3.3 HW 4
Date due and handed in March 18,2010
3.3.1 Problem 3.23 (a)
Write the state variable equation for the following
Solution
Let \(x_{1}\left ( t\right ) \) and \(x_{2}\left ( t\right ) \) be the state variables. Hence from the diagram we see the following
\begin{align*} x_{1}^{\prime }\left ( t\right ) & =ax_{1}\left ( t\right ) +u\left ( t\right ) \\ x_{2}^{\prime }\left ( t\right ) & =bx_{2}\left ( t\right ) +u\left ( t\right ) \end{align*}
And
\[ y\left ( t\right ) =x_{1}\left ( t\right ) +x_{2}\left ( t\right ) \]
Hence
\begin{align*}\begin {pmatrix} x_{1}^{\prime }\left ( t\right ) \\ x_{2}^{\prime }\left ( t\right ) \end {pmatrix} & =\overset {A}{\overbrace {\begin {pmatrix} a & 0\\ 0 & b \end {pmatrix} }}\begin {pmatrix} x_{1}\left ( t\right ) \\ x_{2}\left ( t\right ) \end {pmatrix} +\overset {B}{\overbrace {\begin {pmatrix} 1\\ 1 \end {pmatrix} }}u\left ( t\right ) \\ y\left ( t\right ) & =\overset {C}{\overbrace {\begin {pmatrix} 1 & 1 \end {pmatrix} }}\begin {pmatrix} x_{1}\left ( t\right ) \\ x_{2}\left ( t\right ) \end {pmatrix} \end{align*}