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. In particular the following operations and functions are provided:

   TwoInputs     Change causality of input signals.
   TwoOutputs    Change causality of output signals.
   Gain          Output the input multiplied by a scalar gain
   MatrixGain    Output the product of a gain matrix with the input
   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
   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
   Log10         Output the base 10 logarithm of the input
   RealToInteger Output the nearest Integer value to the input
   IntegerToReal Output the input as Real value
   Max           Output the maximum of the two inputs
   Min           Output the minimum of the two inputs
   Edge          Set output to true at rising edge of the input
   BooleanChange Set output to true when Boolean input changes
   IntegerChange Set output to true when Integer input changes

Main Author:

Martin Otter
Deutsches Zentrum fuer Luft und Raumfahrt e.V. (DLR)
Oberpfaffenhofen
Postfach 1116
D-82230 Wessling
email: Martin.Otter@dlr.de

Release Notes:

• October 21, 2002 by Martin Otter and Christian Schweiger:
  New blocks added: RealToInteger, IntegerToReal, Max, Min, Edge, BooleanChange, IntegerChange.
• August 7, 1999 by Martin Otter:
  Realized (partly based on an existing Dymola library of Dieter Moormann and Hilding Elmqvist).

Copyright © 1999-2002, Modelica Association and DLR.

The Modelica package is free software; it can be redistributed and/or modified under the terms of the Modelica license, see the license conditions and the accompanying disclaimer in the documentation of package Modelica in file "Modelica/package.mo".

Modelica.Blocks.Math.Gain

Information

This block computes the output y=outPort.signal as element-wise product of gain k with the input u = inPort.signal:

    y[i] = k[i] * u[i];

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
k[:]	{1}	Gain vector multiplied element-wise with input vector

Modelica definition

block Gain 
  "Output the element-wise product of a gain vector with the input signal vector"
   
  
  parameter Real k[:]={1} 
    "Gain vector multiplied element-wise with input vector";
protected 
  Real u[size(k, 1)] "Input signals";
  Real y[size(k, 1)] "Output signals";
public 
  Interfaces.InPort inPort(final n=size(k, 1)) "Input signal connector";
  Interfaces.OutPort outPort(final n=size(k, 1)) "Output signal connector";
equation 
  u = inPort.signal;
  y = outPort.signal;
  for i in 1:size(k, 1) loop
    y[i] = k[i]*u[i];
  end for;
end Gain;

Modelica.Blocks.Math.Add3

Information

This block computes the output y=outPort.signal as sum of the three input signals u1=inPort1.signal, u2=inPort2.signal and u3=inPort3.signal:

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

Example:

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

  results in the following equations:

     y[1] = 2 * u1[1] - 3 * u2[1] + u3[1];
     y[2] = 2 * u1[2] - 3 * u2[2] + u3[2];

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
k1	+1	Gain of upper input
k2	+1	Gain of middle input
k3	+1	Gain of lower input
n	1	Dimension of input and output vectors.

Modelica definition

block Add3 "Output the sum of the three inputs" 
  extends Interfaces.BlockIcon;
  
  parameter Real k1=+1 "Gain of upper input";
  parameter Real k2=+1 "Gain of middle input";
  parameter Real k3=+1 "Gain of lower input";
  parameter Integer n=1 "Dimension of input and output vectors.";
  Interfaces.InPort inPort1(final n=n) "Connector 1 of Real input signals";
  Interfaces.InPort inPort2(final n=n) "Connector 2 of Real input signals";
  Interfaces.InPort inPort3(final n=n) "Connector 3 of Real input signals";
  Interfaces.OutPort outPort(final n=n) "Connector of Real output signals";
equation 
  outPort.signal = k1*inPort1.signal + k2*inPort2.signal + k3*inPort3.signal;
end Add3;

Modelica.Blocks.Math.Add

Information

This block computes the output y=outPort.signal as sum of the two input signals u1=inPort1.signal and u2=inPort2.signal:

    y = k1*u1 + k2*u2;

Example:

     parameter:   n = 2, k1= +2, k2= -3

  results in the following equations:

     y[1] = 2 * u1[1] - 3 * u2[1]
     y[2] = 2 * u1[2] - 3 * u2[2]

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.
k1	+1	Gain of upper input
k2	+1	Gain of lower input

Modelica definition

block Add "Output the sum of the two inputs" 
  extends Interfaces.MI2MO;
  parameter Real k1=+1 "Gain of upper input";
  parameter Real k2=+1 "Gain of lower input";
equation 
  y = k1*u1 + k2*u2;
end Add;

Modelica.Blocks.Math.TwoInputs

Information

This block is used to enable assignment of values to variables preliminary defined as outputs (e.g. useful for inverse model generation).

Release Notes:

• June 28, 2001 by Michael Thümmel:
  Realized.

Parameters

Name	Default	Description
n	1	number of input signals

Modelica definition

block TwoInputs 
  "Change causality of input signals (e.g. for inverse models)" 
  
  extends Interfaces.BlockIcon;
  parameter Integer n=1 "number of input signals";
  
  Interfaces.InPort inPort1(n=n);
  Interfaces.InPort inPort2(n=n);
equation 
  inPort1.signal = inPort2.signal;
end TwoInputs;

Modelica.Blocks.Math.TwoOutputs

Information

This block is used to enable assignment of values to variables preliminary defined as inputs (e.g. useful for inverse model generation).

Release Notes:

• June 28, 2001 by Michael Thümmel:
  Realized.

Parameters

Name	Default	Description
n	1	number of input signals

Modelica definition

block TwoOutputs 
  "Change causality of output signals (e.g. for inverse models)" 
  extends Interfaces.BlockIcon;
  parameter Integer n=1 "number of input signals";
  
  Interfaces.OutPort OutPort1(n=n);
  Interfaces.OutPort OutPort2(n=n);
equation 
  OutPort1.signal = OutPort2.signal;
end TwoOutputs;

Modelica.Blocks.Math.MatrixGain

Information

This block computes the output y=outPort.signal as product of the gain matrix K with the input signal vector u = inPort.signal:

    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] |

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
K[:, :]	[1, 0; 0, 1]	Gain matrix which is multiplied with the input

Modelica definition

block MatrixGain 
  "Output the product of a gain matrix with the input signal vector" 
  parameter Real K[:, :]=[1, 0; 0, 1] 
    "Gain matrix which is multiplied with the input";
  extends Interfaces.MIMO(final nin=size(K, 2), final nout=size(K, 1));
equation 
  y = K*u;
end MatrixGain;

Modelica.Blocks.Math.Sum

Information

This block computes the output y=outPort.signal[1] as sum of the elements of the input signal vector u=inPort.signal:

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

Example: With parameter nin = 3 results the following equation:

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

Optionally, the parameter Real k[nin]=ones(nin) could be changed to weight the sum elements in order to calculate the scalar product

    y = k*u = k[1]*u[1] + k[2]*u[2] + ... + k[nin]*u[nin]  .

Release Notes:

• Oct. 25, 2002 by Christian Schweiger:
  Realized an improved implementation with avoidance of an algorithm section, which would lead to expensive function calls.
• Oct. 24, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
nin	1	Number of inputs
k[nin]	ones(nin)	Optional: sum coefficients

Modelica definition

block Sum "Output the sum of the elements of the input vector" 
  extends Interfaces.MISO;
  
  parameter Real k[nin]=ones(nin) "Optional: sum coefficients";
  
equation 
  y = k*u;
end Sum;

Modelica.Blocks.Math.Feedback

Information

This block computes the output y=outPort.signal as difference of the commanded input u1=inPort1.signal and the feedback input u2=inPort2.signal:

    y = u1 - u2;

Example:

     parameter:   n = 2

  results in the following equations:

     y[1] = u1[1] - u2[1]
     y[2] = u1[2] - u2[2]

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	size of input and feedback signal

Modelica definition

block Feedback 
  "Output difference between commanded and feedback input" 
  parameter Integer n=1 "size of input and feedback signal";
  Interfaces.InPort inPort1(final n=n);
  Interfaces.InPort inPort2(final n=n);
  Interfaces.OutPort outPort(final n=n);
equation 
  outPort.signal = inPort1.signal - inPort2.signal;
end Feedback;

Modelica.Blocks.Math.Product

Information

This block computes the output y=outPort.signal element-wise as product of the corresponding elements of the two inputs u1=inPort1.signal and u2=inPort2.signal:

    y[i] = u1[i] * u2[i];

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.

Modelica definition

block Product "Output product of the two inputs" 
  extends Interfaces.MI2MO;
equation 
  for i in 1:n loop
    y[i] = u1[i]*u2[i];
  end for;
end Product;

Modelica.Blocks.Math.Division

Information

This block computes the output y=outPort.signal element-wise by dividing the corresponding elements of the two inputs u1=inPort1.signal and u2=inPort2.signal:

    y[i] = u1[i] / u2[i];

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.

Modelica definition

block Division "Output first input divided by second input" 
  extends Interfaces.MI2MO;
equation 
  for i in 1:n loop
    y[i] = u1[i]/u2[i];
  end for;
end Division;

Modelica.Blocks.Math.Abs

Information

This block computes the output y=outPort.signal element-wise as absolute value of the input u=inPort.signal:

    y[i] = abs( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Abs "Output the absolute value of the input" 
  extends Interfaces.MIMOs;
equation 
  y = abs(u);
end Abs;

Modelica.Blocks.Math.Sign

Information

This block computes the output y=outPort.signal element-wise as sign of the input u=inPort.signal:

            1  if u[i] > 0
    y[i] =  0  if u[i] == 0
           -1  if u[i] < 0

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Sign "Output the sign of the input" 
  extends Interfaces.MIMOs;
equation 
  y = sign(u);
end Sign;

Modelica.Blocks.Math.Sqrt

Information

This block computes the output y=outPort.signal element-wise as square root of the input u=inPort.signal:

    y[i] = sqrt( u[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Sqrt 
  "Output the square root of the input (input >= 0 required)" 
  extends Interfaces.MIMOs;
equation 
  y = sqrt(u);
end Sqrt;

Modelica.Blocks.Math.Sin

Information

This block computes the output y=outPort.signal element-wise as sine of the input u=inPort.signal:

    y[i] = sin( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Sin "Output the sine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.sin(u);
end Sin;

Modelica.Blocks.Math.Cos

Information

This block computes the output y=outPort.signal element-wise as cos of the input u=inPort.signal:

    y[i] = cos( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Cos "Output the cosine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.cos(u);
end Cos;

Modelica.Blocks.Math.Tan

Information

This block computes the output y=outPort.signal element-wise as tan of the input u=inPort.signal:

    y[i] = tan( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Tan "Output the tangent of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.tan(u);
end Tan;

Modelica.Blocks.Math.Asin

Information

This block computes the output y=outPort.signal element-wise as the sine-inverse of the input u=inPort.signal:

    y[i] = asin( u[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Asin "Output the arc sine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.asin(u);
end Asin;

Modelica.Blocks.Math.Acos

Information

This block computes the output y=outPort.signal element-wise as the cosine-inverse of the input u=inPort.signal:

    y[i] = acos( u[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Acos "Output the arc cosine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.acos(u);
end Acos;

Modelica.Blocks.Math.Atan

Information

This block computes the output y=outPort.signal element-wise as the tangent-inverse of the input u=inPort.signal:

    y[i] = atan( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Atan "Output the arc tangent of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.atan(u);
end Atan;

Modelica.Blocks.Math.Atan2

Information

This block computes the output y=outPort.signal element-wise as the tangent-inverse of the input u1=inPort1.signal divided by input u2=inPort2.signal:

    y[i] = atan2( u1[i], u2[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.

Modelica definition

block Atan2 "Output atan(u1/u2) of the inputs u1 and u2" 
  extends Interfaces.MI2MO;
equation 
  y = Modelica.Math.atan2(u1, u2);
end Atan2;

Modelica.Blocks.Math.Sinh

Information

This block computes the output y=outPort.signal element-wise as the hyperbolic sine of the input u=inPort.signal:

    y[i] = sinh( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Sinh "Output the hyperbolic sine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.sinh(u);
end Sinh;

Modelica.Blocks.Math.Cosh

Information

This block computes the output y=outPort.signal element-wise as the hyperbolic cosine of the input u=inPort.signal:

    y[i] = cosh( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Cosh "Output the hyperbolic cosine of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.cosh(u);
end Cosh;

Modelica.Blocks.Math.Tanh

Information

This block computes the output y=outPort.signal element-wise as the hyperbolic tangent of the input u=inPort.signal:

    y[i] = tanh( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Tanh "Output the hyperbolic tangent of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.tanh(u);
end Tanh;

Modelica.Blocks.Math.Exp

Information

This block computes the output y=outPort.signal element-wise as the exponential (of base e) of the input u=inPort.signal:

    y[i] = exp( u[i] );

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Exp "Output the exponential (base e) of the input" 
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.exp(u);
end Exp;

Modelica.Blocks.Math.Log

Information

This block computes the output y=outPort.signal element-wise as the natural (base e) logarithm of the input u=inPort.signal:

    y[i] = log( u[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Log 
  "Output the natural (base e) logarithm of the input (input > 0 required)"
   
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.log(u);
end Log;

Modelica.Blocks.Math.Log10

Information

This block computes the output y=outPort.signal element-wise as the base 10 logarithm of the input u=inPort.signal:

    y[i] = log10( u[i] );

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

Release Notes:

• August 7, 1999 by Martin Otter:
  Realized.

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Log10 
  "Output the base 10 logarithm of the input (input > 0 required)"
   
  
  extends Interfaces.MIMOs;
equation 
  y = Modelica.Math.log10(u);
end Log10;

Modelica.Blocks.Math.RealToInteger

Information

This block computes the output y=outPort.signal element-wise as nearest integer value of the input u=inPort.signal:

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

Parameters

Name	Default	Description
n	1	Number of input and output signals

Modelica definition

block RealToInteger "Convert real to integer signals" 
  extends Interfaces.IntegerBlockIcon;
  parameter Integer n=1 "Number of input and output signals";
protected 
  Real u[n];
public 
  Interfaces.InPort inPort(final n=n);
  Interfaces.IntegerOutPort outPort(final n=n);
equation 
  u = inPort.signal;
  for i in 1:n loop
    outPort.signal[i] = if (u[i] > 0) then integer(floor(u[i] + 0.5)) else 
      integer(ceil(u[i] - 0.5));
  end for;
end RealToInteger;

Modelica.Blocks.Math.IntegerToReal

Information

This block computes the output y=outPort.signal element-wise as Real equivalent of the Integer input u=inPort.signal:

    y[i] = u[i];

where u is of Integer and y of Real type.

Parameters

Name	Default	Description
n	1	Number of input signals (= number of output signals)

Modelica definition

block IntegerToReal "Convert integer to real signals" 
  extends Interfaces.BlockIcon;
  parameter Integer n=1 "Number of input signals (= number of output signals)";
  Interfaces.OutPort outPort(final n=n);
  Interfaces.IntegerInPort inPort(final n=n);
equation 
  outPort.signal = inPort.signal;
end IntegerToReal;

Modelica.Blocks.Math.Max

Information

This block computes the output y=outPort.signal element-wise as maximum of the two inputs u1=inPort1.signal and u2=inPort2.signal:

    y[i] = max ( u1[i] , u2[i] );

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.

Modelica definition

block Max "Pass through the largest signal" 
  extends Interfaces.MI2MO;
equation 
  for i in 1:n loop
    y[i] = max(u1[i], u2[i]);
  end for;
end Max;

Modelica.Blocks.Math.Min

Information

This block computes the output y=outPort.signal element-wise as minimum of the two inputs u1=inPort1.signal and u2=inPort2.signal:

    y[i] = min ( u1[i] , u2[i] );

Parameters

Name	Default	Description
n	1	Dimension of input and output vectors.

Modelica definition

block Min "Pass through the smallest signal" 
  extends Interfaces.MI2MO;
equation 
  for i in 1:n loop
    y[i] = min(u1[i], u2[i]);
  end for;
end Min;

Modelica.Blocks.Math.Edge

Information

This block sets the Boolean output y=outPort.signal element-wise to true, when the Boolean input u=inPort.signal shows a rising edge:

    y[i] = edge( u[i] );

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block Edge "Indicates rising edge of boolean signal" 
  extends Interfaces.BooleanMIMOs;
equation 
  for i in 1:n loop
    y[i] = edge(u[i]);
  end for;
end Edge;

Modelica.Blocks.Math.BooleanChange

Information

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

    y[i] = change( u[i] );

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block BooleanChange "Indicates boolean signal changing" 
  extends Interfaces.BooleanMIMOs;
equation 
  for i in 1:n loop
    y[i] = change(u[i]);
  end for;
end BooleanChange;

Modelica.Blocks.Math.IntegerChange

Information

This block sets the Boolean output y=outPort.signal element-wise to true, when the Integer input u=inPort.signal changes:

    y[i] = change( u[i] );

Parameters

Name	Default	Description
n	1	Number of inputs (= number of outputs)

Modelica definition

block IntegerChange "Indicates integer signal changing" 
  extends Interfaces.IntegerMIBooleanMOs;
equation 
  for i in 1:n loop
    y[i] = change(u[i]);
  end for;
end IntegerChange;