## 2 Modeling of Permanent Magnet Synchronous Motor
\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right],\]  
\[\text{and }\textbf{B}^{k}(\textbf{x}^{k})=\left[ \begin{array}{cccc}T_{s}.\frac{1} {L_{d}}&T_{s}.\frac{1}{L_{d}}\\ -T_{s}.\frac{1}{L_{q}}&T_{s}.\frac{1}{L_{q}}\\ 0&0\\ 0&0\end{array} \right]\]The \(i_{q}\) is the torque-producing component of the stator current and \(i_{d}\) is flux-producing component of the stator current. When \(i_{q}=I_{m}\), then \(i_{d}=0\), which is a condition for maximum torque. The angle \(\alpha\) between \(i_{q}\) and \(i_{d}\) determines the minimum and maximum torque computed as \(\alpha=\tan^{-1}\left(\frac{i_{q}}{i_{d}}\right)\). The \(\alpha\) is the angle that determines the rotor position to compute the torque. Non-Gaussian noise as measurement noise \(v^{k}\) and process noise \(w^{k}\) are also added in the mathematical model for the real-time realization of the mathematical model. Moreover, it is also assumed to have very slow change in load torque compared to the sampling time. The state variables vector, input vector, and output vectors are as follows, respectively [5]:  
\[\mathbf{x}^{k}=\left[ \begin{array}{ccc}i_{d}^{k}&i_{q}^{k}&\omega_{m}^{k}& \theta^{k}\end{array} \right]^{T} \tag{5a}\]  
\[\mathbf{u}^{k}=\left[ \begin{array}{ccc}v_{d}^{k}&v_{q}^{k}\end{array} \right] ^{T} \tag{5b}\]  
\[\mathbf{y}^{k}=\left[ \begin{array}{ccc}i_{d}^{k+1}&i_{q}^{k+1}\end{array}  \right]^{T} \tag{5c}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Notation** & **Description** & **Notation** & **Description** \\ \hline \(B_{\omega}\) & Damping coefficient & \(f(\theta)\) & Normalized trapezoidal function \\ \(\varepsilon_{a,b,c}\) & RMS value of phase back EMF & \(\tilde{\omega}_{r},\omega_{r}^{s}\) & Estimated speed and reference speed \\ \(i_{a,b,c},v_{a,b,c}\) & Phase armature current and Phase & \(i_{d,q}^{s},\tilde{\theta}\) & Updated currents and Estimated position \\ \(i_{d},i_{q}\) & Direct-axis and Quadrature-axis currents & \(\mathbf{u}_{s}\) & Synchronous rotating frame stator \\ \(v_{d},v_{q}\) & Direct axis and Quadrature axis voltages & \(S_{a,b,c}\) & State space vectors \\ \(v_{n}\) & Neutral voltage & \(\mathbf{i}_{s}\) & Synchronous rotating frame stator \\ \(\mathbf{i}_{sm},\mathbf{u}_{sm}\) & Current and voltage constraints & \(e_{s},i_{sm}\) & Current predictive error and Current \\ \(J,\rho\) & Inertia of the rotor and number of poles & \(\mathbf{h},\chi\) & Correction factor and Sigma point matrix \\ \(L_{a,b,c},M\) & Phase self-inductance and Mutual & \(\lambda,P\) & Scaling parameter and Covariance matrix \\ \(L_{d},L_{q}\) &ductance & \(\alpha\) & Spread of sigma points around the mean \\ \(R_{s},T_{s}\) & Phase resistance and Sampling Time & \(k\) & Secondary scaling parameter \\ \(T_{e},T\) & Motor electromagnetic and loading & \(W_{m}^{m},W_{i}^{c}\) & Weights for mean and covariance \\  & torques & & \\ \(\theta\) & Rotor position angle or position & \(\overline{x}^{k},\overline{\mathbf{P}}^{k}\) & Predicted mean and predicted covariance \\ \(\phi_{f}\) & Amplitude of stator permanent magnet & \(\chi_{i}^{k}\) & Transformed sigma points in \\  & flux linkage & & measurement space \\ \(\omega_{e},\mu_{0}\) & Electrical velocity and Permeability in air & \(\overline{x}^{k}\) & Mean in measurement space \\  & gap & & \\ \(\mathbf{Q}_{MPC}\), & Error and control weighting coefficient & \(\phi^{k},\mathbf{K}^{k}\) & Jacobian of the matrix and Kalman gain \\ \(\mathbf{R}_{MPC}\) & \(\mathbf{C}\) & Covariance matrices for process and & \\ \(\mathbf{Q}^{k},\mathbf{R}^{k}\) & measurement noises & \(\mathbf{P}_{yy}^{k}\) & Predicted covariance matrix \\ \(\mathbf{w}^{k},\mathbf{v}^{k}\) & Process noise vector and Measurement & \(\mathbf{P}_{xy}^{k}\) & Cross-correlation matrix between actual \\  & noise vector & & and predicted spaces \\ \(I_{m}\) & Maximum rotor current & \(\mathbf{P}_{yy}^{k}\) & Function that maps our sigma points to \\ \(K_{\tau},K_{\varepsilon}\) & Torque constant and Back EMF constant & \(\mathbf{P}_{yy}^{k}\) & Covariance in measurement space \\ \hline \end{tabular}
\end{table}