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Nonlinear RC Ladder


Description

Nonlinear RC-Ladder

The nonlinear RC-ladder is an electronic test circuit first introduced in [1], and its variant is also introduced in [2]. 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 [3]:


\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 [4]:


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 [1].

Input

As an external input, several alternatives are presented in [5], 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 [6]:


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}
   }

References

  1. 1.0 1.1 Y. Chen, "Model Reduction for Nonlinear Systems", Master Thesis, 1999.
  2. M. Rewienski and J. White, "A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices", IEEE Transactions on computer-aided design of integrated circuits and systems 22(2): 155--170, 2003.
  3. M. Condon and R. Ivanov, "Empirical Balanced Truncation for Nonlinear Systems", Journal of Nonlinear Science 14(5):405--414, 2004.
  4. T. Reis. "Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.
  5. Y. Chen and J. White, "A Quadratic Method for Nonlinear Model Order Reduction", Int. Conference on Modelling and Simulation of Microsystems Semiconductors, Sensors and Actuators, 2000.
  6. M. Condon and R. Ivanov, "Model Reduction of Nonlinear Systems", COMPEL 23(2): 547--557, 2004

Contact

Christian Himpe