## 2 Dynamic Model of FFPMSM
### Healthy Model  
In the original _abcde_ frame, the equations for the stator voltage can be expressed as follows [12]:  
\[v_{s}=R_{s}i_{s}+\frac{d}{dt}(L_{s}i_{s}+\phi_{fm}) \tag{1}\]  
where \(v_{s}\) is a voltage vector of the stator \([v_{sa},v_{sb},v_{sc},v_{sd},v_{se}]^{\top}\), \(i_{s}\) is a current vector of the stator \([i_{sa},i_{sb},i_{sc},i_{sd},i_{se}]^{\top}\), \(R_{s}\) is a diagonal stator resistance \(5\times 5\) matrix, \(L_{s}\) is a stator inductance \(5\times 5\) matrix that has a diagonal, in which each element corresponds to the self-inductance of each phase, and the off-diagonal corresponds to the mutual inductance between each phase. \(\phi_{fm}\) represents the flux-linkage vector of a permanent magnet, including \(\phi_{f1}\) and \(\phi_{f3}\) (first- and third-order harmonic components of back-EMF) in the _abcde_ frame.  
\[\phi_{fm}=\phi_{f1}\left[\begin{array}{c}\cos\theta\\ \cos(\theta-\frac{2\pi}{5})\\ \cos(\theta-\frac{4\pi}{5})\\ \cos(\theta-\frac{6\pi}{5})\\ \cos(\theta-\frac{8\pi}{5})\end{array}\right]+\phi_{f3}\left[\begin{array}[] {c}\cos 3\theta\\ \cos(3\theta-\frac{6\pi}{5})\\ \cos(3\theta-\frac{12\pi}{5})\\ \cos(3\theta-\frac{18\pi}{5})\\ \cos(3\theta-\frac{24\pi}{5})\end{array}\right] \tag{2}\]  
When applying the Clarke and Park transformation (A1), the \(dq_{p}\) and \(dq_{s}\) frame of the stator voltage equation is given by [13; 25]:  
\[\left\{\begin{aligned} v_{dp}&=R_{s}i_{dp}+L_{dp} \frac{di_{dp}}{dt}-n_{p}\Omega L_{ap}i_{ap}\\ v_{qp}&=R_{s}i_{qp}+L_{qp}\frac{di_{qp}}{dt}+n_{p} \Omega(L_{dp}i_{dp}+\phi_{f1})\\ v_{ds}&=R_{s}i_{ds}+L_{ds}\frac{di_{ds}}{dt}-3n_{p} \Omega L_{qs}i_{qs}\\ v_{qs}&=R_{s}i_{qs}+L_{qs}\frac{di_{qs}}{dt}+3n_{p} \Omega(L_{ds}i_{ds}+\phi_{f3})\end{aligned}\right. \tag{3}\]  
where \(L_{dp}\), \(l_{qp}\), \(i_{dp}\), \(i_{qp}\), \(v_{dp}\), and \(v_{qp}\) are the inductance, current, and voltage in the \(d_{p}\)-\(q_{p}\) frame affected by the first harmonic, respectively; \(L_{ds}\), \(L_{qs}\), \(i_{ds}\), \(i_{qs}\), \(v_{ds}\), and \(v_{qs}\) are the inductance, current, and voltage in the \(d_{s}\)-\(q_{s}\) frame affected by the third harmonic, respectively; and \(\Omega\) and \(n_{p}\) are the mechanical speed and the number of pole pairs, respectively.  
The torque produced by the FFPMSM is given as follows [25]:  
\[T_{e}=\frac{5}{2}n_{p}[\phi_{f1}i_{qp}+(L_{dp}-L_{qp})i_{dp}i_{qp}+3\phi_{f3}i _{qs}+3(L_{ds}-L_{qs})i_{ds}i_{qs}] \tag{4}\]  
In order to assist the design of control laws for the FFPMSM and improve the comprehension of the model, the voltages (3) can be expressed as follows:  
\[\left\{\begin{aligned} \dot{i}_{dp}&=c_{1}i_{dp}+c_{2} \Omega i_{qp}+b_{1}v_{dp}\\ i_{qp}&=c_{3}i_{qp}-c_{4}\Omega i_{dp}-c_{5}\Omega+b_{2}v_{ qp}\\ i_{ds}&=c_{6}i_{ds}+c_{7}\Omega i_{qs}+b_{3}v_{ds}\\ i_{qs}&=c_{8}i_{qs}-c_{9}\Omega i_{ds}-c_{10}\Omega+b_{4}v_{ qs}\\ \dot{\Omega}&=a_{1}T_{e}-a_{1}T_{L}-a_{2}\Omega\end{aligned}\right. \tag{5}\]  
where the machine parameters describe the following FFPMSM model components as follows: \(c_{1}=-\frac{R_{s}}{L_{dp}}\), \(c_{2}=\frac{L_{qp}}{L_{dp}}n_{p}\), \(c_{3}=-\frac{R_{s}}{L_{dp}}\), \(c_{4}=\frac{L_{dp}}{L_{qp}}n_{p}\), \(c_{5}=\frac{\dot{\phi}_{f1}}{L_{qp}}n_{p}\), \(c_{6}=-\frac{R_{s}}{L_{s}}\), \(c_{7}=\frac{3L_{qs\[c_{8}=-\frac{R_{s}}{L_{qs}}\,c_{9}=\frac{3L_{qs}}{L_{qs}}n_{p},c_{10}=\frac{3 \phi_{f3}}{L_{qs}}n_{p},b_{1}=\frac{1}{L_{lp}},b_{2}=\frac{1}{L_{qp}},b_{3}=\frac{1 }{L_{bs}},b_{4}=\frac{1}{L_{pq}},a_{1}=\frac{1}{7},a_{2}=\frac{F}{7}.\]