Nonlinear RC Ladder

Description

Nonlinear RC-Ladder

The nonlinear RC-ladder is an electronic test circuit first introduced in , and its variant is also 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.

Model

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

\[ \dot{x}(t) = \begin{pmatrix} -g(x_1(t)) - g(x_1(t) - x_2(t)) \\ g(x_1(t)-x_2(t)) - g(x_2(t)-x_3(t)) \\ \vdots \\ g(x_{k-1}(t) - x_k(t)) - g(x_k(t) - x_{x+1}(t)) \\ \vdots \\ g(x_{N-1}(t) - x_N(t)) \end{pmatrix}+\begin{pmatrix}u(t) \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \]

\[ y(t) = x_1(t), \]

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

\[ g(x_i) = g_D(x_i) + x_i, \]

which combines the effect of a diode and a resistor.

Nonlinearity

The nonlinearity \(g_D\) 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 an external input, several alternatives are presented in , which are listed next. A simple step function is given by: \[ u_1(t)=\begin{cases}0 & t < 4 \\ 1 & t \geq 4 \end{cases}, \]

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: \[ \begin{align} \dot{x}(t) &= f(x(t)) + Bu(t), \\ y(t) &= Cx(t). \end{align} \]

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_modNonRCL,
    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