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 . These nonlinear first-order system model a resistor-capacitor network that exhibits a distinct nonlinear behaviour caused by either the nonlinear resistors consisting of a parallel connected resistor with a diode (see the right figure) or the nonlinear resistors connected parallel to the capacitor (see Fig. 7 in ).

Model 1

First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, 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 .

Model 2

Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in ). For this, the underlying model is also given by a SISO gradient system of the form :

\[ \dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x), \]

and the output function:

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

where the \(g\) is a mapping \(g(x_i):\mathbb{R} \to \mathbb{R}\), respresenting the effect of a nonlinear resistor.

\[ g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2 \]

which represents the effect of a nonlinear resistor, and sgn denotes the sign function.

Nonlinearity

The nonlinearity \(g_D\) models a diode as a nonlinear resistor :

\[ g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2 \]

which represents the effect of a nonlinear resistor, and sgn denotes the sign function.

For this benchmark the parameters are selected as\[a = 1\] as in .

Alternatively, the variant from can be used, featuring the nonlinearity:

\[ g(x_i) = \frac{x_i^2}{2} + \frac{x_i^3}{3}. \]

This represents practically a resistor-inductor cascade with nonlinear resistors.

Model 3

Third, a circuit of chained inverter gates, a so-called inverter chain , is presented. This is a SISO system, given by:

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

and the output function:

\[ y(t) = x_N(t). \]

Nonlinearity

The nonlinear function \(g\) is a mapping \(g(x_i):\mathbb{R} \to \mathbb{R}\) describing the inverter characteristic:

\[ g(x_i) = v \tanh(a x_i), \]

being a parameterized hyperbolic tangent with the supply voltage \(v=1\) and a physical parameter \(a\), i.e. \(a=5\).

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 for order \(N\) for all three models is given by:

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

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

  switch lower(model)

    case 'shockley'

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

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

      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);

      f = @(x) -g(A0*x) + g(A1*x) - g(A2*x);

    case 'sign'

      A = gallery('tridiag',N,1,-2,1);

      f = @(x) A*x - sign(x).*(x.*x);

    case 'ind'

      A = gallery('tridiag',N,1,-2,1);

      f = @(x) A*x - ((x.^2)./2 + (x.^3)./3);

    case 'inv'

      f = @(x) [0;tanh(x(2:end))] - x;

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

    otherwise

      error('Choose shockley, sign, ind or inv');
  end
end

Here the vector field 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 these benchmarks, 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 =          {https://modelreduction.org/morwiki/Nonlinear_RC_Ladder},
  year =         2018
}

• For the background on the benchmarks:
  • for Model 1 morChe99 (BibTeX)
  • for Model 2 morRew03 (BibTeX)
  • for Model 3 morGu12 (BibTeX)

Contact

Christian Himpe