Consider the system U(s)G(s) 20 Y(s) with transfer function G(s) = . . Let u(t) and...

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Consider the system U(s)G(s) 20 Y(s) with transfer function G(s) = . . Let u(t) and y(t) be the s+2s+10 inverse Laplace transform of U(s) and Y(s), respectively. (2a) Find the differential equation corresponding to the transfer function G(s). (2%) (2b) Find the characteristic equation, the system poles of the system, and explain how a and @ of the system poles ajo will affect the step response. (4%) (2c) Find the damping ratio 5, the natural frequency @, and the steady-state step response x of the system, and explain how they will affect the step response of the system. (4%) (2d) Assume u(t) = u(t) is the unit step function, find the step response y(t). (7%) 20 2 Hint: Y(s)= + + 1.054e/162 1.054e-/162 Find the inverse Laplace transform of Y(s). s(s+2s+10) S s+1-j3 s+1+j3 (2e) Use the values of 5, w, and x, obtained in (2c) to compute and then plot the step response according to the formula: y(t) = xss -at e sin(@t+0) where = 5, @= 1-5, and 0 = cos 5. (3%) Consider the system U(s)G(s) 20 Y(s) with transfer function G(s) = . . Let u(t) and y(t) be the s+2s+10 inverse Laplace transform of U(s) and Y(s), respectively. (2a) Find the differential equation corresponding to the transfer function G(s). (2%) (2b) Find the characteristic equation, the system poles of the system, and explain how a and @ of the system poles ajo will affect the step response. (4%) (2c) Find the damping ratio 5, the natural frequency @, and the steady-state step response x of the system, and explain how they will affect the step response of the system. (4%) (2d) Assume u(t) = u(t) is the unit step function, find the step response y(t). (7%) 20 2 Hint: Y(s)= + + 1.054e/162 1.054e-/162 Find the inverse Laplace transform of Y(s). s(s+2s+10) S s+1-j3 s+1+j3 (2e) Use the values of 5, w, and x, obtained in (2c) to compute and then plot the step response according to the formula: y(t) = xss -at e sin(@t+0) where = 5, @= 1-5, and 0 = cos 5. (3%)