Modelica.Blocks.Math

Library of Real mathematical functions as input/output blocks

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.

Extends from Modelica.Icons.Package (Icon for standard packages).

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
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
ContinuousMean	Calculates the empirical expectation (mean) value of its input signal
RootMeanSquare	Calculate root mean square over period 1/f
Variance	Calculates the empirical variance of its input signal
StandardDeviation	Calculates the empirical standard deviation of its input signal
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

Construct inverse model by requiring that two inputs and two outputs are identical

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

Modelica.Blocks.Math.Gain

Output the product of a gain value with the input signal

Information

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

    y = k * u;

Parameters

Connectors

Modelica.Blocks.Math.MatrixGain

Output the product of a gain matrix with the input signal vector

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] |

Extends from Interfaces.MIMO (Multiple Input Multiple Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.MultiSum

Sum of Reals: y = k[1]*u[1] + k[2]*u[2] + ... + k[n]*u[n]

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.

Extends from Modelica.Blocks.Interfaces.PartialRealMISO (Partial block with a RealVectorInput and a RealOutput signal).

Parameters

Connectors

Modelica.Blocks.Math.MultiProduct

Product of Reals: y = u[1]*u[2]* ... *u[n]

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.

Extends from Modelica.Blocks.Interfaces.PartialRealMISO (Partial block with a RealVectorInput and a RealOutput signal).

Parameters

Connectors

Modelica.Blocks.Math.MultiSwitch

Set Real expression that is associated with the first active input signal

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

Connectors

Modelica.Blocks.Math.Sum

Output the sum of the elements of the input vector

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];

Extends from Interfaces.MISO (Multiple Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Feedback

Output difference between commanded and feedback input

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

Modelica.Blocks.Math.Add

Output the sum of the two inputs

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

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Add3

Output the sum of the three inputs

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;

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Blocks.Math.Product

Output product of the two inputs

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;

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Division

Output first input divided by second input

Information

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

    y = u1 / u2;

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Abs

Output the absolute value of the input

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Sign

Output the sign of the input

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Sqrt

Output the square root of the input (input >= 0 required)

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Sin

Output the sine of the input

Information

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

    y = sin( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Cos

Output the cosine of the input

Information

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

    y = cos( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Tan

Output the tangent of the input

Information

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

    y = tan( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Asin

Output the arc sine of the input

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Acos

Output the arc cosine of the input

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Atan

Output the arc tangent of the input

Information

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

    y= atan( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Atan2

Output atan(u1/u2) of the inputs u1 and u2

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.

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Sinh

Output the hyperbolic sine of the input

Information

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

    y = sinh( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Cosh

Output the hyperbolic cosine of the input

Information

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

    y = cosh( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Tanh

Output the hyperbolic tangent of the input

Information

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

    y = tanh( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Exp

Output the exponential (base e) of the input

Information

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

    y = exp( u );

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Log

Output the natural (base e) logarithm of the input (input > 0 required)

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Log10

Output the base 10 logarithm of the input (input > 0 required)

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.

Extends from Interfaces.SISO (Single Input Single Output continuous control block).

Connectors

Modelica.Blocks.Math.RealToInteger

Convert Real to Integer signal

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;

Extends from Modelica.Blocks.Icons.IntegerBlock (Basic graphical layout of Integer block).

Connectors

Modelica.Blocks.Math.IntegerToReal

Convert Integer to Real signals

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.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Connectors

Modelica.Blocks.Math.BooleanToReal

Convert Boolean to Real signal

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.

Extends from Interfaces.partialBooleanSI (Partial block with 1 input Boolean signal).

Parameters

Connectors

Modelica.Blocks.Math.BooleanToInteger

Convert Boolean to Integer signal

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.

Extends from Interfaces.partialBooleanSI (Partial block with 1 input Boolean signal).

Parameters

Connectors

Modelica.Blocks.Math.RealToBoolean

Convert Real to Boolean signal

Information

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

    y = u ≥ threshold;

where threshold is a parameter.

Extends from Interfaces.partialBooleanSO (Partial block with 1 output Boolean signal).

Parameters

Connectors

Modelica.Blocks.Math.IntegerToBoolean

Convert Integer to Boolean signal

Information

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

    y = u ≥ threshold;

where threshold is a parameter.

Extends from Interfaces.partialBooleanSO (Partial block with 1 output Boolean signal).

Parameters

Connectors

Modelica.Blocks.Math.RectangularToPolar

Convert rectangular coordinates to polar coordinates

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)

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Connectors

Modelica.Blocks.Math.PolarToRectangular

Convert polar coordinates to rectangular coordinates

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 )

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Connectors

Modelica.Blocks.Math.Mean

Calculate mean over period 1/f

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.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.RectifiedMean

Calculate rectified mean over period 1/f

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.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.ContinuousMean

Calculates the empirical expectation (mean) value of its input signal

Information

This block continuously calculates the mean value of its input signal. It uses the function:

    integral( u over time)
y = ----------------------
      time - startTime

This can be used to determine the empirical expectation value of a random signal, such as generated by the Noise blocks.

The parameter t_eps is used to guard against division by zero (the mean value computation starts at startTime + t_eps and before that time instant y = u).

See also the Mean block for a sampled implementation.

This block is demonstrated in the examples UniformNoiseProperties, NormalNoiseProperties and WeibullNoiseProperties.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Blocks.Math.RootMeanSquare

Calculate root mean square over period 1/f

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.

Extends from Modelica.Blocks.Interfaces.SISO (Single Input Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Variance

Calculates the empirical variance of its input signal

Information

This block calculates the empirical variance of its input signal. It is based on the formula (but implemented in a more reliable numerical way):

y = mean(  (u - mean(u))^2  )

The parameter t_eps is used to guard against division by zero (the variance computation starts at startTime + t_eps and before that time instant y = 0).

The variance of a signal is also equal to its mean power.

This block is demonstrated in the examples UniformNoiseProperties, NormalNoiseProperties and WeibullNoiseProperties.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Blocks.Math.StandardDeviation

Calculates the empirical standard deviation of its input signal

Information

This block calculates the standard deviation of its input signal. The standard deviation is the square root of the signal's variance:

y = sqrt( variance(u) )

The Variance block is used to calculate variance(u).

The parameter t_eps is used to guard against division by zero (the computation of the standard deviation starts at startTime + t_eps and before that time instant y = 0).

This block is demonstrated in the examples UniformNoiseProperties, NormalNoiseProperties and WeibullNoiseProperties.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Blocks.Math.Harmonic

Calculate harmonic over period 1/f

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)

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Blocks.Math.Max

Pass through the largest signal

Information

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

    y = max ( u1 , u2 );

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica.Blocks.Math.Min

Pass through the smallest signal

Information

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

    y = min ( u1 , u2 );

Extends from Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Connectors

Modelica.Blocks.Math.MinMax

Output the minimum and the maximum element of the input vector

Information

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

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Connectors

Modelica.Blocks.Math.LinearDependency

Output a linear combination of the two inputs

Information

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

Note, for y0=0 the output is always zero.

To improve the implementation, the formula will be changed (non-backwards compatible) in the future: y = y0 + k1*u1 + k2*u2

Extends from Modelica.Blocks.Interfaces.SI2SO (2 Single Input / 1 Single Output continuous control block).

Parameters

Connectors

Modelica.Blocks.Math.Edge

Indicates rising edge of Boolean signal

Information

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

    y = edge( u );

Extends from Interfaces.BooleanSISO (Single Input Single Output control block with signals of type Boolean).

Connectors

Modelica.Blocks.Math.BooleanChange

Indicates Boolean signal changing

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 );

Extends from Interfaces.BooleanSISO (Single Input Single Output control block with signals of type Boolean).

Connectors

Modelica.Blocks.Math.IntegerChange

Indicates integer signal changing

Information

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

    y = change( u );

Extends from Interfaces.IntegerSIBooleanSO (Integer Input Boolean Output continuous control block).

Connectors