In an effort to open up the far side of the Moon to exploration, studies have been conducted to determine the feasibility of operating a communication satellite around the translunar equilibrium point in the Earth-Sun-Moon system. The desired satellite orbit, known as a halo orbit. The objective of the controller is to keep the satellite on a halo orbit trajectory that can be seen from the Earth so that the lines of communication are accessible at all times. The communication link is from the Earth to the satellite and then to the far side of the Moon. The linearized (and normalized) equations of motion of the satellite around the translunar equilibrium point are [25]

The state vector x is the satellite position and velocity, and the inputs «,, i - 1,2,3, are the engine thrust accelerations in the £, 77, and £ directions, respectively.
(a) Is the translunar equilibrium point a stable location? (b) Is the system controllable from ux alone? (c) Repeat part (b) for u2. (d) Repeat part (b) for «3. (e) Suppose that we can observe the position in the 17 direction. Determine the transfer function from
«2 to 77. (Hint: Let y = [0 1 0 0 0 0]x. ) (f) Compute a state-space representation of the transfer function in part (e) using the ss function. Verify that the system is controllable, (g) Using state feedback
u2 = -Kx,
design a controller (i.e., find K) for the system in part (f) such that the closed-loop system poles are at s1,2 = -1 ± j and s3-4 = -10.

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