Quiz 4

\underline{Problem 1}: Consider the following linearized model in deviation form:

\frac{dX}{dt} = AX + BU  (1)

where

X = [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ]  (2)
A = [ \begin{matrix}−1 & 0 \\ 2 & −2 \end{matrix} ]  (3)
B = [ \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ]  (4)

and

U = [ \begin{matrix}U_{1} \\ U_{2} \end{matrix} ]  (5)

Suppose the input undergoes the following change:

U_{1} = \{ \begin{matrix}0 & t < 0 \\ 3 & t ≥ 0 \end{matrix}  (6)
U_{2} = \{ \begin{matrix}0 & t < 0 \\ 3.t & t ≥ 0 \end{matrix}  (7)

Compute how the X vector changes with time when t ≥ 0. Assume that the system is at steady state when t = 0.