5) Consider the system defined by the equations

i(t) = Ar(t) + bu(t),

y(t) = cx(t),

where x(t) = R³, u(t) = R, and y(t) = R;

2.75

0.25 0.625

1 2 0.5

1.5 -1.5 1.25

A =

b = (1, 2, -2),

and c= (-0.5, 0.5, 0.25).

Compute a matrix V € R³x3 whose columns provide the Kalman decomposition of the

system; compute à := V-¹AV, 6 = V-¹b, and c = cV; and use the Kalman decomposition

to derive the input-output transfer function of the system.