The following equation of motion describes the dynamics of a proportional controller forthe attitude of a satellite that is subject to a damping moment: \ddot{y}(t)+7 \dot{y}(t)+10 K y(t)=10 K r(t) Here, y(t) is the angular position of the satellite, r(t) is the reference angular position, and K is a constant gain. (a) Use this equation of motion to derive the closed-loop transfer function, Hc1(s)Y(s)/R(s).= (b) Determine the range of the gain K that can be used to meet both of these specifications simultaneously: (1) maximum overshoot Mp ≤ 0.043; (2) rise time tr≤ 0.45 sec. (c) Let K = 1. The satellite starts at rest (ỷ(0) = 0) at the angle y(0) = 0 degrees. A reference angle r(t) = 5uç(t) degrees is commanded, where uç(t) is the unit step function. Using your expression for Hel(s) from part (a), explain whether the Final Value Theorem can be applied. If so, apply it to find the steady-state value of y(t).

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