To find the Jacobian matrices, we have to take partial derivatives of the differential equations according to state variables. For example from Wikipedia,
In my system, i have 8 differential equations, but number of state variables is 9. So, Jacobian is not square matrix. As a result, i cannot use this matrix in state space modelling.
How can i fix my problem? Should i define new differential equation or change the state variables?
Thanks,
I would guess that the 9th variable is not a state, but an external input, e.g., the control signal. Consider the example $$\begin{aligned}\dot{x}_1 &= f_1(x_1,x_2,x_3),\\ \dot{x}_2 &= f_2(x_1,x_2,x_3).\end{aligned}$$ Then your linearization (I assume here that you linearize around the origin that is the equilibrium) is $$ \begin{aligned} \begin{bmatrix}\dot{x}_1 \\ \dot{x}_2\end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x_1}f_1 & \frac{\partial}{\partial x_2}f_1 \\ \frac{\partial}{\partial x_1}f_2 & \frac{\partial}{\partial x_2}f_2\end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} +\begin{bmatrix}\frac{\partial}{\partial x_3}f_1 \\ \frac{\partial}{\partial x_3}f_2 \end{bmatrix} x_3 \end{aligned} $$ I.e., the Jacobian is $2\times 3$, but the linearization state matrix is only $2 \times 2$ as it includes only those state for which you have the differential equations.