For the follwing nonlinear system: $$\dot x_{1} = 2/b \cdot [(c_{1}-u_{1}) \dot u_{1} + (r_{1}-u_{2}) \dot u_{2}]x_{1} \\ \dot x_{2} = 2/b \cdot [(c_{2}-u_{1}) \dot u_{1} + (r_{2}-u_{2}) \dot u_{2}]x_{2} \\ \dot x_{3} = 2/b \cdot [(c_{1}-u_{3}) \dot u_{3} + (r_{1}-u_{4}) \dot u_{4}]x_{3} \\ \dot x_{4} = 2/b \cdot [(c_{2}-u_{3}) \dot u_{3} + (r_{2}-u_{4}) \dot u_{4}]x_{4} \\ \\ \\ y_{1} = x_{1} + x_{2} \\ y_{2} = x_{3} + x_{4} $$ I am looking for a state space representation in the form:
$$ \mathbf{\dot x} = A(\mathbf{x})\mathbf{x} + B(\mathbf{x})\mathbf{u} \\ y = C\mathbf{x}$$ Is it possible to transform the system in a way that this is possible?
All time derivatives are linearly dependent on $\dot{u}_1$ through $\dot{u}_4$, so if one defines the input vector as
$$ \mathbf{u} = \begin{bmatrix} \dot{u}_1 \\ \dot{u}_2 \\ \dot{u}_3 \\ \dot{u}_4 \end{bmatrix}, $$
one also has to include $u_1$ through $u_4$ as states of the system. This would also imply that $A(\mathbf{x})=0$. The required $B(\mathbf{x})$ should then follow relatively straightforward, i.e. the Jacobian of $\mathbf{x}$ with respect to $\mathbf{u}$.