# An Adaptive Model Predictive Voltage Control for LC-Filtered Voltage Source Inverters
## 3 Conventional Model Predictive Control (MPC)  
In the conventional MPC, Figure 1a, the main goal is selecting the inverter voltage vector that minimizes a cost function. Indeed, the output voltage (\(v_{o}\)) is predicted for all possible inverter voltage vectors (\(v_{inv}\)) based on the state-space model (3) and measured signals. Then, the inverter voltage vector produces the output voltage with minimum error to the reference one is selected, saved, and applied in the next sampling period. And this procedure is repeated in each sampling period [1; 2; 11; 12; 13].  
To compensate for one sample control delay, the predicted output voltage at the beginning of the (\(k\)+2)th period is required, which is calculated based on the predictive model (3) for different available converter output voltage vectors given as:  
\[v_{o}(k+2)=A_{d}(2,1)i_{f}(k+1)+A_{d}(2,2)v_{o}(k+1)+B_{d}(2,1)v_{inv}(k+1)+D_{ d}(2,1)i_{o}(k+1) \tag{4}\]  
In the above equation, the sample (\(k\)+1) system states are estimated via the state-space model in (3) and measured signals at sample \(k\). Moreover, it is usually assumed that the load-current variations in two consecutive samples are negotiable, i.e., \(i_{o}(k\)+1)\approx i_{o}\) (\(k\)).  
The predicted voltages in (4) are examined in the following cost function, then the optimum voltage vector (\(v_{inv}\)) that minimizes the cost function is selected.  
\[\left\{\begin{array}{l}\vspace{0.2 cm}g=g_{cro}{}^{2}+\lambda_{sw}g_{sw}{}^{ 2}+g_{if}\\ \vspace{0.2 cm}g_{evo}=\left\lvert v_{o,ref}(k+1)-v_{o}(k+2)\right\rvert\\ \vspace{0.2 cm}g_{sw}=\left\lvert SW(k+1)-SW(k)\right\rvert\\ \vspace{0.2 cm}g_{if}=\left\{\begin{array}{l}0\quad\text{if}\ \left\lvert i_{f}(k)\right\rvert\ \leq\ i_{f,\max}\\ \text{inf}\ \left\lvert i_{f}(k)\right\rvert\ >\ i_{f,\max}\\ \end{array}\right.\end{array}\right. \tag{5}\]  
Figure 1: Single-phase diagram of three-phase LC-filtered voltage source inverter (VSI) under (**a**) conventional model predictive control (MPC), (**b**) proposed adaptive MPC.  
As can be seen in (5), the proposed cost function includes different terms. In summary, the main goal of \(g_{euv}\) is the minimization of the voltage tracking error. \(g_{sw}\) has switching efforts to control switching frequency and switching losses, and finally, \(g_{\sharp}\) exposes the inverter (inductor) current constraint. In (5), \(SW\) and \(\lambda_{sw}\) are gating signals and the constant weighting factor. Weighting factor is usually tuned by trial and error and simulation and experiment to compromise the tracking error and average switching frequency and, consequently, switching losses. However, many efforts have been made to systematically choose weighting factors, as discussed in [31].  
The conventional MPC has a simple concept and an appropriate steady-state performance and fast, dynamic response. However, as can be concluded from (4), it depends on the system model's accuracy. Therefore, uncertainties in the system parameters may degrade the expected system performance or even cause system instability. Moreover, for good disturbance rejection, MPC needs an extra current sensor to measure load currents. Although it provides a simple and good disturbance rejection, it increases the system cost and volume. Also, it decreases the system reliability due to extra components that are subject to damage. To overcome the mentioned issues in the following section, an adaptive MPC is proposed based on the augmented state-space model, which includes all system uncertainties and disturbances. Moreover, a proper adaptive observer is presented to identify this augmented state-space model.