## 2 Port-Hamiltonian Systems
### Formalism and Property
#### 2.1.2. Conservative Interconnection  
The interconnection of the components is achieved by relating all the voltages and currents through the application of the [PERSON]'s laws to the interconnection network (schematic). This defines a conservative interconnection, according to [PERSON]'s theorem recalled below (see also [26] and [27] SS9.4). Theorem 1 ([PERSON]): _Consider an electronic circuit made of \(N\) edges defined in same convention (here receiver), with individual voltages \(v=(v_{1},\cdots,v_{N})^{\intercal}\) and currents \(i_{n}=(i_{1},\cdots,i_{N})^{\intercal}\) which comply with the [PERSON]'s laws. Then_  
\[\mathbf{v}^{\intercal}\cdot\mathbf{i}=0. \tag{2}\]  
A direct consequence of (2) is that no power is created nor lost in the structure: \(\mathbf{v}^{\intercal}\cdot\mathbf{i}=\sum_{i=1}^{N}\mathtt{P}_{n}=0\), with \(\mathtt{P}_{n}=v_{n}\cdot i_{n}\) the power received by edge \(n\), thus defining a _conservative interconnection_ ([PERSON]'s theorem is a special case of a more general interconnection structure, namely, the _Dirac structure_ (see [23] SS2.1.2 for details)). Now, denote \((\mathbf{v_{s}},\mathbf{i_{s}})\), \((\mathbf{v_{d}},\mathbf{i_{d}})\), and \((\mathbf{v_{p}},\mathbf{i_{p}})\) the sets of all the power variables associated with storage components, dissipative components, and sources(respectively), and \(\mathbf{v}=(\mathbf{v}_{\mathbf{s}}^{\intercal},\mathbf{v}_{\mathbf{d}}^{ \intercal},\mathbf{v}_{\mathbf{p}}^{\intercal})^{\intercal}\), \(\mathbf{i}=(\mathbf{i}_{\mathbf{s}}^{\intercal},\mathbf{i}_{\mathbf{d}}^{ \intercal},\mathbf{i}_{\mathbf{p}}^{\intercal})^{\intercal}\) the vectors of all the power variables. Then, [PERSON]'s theorem restores the power balance (1) with  
\[\mathbf{v}^{\intercal}\cdot\mathbf{i} = \mathbf{v}_{\mathbf{s}}^{\intercal}\cdot\mathbf{i}_{\mathbf{s}}+ \mathbf{v}_{\mathbf{d}}^{\intercal}\cdot\mathbf{i}_{\mathbf{d}}+\mathbf{v}_{ \mathbf{p}}^{\intercal}\cdot\mathbf{i}_{\mathbf{p}} \tag{3}\] \[= \underbrace{\
abla\mathcal{H}^{\intercal}(\mathbf{x})\cdot \frac{\mathbf{a}\mathbf{x}}{\mathrm{d}t}}_{\frac{\mathbf{a}\mathbf{x}}{ \mathrm{d}t}}+\underbrace{\mathbf{z}(\mathbf{w})^{\intercal}\cdot\mathbf{w}}_ {\text{D}}-\underbrace{\mathbf{u}^{\intercal}\cdot\mathbf{y}}_{\text{S}},\]  
where \(\
abla\mathcal{H}:\mathbb{R}^{n_{S}}\rightarrow\mathbb{R}^{n_{S}}\) denotes the gradient of the total energy \(\mathsf{E}=\mathcal{H}(\mathbf{x})=\sum_{\mathbf{s}=1}^{n_{S}}h_{s}(x_{s})\) with respect to (w.r.t.) the vector of the states \([\mathbf{x}]_{s}=x_{s}\), and function \(\mathbf{z}:\mathbb{R}^{n_{D}}\rightarrow\mathbb{R}^{n_{D}}\) denotes the collection of functions \(z_{d}\) w.r.t. the vector \(\mathbf{w}\in\mathbb{R}^{n_{D}}\) of \([\mathbf{w}]_{d}=w_{d}\) so that \(\mathbf{z}(\mathbf{w})^{\intercal}\cdot\mathbf{w}=\sum_{d=1}^{n_{D}}\mathsf{ D}_{d}(w_{d})\) is the total dissipated power. The above description of storage components, dissipative components, and source, along with the conservative interconnection stated by the [PERSON]'s laws, constitute the minimal definition of a port-Hamiltonian system (PHS) (see [23] SS2.2). In this work, we focus on circuits that admit an _explicit realization_ of PHS, for which the quantities \(\mathbf{b}=(b_{1},\cdots,b_{N})^{\intercal}=(\frac{\mathrm{d}\mathbf{x}}{ \mathrm{d}t},\mathbf{w},-\mathbf{y})^{\intercal}\) (with \(b_{n}=v_{n}\) or \(b_{n}=i_{n}\)) can be expressed as linear combinations of the remaining \(N\) powers variables organized in the dual vector \(\mathbf{a}=(a_{1},\cdots,a_{N})^{\intercal}=(\
abla\mathcal{H}(\mathbf{x}), \mathbf{z}(\mathbf{w}),\mathbf{u})^{\intercal}\) (with \(a_{n}=i_{n}\) if \(b_{n}=v_{n}\) or \(a_{n}=v_{n}\) if \(b_{n}=i_{n}\)):  
\[\mathbf{b}=\mathbf{J}\cdot\mathbf{a}. \tag{4}\]  
Then, \(\mathbf{a}^{\intercal}\cdot\mathbf{b}=\mathbf{a}^{\intercal}\cdot\mathbf{J} \cdot\mathbf{a}=0\) from [PERSON]'s theorem, so that the matrix \(\mathbf{J}\) is necessarily skew-symmetric (\(\mathbf{J}^{\intercal}=-\mathbf{J}\)).