A linear time-invar-iant system is descr-ibed by the differ--

ential equation.

d3y(t) 3d2 y. (t) dy(t) ()- ·()

d 3 + d2 +3d +yt-rt. t t .t

(a) Let the state var-iables be defined as x1 = y, Xz = dyjdt, x 3 =

d2 y / dt 2 . Wr-ite the state equations of the system in vector-matrix for-m.

(b) Find the state-tmnsition matr-ix cp(t) of A.

(c) Let y(O) = 1, dy(O)/dt = 0, d2 y(O)/dt 2 = 0, and r-(t) = u 5 (t).

Find the state tmnsition equation of the system.

(d) Find the chamcteTistic equation and the eigenvalues of A.

Problem