Induction Motor

Induction motor is represented by fifth order nonlinear differential equation as

$$\begin{align*} \dot{i}_{sa} &={MR_{r} \over \sigma L_{s}L_{r}^{2}}\phi_{ra}+{n_{p} M\over \sigma L_{s}L_{r}} \omega\phi_{rb}- \gamma i_{sa}+{1\over \sigma L_{s}}u_{sa} \\ \dot{i}_{sb} &={MR_{r} \over \sigma L_{s}L_{r}^{2}}\phi_{rb}- {n_{p} M \over \sigma L_{s}L_{r}}\omega\phi_{ra}-\gamma i_{sb}+{1\over \sigma L_{s}}u_{sb} \\ \dot{\phi}_{ra} &=-{R_{r}\over L_{r}}\phi_{ra}-n_{p}\omega \phi_{rb}+{MR_{r} \over L_{r}}i_{sa} \\ \dot{\phi}_{rb} &=n_{p}\omega\phi_{ra}-{R_{r}\over L_{r}}\phi_{rb}+{MR_{r}\over L_{r}} i_{sb}\cr \dot{\omega} &={n_{p} M \over JL_{r}}(\phi_{ra}i_{sb}-\phi_{rb}i_{sa})-{fv\over J}\omega-{1\over J}T_{l}, \end{align*}$$

where $i_{sa}, i_{sb}, \phi_{ra},\phi_{rb}$ and $\omega$ denote stator currents, rotor fluxes, and angular velocity, respectively, and $u_{sa}$ and $u_{sb}$ denote stator voltage inputs. The parameters $\sigma$ and $\gamma$ are defined as $\sigma = 1-M^2/L_sL_r, \gamma = (L_r^2r_s+M^2R_r)/\sigma L_s L_r^2 \cdot M, L_s, L_r, R_s$ and $R_r$ denote the mutual inductance, the self-inductances, the resistances, respectively. The subscript $a$ and $b$ denote the components of a vector with respect to a fixed stator reference frame and $s, r$ stand for stator and rotor of motor. $n_p, f_v, J, T_l$ are the number of pole-pair, the co-efficient of viscous damping, the inertia of rotor, and the load torque. We assume that the state variables $i_{sa}, i_{sb}, \omega$ are available for measurement and $T_l$ has a unknown constant value, that is, $\dot{T}_l=0$ . As a result, the model of induction motor can be rewritten into the form

$\eqalignno{ \dot{x}_{i} & =A_{i} (u, y_{i+1}, \cdots,y_{p})x_{i}\cr & +g_{i}(x_{1}, \cdots, x_{i},; u; y_{i+1}, \cdots, y_{p}) \cr & y_{i}=C_{i}x_{i}, 1\leq i \leq p}$

as follows:

$\eqalignno{ & \dot{x}_{1}=\left(\matrix{ 0 & A_{11}(y_{2})\cr 0 & 0}\right)x_{1}+g_{1}(x_{1}, u, y_{2})\cr & \dot{x}_{2}=\left(\matrix{ 0 & A_{21}\cr 0 & 0}\right) x_{2}+g_{2}(x_{1}, x_{2}, u)\cr & y_{1}=C_{1}x_{1}\cr & y_{2}=C_{2}x_{2},}$

where $x_1=[i_{sa},i_{sb},\phi_{ra},\phi_{rb}]^T$, $x_2=[\omega,T_l]^T$, $y_1=[i_{sa},i_{sb}]^T$, $y_2=\omega$, $u=[u_{sa},u_{sb}]^T$ and

$\eqalignno{ & A_{11}= \left(\matrix{ MR_{r}/ \sigma L_{s}L_{r}^{2} & (n_{p}M/\sigma L_{s}L_{r})y_{2}\cr -(n_{p}M/ \sigma L_{s}L_{r})y_{2} & M R_{r}/\sigma L_{s}L_{r}^{2}}\right)\cr & A_{21}=\left(\matrix{ -{1\over J}}\right)\cr & g_{1}=\left(\matrix{ -\gamma i_{sa} +(1/ \sigma L_{s})u_{sa}\cr -\gamma i_{sb} + (1/\sigma L_{s})u_{sb}\cr -(R_{r}/L_{r})\phi_{ra}-n_{p}y_{2}\phi_{rb}+(MR_{r}/L_{r})i_{sa}\cr n_{p}y_{2}\phi_{ra} -(R_{r}/L_{r})\phi_{rb}+(MR_{r}/L_{r})i_{sb}}\right)\cr & g_{2}=\left(\matrix{(n_{p}M/JL_{r})(\phi_{ra}i_{sb})-(\phi_{rb} i_{sa}) - (f_{v}/J)\omega)\cr 0}\right)}$