I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ .
The function $f_i(\omega, \alpha)$ can be written as two independent functions containing only $\omega$ or $\alpha$: $f_i(\omega, \alpha) = g_i(\omega) + h_i(\alpha)$. Here, $h_i(\alpha)$ is linear in $\alpha$, $g_i(\omega)$ is quadratic in $\omega$.
The given equation set can be transformed into a (descriptor) state-space representation: $E\dot{x} = A(x) + Bu$, where $x = \omega$, $\dot{x} = \alpha$ and $u = T$. Note that the state and input vectors are all members of $\mathbb{R}^4$. Because $A(x)$ has the property of it being quadratic in $\omega$, one can define $g_i(\omega) = \omega^TA_i\omega$, where $A$ is an upper-triangular matrix.
$A(x)$ now becomes $\left[\matrix{\omega^TA_1\omega\\ \omega^TA_2\omega\\ \omega^TA_3\omega\\ \omega^TA_4\omega }\right]$ which leads to $\left[\matrix{\omega^TA_1\\ \omega^TA_2\\ \omega^TA_3\\ \omega^TA_4 }\right]\omega$. But I cannot remove extract $\omega^T$ without breaking the dimensions of the whole term.
How can I extract both $\omega$ terms to end with a conventional quadratic matrix equation?
Personally, I am thinking of adding an extra matrix $P$ in between $\omega^T$ and $A$, but I do not know for sure if this alters the system equations and also what the entries of this matrix should be.