## 3 Improved Passivity-Based Control  
At the beginning of investigation for that type of control, the object (three-phase inverter) should be described using Euler-Lagrange equations or Port-Hamiltonian energy conversion system analysis [18]. The PCH model ensures interconnection structure related to energy shaping, so it is more beneficial for the PBC controller. In case of passivity control we are focusing on different rules for energy shaping [19; 20]. The examined system will be \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline DC link voltage & \(V_{dc}\) & 600 V \\ Switching frequency & \(f_{c}\) & 12,800 Hz \\ Nonlinear load (diode rectifier) & \(C_{L},R_{L}\) & 460 \(\mu\)F, 35 \(\Omega\) \\ Sampling time & \(T_{s}\) & 39 \(\mu\)s \\ Nominal RMS output voltage (L-L) & \(v_{o,ref}\) & 173 V \\ Reference wave frequency & \(f_{ref}\) & 50 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the studied experimental system. Figure 1: Control schematic for MPC/PBC controller with the RC load type. also further referred to as Port-Hamiltonian type. The simplest case of PBC conventional algorithm is based on the linearization of the system equation and its Lagrangian, which is not the purpose of this paper. The IPBC2-used controller is based on IDA-PBC methodology, which consists of the desired interconnections and damping matrices [21]. The next stage is solving the partial differential equation (PDE). The PBC control is the sum of total energy shaping and additionally assumed damping. This type of control is designated for all systems that could store energy on their elements--a typical feedback system that could use energy shaping principle. In the first step, the number of phases should be identified and internal structure of the bridge analyzed--in this case, H type full bridge 2L-VSI. This allows indicating the coefficient of the components in the input matrix. In the next step, based on the equations for each phase, the natural energy must be changed via adding additional resistance \(R_{i}\), named dissipative factor. The controller should track the reference error of the state variable \(x\). The process was first mentioned by [PERSON] [1]. The system is very often represented in the \(dq\) rotationary frame. For investigation, an improvement of the IDA-PBC [22; 23] method was investigated--IPBC2 presented in [24], but based on the non-interconnection equations in the matrix \(J\). In that solution, the controlled variables were introduced to the stationary \(\alpha\beta\) frame. This is connected to the fact that \(\alpha\beta\) leads to obtaining additional derivative factor over the capacitor variable. The only tunable parameters are voltage gain and dissipation factor. That additional tracking affects \(K_{v}\), which is the gain of the output voltage included in the control of each phase. In the calculation it depends on switching frequency, because of the occurrence of the derivatives. This type of control was previously used in other papers, e.g., with the dynamic resistive \(R\) load during the unbalanced defects in the delta connected inverter. The system should be described in the \(abc\) rotationary frame, taking into account the coefficient depending on the load type, star or delta, and the modulation technique [16; 21]. In this study, the star load connection was implemented both for MPC and PBC. The internal matrix coefficients for the Clarke transformation vary in case of the used load type [16]. The load current is treated as an independent disturbance. In case of application of the passivity-based control, the system must fulfill two conditions [1; 19]:  
* System without the feedback must be stable. * Stored energy must be lower than supplied energy. The problem formulation comes down to description of the system by the following Equations (1) and (2). The IPBC2 problem formulations focused on the difference between interconnection matrix and dissipation matrix and handling the error between current and previous states.