State-space description of system with two differential equations

Question

The dynamics of my system are described by the following equations. $$J_1\ddot{\phi_1}+c(\dot{\phi_1}-\dot{\phi_2})+k(\phi_1-\phi_2)=k_I I \\ J_2\ddot{\phi_2}+c(\dot{\phi_2}-\dot{\phi_1})+k(\phi_2-\phi_1)=T_d$$ I have to find a state-space description $\dot{x}=Ax+Bu, y=Cx+Du$ of this system with input and output $$u=\begin{bmatrix}I \\ T_d\end{bmatrix} \text{and } y=\begin{bmatrix}\phi_1 \\ \phi_2\end{bmatrix}$$ My problem is that I don't understand how I can write this system with only two states. How can I start on this problem without writing $x=\begin{bmatrix}\phi_1 \\ \phi_2 \\ \dot{\phi_1} \\ \dot{\phi_2} \\ \ddot{\phi_1} \\ \ddot{\phi_2}\end{bmatrix}$?

Answer 1

$$\dot{x} = \frac{d}{dt}\begin{bmatrix}\phi_1 \\ \dot{\phi}_1 \\ \phi_2 \\ \dot{\phi}_2\end{bmatrix} = \begin{bmatrix}\dot{\phi}_1 \\ \ddot{\phi}_1 \\ \dot{\phi}_2 \\ \ddot{\phi}_2\end{bmatrix} = \begin{bmatrix}0 & 1 & 0 & 0 \\ -k/J_1 & -c/J_1 & k/J_1 & c/J_1 \\ & & & \\ & & & \end{bmatrix} \begin{bmatrix}\phi_1 \\ \dot{\phi}_1 \\ \phi_2 \\ \dot{\phi}_2\end{bmatrix} + \begin{bmatrix} 0 & 0 \\&\\&\\&\end{bmatrix}\begin{bmatrix}I\\T_d\end{bmatrix}$$ $$y=\begin{bmatrix}\phi_1\\\phi_2\end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 & 0 \\ & & & \end{bmatrix}\begin{bmatrix}\phi_1 \\ \dot{\phi}_1 \\ \phi_2 \\ \dot{\phi}_2\end{bmatrix}$$

Can you fill out the rest? You see, you need 1 state for each derivative degree.