Handout organization: Approximately one HW per week. About 9 in total.
Will give back detailed key solution. Requires using Matlab and simulink. See handouts.
What is state space model? Given \(m\) inputs to system and \(r\) outputs. The input are the controls (since we can manipulate them). This is the vector \(u(t)\). The states of the system are \(x(t)\) and the output is \(y(t)\). A simple example of spring mass damper is now given.
The diļ¬erential equation is
\[ m y'' + c y' + k y = f(t) \]
Let \(x_1= y, x_2=y'\) be the two states, hence \(x_{1}'= x_2\) and \(x_{2}' = \frac{f(t)}{m} - \frac{k}{m} x_1- \frac{c}{m} x_2\). Therefore the state space representation is
\begin{align*} \begin{pmatrix} x_{1}'\\ x_{2}' \end{pmatrix} & = \begin{pmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{c}{m} \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} + \begin{pmatrix} 0\\ \frac{1}{m} \end{pmatrix} f(t) \\ y & = \begin{pmatrix} 1 & 0 \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix} + [0] f(t) \end{align*}
More generally,
\begin{align*} x' &= A x + B u\\ y &= Cx + D u \end{align*}
Reader: Find LTI for \(y'''+6 y'- 2 y = 2 u(t)\)
Answer:
\begin{align*} \begin{pmatrix} x_{1}'\\ x_{2}'\\ x_{3}'\end{pmatrix} & = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 2 & -6 & 0 \end{pmatrix}\begin{pmatrix} x_1\\ x_2\\ x_3\end{pmatrix} + \begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} u(t) \\ y & =\begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\begin{pmatrix} x_1\\ x_2\\ x_3\end{pmatrix} +[0] u(t) \end{align*}
The matrix \(A\) has one on the superdiagonal. What if the input had additional term? Such as \(\frac{du}{dt}\) or \(\frac{d^2 u}{dt^2}\) in it? We need a state space realization to handle this.