Nonlinear RC Ladder: Difference between revisions

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Description

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

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 ^[3]:

\dot{x} (t) = (\begin{matrix} - 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{matrix}) + (\begin{matrix} u (t) \\ 0 \\ ⋮ \\ 0 \\ ⋮ \\ 0 \end{matrix}),

y (t) = x_{1} (t),

where the $g$ is a mapping $g (x_{i}) : ℝ \to ℝ$ :

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].

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 ^[3]:

\dot{x} (t) = (\begin{matrix} - 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{matrix}) + (\begin{matrix} u (t) \\ 0 \\ ⋮ \\ 0 \\ ⋮ \\ 0 \end{matrix}),

y (t) = x_{1} (t),

where the $g$ is a mapping $g (x_{i}) : ℝ \to ℝ$ :

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 π 50 t) + \sin (2 π 1000 t),

u_{4} (t) = \sin (2 π 50 t) \sin (2 π 1000 t) .

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{aligned} \dot{x} (t) & = f (x (t)) + B u (t), \\ y (t) & = C x (t) . \end{aligned}

System dimensions:

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

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)

References

↑ ^1.0 ^1.1 ^1.2 Y. Chen, "Model Reduction for Nonlinear Systems", Master Thesis, 1999.
↑ ^2.0 ^2.1 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.
↑ ^3.0 ^3.1 M. Condon and R. Ivanov, "Empirical Balanced Truncation for Nonlinear Systems", Journal of Nonlinear Science 14(5):405--414, 2004.
↑ ^4.0 ^4.1 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.
↑ 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.
↑ M. Condon and R. Ivanov, "Model Reduction of Nonlinear Systems", COMPEL 23(2): 547--557, 2004

Contact

Christian Himpe

[chen99-1] 1.0 ^1.1 ^1.2 Y. Chen, "Model Reduction for Nonlinear Systems", Master Thesis, 1999.

[RewW03-2] 2.0 ^2.1 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.

[condon04-3] 3.0 ^3.1 M. Condon and R. Ivanov, "Empirical Balanced Truncation for Nonlinear Systems", Journal of Nonlinear Science 14(5):405--414, 2004.

[reis14-4] 4.0 ^4.1 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.

[chen00-5] 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.

[condon04a-6] M. Condon and R. Ivanov, "Model Reduction of Nonlinear Systems", COMPEL 23(2): 547--557, 2004

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@@ Line 9: / Line 9: @@
 The nonlinear RC-ladder is an electronic test circuit first introduced in <ref name="chen99"/>, and its variant is also introduced in <ref name="RewW03"/>.
 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 <ref name = "RewW03"/>).
+===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 ([[List_of_abbreviations#SISO|SISO]]) gradient system of the form <ref name="condon04"/>:
+:<math>
+\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},
+</math>
+:<math>
+y(t) = x_1(t),
+</math>
+where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>:
+:<math>
+g(x_i) = g_D(x_i) + x_i,
+</math>
+which combines the effect of a [[wikipedia:Diode|diode]] and a resistor.
+====Nonlinearity====
+The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor,
+based on the [[wikipedia:Diode_modelling#Shockley_diode_model|Shockley model]] <ref name="reis14"/>:
+:<math>
+g_D(x_i) = i_S (\exp(u_P x_i) - 1),
+</math>
+with material parameters <math>i_S > 0</math> and <math>u_P > 0</math>.
+For this benchmark the parameters are selected as: <math>i_S = 1</math> and <math>u_P = 40</math> as in <ref name="chen99"/>.
 ===Model 1===