Question

Problem #4 [15 Points] Given -1001 00-1」 11 0 1 00 (a) Find the impulse response h(t) and the transfer function matrix H(s). Which modes are present in h(t)? What are the poles of H(s)? b) Is the system asymptotically stable? Is it BIBO stable? Explain. (c) ls it possible to determine a linear state feedback control law u = Fr+r so that all closed loop eigenvalues are at -1? If yes, find such F

Answer 1

H(s) = [S(SI - A)^-1 B] = [1 1 1 0 0 0] [s + 1 0 0 0 s - 2 0 0 0 s + 1]^-1 [1 0 0 0 1 0] = [1 1 1 0 0 0] (s + 1)^2 (s - 2) [(s - 2) (s + 1) 0 0 0 (s + 1)^2 0 0 0 (s + 1) (s - 2)]^-1 [1 0 0 0 1 0] = [1 1 1 0 0 0] [(s - 2) (s + 1) 0 0 0 (s + 1)^2 0] times 1/(s + 1)^2 (s - 2) = [(s - 2) (s + 1) (s - 2) (s + 1) (s + 1)^2 0] times 1/(s + 1)^2 (s - 2) L^_1 [[1/s + 1 1/s + 1 1/s - 2 0]] h(t) = [e^-t e^-t e^2t 0] poles of H(s) = s = 2, -1, -1 (s + 1)^2 (s - 2) = (s^2 + 1 + 2s) (s - 2) rightarrow s^3 - 2s^2 + s + 2s^2 - 4s = 0 s^3 - 3s = 0 ....... (1)