Date due and handed in March 18,2010
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*}
