# Continuous Control Set Predictive Current Control for Induction Machine
## 4 Stability and Controllability of the System
### Stability Analysis  
To prove stability of the control law, one must know the closed loop dynamics of the regulated system. The control law (17) derived in the previous section can be plugged into (13) so that closed loop system dynamics can be derived as shown in Equation (24):  
\[\begin{gathered}\mathbf{x}[k+1]=\mathbf{A^{\prime}}(\omega_{r})\mathbf{x}[k]+ \mathbf{B_{d}}\mathbf{E_{d}}\mathbf{i}_{\alpha\beta}^{*}[k+1]\\ \mathbf{A^{\prime}}(\omega_{r})=\mathbf{A}_{d}(\omega_{r})+\mathbf{B_{d}}\mathbf{ K_{d}}(\omega_{r})\end{gathered} \tag{24}\]  
One can observe that derived control law and closed loop dynamic system resemble classic state feedback control method, where \(\mathbf{K}_{d}(\omega_{r})\) can be considered as feedback gain matrix. If the term \(\mathbf{A^{\prime}}\) can be proven to be Hurwitz, i.e., all eigenvalues of the \(\mathbf{A^{\prime}}\) matrix lie in the unit circle, the system can be considered stable. In this case, state matrix \(\mathbf{A}_{d}(\omega_{r})\) and feedback gain matrix \(\mathbf{K}_{d}(\omega_{r})\) depend on the instantaneous value of the rotor speed \(\omega_{r}\), and the expressions for the calculation and analytical representation of eigenvalues of such system become extremely complicated. Therefore, the authors conducted numerical solutions of the system poles for the range of the possible rotor speed values, and graphically presented the poles. This kind of approach was not considered in the referenced papers that deal with similar topics. In [27, 28] only the performance is presented, in [29], the stability for the finite control set _MPC_ is achieved by calculating sets of weighing matrices, but no approach for stability of continuous control set _MPC_. In [30] there is a stability consideration, however the weighing factor calculation is included and discrete Lyapunov function is used.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Machine Parameter** & **Value** & **Control Parameter** & & **Value** \\ \hline Stator resistance & \(R_{s}\) (\(\Omega\)) & 1.1507 & MRAS proportional gain & \(K_{P_{max}}\) & 1000 \\ Rotor resistance & \(R_{r}\) (\(\Omega\)) & 1.0107 & MRAS integrator gain & \(K_{I_{max}}\) & 10,000 \\ Stator inductance & \(L_{s}\) (H) & 0.1315 & Speed controller proportional gain & \(K_{P\omega}\) & 10 \\ Rotor inductance & \(L_{r}\) (H) & 0.1315 & Speed controller integrator gain & \(K_{I\omega}\) & 100 \\ Mutual inductance & \(L_{m}\) (H) & 0.126 & Simulation time step & \(T_{sim}\) (s) & \(10^{-5}\) \\ Pole pairs & \(p\) & 2 & DC link voltage & \(v_{dc}\) (V) & 565 \\ Inertia moment & \(J\) (kgm\({}^{2}\)) & 0.129 & Inverter switching frequency & \(f_{sw}\) (Hz) & \(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Induction machine parameters and control parameters.  
Figure 1: Block diagram of the control method.  
Figure 2 shows discrete poles of the closed loop system for the rotor speed range: \(-157\;\text{rad}\,\text{s}^{-1}\leqslant\omega_{r}\leqslant 157\;\text{rad}\, \text{s}^{-1},\omega_{r}\in\mathbb{Z}\). It can be seen that the closed loop poles stay inside the unit circle throughout the whole speed range, which suggest that the system is stable.