Design a controller for the system so that the closed-loop system performs satisfactorily. The system to be controlled is the ball-suspension system shown in the figure below. The steel ball is suspended in the air by the electromagnetic force generated by the electromagnet. The control objective is to keep the metal ball suspended at the nominal equilibrium position by controlling the current in the magnet with voltage e(t). The resistance of the coil is R and the inductance is L(y) L/y (depending on y). The force generated by the electromagnet is Ki^2/y^2. 1. Model the system by deriving the state equations. Let E be the nominal value of e(t). Find the nominal values of y and dy/dt at equilibrium. 2. 3. Linearize the system around its equilibrium. Assume R=1~292, L=0.4-0.6H, E=10-15V, K=0.01 ~0.1, g-9.8, M-0.2-0.3kg. 4. Assume that we can observe the position of the ball. Check if the system is controllable and observable. 5. Assume that we can measure all the states. Design a state feedback so that the closed-loop M system performs satisfactory (you select the pole locations). R Mg Steel Ball m L Electromagnet i e(t)