.Modelica_LinearSystems2.Controllers.Internal.random

Information

Random generates a sequence of uniform distributed pseudo-random numbers. The algorithm is a variant of the multiplicative congruential algorithm, known as the Wichmann-Hill generator:

   x(k) = (a1*x(k-1)) mod m1
   y(k) = (a2*y(k-1)) mod m2
   z(k) = (a3*z(k-1)) mod m3
   U(k) = (x(k)/m1 + y(k)/m2 + z(k)/m3) mod 1

This generates pseudo-random numbers U(k) uniformly distributed in the interval (0,1). There are many forms of generators depending on the parameters m (prime numbers) and a. The sequence needs an initial Integer vector {x,y,z} known as the seed. The use of the same seed will lead to the same sequence of numbers.

Remarks

Random number generators (RNG) are pseudo-functions which are not true functions but algorithms which deliver a fixed sequence of (usually Integer) numbers which should have a very large period before they repeat itself and appropriate statistic properties such that the sequence appears to be a random draw. For real-valued random numbers, the integers are scaled to the real interval 0.0-1.0. They result in a uniformly distributed random variate between 0-1, which has to be transformed to give a random variate of a wanted distribution. There are two types of techniques for transforming random variates:

• Acceptance-Rejection techniques
• Transformation techniques

Acceptance-Rejection techniques throw away some of the generated variates and are thus less efficient. They can not be avoided for all distributions. A good summary about random number generation and most of the transformation techniques used below is given in:

Some of the other references are quoted below.

   WICHMANN-HILL RANDOM NUMBER GENERATOR
   Wichmann, B. A. & Hill, I. D. (1982)
     Algorithm AS 183:
     An efficient and portable pseudo-random number generator
     Applied Statistics 31 (1982) 188-190
   see also:
     Correction to Algorithm AS 183
     Applied Statistics 33 (1984) 123
   McLeod, A. I. (1985)
     A remark on Algorithm AS 183
     Applied Statistics 34 (1985),198-200
   In order to completely avoid external functions, all seeds are
   set via parameters. For simulation purposes this is almost
   always the desired behaviour.
   Translated by Hubertus Tummescheit from Python source provided by
   Guido van Rossum translated from C source by Adrian Baddeley.
   http://www.python.org/doc/current/lib/module-random.html
   R A N D O M   V A R I A B L E   G E N E R A T O R S
   distributions on the real line:
   ------------------------------
     normal (Gaussian) 2 versions

Reference Literature:

• function random: Wichmann, B. A. & Hill, I. D. (1982), Algorithm AS 183:
  An efficient and portable pseudo-random number generator, Applied Statistics 31 (1982) 188-190
  see also: Correction to Algorithm AS 183, Applied Statistics 33 (1984) 123
  McLeod, A. I. (1985), A remark on Algorithm AS 183, Applied Statistics 34 (1985),198-200
• function normalvariate: Kinderman, A.J. and Monahan, J.F., 'Computer generation of random variables using the ratio of uniform deviates', ACM Trans Math Software, 3, (1977), pp257-260.
• function gaussianvariate: Discrete Event Simulation, Jerry Banks and John S. Carson II,
  Prentice Hall Inc. Englewood Cliffs, New Jersey, page 315/316

Copyright © Hubertus Tummescheit and Department of Automatic Control, Lund University, Sweden.

This Modelica function 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 BSD 3-Clause License.

Interface

function random
  extends Modelica.Icons.Function;
  input Integer seedIn[3] "Integer vector defining random number sequence, e.g., {23,87,187}";
  output Real x "Random number between 0 and 1";
  output Integer seedOut[3] "Modified seed to be used for next call of random()";
end random;