Home Assignment

\underline{Problem 1}: Compute e^{At} 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}1 & 2 & 3 \\ 4 & −5 & 7 \\ −6 & −3 & −2 \end{matrix} ]

4.

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

5.

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

\underline{Problem 2}: A process in modeled in deviation form as

\frac{dX}{dt} = AX + BU

Compute the unforced dynamics for each A matrix given in Problem 1. Assume that at t = 0, each element of the vector X(0) is equal to 1.