52. Consider a system with transfer function Y(s) U(s) s+10 = G(s) = s2 +1 A. Find the differential equation relating u and y. B. Find the response, y(t), to a unit step input, u(t) = 1(t). Assume the system is initially at rest. Sketch y(t). To verify your result, calculate the response using MATLAB. C. Sketch the Bode plot for the system. Clearly indicate all corners and asymptotes on the plot. Verify your result using MATLAB. D. Find the open loop poles and zeros of the system. Show the pole and zero locations on a sketch of the s-plane. For parts E though M, consider the proportional feedback control of the system as shown below where r(t) is the response input, e(t) is the error and d(t) is a disturbance. |d(t) r(t) u Kp G(s) E. Find the closed-loop transfer function Y(s) R(s)' F. Find the characteristic equation, A(s), for the closed-loop system. Find expressions for the undamped natural frequency and damping ratio, o, and 3, as a function of Kp. G. What value of Kp gives critical damping, that is, 31? What is the resulting undamped natural frequency, con? H. Find the steady-state error, e(co), for a unit step in r(t) and the steady-state error for a unit step in d(t). I. As a preliminary step in calculating the root locus, put the loop transfer function in Bode form, that is, a(s) K Identify K, a(s) and b(s). b(s)' J. Sketch the root locus of the system. Clearly identify the open loop poles and zeros and any break-in or break-away points on the root locus. K. From your root locus sketch, identify the point on the locus for critical damping. (Hint: Critical damping occurs at a break-in point). Find the gain, Kp, and the undamped natural frequency, n, from the root locus. Your answer should agree with part G. L. Use MATLAB to create the root locus. Verify the values of K, and co, for critical damping. M. Use your MATLAB program to identify points on the root locus having damping ratio of 3 = 0.5 and 30.7. There may be more than one point that meets three requirements. For each case you locate, find Kp and on. Note: For each section that uses MATLAB provide a printout of your code and relevant plots as part of your solution.