7.46 The linearized equations of motion of the simple pendulum in Fig. 7.96 are 8+ 030 = 1. (a) Write the equations of motion in state-space form. (b) Design an estimator (observer) that reconstructs the state of the pendulum given measurements of 6. Assume @ = 5 rad/sec, and pick the estimator roots to be at $ = -10 # 10j.Reference longitude Desired location on orbit (c) Write the transfer function of the estimator between the measured value of o and the estimated value of e. (d) Design a controller (that is, determine the state feedback gain K) so that the roots of the closed-loop characteristic equation are at s = -4 1 4j