## 2 Port-Hamiltonian Systems
### Formalism and Property
#### 2.1.2. Conservative Interconnection
More precisely, we consider the following algebraic-differential system of equations  
\[\underbrace{\left( \begin{array}{c|c|c}\frac{\mathrm{d}\mathbf{x}}{\mathrm{d }t}\\ \hline\mathbf{w}\\ \hline-\mathbf{y}\end{array} \right)}_{\mathbf{b}}=\underbrace{\left( \begin{array} []{c|c}\mathbf{J}_{\mathbf{x}}&-\mathbf{K}&-\mathbf{G}_{\mathbf{x}}\\ \hline\mathbf{K}^{\intercal}&\mathbf{J}_{\mathbf{w}}&-\mathbf{G}_{\mathbf{w}}\\ \hline\mathbf{G}_{\mathbf{x}}^{\intercal}&\mathbf{G}_{\mathbf{w}}^{\intercal} &\mathbf{J}_{\mathbf{y}}\end{array} \right)}_{\mathbf{a}}. \tag{5}\]  
where matrices \(\mathbf{J}_{\mathbf{x}},\mathbf{J}_{\mathbf{w}},\mathbf{J}_{\mathbf{y}}\) are skew-symmetric. The significance of the structure matrices is the following:  
\(\mathbf{J}_{\mathbf{x}}\in\mathbb{R}^{n_{S}\times n_{S}}\) expresses the conservative power exchanges between storage components (this corresponds to the so-called \(\mathbf{J}\) matrix in classical Hamiltonian systems);  
\(\mathbf{J}_{\mathbf{w}}\in\mathbb{R}^{n_{D}\times n_{D}}\) expresses the conservative power exchanges between dissipative components;  
\(\mathbf{J}_{\mathbf{y}}\in\mathbb{R}^{n_{P}\times n_{P}}\) expresses the conservative power exchanges between ports (direct connections of inputs to outputs);  
\(\mathbf{K}\in\mathbb{R}^{n_{S}\times n_{D}}\) expresses the conservative power exchanges between the storage components and the dissipative components;  
\(\mathbf{G}_{\mathbf{x}}\in\mathbb{R}^{n_{S}\times n_{P}}\) expresses the conservative power exchanges between ports and storage components (input gain matrix);  
\(\mathbf{G}_{\mathbf{w}}\in\mathbb{R}^{n_{D}\times n_{P}}\) expresses the conservative power exchanges between ports and dissipative components (input gain matrix). The PHS (5) fulfills the definition of passivity (e.g., [16]) according to the following property. **Property 1** (**Power Balance**). : _The variation of the total energy \(\mathsf{E}=\mathcal{H}\big{(}\mathbf{x}\big{)}\) of a system governed by (5) is given by (1), with \(\mathsf{D}=\mathbf{z}(\mathbf{w})^{\intercal}\cdot\mathbf{w}\geq 0\) the total dissipated power, and \(\mathsf{S}=\mathbf{u}^{\intercal}\cdot\mathbf{y}\) the total power incoming on external ports._  
Proof. : We have \(\mathbf{a}^{\intercal}\cdot\mathbf{b}=\frac{\mathrm{d}\mathbf{E}}{\mathrm{d}t} +\mathsf{D}-\mathbf{S}\). Now \(\mathbf{a}^{\intercal}\cdot\mathbf{b}=\mathbf{a}^{\intercal}\cdot\mathbf{J} \cdot\mathbf{a}=0\) since \(\mathbf{J}\) is skew-symmetric. **Remark 1** (Power variables). : _This work is devoted to the treatment of electronic circuits for which power variables are chosen as current and voltage. However, all the aforementioned definitions apply equally to multiphysical systems, provided an adapted set of power variables, generically denoted by flux (currents, velocities, magnetic flux variations) and efforts (voltages, forces, magnetomotive force), the product of which is a power (see [23] Table 1.1). This follows the bond-graph modeling approach [28; 29], on which the PHS formalism is built (see [23] SS1.6 and 2.1). The treatment of multiphysical audio systems in the PHS formalism can be found in [30] (electromechanical piano that includes mechanical, electrical, and magnetic phenomena) and [31] (SIII.B) (modulated air flow for musical acoustics applications that includes mechanical and acoustical phenomena)._