Nonlinear RC Ladder

Description

The nonlinear RC-ladder is an electronic test circuit introduced in . This nonlinear first-order system models a resistor-capacitor network that exhibits a distinct nonlinear behavior caused by the nonlinear resistors consisting of a parallel connected resistor with a diode.

Nonlinear RC-Ladder

Model

The underlying model is given by a (SISO) gradient system of the form :

$\dot{x}(t)=\left(\begin{array}{c}-g(x_1(t))-g(x_1(t)-x_2(t))\\ g(x_1(t)-x_2(t))-g(x_2(t)-x_3(t))\\ ⋮\\ g(x_{k-1}(t)-x_k(t))-g(x_k(t)-x_{x+1}(t))\\ ⋮\\ g(x_{N-1}(t)-x_N(t))\end{array}\right)+\left(\begin{array}{c}u(t)\\ 0\\ ⋮\\ 0\\ ⋮\\ 0\end{array}\right),$

$y(t)=x_1(t),$

where the $g$ is a mapping $g(x):\mathbb{ℝ}\to \mathbb{ℝ}$:

$g(x_i)=g_D(x_i)-1+x_i,$

which combines the effect of a diode and a resistor.

Nonlinearity

The nonlinearity $g_S$ models a diode as a nonlinear resistor, based on the Shockley model :

$g_D(x_i)=i_S(\exp (u_P{x}_i)-1),$

with material parameters $i_S>0$ and $u_P>0$.

For this benchmark the parameters are selected as: $i_S=1$ and $u_P=40$ as in .

Input

As external input several alternatives are presented in , which are listed next. A simple step function is given by:

$u_1(t)=\{\begin{array}{ll}0& t<4\\ 1& t\ge 4\end{array},$

an exponential decaying input is provided by:

$u_2(t)=e^{-t}.$

Additional input sources are given by conjunction of sine waves with different periods :

$u_3(t)=\sin (2\pi 50t)+\sin (2\pi 1000t),$

$u_4(t)=\sin (2\pi 50t)\sin (2\pi 1000t).$

Data

A sample procedural MATLAB implementation of order $N$ is given by:

function [f,B,C] = nrc(N)
%% Procedural generation of "Nonlinear RC Ladder" benchmark system

  % nonlinearity
  g = @(x) exp(40.0*x) + x - 1.0;

  A0 = sparse(N,N);
  A0(1,1) = 1;

  A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);
  A1(1,1) = 0;

  A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);

  % input matrix
  B = sparse(N,1);
  B(1,1) = 1;

  % output matrix
  C = sparse(1,N);
  C(1,1) = 1;

  % vector field and output functional
  f = @(x) -g(A0*x) + g(A1*x) - g(A2*x);

end

Here the nonlinear part of the vectorfield is realized in a vectorized form as a closure.

Dimensions

System structure:

$\begin{array}{rl}\dot{x}(t)& =f(x(t))+Bu(t)\\ y(t)& =Cx(t)\end{array}$

System dimensions:

$f:{\mathbb{ℝ}}^N\to {\mathbb{ℝ}}^N$, $B\in {\mathbb{ℝ}}^{N\times 1}$, $C\in {\mathbb{ℝ}}^{1\times N}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

The MORwiki Community. Nonlinear RC Ladder. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_RC_Ladder

   @MISC{morwiki_modgyro,
    author = {The {MORwiki} Community},
    title = {Nonlinear RC Ladder},
    howpublished = {{MORwiki} -- Model Order Reduction Wiki},
    url = {http://modelreduction.org/index.php/Nonlinear_RC_Ladder},
    year = {2018}
   }

• For the background on the benchmark: morChe99 (BibTeX)

Contact

Christian Himpe