RCL Ladder Network (ODE)

Description

This benchmark considers a linear, time-invariant pH-ODE model for electrical ladder networks as described and used for numerical examples in [1, 3].

The network consists of $N=\tfrac{n}{2}$ cells, where each cell $i=1,\ldots,N$ contains a resistor with resistance $R_i$, capacitor with capacity $C_i$, and inductor with inductance $L_i$. The last cell contains one additional resistor with resistance $R_{\tfrac{n}{2}+1}$.

The input of the system is chosen as the current $I$ and the output is the voltage over the first capacitor $U_{C_1}$. The state variables are alternating the charge $q_i$ of the capacitors and the flux $\phi_i$ of the inductors, i.e.,

\[ x(t) = \begin{bmatrix} q_1(t), \phi_1(t), \ldots, q_{\tfrac{n}{2}}(t), \phi_{\tfrac{n}{2}}(t) \end{bmatrix} \in \R^{n}.\]

This choice yields the following pH system matrices

\[ \begin{aligned} J &= \begin{bmatrix} 0 & -1 & 0 & 0\\ 1 & \ddots & \ddots & 0\\ 0 & \ddots & \ddots & -1\\ 0 & 0 & 1 & 0 \end{bmatrix}, & R &= \mathrm{diag}(0, R_1, 0, R_2, \cdots, R_{\tfrac{n}{2}} + R_{\tfrac{n}{2}+1}),\\ Q &= \mathrm{diag}(C_1^{-1}, L_1^{-1}, C_2^{-1}, L_2^{-1}, \cdots, C_{\tfrac{n}{2}}^{-1}, L_{\tfrac{n}{2}}^{-1}), & G^\top &= \begin{bmatrix} 1 & \cdots & 0 \end{bmatrix}. \end{aligned}\]

Parameters

The number of cells $N = \tfrac{n}{2} \in \N$ can be chosen by the user and the input dimension can be set to $m = [1, 2]$. The remaining parameters $R_i$, $C_i$, and $L_i$ can be chosen as positive real numbers. Following [3], the default parameters are

\[\begin{aligned} N &= 100, \\ m &= 1, \\ R_1 &= \ldots = R_{50} = 0.2, \\ R_{51} &= 0.4, \\ C_1 &= \ldots = C_{50} = 1, \\ L_1 &= \ldots = L_{50} = 1. \\ \end{aligned}\]

Interface

The system matrices can be obtained via the RCLLadderConfig.

RCLLadderConfig(id::String)

In more detail, the system matrices from [3] can be obtained as follows:

using PortHamiltonianBenchmarkSystems
config = RCLLadderConfig("PS10")
J, R, Q, G = construct_system(config)

Alternatively, the parameters can be specified as follows:

using PortHamiltonianBenchmarkSystems
config = RCLLadderConfig(2, 2, [1, 2, 3], [4, 5], 6)
J, R, Q, G = construct_system(config)

References