State-space representation of a nonlinear MIMO system

Question

Question: Obtain a state-space representation of nonlinear multiple-input multiple-output (MIMO) system:

$$\dddot{y}_1 + 2\dot{y_1} + 3y_2 + 2 = u_1 y_2 \tag{1}$$ $$\ddot{y}_2 - 2 \dot{y}_2 + \dot{y}_1^3 + y_2 + y_1 = (u_2 - u_3)y_1 \tag{2}$$

I find it difficult solving the above equations. I have the following queries:

1. What do I do with $(dy_1/dt)^3$? How do I represent it in state space model?
2. Are $u_1, u_2$ and $u_3$ control inputs or just constants (coefficients of $y_1$ and $y_2$)?
3. Does the constant $2$ in equation come in $\mathbf B$ (i.e., $\mathbf A x + \mathbf B u)$?
4. Do I have to convert these equations into linear equations?

Answer 1

Let $y_3=y_1'$, $y_4=y_2'$, $y_5=y_3'$. Then the state space equations look like

$$y_1'=y_3$$ $$y_2'=y_4$$ $$y_3'=y_5$$ $$y_4'-2y_4+y_3^3+y_2+y_1=(u_2-u_3)y_1$$ $$y_5'+2y_3+3y_2+2=u_1y_2$$