my question is very easy: I started study dynamic systems and I have this system:
$Mz_1''(t) +\beta z_1'(t) = F(t)$
$Mz_2''(t) - \beta z_2'(t)+kz_2(t)=\beta z_1'(t)$
where $ M, \beta, k $ are constants and ' is the order of diff. equation.
Now I have to choose the state variables, that they are as the order of the system. Which is the order of this system? 4 or 2? So how much state variables can I choose?
Regards
Generally each differentiation can be selected as a state variable. You have two second order differential equations, which should give 4 state variables. There are of course infinitely many choices of state variables, but the easiest one is
$$ x_1 = z_1, x_2 = z_1', x_3 = z_2, x_4 = z_2' $$
Now, can you write this in matrix form as follows:
$$\begin{bmatrix}x_1' \\ x_2' \\ x_3' \\ x_4'\end{bmatrix} = \begin{bmatrix}* & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * \end{bmatrix} \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4\end{bmatrix} + \begin{bmatrix}* \\ * \\ * \\ *\end{bmatrix}$$
Can you find what each * should be?