This icon shall be used for a package that contains internal classes not to be directly utilized by a user.
Extends from Modelica.Icons.InternalPackage
(Icon for an internal package (indicating that the package should not be directly utilized by user)).
Name | Description |
---|---|
bandPass | Return band pass filter roots as needed for block for given cut-off frequency |
bandStop | Return band stop filter roots as needed for block for given cut-off frequency |
highPass | Return high pass filter roots as needed for block for given cut-off frequency |
lowPass | Return low pass filter roots as needed for block for given cut-off frequency |
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
Extends from Modelica.Icons.Function
(Icon for functions).
Type | Name | Description |
---|---|---|
Real | cr_in[:] | Coefficients of real poles of base filter |
Real | c0_in[:] | Coefficients of s^0 term of base filter if conjugate complex pole |
Real | c1_in[size(c0_in, 1)] | Coefficients of s^1 term of base filter if conjugate complex pole |
Frequency | f_cut | Cut-off frequency |
Type | Name | Description |
---|---|---|
Real | r[size(cr_in, 1)] | Real eigenvalues |
Real | a[size(c0_in, 1)] | Real parts of complex conjugate eigenvalues |
Real | b[size(c0_in, 1)] | Imaginary parts of complex conjugate eigenvalues |
Real | ku[size(c0_in, 1)] | Input gain |
The goal is to implement the filter in the following form:
// real pole: der(x) = r*x - r*u y = -x + u // complex conjugate poles: der(x1) = a*x1 - b*x2 + ku*u; der(x2) = b*x1 + a*x2; y = k1*x1 + k2*x2 + u; ku = (a^2 + b^2)/b k1 = 2*a/ku k2 = (a^2 - b^2) / (b*ku) = (a^2 - b^2) / (a^2 + b^2) = (1 - (b/a)^2) / (1 + (b/a)^2)
This representation has the following transfer function:
// real pole: s*x = r*x - r*u or (s-r)*x = -r*u -> x = -r/(s-r)*u or y = r/(s-r)*u + (s-r)/(s-r)*u = (r+s-r)/(s-r)*u = s/(s-r)*u comparing coefficients with y = s/(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 = k1*x1 + k2*x2 + u = (k1*ku*(s-a) + k2*b*ku + s^2 - 2*a*s + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = (s^2 + (k1*ku - 2*a)*s + k2*b*ku - k1*ku*a + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = (s^2 + (2*a-2*a)*s + a^2 - b^2 - 2*a^2 + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = s^2 / (s^2 - 2*a*s + a^2 + b^2)*u comparing coefficients with y = s^2/(s^2 + c1*s + c0)*u -> a = -c1/2 b = sqrt(c0 - a^2) 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
Extends from Modelica.Icons.Function
(Icon for functions).
Type | Name | Description |
---|---|---|
Real | cr_in[:] | Coefficients of real poles of base filter |
Real | c0_in[:] | Coefficients of s^0 term of base filter if conjugate complex pole |
Real | c1_in[size(c0_in, 1)] | Coefficients of s^1 term of base filter if conjugate complex pole |
Frequency | f_cut | Cut-off frequency |
Type | Name | Description |
---|---|---|
Real | r[size(cr_in, 1)] | Real eigenvalues |
Real | a[size(c0_in, 1)] | Real parts of complex conjugate eigenvalues |
Real | b[size(c0_in, 1)] | Imaginary parts of complex conjugate eigenvalues |
Real | ku[size(c0_in, 1)] | Gains of input terms |
Real | k1[size(c0_in, 1)] | Gains of y = k1*x1 + k2*x + u |
Real | k2[size(c0_in, 1)] | Gains of y = k1*x1 + k2*x + u |
The goal is to implement the filter in the following form:
// complex conjugate poles: der(x1) = a*x1 - b*x2 + ku*u; der(x2) = b*x1 + a*x2; y = k1*x1 + k2*x2; ku = (a^2 + b^2)/b k1 = cn/ku k2 = cn*a/(b*ku)
This representation has the following transfer function:
// 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 = k1*x1 + k2*x2 = (k1*ku*(s-a) + k2*b*ku) / (s^2 - 2*a*s + a^2 + b^2)*u = (k1*ku*s + k2*b*ku - k1*ku*a) / (s^2 - 2*a*s + a^2 + b^2)*u = (cn*s + cn*a - cn*a) / (s^2 - 2*a*s + a^2 + b^2)*u = cn*s / (s^2 - 2*a*s + a^2 + b^2)*u comparing coefficients with y = cn*s / (s^2 + c1*s + c0)*u -> a = -c1/2 b = sqrt(c0 - a^2) 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
Extends from Modelica.Icons.Function
(Icon for functions).
Type | Name | Description |
---|---|---|
Real | cr_in[:] | Coefficients of real poles of base filter |
Real | c0_in[:] | Coefficients of s^0 term of base filter if conjugate complex pole |
Real | c1_in[size(c0_in, 1)] | Coefficients of s^1 term of base filter if conjugate complex pole |
Frequency | f_min | Band of band pass filter is f_min (A=-3db) .. f_max (A=-3db) |
Frequency | f_max | Upper band frequency |
Type | Name | Description |
---|---|---|
Real | a[size(cr_in, 1) + 2 * size(c0_in, 1)] | Real parts of complex conjugate eigenvalues |
Real | b[size(cr_in, 1) + 2 * size(c0_in, 1)] | Imaginary parts of complex conjugate eigenvalues |
Real | ku[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of input terms |
Real | k1[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of y = k1*x1 + k2*x |
Real | k2[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of y = k1*x1 + k2*x |
The goal is to implement the filter in the following form:
// complex conjugate poles: der(x1) = a*x1 - b*x2 + ku*u; der(x2) = b*x1 + a*x2; y = k1*x1 + k2*x2 + u; ku = (a^2 + b^2)/b k1 = 2*a/ku k2 = (c0 + a^2 - b^2)/(b*ku)
This representation has the following transfer function:
// 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 = k1*x1 + k2*x2 + u = (k1*ku*(s-a) + k2*b*ku + s^2 - 2*a*s + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = (s^2 + (k1*ku-2*a)*s + k2*b*ku - k1*ku*a + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = (s^2 + c0 + a^2 - b^2 - 2*a^2 + a^2 + b^2) / (s^2 - 2*a*s + a^2 + b^2)*u = (s^2 + c0) / (s^2 - 2*a*s + a^2 + b^2)*u comparing coefficients with y = (s^2 + c0) / (s^2 + c1*s + c0)*u -> a = -c1/2 b = sqrt(c0 - a^2) 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
Extends from Modelica.Icons.Function
(Icon for functions).
Type | Name | Description |
---|---|---|
Real | cr_in[:] | Coefficients of real poles of base filter |
Real | c0_in[:] | Coefficients of s^0 term of base filter if conjugate complex pole |
Real | c1_in[size(c0_in, 1)] | Coefficients of s^1 term of base filter if conjugate complex pole |
Frequency | f_min | Band of band stop filter is f_min (A=-3db) .. f_max (A=-3db) |
Frequency | f_max | Upper band frequency |
Type | Name | Description |
---|---|---|
Real | a[size(cr_in, 1) + 2 * size(c0_in, 1)] | Real parts of complex conjugate eigenvalues |
Real | b[size(cr_in, 1) + 2 * size(c0_in, 1)] | Imaginary parts of complex conjugate eigenvalues |
Real | ku[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of input terms |
Real | k1[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of y = k1*x1 + k2*x |
Real | k2[size(cr_in, 1) + 2 * size(c0_in, 1)] | Gains of y = k1*x1 + k2*x |