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\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)
B = [ \begin{matrix}1 \\ 2 \end{matrix} ]  (3)

and U is a scalar. Suppose the input undergoes the following step change:

U = \{ \begin{matrix}0 & t < 0 \\ 3 & t ≥ 0 \end{matrix}  (4)

Compute how the X vector changes with time for the following A matrices:

1.

A = [ \begin{matrix}1 & 2 \\ 3 & 8 \end{matrix} ]

2.

A = [ \begin{matrix}1 & −3 \\ 2 & 2 \end{matrix} ]

3.

A = [ \begin{matrix}k_{1} & 0 \\ k_{1} & k_{2} \end{matrix} ]

\underline{Problem 2}: Consider the following linearized model in deviation form:

\frac{dX}{dt} = AX + BU  (5)

where

X = [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ]  (6)
B = [ \begin{matrix}1 \\ 2 \\ 0 \end{matrix} ]  (7)

and U is a scalar. Suppose the input undergoes the following pulse change:

U = \{ \begin{matrix}0 & t < 0 \\ 3 & 0 ≤ t < 1 \\ 0 & t ≥ 1 \end{matrix}  (8)

Compute how the X vector changes with time for the following A matrices:

1.

A = [ \begin{matrix}1 & 2 & 3 \\ 4 & −5 & 7 \\ −6 & −3 & −2 \end{matrix} ]

2.

A = [ \begin{matrix}2 & 0 & 0 \\ 5 & 2 & 0 \\ 8 & 4 & 2 \end{matrix} ]

\underline{Problem 3}: Consider the following linearized model in deviation form:

\frac{dX}{dt} = AX + BU  (9)

where

X = [ \begin{matrix}X_{1} \\ X_{2} \\ X_{3} \end{matrix} ]  (10)
B = [ \begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} ]  (11)
U = [ \begin{matrix}U_{1} \\ U_{2} \end{matrix} ]  (12)

Suppose the input undergoes the following change:

U_{1} = \{ \begin{matrix}0 & t < 0 \\ 3 & t ≥ 0 \end{matrix}  (13)
U_{2} = \{ \begin{matrix}0 & t < 0 \\ 3.t & t ≥ 0 \end{matrix}  (14)

Compute how the X vector changes with time for the following A matrix:

1.

A = [ \begin{matrix}1 & 2 & 3 \\ 4 & −5 & 7 \\ −6 & −3 & −2 \end{matrix} ]