20 Problem 2: Consider the system R(s)G(s) X(s) with transfer function G(s) s +2s +10 x(1)...

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20 Problem 2: Consider the system R(s)G(s) X(s) with transfer function G(s) s +2s +10 x(1) be the inverse Laplace transform of R(s) and X(s), respectively. (Similar to Example 3.10) 2a. Find the differential equation corresponding to the transfer function G(s). (5%) 2b. Find the characteristic equation and the poles of the system. (5%) 2c. Find the damping ratio , the natural frequency, and the steady-state step response x, of the system. (5%) 2d. Is the system overdamped? critically damped? underdamped? or undamped? (5%) 2e. Assume the initial conditions x(0) = 0 and x(0)=0, and let the input r(t) be an unit step function. Find the output response x(1). (15%) . Let r(t) and 2f. Plot the constant term, the decaying sinusoidal term of x(t) and the sum of these two terms versust on the same graph. Discuss how decaying sinusoidal term relates to the system poles and the damping ratio, and how the constant term relates to x,. (15%) 20 Problem 2: Consider the system R(s)G(s) X(s) with transfer function G(s) s +2s +10 x(1) be the inverse Laplace transform of R(s) and X(s), respectively. (Similar to Example 3.10) 2a. Find the differential equation corresponding to the transfer function G(s). (5%) 2b. Find the characteristic equation and the poles of the system. (5%) 2c. Find the damping ratio , the natural frequency, and the steady-state step response x, of the system. (5%) 2d. Is the system overdamped? critically damped? underdamped? or undamped? (5%) 2e. Assume the initial conditions x(0) = 0 and x(0)=0, and let the input r(t) be an unit step function. Find the output response x(1). (15%) . Let r(t) and 2f. Plot the constant term, the decaying sinusoidal term of x(t) and the sum of these two terms versust on the same graph. Discuss how decaying sinusoidal term relates to the system poles and the damping ratio, and how the constant term relates to x,. (15%)