## 3 Improved Passivity-Based Control
Based on [21; 24], the interconnection does not occur and \(J\) is just the identity matrix. \(R(x)\) represents damping matrix, which is the sum of the \(R_{Lf}\) and dissipating resistance \(R_{i}\), which is 10. \(G(x)\) represents input matrix, where \(u\) is the input vector and \(\xi\) is the system disturbance. Input matrix is the DC voltage. During experiments, proper adjustment of the modulation index must be ensured. The system should prevent PWM modulator saturation due to the high-voltage gain response of the output control. \[\dot{x}=[J(x)-R(x)]\frac{\partial H}{\partial x}(x)+g(x)u+\xi \tag{1}\]  
The \(H(x)\) represents total stored energy [18; 20]; \(u\) and \(y\) are the input control vector and output voltage, respectively. \[y=g^{\top}(x)\frac{\partial H}{\partial x}(x)+D(x)u \tag{2}\]  
\[x_{\alpha\beta}=[i_{Lf\alpha\beta}\quad i_{\alpha\beta}\quad v_{\alpha\beta}] \tag{3}\]  
\[y_{\alpha\beta}=[v_{\alpha\alpha}\quad v_{\alpha\beta}] \tag{4}\]The total energy is represented by the Hamiltonian \(m\) external port system equation concerning \(LC\) circuits [1]. Hence, to the linear system consideration, Hamiltonian has a quadratic form. \[H(x)=\frac{1}{2}x^{\top}Qx \tag{5}\]  
The state variable could be then rewritten basing on the PH system including necessary interconnection and dissipation matrices \(J_{\alpha\beta}\) and \(R_{\alpha\beta}\)[1,19]. \[\dot{x}_{\alpha\beta}=[J_{\alpha\beta}+R_{\alpha\beta}]P^{-1}x_{\alpha\beta}+ G_{\alpha\beta}+D_{\alpha\beta}d_{\alpha\beta} \tag{6}\]  
\[A_{\alpha\beta}= \begin{bmatrix}-R_{Lf}/L_{f}&0&-1/L_{f}\\ 0&0&0\\ 1/C_{f}&-1/C_{f}&0\end{bmatrix}  \tag{7}\]  
\[B_{\alpha,\beta}= \begin{bmatrix}1/L_{f}\\ 0\\ 0\end{bmatrix} ,\quad C_{\alpha,\beta}= \begin{bmatrix}0&0&1&0\end{bmatrix} . \tag{8}\]  
\[R_{\alpha\beta}= \begin{bmatrix}R_{i}+R_{Lf}&0&0&0\\ 0&R_{i}+R_{Lf}&0&0\\ 0&0&K_{v}&0\\ 0&0&0&K_{v}\end{bmatrix}  \tag{9}\]  
\[P= \begin{bmatrix}L_{f}&0&0&0\\ 0&L_{f}&0&0\\ 0&0&C_{f}&0\\ 0&0&0&C_{f}\end{bmatrix} ,J_{\alpha\beta}= \begin{bmatrix}0&0&-1&0\\ 0&0&0&-1\\ 1&0&0&0\\ 0&1&0&0\end{bmatrix}  \tag{10}\]  
\[G_{\alpha\beta}= \begin{bmatrix}V_{dc}&0\\ 0&V_{dc}\\ 0&0\\ 0&0\\ 0&0\end{bmatrix} ,\quad D_{\alpha\beta}= \begin{bmatrix}0&0\\ 0&0\\ -1&0\\ 0&-1\end{bmatrix}  \tag{11}\]  
\[d_{\alpha\beta}=[i_{\alpha\beta}&i_{\alpha\beta}] \tag{12}\]  
The total energy in the presented system is the overall sum of the energy stored in the output filters components [24]. \[G_{\alpha\beta,d\eta}=\frac{L_{f}di_{L/\alpha\beta prev}}{dt}+R_{Lf}i_{L \alpha\beta prev}- \tag{13}\]  
\[-R_{i}(i_{Lf\alpha\beta}-i_{Lf\alpha\beta prev})+v_{\alpha prev}\]  
For calculation of the IPBC2 controller [24], the value of the reference filter current is needed--\(i_{L\alpha\beta prev}\). In this element, it is necessary to take into account the value of the difference between output and reference voltage, so the results should be more robust not only for the output filter current but also for the output filter voltage. \[K_{v}(v_{\alpha\beta}-v_{\alpha\beta prev}) \tag{14}\]  
The control law for the stationary \(\alpha\beta\) frame for the \(K_{v}=0\) is the same as the one for the conventional PBC [25]. Introducing \(K_{v}\) (the control of the output voltage error) makes the control law IPBC version. The final form consists of the derivative over the output voltage and voltage gain--IPBC2. On the other hand, the derivative over the \(C_{f}\) makes the system stabilize even longer and it is an important factor in the process of decreasing the \(THD_{v}\). During implementation, the system should behave passively. Parameters in Table 2 maintain stability and passiveness of control. The condition is to store energy less than or equal to the total supplied energy to the system [1]. Additionally, it must be taken into account that passive controllers could modify only destabilizing forces--the forces acting on the system are divided into stabilizing, destabilizing, and supporting ones [1].