AAE 450 Spring 2008 Adam Waite 2/7/08 Dynamics and Control Gaussian Probability Function Collaboration with Mike Walker

AAE 450 Spring 2008 Gaussian Probability Method  Inputs  Average Value of Parameter (ex. Isp, wind speed, mass flow rate)  Standard Deviation  Create an array (ex. input = [AvWindSpeed, StdDevWind])  Output  Randomized value for the Parameter based on Gaussian Probability Model ex. WindSpeed = Gauss_Density(input) Dynamics and Control Num. of Iterations Percent Error Example using WindSpeed equal to 20 m/s with StdDev = 1 m/s By Adam Waite

AAE 450 Spring 2008 Implementing Gaussian Function  Currently used by the simulator for thrust, mass flow rate, and wind speed  Need standard deviation values for: -Isp, mdot, wind speed Dynamics and Control Future Work  Finalize controller  Set up program for Monte Carlo simulation  Investigate hardware necessary to implement controller Simulink by Mike Walker and Adam Waite Location of Gauss Function is in DNC subgroup in Sim 2.05 folder

AAE 450 Spring 2008 References  Smith, Julius O. Spectral Audio Signal Processing, March 2007 Draft, Stanford University  The MathWorks Inc., “gaussmf help file,” 2007 Dynamics and Control

AAE 450 Spring 2008 Derivation of Gaussian Function  This is the main Gaussian equation  Rearrange the function to solve for the variable x  Matlab is then used to pick a random number between 0 and 2 to put as the function f  If the number is between 0 and 1, the negative square root is used  If the number is between 1 and 2, the positive square root is used  This is the Gauss_Density function with average value and sigma as inputs Derivation by Mike Walker

Gauss_Density Function AAE 450 Spring 2008 %ISP Variance function for integration with DNC SIM %By: Mike Walker %January function Value = Gauss_Density(inp) %sigma is sqrt(9) = 3 format long g MEAN=inp(1); sig=inp(2); num_rand = 2.*rand; if(num_rand <=1) r= num_rand; x1 = -sqrt(-2*sig*sig*log(r))+MEAN; else r=num_rand -1; x1 = sqrt(-2*sig*sig*log(r))+MEAN; end Value = x1; Function by Mike Walker

Test File AAE 450 Spring 2008 % Test Run for Wind Speed Values % Adam Waite clear clc close all AvWindSpeed = 20; sigma = 1; input = [AvWindSpeed, sigma]; for t = 1:1000 WindSpeed(t) = Gauss_Density(input); end % For infinite iterations, the averaged values for WindSpeed % Should be equal to the given average value number = sum(WindSpeed)/length(WindSpeed) Test File for Gauss_Density function by Adam Waite

Old Method Using Simulink Block AAE 450 Spring 2008 Simulink by Adam Waite  Used a randomizer block that needed the average value and the variance  Can be seen in the above model as Thrust Value Randomizer  Used for thrust and wind direction

Old Method (cont.) AAE 450 Spring 2008 Simulink by Adam Waite  Reasons for Not Using this Method  Simulink block also required a seed number which produced the same “randomizations” for every run  Required a for loop in Matlab to run through different seed numbers for each use of the block which would slow down the program  Did not use Gaussian method