4) Consider the system defined by the equation

July 29, 2023

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

y(t) = cx(t),

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

-1.2

1

0.8

0.4 -3 -1.6

-0.3

1.5 0.2

A =

DRAFT

208

b=(3, 0, -0.5),

and c= (1, 1, 2).

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

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

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