Example 3.3-2. Station Keeplng Control for a Libration-Point Satelite (Bryson) O

Question

Example 3.3-2. Station Keeplng Control for a Libration-Point Satelite (Bryson) On the line connecting the center of the earth to the center of the moon, there is a so-called libration point Li where the pull of the earth on a satellite (in an orbit about the earth with the same perlod as the moon's orbit) exactly equals the pull of the moon plus the centrifugal force. However, we shall see that this point is an unstable equilibrium point. Then we shall show that by using state feedhack (via a small reaction engine) a satellite at that point can be stabilized. The dynamic equations for small deviations in position away from the libration point can be shown to be (see the figure) L1 Moon Earth where x = radial position perturbation, y-azimuthal position perturbation, u = F1m2, F-engine thrust in the y direction, m = satellite mass, and = 2T/29 rad day.. With -0, show that the equilibrium point x-y-0 s unstable. 2. To stabilize the position, use state-variable feedback

Explanation / Answer

converting the Set of 2 second order differential equations to 4 first order differential equations, we can find the Jacobian of this system. The Determinant of this Jacobian turns out to be less than 0. Hence, according to Poincare-Lyapunov Theorem the equilibrium point (0,0) i.e. L1 is unstable saddle point.

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