# Multi-Phase Stator Current Tracking with Gradual Penalization of Commutations
## 2 Multi-Phase Predictive Stator Current Control  
The Predictive Stator Current Control (PSCC) for multi-phase drives is applied considering the Clarke projection of stator currents into the \(\alpha-\beta\) and \(x_{j}-y_{j}\) planes, as shown in Figure 1. For a 5 pIM, just one \(x-y\) plane is produced. The torque-producing currents must follow a reference set by the torque/speed control loop. The \(x-y\) current references are set to zero since they contribute to losses. The scheme is presented in Figure 1, where flux and torque are independently regulated. The flux set point is provided by \(i_{d}^{*}\), whereas reference \(i_{q}^{*}\) is used for the electrical torque. These references are projected to the \(\alpha-\beta\) space using the Park transformation, obtaining a reference for stator current in the \(\alpha-\beta\) plane as \(I_{\alpha-\beta}^{*}=D\left(i_{d}^{*},i_{q}^{*}\right)^{\intercal}\), where matrix \(D\) is given by  
\[D= \begin{pmatrix}\cos\ \theta_{a}&\sin\ \theta_{a}\\ -\sin\ \theta_{a}&\cos\ \theta_{a}\end{pmatrix}  \tag{1}\]  
The flux angle \(\theta_{a}\) is obtained as \(\theta_{a}=\int\omega_{e}dt\). The reference signal (\(r\)) is set by the speed/torque controller, where the \(\alpha-\beta\) plane receives a sinusoidal wave with amplitude \(I^{*}\) and frequency \(f_{e}\) and the \(x-y\) plane is set to zero. As a result, the reference vector for the \(\alpha-\beta-x-y\) axes is \(r(k)=i_{s}^{*}(k)\) defined by  
\[r=(I^{*}\sin 2\pi f_{e}kT_{s},\ I^{*}\cos 2\pi f_{e}kT_{s},\ 0,\ 0)^{\intercal}. \tag{2}\]  
In single-vector PSCC, the control action is the state of the VSI \(u=(K_{a},K_{b},\cdots,K_{e})^{\top}\), where the values \(K_{j}\) indicate the state of the corresponding VSI switch for each phase a to \(\mathbf{e}\). For a 5 pIM, there are 32 possible VSI configurations, each producing a VV [28]. The stator voltages provided by each VSI state can be found as \(V(k)=V_{DC}TMu(k)\), where \(V_{DC}\) is the DC-link voltage and  
\[T=\frac{1}{5} \begin{pmatrix}4&-1&-1&-1&-1\\ -1&4&-1&-1&-1\\ -1&-1&4&-1&-1\\ -1&-1&-1&4&-1\\ -1&-1&-1&-1&4\end{pmatrix} , \tag{3}\]  
\[M=\frac{2}{5} \begin{pmatrix}1&\gamma_{1}^{\zeta}&\gamma_{2}^{\zeta}&\gamma_{3 }^{\zeta}&\gamma_{4}^{\zeta}\\ 0&\gamma_{1}^{\zeta}&\gamma_{2}^{\zeta}&\gamma_{3}^{\zeta}&\gamma_{4}^{\zeta} \\ 1&\gamma_{2}^{\zeta}&\gamma_{4}^{\zeta}&\epsilon_{\theta}&\gamma_{3}^{\zeta} \\ 0&\gamma_{2}^{\zeta}&\gamma_{4}^{\zeta}&\gamma_{1}^{\zeta}&\gamma_{3}^{\zeta} \\ 1/2&1/2&1/2&1/2&1/2\end{pmatrix}  \tag{4}\]  
where \(\gamma_{h}^{\zeta}=\cos h\vartheta\), \(\gamma_{h}^{\zeta}=\sin h\vartheta\), and \(\vartheta=2\pi/5\). Figure 1: Diagram of PSCC for a five-phase induction motor. The PSCC uses a predictive model to link future stator currents to actual voltages. This model is often a set of discrete-time state-space equations as follows:  
\[\hat{i}(k+1)=(C(\omega)+G)i(k)+BV(k) \tag{5}\]  
where \(i\) contains the \(\alpha\), \(\beta\), \(x\), and \(y\) stator currents; \(\hat{i}\) is the prediction of \(i\); and \(\omega\) is the angular speed. Matrices \(C\) and \(B\) are obtained from first principles applying time discretization with sampling time \(T_{s}\). This gives \(C=(I+A_{c}T_{s})\) and \(B=T_{s}B_{c}\), where  
\[A_{c}= \begin{pmatrix}a_{2}&-a_{4}&0&0\\ a_{4}&a_{2}&0&0\\ 0&0&a_{3}&0\\ 0&0&0&a_{3}\end{pmatrix} , \tag{6}\]  
\[B_{c}= \begin{pmatrix}c_{2}&0&0&0\\ 0&c_{2}&0&0\\ 0&0&c_{3}&0\\ 0&0&0&c_{3}\end{pmatrix} . \tag{7}\]  
The term \(G\) accounts for the effect of the rotor currents, which are usually unmeasured variables. The technique of [31] is used to directly estimate the effect of rotor currents on the evolution of stator currents, then \(G\) can be written as  
\[G_{\alpha-\beta}(k)= \begin{pmatrix}I+T_{s}\begin{pmatrix}a_{\pi 4}&a_{\pi 4}\\ -a_{\pi 4}&a_{\pi 4}\end{pmatrix}\end{pmatrix} \cdot \begin{pmatrix}\cos\delta&- \sin\delta\\ \sin\delta&\cos\delta\end{pmatrix} , \tag{8}\]  
with \(G_{x-y}(k)=0\) since rotor currents do not induce \(x-y\) components in the stator.