Question

A laser printer uses a laser beam to print copy rapidly for a computer. The laser is positioned by a control input r(t), so that we have Y(s) = 4(s + 50)/s² + 30s + 200 R(s). The input r(t) represents the desired position of the laser beam.

a)If r(t) is a unit step input, find the output y(t).
b)What is the final value of y(t)?

Answer 1

The question requires finding the response of a control system for a laser printer to a unit step input, and determining the final value of the output, which involves control theory concepts and Laplace transforms.

Step-by-step explanation:

The question involves solving a control system problem where a laser printer uses a laser beam to create a photoconductive image on a drum, employing xerographic processes to print high-quality images. When given a unit step input as the control input r(t), we need to determine the output y(t) of the system described by the transfer function Y(s) = 4(s + 50)/(s² + 30s + 200) R(s).

a) With a unit step input, R(s) = 1/s, so the output in the Laplace domain is Y(s) = 4(s + 50)/(s(s + 10)(s + 20)). By applying partial fraction decomposition and inverse Laplace transform, we can find y(t) in the time domain.

b) The final value theorem in control theory allows us to find the final value of y(t) as t approaches infinity. The final value of y(t) can be calculated by taking the limit of s · Y(s) as s approaches zero.