Consider a satellite whose thrusters are aligned with the coordinate axes; we can then decouple the satellite's dynamics along each coordinate direction. The simplest model of the dynamics in each direction is F = M, where F is the net thruster force applied (in N), M 0 is the satellite mass (in kg), and is the displacement in that direction (in m). If only the satellite displacement can be measured and not the satellite velocity v, then 0 1 - [8 6] x + [M] F x = y = [10]x is a valid state-space model for this system, where x = [x v]. (a) Suppose we want to apply full-state feedback to the system, of the form F = -Kx. Find K such that the cost function J = [ (2 + p F) dt is minimized, where p > 0 is some parameter. Your answer should be in terms of p and M. (b) What happens to your feedback gains as p is varied? Why does this make sense? In parts (c) through (e), assume that p = 1 and M = 8 kg. (c) Confirm that the optimal feedback strategy from part (a) is F = -x-4v. (d) Suppose that at time t = 0 s, the satellite displacement is x (0) = 0 m and the satellite velocity is v(0) = 10 m/s. What will be the displacement and velocity at time t = 5 s if: No feedback is applied? (open-loop) The optimal feedback from part (c) is applied? (closed-loop) (e) Using Matlab, plot the closed-loop state response to the initial condition x(0) = [0 10 ]. (This should match your answers in part (d) at time t = 5 s.)