A coupled-drive apparatus is shown. The coupled drives consist of two pulleys connected via an elastic belt, which is tensioned by a third pulley mounted on springs providing an under damped dynamic mode. One of the main pulleys, pulley A, is driven by an electric DC motor. Both pulleys A and B are fitted with tachometers that generate measurable voltages proportional to the rate of rotation of the pulley. When a voltage is applied to the DC motor, pulley A will accelerate at a rate governed by the total inertia experienced by the system. Pulley B, at the other end of the elastic belt, will also accelerate owing to the applied voltage or torque, but with a lagging effect caused by the elasticity of the belt. Integration of the velocity signals measured at each pulley will provide an angular position estimate for the pulley [23]. The second-order model of a coupled-drive is

and y = x±.
(a) Design a state variable feedback controller that will yield a step response with deadbeat response and a settling time (with a 2% criterion) less than 0.5 second.
(b) Design an observer for the system by placing the observer poles appropriately in the left half-plane.
(c) Draw the block diagram of the system including the compensator with the observer and state feedback.
(d) Simulate the response to an initial state at x(0) = [1 Of and x(0) = [0 Of.

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