In case of a direct torque control we can use the rotor fixed frame, which means that the $\omega_\gamma=0$. In this case we can describe the motor with the following equations(you can find the deduction under the following link)
$$
\begin{bmatrix} u_{sd} \\ u_{sq} \end{bmatrix}=R_r \begin{bmatrix} i_{sd} \\ i_{sq} \end{bmatrix} +\frac{d}{dt}\begin{bmatrix} \psi_{sd} \\ \psi_{sq} \end{bmatrix}\tag{1}
$$
$$\begin{bmatrix} u_{rd} \\ u_{rq} \end{bmatrix}=R_r \begin{bmatrix} i_{rd} \\ i_{rq} \end{bmatrix} +\frac{d}{dt}\begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix}+\omega_r\begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}\tag{2}$$
The motor torque can be expressed the rotor flux magnitude $\psi_r$ and stator current component $i_s$
$$
T_e = \frac{3p L_m}{2 L_r}\left(\psi_{rd}i_{sq}-\psi_{rq}i_{sd}\right)\tag{3}
$$
and the flux equations are the following
$$\begin{bmatrix}\psi_{sd} \\ \psi_{sq}\end{bmatrix}=L_s\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}+L_m\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}\tag{4}$$
$$\begin{bmatrix}\psi_{rd} \\ \psi_{rq}\end{bmatrix}=L_m\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}+L_r\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}\tag{5}$$
I would like to estimate the rotor fluxes and the electrical motor speed $\omega_r$. In this case there are five state variables $\begin{bmatrix}i_{sd} & i_{sq} & \psi_{rd} & \psi_{rq} & \omega_r \end{bmatrix}$, to get the state variables we have to make a few arithmetic operations.
We would like to get the flux equations, after sorting the $2^{nd}$ equation we will end up with the following
$$\frac{d}{dt}\begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = - R_r \begin{bmatrix} i_{rd} \\ i_{rq} \end{bmatrix} - \omega_r\begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}+\begin{bmatrix} u_{rd} \\ u_{rq} \end{bmatrix}\tag{6}$$
We can't measure the rotor currents, but we can express the rotor currents using the $5^{th}$ equations.
$$\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}=\frac{1}{L_r}\begin{bmatrix}\psi_{rd} \\ \psi_{rq}\end{bmatrix}-\frac{L_m}{L_r}\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\tag{7}$$
Replacing the rotor currents in the $6^{th}$ expression we will get the rotor flux equations
$$\frac{d}{dt} \begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = -R_r \left(\frac{1}{L_r} \begin{bmatrix}\psi_{rd} \\ \psi_{rq} \end{bmatrix}-\frac{L_m}{L_r} \begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\right) - \omega_r \begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix} $$ $$ \frac{d}{dt} \begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = \frac{L_m}{\tau_r} \begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix} + \begin{bmatrix} -\frac{1}{\tau_r} & -\omega_r \\ \omega_r & -\frac{1}{\tau_r} \end{bmatrix} \begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}\tag{8} $$
After those arithmetic operation we got the rotor's flux equations, to get the stator's current equations we replace the $1^{st}$ equation's stator fluxes using the $4^{th}$ expressions, also we have to replace in the $4^{th}$ expression the rotor currents, by using the $7^{th}$ equations.
After some arithmetical operation we will get the stator currents
$$ \frac{d}{dt} \begin{bmatrix} i_{sd} \\ i_{sq} \end{bmatrix} = \begin{bmatrix} -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma}& 0 \\ 0 & -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} \end{bmatrix}\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\\+\begin{bmatrix} \frac{L_m}{L_s\sigma L_r\tau_r} & \frac{\omega_rL_m}{L_s\sigma L_r} \\ -\frac{\omega_rL_m}{L_s\sigma L_r} & \frac{L_m}{L_s\sigma L_r\tau_r} \end{bmatrix}\begin{bmatrix}\psi_{rd} \\ \psi_{rq} \end{bmatrix}+\begin{bmatrix} \frac{1}{L_s\sigma} & 0 \\ 0 & \frac{1}{L_s\sigma} \end{bmatrix}\begin{bmatrix} u_{sd} \\ u_{sq} \end{bmatrix}\tag{9} $$
Where
$$
\tau_r=\frac{L_r}{R_r}
$$
and
$$
\sigma=1-\frac{L_m^2}{L_sL_r}
$$
The equations above combined with the Extended Kalman filte can be used to estimate the fluxes and rotor speed. The dynamic model for an induction motor by choosing the stator currents $i_sd$, $i_sq$ and rotor fluxes $\psi_rd$, $\psi_rq$ as state variables, is as follows
$$
\frac{dx}{dt}=Ax(t)+Bu(t)\tag{10}
$$
$$
y(t)=Cx(t)+Du(t)\tag{11}
$$
where
$$
A= \begin{bmatrix} -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} & 0 & \frac{L_m}{L_s\sigma L_r\tau_r} & \frac{\omega_rL_m}{L_s\sigma L_r} \\ 0 & -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} & -\frac{\omega_rL_m}{L_s\sigma L_r} & \frac{L_m}{L_s\sigma L_r\tau_r} \\ \frac{L_m}{\tau_r} & 0 & -\frac{1}{\tau_r} & -\omega_r \\ 0 & \frac{L_m}{\tau_r} & \omega_r & -\frac{1}{\tau_r} \end{bmatrix}\tag{12}
$$
$$B=\begin{bmatrix} \frac{1}{L_s\sigma} & 0 \\ 0 & \frac{1}{L_s\sigma} \\ 0 & 0 \\ 0 & 0\end{bmatrix}\tag{13}$$
$$C=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}\tag{14}$$
$$D=\begin{bmatrix} 0 \end{bmatrix} \tag{15}$$
$$
\begin{bmatrix} u_{sd} \\ u_{sq} \end{bmatrix}=R_r \begin{bmatrix} i_{sd} \\ i_{sq} \end{bmatrix} +\frac{d}{dt}\begin{bmatrix} \psi_{sd} \\ \psi_{sq} \end{bmatrix}\tag{1}
$$
$$\begin{bmatrix} u_{rd} \\ u_{rq} \end{bmatrix}=R_r \begin{bmatrix} i_{rd} \\ i_{rq} \end{bmatrix} +\frac{d}{dt}\begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix}+\omega_r\begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}\tag{2}$$
The motor torque can be expressed the rotor flux magnitude $\psi_r$ and stator current component $i_s$
$$
T_e = \frac{3p L_m}{2 L_r}\left(\psi_{rd}i_{sq}-\psi_{rq}i_{sd}\right)\tag{3}
$$
and the flux equations are the following
$$\begin{bmatrix}\psi_{sd} \\ \psi_{sq}\end{bmatrix}=L_s\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}+L_m\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}\tag{4}$$
$$\begin{bmatrix}\psi_{rd} \\ \psi_{rq}\end{bmatrix}=L_m\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}+L_r\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}\tag{5}$$
I would like to estimate the rotor fluxes and the electrical motor speed $\omega_r$. In this case there are five state variables $\begin{bmatrix}i_{sd} & i_{sq} & \psi_{rd} & \psi_{rq} & \omega_r \end{bmatrix}$, to get the state variables we have to make a few arithmetic operations.
We would like to get the flux equations, after sorting the $2^{nd}$ equation we will end up with the following
$$\frac{d}{dt}\begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = - R_r \begin{bmatrix} i_{rd} \\ i_{rq} \end{bmatrix} - \omega_r\begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}+\begin{bmatrix} u_{rd} \\ u_{rq} \end{bmatrix}\tag{6}$$
We can't measure the rotor currents, but we can express the rotor currents using the $5^{th}$ equations.
$$\begin{bmatrix}i_{rd} \\ i_{rq} \end{bmatrix}=\frac{1}{L_r}\begin{bmatrix}\psi_{rd} \\ \psi_{rq}\end{bmatrix}-\frac{L_m}{L_r}\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\tag{7}$$
Replacing the rotor currents in the $6^{th}$ expression we will get the rotor flux equations
$$\frac{d}{dt} \begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = -R_r \left(\frac{1}{L_r} \begin{bmatrix}\psi_{rd} \\ \psi_{rq} \end{bmatrix}-\frac{L_m}{L_r} \begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\right) - \omega_r \begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix} $$ $$ \frac{d}{dt} \begin{bmatrix} \psi_{rd} \\ \psi_{rq} \end{bmatrix} = \frac{L_m}{\tau_r} \begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix} + \begin{bmatrix} -\frac{1}{\tau_r} & -\omega_r \\ \omega_r & -\frac{1}{\tau_r} \end{bmatrix} \begin{bmatrix} \psi_{rd} \\ -\psi_{rq} \end{bmatrix}\tag{8} $$
After those arithmetic operation we got the rotor's flux equations, to get the stator's current equations we replace the $1^{st}$ equation's stator fluxes using the $4^{th}$ expressions, also we have to replace in the $4^{th}$ expression the rotor currents, by using the $7^{th}$ equations.
After some arithmetical operation we will get the stator currents
$$ \frac{d}{dt} \begin{bmatrix} i_{sd} \\ i_{sq} \end{bmatrix} = \begin{bmatrix} -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma}& 0 \\ 0 & -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} \end{bmatrix}\begin{bmatrix}i_{sd} \\ i_{sq} \end{bmatrix}\\+\begin{bmatrix} \frac{L_m}{L_s\sigma L_r\tau_r} & \frac{\omega_rL_m}{L_s\sigma L_r} \\ -\frac{\omega_rL_m}{L_s\sigma L_r} & \frac{L_m}{L_s\sigma L_r\tau_r} \end{bmatrix}\begin{bmatrix}\psi_{rd} \\ \psi_{rq} \end{bmatrix}+\begin{bmatrix} \frac{1}{L_s\sigma} & 0 \\ 0 & \frac{1}{L_s\sigma} \end{bmatrix}\begin{bmatrix} u_{sd} \\ u_{sq} \end{bmatrix}\tag{9} $$
Where
$$
\tau_r=\frac{L_r}{R_r}
$$
and
$$
\sigma=1-\frac{L_m^2}{L_sL_r}
$$
The equations above combined with the Extended Kalman filte can be used to estimate the fluxes and rotor speed. The dynamic model for an induction motor by choosing the stator currents $i_sd$, $i_sq$ and rotor fluxes $\psi_rd$, $\psi_rq$ as state variables, is as follows
$$
\frac{dx}{dt}=Ax(t)+Bu(t)\tag{10}
$$
$$
y(t)=Cx(t)+Du(t)\tag{11}
$$
where
$$
A= \begin{bmatrix} -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} & 0 & \frac{L_m}{L_s\sigma L_r\tau_r} & \frac{\omega_rL_m}{L_s\sigma L_r} \\ 0 & -\frac{R_s+\frac{L_m^2R_r}{L_r^2}}{L_s\sigma} & -\frac{\omega_rL_m}{L_s\sigma L_r} & \frac{L_m}{L_s\sigma L_r\tau_r} \\ \frac{L_m}{\tau_r} & 0 & -\frac{1}{\tau_r} & -\omega_r \\ 0 & \frac{L_m}{\tau_r} & \omega_r & -\frac{1}{\tau_r} \end{bmatrix}\tag{12}
$$
$$B=\begin{bmatrix} \frac{1}{L_s\sigma} & 0 \\ 0 & \frac{1}{L_s\sigma} \\ 0 & 0 \\ 0 & 0\end{bmatrix}\tag{13}$$
$$C=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}\tag{14}$$
$$D=\begin{bmatrix} 0 \end{bmatrix} \tag{15}$$
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