Question

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Problem 3.27: You must find K and a. In addition: (i) plot the admissible domain corresponding to the specs, and (ii) plot, using Matlab, the step response of your closed loop system with K and a that you computed (don't forget to title your plot with your name and mark your axes)

Answer 1

unity feedback system rightarrow connect the cascade blocks eliminate the feedback path y (s) /R (s) = k (100)/(s + a) (s + 25)/1 + k (100)/(s + a) (s + 25 0 = (100 k) (s + a) (s + 25)/((s + a) (s + 25) + 100 k)/(s + a) (s + 25 0 = 100 k/(s + a) (s + 25) + 100 k y (s)/R (s) = 100 k /s^2 + (a + 25) s + (100 k + 25 a) \r\n characteristic equation of the closed loop transfer function 1 + G (s) H (s) = 0 Therefore s^2 + (a + 25)s + (100 k + 25 a) = 0 compare this equation with standard 2nd sides equation s^2 + 2e w_n s + w^2_n = 0 Therefore 100 k + 25 a = w^2_n w_n = squareroot 100 k + 25 a rightarrow (20 2 w_n = s = a + 25 epsilon = (a + 25)/2 squareroot 100 k + 25 a rightarrow (3) given problem peak over shoot = 25 % getting time t_s = 0.1 sec m_p = e^- pi epsilon/squareroot 1- epsilon^v = 0.25