# An Adaptive Model Predictive Voltage Control for LC-Filtered Voltage Source Inverters
## 5 Simulation and Experimental Results  
The experimental setup in Figure 2 was prepared to check the theoretical analysis. The laboratory setup includes a three-phase 5 kW VSI, a three-phase resistive load, and an LC-type output filter. The DS1007 dSPACE system platform is also employed to drive the VSI and realize both control methods. Moreover, to generate switching pulses and digitalize the measured currents and voltages, a DS5101 digital waveform output board and an Analog-to-Digital DS2004 board are used, respectively. The parameters of three-phase LC-filtered VSI are given in Table 1.  
The steady-state performance of the conventional and the proposed adaptive MPC under linear and nonlinear loads are shown in Figures 3 and 4, respectively, which includes output voltages and current and voltage tracking error. Figures confirm the acceptable steady-state error and low total harmonic distortion (THD) of the output voltage under both control methods and different load conditions. It is worth noting that the average switching frequency of the proposed control method and the conventional one under linear (nonlinear) load are 5.7 kHz (5.3 kHz) and 5.8 kHz (5.3 kHz), respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Voltage Source Inverter(VSI) and Inductor and Capacitor (LC) Filter** \\ \hline Nominal power & 5 (kW) \\ Line voltage (rms) & 400 (V) \\ Output frequency (f) & 50 (Hz) \\ Inductor (\(L_{\beta}\)) & 4 (mH) \\ Capacitor (\(\hat{C}_{\beta}\)) & 20 (\(\mu\)F) \\ DC-link voltage (\(V_{dc}\)) & 700 (V) \\ Sampling time (\(T_{S}\)) & 25 (\(\mu\)s) \\ \hline \multicolumn{2}{c}{**Linear Load**} \\ \hline Three-phase resistive load & 30 (\(\Omega\)) \\ \hline \multicolumn{2}{c}{**Nonlinear Load**} \\ \hline Three-phase diode-bridge rectifier & \\ Resistive load & 60 (\(\Omega\)) \\ \hline \multicolumn{2}{c}{**Control Parameters**} \\ \hline \(\lambda_{SW}\) & 0.5 \\ Observer Poles & 0.03, 0.05, 0.35, 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 2: Experimental setup to implement the conventional model predictive control (MPC) and the proposed adaptive MPC on a three-phase inductor and capacitor filter (LC)-filtered voltage source inverter (VSI). Moreover, the transient performance of both control methods under a step change of resistive load is shown in Figure 5. In this figure, the load power has been suddenly changed from zero to nominal ones. This figure shows excellent transient response and also disturbance rejection using the model predictive control method. Figure 4: Simulation results showing steady-state performance under nominal nonlinear load (5 kW), (**a**) the proposed adaptive MPC, (**b**) conventional MPC. Figure 3: Simulation results showing steady-state performance under nominal linear load (5 kW), (**a**) the proposed adaptive MPC, (**b**) conventional MPC. In the previous tests, both control methods have almost the same performance since the filter parameters' nominal values are used. However, the conventional MPC employs extra load-current sensors, while the proposed MPC only uses the estimated signals provided by the adaptive observer. The effect of filter parameter uncertainties is investigated in Figure 6, where a 75% error in the filter capacitance is considered as an example. As can be seen, under this condition, the THD of the output voltage under conventional and the proposed MPCs are 7.8% and 3%, respectively. Moreover, the steady-state error under conventional MPC is about twice the proposed ones. Therefore, this parameter uncertainty significantly degrades the performance of the conventional MPC, while the proposed adaptive MPC based on the augmented state-space model and the adaptive observer can adequately tackle this issue. The adaptive observer outputs are shown in Figure 7, including estimated capacitor voltages, inductor currents, lumped disturbances, and estimation errors.