in 4 and sketch the root locus.  Now find Kp to make the system stable.
6 -  Use bode diagrams to analyze the stability margins of the controlled system obtained in 4
7 -  Design a suitable controller Gc(s) to make the system have a phase margin of 45o and a gain margin of not less than 8 dB.  It must also have a velocity error constant of 1/100

Each of the follow up parts for the post will contain steps to complete the tasks set forward here.

Hope you will join me in solving these problem statements so we can learn from each other.

Post questions as comments on this post, or post it on our Google+ page or email me at mustang.jurie@gmail.com