Linearization: Selecting of state variables to make the Jacobian matrix square

Question

I have 5 equations like:

My state variables are:

I cannot make Jacobi matrix square according the this state vector. What do you suggest for this operation? Should i change state vector, or can i produce 7 equations by using this vector?

Thanks,

NOTE: Arastas' approach.

Answer 1

$\dot{w}_z$ and $\dot{w}_x$ are not state variables since you do not have equations with their derivatives, namely $\ddot{w}_z$ and $\ddot{w}_x$. Remove them, and you have five equations for five states.

UPD

Define $x = [u,\ v,\ w_z,\ w_x,\ \theta]^\top$. Then the system can be written as (check it, typos are possible) $$ \begin{bmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 5 & -1 & -1 & 0 \\ 0 & -10 & 4 & 3 & 0 \\ 0 & -1 & 14 & 10 & -1 \\ 0& 0& 0& 0& 1\end{bmatrix} \begin{bmatrix}\dot{x}_1 \\ \dot{x}_2\\ \dot{x}_3\\ \dot{x}_4\\ \dot{x}_5\end{bmatrix} = \begin{bmatrix}x_3^2 + x_3x_4 + 2x_3x_2 \\ -5x_1x_3 \\ 10x_1x_3 \\ x_1x_3 + x_5\\ x_4\end{bmatrix} $$ that is $$ T \dot{x} = \psi(x). $$ The matrix $T$ is invertible, and thus your model is $$ \dot{x} = T^{-1}\psi(x) = f(x). $$ Then you can compute the Jacobian.