Electric drive with a four-pole squirrel-cage induction motor

The motor dynamics are mapped by aset of five highly coupled nonlinear differential equations as given by

$$\begin{align*} \dfrac{{\mathrm d}i_{qs}^r}{{\mathrm d}t} &= \dfrac{1}{L_{\Sigma}} \left[-\dfrac{L_{RM}^2r_s+M^2r_r}{L_{RM}}i_{qs}^r + \dfrac{r_rM}{L_{RM}}\Psi_{qr}^r - (L_\Sigma i_{ds}^r + M \Psi_{dr}^r) \omega_r + L_{RM}u_{qs}^r\right] \\ \dfrac{{\mathrm d}i_{ds}^r}{{\mathrm d}t} &= \dfrac{1}{L_{\Sigma}} \left[-\dfrac{L_{RM}^2r_s+M^2r_r}{L_{RM}}i_{ds}^r + \dfrac{r_rM}{L_{RM}}\Psi_{dr}^r + (L_\Sigma i_{qs}^r + M \Psi_{qr}^r) \omega_r + L_{RM}u_{ds}^r\right] \\ \dfrac{{\mathrm d}\Psi_{qr}^r}{{\mathrm d}t} &= \dfrac{r_r M}{L_{RM}}i_{qs}^r - \dfrac{r_r}{L_{RM}}\Psi_{qr}^r \\ \dfrac{{\mathrm d}\Psi_{dr}^r}{{\mathrm d}t} &= \dfrac{r_r M}{L_{RM}}i_{ds}^r - \dfrac{r_r}{L_{RM}}\Psi_{dr}^r \\ \dfrac{{\mathrm d}\omega_r}{{\mathrm d}t} &= -\dfrac{B_m}{J}\omega_r + \dfrac{P}{2J}\left[\dfrac{P}{2}\dfrac{M}{L_{RM}}(i_{qs}^r\Psi_{dr}^r - i_{ds}^r-\Psi_{qr})-T_L\right], \end{align*}$$

where $L_{\Sigma} = L_{SM}L_{RM}-M^2$. For the load torque we assume the expression $T_L=c_2\omega_r^2 + c_3\omega_r^3$.

The state vector is given by

$x(t) = [i_{qs}^r, i_{ds}^r, \Psi_{qr}^r, \Psi_{dr}^r, \omega_r]^{\mathrm T} = [x_1, x_2, x_3, x_4, x_5]^{\mathrm T}.$