RLC Ladder (Gugercin/Antoulas 2004)

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RCL Circuit Equations

Background
Benchmark ID	rlc_ladder_GA_n?m1q1
Category	morwiki
System-Class	LTI-FOS
Parameters
nstates	arbitrary
ninputs	1
noutputs	1
nparameters	0
components	A, B, C, D
Copyright
License	MIT
Creator	Serkan Gugercin
Editor	Jens Saak
Location
NA

Description

Figure 1: One section of the circuit.

This model represents an electic circuit made of a cascade of interconnected sections. Each section consists of a simple setup as depicted in Figure 1. The input is the voltage \(V\) applied to the first section; the output is the current \(I\) of the first section.

\[ \begin{align} \dot x(t) &= A x(t) + B u(t), \\ y(t) &= C x(t) + D u(t), \\ x(0) &= x_0, \end{align} \]

with \(t>0\).

Origin

This model was used in the numerical experiments section of .

Data

The following Matlab code assembles the above described \(A\), \(B\), \(C\), and \(D\), matrices of requested dimensions. The input parameters R, Rbar, Cap and L represent the two resistances, the capacitance and the inductivity of each section of the circuit, as depicted in Figure 1. They default to the values from , when omitted.

function [A, B, C, D, sys_rlc] = circuit_sparse(N, R , Rbar, Cap, L)
% circuit_sparse generates a SISO RLC ladder model of prescribed dimensions
%
% The model follows the descriptioin in DOI:10.1080/00207170410001713448
% and the input parameters R, Rbar, Cap and L default to the values given
% there, when omitted. If also N is omitted the original 100 dimensional
% realization is created.
%
% The code is a sparse adaption of the original dense generator function by
% Serkan Gugercin.
%

%
% MIT License
%
% Copyright (c) 2026 The MORwiki Community
%
% Permission is hereby granted, free of charge, to any person
% obtaining a copy of this software and associated documentation files
% (the "Software"), to deal in the Software without restriction,
% including without limitation the rights to use, copy, modify, merge,
% publish, distribute, sublicense, and/or sell copies of the Software,
% and to permit persons to whom the Software is furnished to do so,
% subject to the following conditions:
%
% The above copyright notice and this permission notice shall be
% included in all copies or substantial portions of the Software.
%
% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
% EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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% NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
% BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
% ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
% CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
% SOFTWARE.
%

if nargin < 1
    N = 100;
end

if nargin < 2
    % parameter values from DOI:10.1080/00207170410001713448
    R    = 0.1;
    Rbar = 1;
    Cap  = 0.1;
    L    = 0.1;
end

z1 =  1 / L                         * ones(N, 1);
z2 = -1 / L * Rbar * R / (Rbar + R) * ones(N, 1);
z3 = -1 / L * Rbar / (Rbar + R)     * ones(N, 1);

% Every second row needs to be adapted for the second ladder step
z3(2:2:N) = -1 / Cap;
z2(1:2:N) = -1 / Cap / (Rbar + R);
z1(2:2:N) =  1 / Cap * Rbar / (Rbar + R);

% start end end are exceptional
z2(1)     = -1 / R / Cap; % sets A(1, 1)
z3(1)     = -1 / Cap;     % sets A(1, 2)
z2(N)     = -Rbar / L;    % sets A(N, N)
z1(N - 1) =  1 / L;       % sets A(N, N-1)

A = spdiags([z1 z2 z3], -1:1, N, N);
B = [(1 / R / Cap); sparse(N - 1, 1)];
C = [(-1 / R)       sparse(1, N - 1)];
D = 1 / R;

if exist('sparss', 'file')
    sys_rlc = sparss(A, B, C, D);
else
    sys_rlc = [];
end

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

RLC Ladder (Gugercin/Antoulas 2004). hosted at MORwiki - Model Order Reduction Wiki, 2026. http://modelreduction.org/index.php/RLC_Ladder_(Gugercin/Antoulas_2004)

@MISC{morwiki_steel,
  author =       {{The MORWiki Community}},
  title =        {RLC Ladder (Gugercin/Antoulas 2004)}
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/RLC_Ladder_(Gugercin/Antoulas_2004)},
  year =         2026
}

• For the background on the benchmark:

@Article{morGugA04,
 author =       {Gugercin, S. and Antoulas, A.~C.},
 title =        {A survey of model reduction by balanced truncation and some
                 new results},
 journal =      IntControl,
 year =         2004,
 volume =       77,
 number =       8,
 pages =        {748--766},
 doi =          {10.1080/00207170410001713448}

}