Modelica.Blocks.Math

Information

This package contains basic mathematical operations, such as summation and multiplication, and basic mathematical functions, such as sqrt and sin, as input/output blocks. All blocks of this library can be either connected with continuous blocks or with sampled-data blocks.

Package Content

Name	Description
UnitConversions	Conversion blocks to convert between SI and non-SI unit signals
InverseBlockConstraints	Construct inverse model by requiring that two inputs and two outputs are identical (replaces the previously, unbalanced, TwoInputs and TwoOutputs blocks)
Gain	Output the product of a gain value with the input signal
MatrixGain	Output the product of a gain matrix with the input signal vector
MultiSum	Sum of Reals: y = k[1]*u[1] + k[2]*u[2] + ... + k[n]*u[n]
MultiProduct	Product of Reals: y = u[1]*u[2]* ... *u[n]
MultiSwitch	Set Real expression that is associated with the first active input signal
Sum	Output the sum of the elements of the input vector
Feedback	Output difference between commanded and feedback input
Add	Output the sum of the two inputs
Add3	Output the sum of the three inputs
Product	Output product of the two inputs
Division	Output first input divided by second input
Abs	Output the absolute value of the input
Sign	Output the sign of the input
Sqrt	Output the square root of the input (input >= 0 required)
Sin	Output the sine of the input
Cos	Output the cosine of the input
Tan	Output the tangent of the input
Asin	Output the arc sine of the input
Acos	Output the arc cosine of the input
Atan	Output the arc tangent of the input
Atan2	Output atan(u1/u2) of the inputs u1 and u2
Sinh	Output the hyperbolic sine of the input
Cosh	Output the hyperbolic cosine of the input
Tanh	Output the hyperbolic tangent of the input
Exp	Output the exponential (base e) of the input
Log	Output the natural (base e) logarithm of the input (input > 0 required)
Log10	Output the base 10 logarithm of the input (input > 0 required)
RealToInteger	Convert Real to Integer signal
IntegerToReal	Convert Integer to Real signals
BooleanToReal	Convert Boolean to Real signal
BooleanToInteger	Convert Boolean to Integer signal
RealToBoolean	Convert Real to Boolean signal
IntegerToBoolean	Convert Integer to Boolean signal
RectangularToPolar	Convert rectangular coordinates to polar coordinates
PolarToRectangular	Convert polar coordinates to rectangular coordinates
Mean	Calculate mean over period 1/f
RectifiedMean	Calculate rectified mean over period 1/f
RootMeanSquare	Calculate root mean square over period 1/f
Harmonic	Calculate harmonic over period 1/f
Max	Pass through the largest signal
Min	Pass through the smallest signal
MinMax	Output the minimum and the maximum element of the input vector
LinearDependency	Output a linear combination of the two inputs
Edge	Indicates rising edge of Boolean signal
BooleanChange	Indicates Boolean signal changing
IntegerChange	Indicates integer signal changing

Modelica.Blocks.Math.InverseBlockConstraints

Information

Exchange input and output signals of a block, i.e., the previous block inputs become block outputs and the previous block outputs become block inputs. This block is used to construct inverse models. Its usage is demonstrated in example: Modelica.Blocks.Examples.InverseModel.

Note, if a block shall be inverted that has several input and output blocks, then this can be easily achieved by using a vector of InverseBlockConstraints instances:

   InverseBlockConstraint invert[3];  // Block to be inverted has 3 input signals

Connectors

Name	Description
u1	Input signal 1 (u1 = u2)
u2	Input signal 2 (u1 = u2)
y1	Output signal 1 (y1 = y2)
y2	Output signal 2 (y2 = y2)

Modelica.Blocks.Math.Gain

Information

This block computes output y as product of gain k with the input u:

    y = k * u;

Parameters

Name	Description
k	Gain value multiplied with input signal [1]

Connectors

Name	Description
u	Input signal connector
y	Output signal connector

Modelica.Blocks.Math.MatrixGain

Information

This blocks computes output vector y as product of the gain matrix K with the input signal vector u:

    y = K * u;

Example:

   parameter: K = [0.12 2; 3 1.5]

   results in the following equations:

     | y[1] |     | 0.12  2.00 |   | u[1] |
     |      |  =  |            | * |      |
     | y[2] |     | 3.00  1.50 |   | u[2] |

Parameters

Name	Description
K[:, :]	Gain matrix which is multiplied with the input
nin	Number of inputs
nout	Number of outputs

Connectors

Name	Description
u[nin]	Connector of Real input signals
y[nout]	Connector of Real output signals

Modelica.Blocks.Math.MultiSum

Information

This blocks computes the scalar Real output "y" as sum of the elements of the Real input signal vector u:

y = k[1]*u[1] + k[2]*u[2] + ... k[N]*u[N];

The input connector is a vector of Real input signals. When a connection line is drawn, the dimension of the input vector is enlarged by one and the connection is automatically connected to this new free index (thanks to the connectorSizing annotation).

The usage is demonstrated, e.g., in example Modelica.Blocks.Examples.RealNetwork1.

If no connection to the input connector "u" is present, the output is set to zero: y=0.

Parameters

Name	Description
k[nu]	Input gains
Advanced
significantDigits	Number of significant digits to be shown in dynamic diagram layer for y

Connectors

Name	Description
u[nu]
y

Modelica.Blocks.Math.MultiProduct

Information

This blocks computes the scalar Real output "y" as product of the elements of the Real input signal vector u:

y = u[1]*u[2]* ... *u[N];

The input connector is a vector of Real input signals. When a connection line is drawn, the dimension of the input vector is enlarged by one and the connection is automatically connected to this new free index (thanks to the connectorSizing annotation).

The usage is demonstrated, e.g., in example Modelica.Blocks.Examples.RealNetwork1.

If no connection to the input connector "u" is present, the output is set to zero: y=0.

Parameters

Name	Description
Advanced
significantDigits	Number of significant digits to be shown in dynamic diagram layer for y

Connectors

Name	Description
u[nu]
y

Modelica.Blocks.Math.MultiSwitch

Information

This block has a vector of Boolean input signals u[nu] and a vector of (time varying) Real expressions expr[nu]. The output signal y is set to expr[i], if i is the first element in the input vector u that is true. If all input signals are false, y is set to parameter "y_default".

  // Conceptual equation (not valid Modelica)
  i = 'first element of u[:] that is true';
  y = if i==0 then y_default else expr[i];

The input connector is a vector of Boolean input signals. When a connection line is drawn, the dimension of the input vector is enlarged by one and the connection is automatically connected to this new free index (thanks to the connectorSizing annotation).

The usage is demonstrated, e.g., in example Modelica.Blocks.Examples.RealNetwork1.

Parameters

Name	Description
expr[nu]	y = if u[i] then expr[i] else y_default (time varying)
y_default	Default value of output y if all u[i] = false
Advanced
precision	Number of significant digits to be shown in dynamic diagram layer for y

Connectors

Name	Description
u[nu]	Set y = expr[i], if u[i] = true
y	Output depending on expression

Modelica.Blocks.Math.Sum

Information

This blocks computes output y as sum of the elements of the input signal vector u:

    y = u[1] + u[2] + ...;

Example:

     parameter:   nin = 3;

  results in the following equations:

     y = u[1] + u[2] + u[3];

Parameters

Name	Description
nin	Number of inputs
k[nin]	Optional: sum coefficients

Connectors

Name	Description
u[nin]	Connector of Real input signals
y	Connector of Real output signal

Modelica.Blocks.Math.Feedback

Information

This blocks computes output y as difference of the commanded input u1 and the feedback input u2:

    y = u1 - u2;

Example:

     parameter:   n = 2

  results in the following equations:

     y = u1 - u2

Connectors

Name	Description
u1
u2
y

Modelica.Blocks.Math.Add

Information

This blocks computes output y as sum of the two input signals u1 and u2:

    y = k1*u1 + k2*u2;

Example:

     parameter:   k1= +2, k2= -3

  results in the following equations:

     y = 2 * u1 - 3 * u2

Parameters

Name	Description
k1	Gain of upper input
k2	Gain of lower input

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Add3

Information

This blocks computes output y as sum of the three input signals u1, u2 and u3:

    y = k1*u1 + k2*u2 + k3*u3;

Example:

     parameter:   k1= +2, k2= -3, k3=1;

  results in the following equations:

     y = 2 * u1 - 3 * u2 + u3;

Parameters

Name	Description
k1	Gain of upper input
k2	Gain of middle input
k3	Gain of lower input

Connectors

Name	Description
u1	Connector 1 of Real input signals
u2	Connector 2 of Real input signals
u3	Connector 3 of Real input signals
y	Connector of Real output signals

Modelica.Blocks.Math.Product

Information

This blocks computes the output y (element-wise) as product of the corresponding elements of the two inputs u1 and u2:

    y = u1 * u2;

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Division

Information

This block computes the output y (element-wise) by dividing the corresponding elements of the two inputs u1 and u2:

    y = u1 / u2;

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Abs

Information

This blocks computes the output y as absolute value of the input u:

    y = abs( u );

The Boolean parameter generateEvent decides whether Events are generated at zero crossing (Modelica specification before 3) or not.

Parameters

Name	Description
generateEvent	Choose whether events shall be generated

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Sign

Information

This blocks computes the output y as sign of the input u:

         1  if u > 0
    y =  0  if u == 0
        -1  if u < 0

The Boolean parameter generateEvent decides whether Events are generated at zero crossing (Modelica specification before 3) or not.

Parameters

Name	Description
generateEvent	Choose whether events shall be generated

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Sqrt

Information

This blocks computes the output y as square root of the input u:

    y = sqrt( u );

All elements of the input vector shall be zero or positive. Otherwise an error occurs.

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Sin

Information

This blocks computes the output y as sine of the input u:

    y = sin( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Cos

Information

This blocks computes the output y as cos of the input u:

    y = cos( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Tan

Information

This blocks computes the output y as tan of the input u:

    y = tan( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Asin

Information

This blocks computes the output y as the sine-inverse of the input u:

    y = asin( u );

The absolute values of the elements of the input u need to be less or equal to one (abs( u ) <= 1). Otherwise an error occurs.

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Acos

Information

This blocks computes the output y as the cosine-inverse of the input u:

    y = acos( u );

The absolute values of the elements of the input u need to be less or equal to one (abs( u ) <= 1). Otherwise an error occurs.

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Atan

Information

This blocks computes the output y as the tangent-inverse of the input u:

    y= atan( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Atan2

Information

This blocks computes the output y as the tangent-inverse of the input u1 divided by input u2:

    y = atan2( u1, u2 );

u1 and u2 shall not be zero at the same time instant. Atan2 uses the sign of u1 and u2 in order to construct the solution in the range -180 deg ≤ y ≤ 180 deg, whereas block Atan gives a solution in the range -90 deg ≤ y ≤ 90 deg.

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Sinh

Information

This blocks computes the output y as the hyperbolic sine of the input u:

    y = sinh( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Cosh

Information

This blocks computes the output y as the hyperbolic cosine of the input u:

    y = cosh( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Tanh

Information

This blocks computes the output y as the hyperbolic tangent of the input u:

    y = tanh( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Exp

Information

This blocks computes the output y as the exponential (of base e) of the input u:

    y = exp( u );

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Log

Information

This blocks computes the output y as the natural (base e) logarithm of the input u:

    y = log( u );

An error occurs if the elements of the input u are zero or negative.

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Log10

Information

This blocks computes the output y as the base 10 logarithm of the input u:

    y = log10( u );

An error occurs if the elements of the input u are zero or negative.

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.RealToInteger

Information

This block computes the output y as nearest integer value of the input u:

    y = integer( floor( u + 0.5 ) )  for  u > 0;
    y = integer( ceil ( u - 0.5 ) )  for  u < 0;

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Integer output signal

Modelica.Blocks.Math.IntegerToReal

Information

This block computes the output y as Real equivalent of the Integer input u:

    y = u;

where u is of Integer and y of Real type.

Connectors

Name	Description
u	Connector of Integer input signal
y	Connector of Real output signal

Modelica.Blocks.Math.BooleanToReal

Information

This block computes the output y as Real equivalent of the Boolean input u:

    y = if u then realTrue else realFalse;

where u is of Boolean and y of Real type, and realTrue and realFalse are parameters.

Parameters

Name	Description
realTrue	Output signal for true Boolean input
realFalse	Output signal for false Boolean input

Connectors

Name	Description
u	Connector of Boolean input signal
y	Connector of Real output signal

Modelica.Blocks.Math.BooleanToInteger

Information

This block computes the output y as Integer equivalent of the Boolean input u:

    y = if u then integerTrue else integerFalse;

where u is of Boolean and y of Integer type, and integerTrue and integerFalse are parameters.

Parameters

Name	Description
integerTrue	Output signal for true Boolean input
integerFalse	Output signal for false Boolean input

Connectors

Name	Description
u	Connector of Boolean input signal
y	Connector of Integer output signal

Modelica.Blocks.Math.RealToBoolean

Information

This block computes the Boolean output y from the Real input u by the equation:

    y = u ≥ threshold;

where threshold is a parameter.

Parameters

Name	Description
threshold	Output signal y is true, if input u >= threshold

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Boolean output signal

Modelica.Blocks.Math.IntegerToBoolean

Information

This block computes the Boolean output y from the Integer input u by the equation:

    y = u ≥ threshold;

where threshold is a parameter.

Parameters

Name	Description
threshold	Output signal y is true, if input u >= threshold

Connectors

Name	Description
u	Connector of Integer input signal
y	Connector of Boolean output signal

Modelica.Blocks.Math.RectangularToPolar

Information

The input values of this block are the rectangular components u_re and u_im of a phasor in two dimensions. This block calculates the length y_abs and the angle y_arg of the polar representation of this phasor.

  y_abs = abs(u_re + j*u_im) = sqrt( u_re2 + u_im2 )
  y_arg = arg(u_re + j*u_im) = atan2(u_im, u_re)

Connectors

Name	Description
u_re	Real part of rectangular representation
u_im	Imaginary part of rectangular representation
y_abs	Length of polar representation
y_arg	Angle of polar representation

Modelica.Blocks.Math.PolarToRectangular

Information

The input values of this block are the polar components uabs and uarg of a phasor. This block calculates the components y_re and y_im of the rectangular representation of this phasor.

   y_re = u_abs * cos( u_arg )
   y_im = u_abs * sin( u_arg )

Connectors

Name	Description
u_abs	Length of polar representation
u_arg	Angle of polar representation
y_re	Real part of rectangular representation
y_im	Imaginary part of rectangular representation

Modelica.Blocks.Math.Mean

Information

This block calculates the mean of the input signal u over the given period 1/f:

1 T
- ∫ u(t) dt
T 0

Note: The output is updated after each period defined by 1/f.

If parameter yGreaterOrEqualZero in the Advanced tab is true (default = false), then the modeller provides the information that the mean of the input signal is guaranteed to be ≥ 0 for the exact solution. However, due to inaccuracies in the numerical integration scheme, the output might be slightly negative. If this parameter is set to true, then the output is explicitly set to 0.0, if the mean value results in a negative value.

Parameters

Name	Description
f	Base frequency [Hz]
x0	Start value of integrator state
Advanced
yGreaterOrEqualZero	=true, if output y is guaranteed to be >= 0 for the exact solution

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.RectifiedMean

Information

This block calculates the rectified mean of the input signal u over the given period 1/f, using the mean block.

Note: The output is updated after each period defined by 1/f.

Parameters

Name	Description
f	Base frequency [Hz]

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.RootMeanSquare

Information

This block calculates the root mean square of the input signal u over the given period 1/f, using the mean block.

Note: The output is updated after each period defined by 1/f.

Parameters

Name	Description
f	Base frequency [Hz]

Connectors

Name	Description
u	Connector of Real input signal
y	Connector of Real output signal

Modelica.Blocks.Math.Harmonic

Information

This block calculates the root mean square and the phase angle of a single harmonic k of the input signal u over the given period 1/f, using the mean block.

Note: The output is updated after each period defined by 1/f.

Note: The harmonic is defined by √2 rms cos(k 2 π f t - arg)

Parameters

Name	Description
f	Base frequency [Hz]
k	Order of harmonic

Connectors

Name	Description
u
y_rms	Root mean square of polar representation
y_arg	Angle of polar representation

Modelica.Blocks.Math.Max

Information

This block computes the output y as maximum of the two Real inputs u1 and u2:

    y = max ( u1 , u2 );

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Min

Information

This block computes the output y as minimum of the two Real inputs u1 and u2:

    y = min ( u1 , u2 );

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.MinMax

Information

Determines the minimum and maximum element of the input vector and provide both values as output.

Connectors

Name	Description
u[nu]
yMax
yMin

Modelica.Blocks.Math.LinearDependency

Information

Determine the linear combination of the two inputs: y = y0*(1 + k1*u1 + k2*u2)

Parameters

Name	Description
y0	Initial value
k1	u1 dependency
k2	u2 dependency

Connectors

Name	Description
u1	Connector of Real input signal 1
u2	Connector of Real input signal 2
y	Connector of Real output signal

Modelica.Blocks.Math.Edge

Information

This block sets the Boolean output y to true, when the Boolean input u shows a rising edge:

    y = edge( u );

Connectors

Name	Description
u	Connector of Boolean input signal
y	Connector of Boolean output signal

Modelica.Blocks.Math.BooleanChange

Information

This block sets the Boolean output y to true, when the Boolean input u shows a rising or falling edge, i.e., when the signal changes:

    y = change( u );

Connectors

Name	Description
u	Connector of Boolean input signal
y	Connector of Boolean output signal

Modelica.Blocks.Math.IntegerChange

Information

This block sets the Boolean output y to true, when the Integer input u changes:

    y = change( u );

Connectors

Name	Description
u	Connector of Integer input signal
y	Connector of Boolean output signal