.Modelica.Blocks.Continuous.Internal.Filter.roots.lowPass

Information

The goal is to implement the filter in the following form:

  // real pole:
   der(x) = r*x - r*u
       y  = x

  // complex conjugate poles:
  der(x1) = a*x1 - b*x2 + ku*u;
  der(x2) = b*x1 + a*x2;
       y  = x2;

            ku = (a^2 + b^2)/b

This representation has the following transfer function:

// real pole:
    s*y = r*y - r*u
  or
    (s-r)*y = -r*u
  or
    y = -r/(s-r)*u

  comparing coefficients with
    y = cr/(s + cr)*u  ->  r = -cr      // r is the real eigenvalue

// complex conjugate poles
    s*x2 =  a*x2 + b*x1
    s*x1 = -b*x2 + a*x1 + ku*u
  or
    (s-a)*x2               = b*x1  ->  x2 = b/(s-a)*x1
    (s + b^2/(s-a) - a)*x1 = ku*u  ->  (s(s-a) + b^2 - a*(s-a))*x1  = ku*(s-a)*u
                                   ->  (s^2 - 2*a*s + a^2 + b^2)*x1 = ku*(s-a)*u
  or
    x1 = ku*(s-a)/(s^2 - 2*a*s + a^2 + b^2)*u
    x2 = b/(s-a)*ku*(s-a)/(s^2 - 2*a*s + a^2 + b^2)*u
       = b*ku/(s^2 - 2*a*s + a^2 + b^2)*u
    y  = x2

  comparing coefficients with
    y = c0/(s^2 + c1*s + c0)*u  ->  a  = -c1/2
                                    b  = sqrt(c0 - a^2)
                                    ku = c0/b
                                       = (a^2 + b^2)/b

  comparing with eigenvalue representation:
    (s - (a+jb))*(s - (a-jb)) = s^2 -2*a*s + a^2 + b^2
  shows that:
    a: real part of eigenvalue
    b: imaginary part of eigenvalue

  time -> infinity:
    y(s=0) = x2(s=0) = 1
             x1(s=0) = -ku*a/(a^2 + b^2)*u
                     = -(a/b)*u

Interface

function lowPass
  extends Modelica.Icons.Function;
  input Real cr_in[:] "Coefficients of real poles of base filter";
  input Real c0_in[:] "Coefficients of s^0 term of base filter if conjugate complex pole";
  input Real c1_in[size(c0_in, 1)] "Coefficients of s^1 term of base filter if conjugate complex pole";
  input Modelica.SIunits.Frequency f_cut "Cut-off frequency";
  output Real r[size(cr_in, 1)] "Real eigenvalues";
  output Real a[size(c0_in, 1)] "Real parts of complex conjugate eigenvalues";
  output Real b[size(c0_in, 1)] "Imaginary parts of complex conjugate eigenvalues";
  output Real ku[size(c0_in, 1)] "Input gain";
end lowPass;