Question

1) The following state-space system is dominated by a pair of lightly damped poles, 0 -1 (t)1-2 2 (1u(t) 0 2 2 y(t)0 11(t) Do the following: i) Verify that the system is controllable by computing the determinant of the con- trollability matrix. Use pole-placement to design a regulator K that makes the closed-loop damping () of the dominant poles 10 times that of the open-loop while keeping the natural frequency (wn) the same, Make a reasonable choice for the location of the third eigenvalue. Use the transformation technique described in class ii) ii) Use Matlab's acker or place command to check your answer in i). iv) Verify that the closed-loop eigenvalues are in the desired locations.

Answer 1

x = A x + B u y = c x A = [0 1 0 - 1 - 2 2 0 2 - 2] B = [1 1 - 1] controllability matrix = [B AB A^2 B] [A] [B] = [- 1 - 3 4] [A^2] [B] = [3 13 - 14] controllability matrix = [1 1 - 1 - 1 - 3 4 3 13 - 14] \r\n determinate of controllability matrix = -8 rightarrow hence the system is controllable (ii) characteristic equation of open transfer function is Vertical-bar AI - A Vertical-bar = 0 [s 0 0 0 s 0 0 0 s] - [0 1 0 - 1 - 2 2 0 2 - 2] = [s - 1 0 1 s + 2 - 2 0 - 2 s + 2) = 0] s [(s + 2)^2 - 4] - 1 [- 1 (s + 2)] s (s + 2)^2 - 4s + 1 (s + 2) = 0 s (s^2 + 4 + 4s) - 4s + s + 2 = 0 s^3 + 4s + 4s^2 - 4s + s + 2 = 0 s^3 + 4s^2 + s + 2 = 0 \r\n 1 + 2/s^3 + 4s^2 + s = 0

after row transformation we take the determinant

comparing the coefficient of both characteristic equation will give the value of 'K' matrix