Parallel Calculation Sensitivity Function for Multi Tasking Environments

Abstract

Problem statement: Calculating sensitive functions for a large dimension control system to find the unknowns vectors for a linear system in both single and multi processors, is not considered internally compatible with multi tasking environments, so breaking the process can cost time and memory and it couldn't be paused, resumed and saved as patterns for later continuity. This study is an attempt to solve this problem in parallel to reduce the time factor needed and increase the efficiency by using parallel calculation sensitivity function for multi tasking environments (PSME) algorithm. Approach: calculate in parallel sensitivity function using n-1 processors where n is a number of linear equations which can be represented as TX = W, where T is a matrix of size n₁×n₂, X = T^-1W, is a vector of unknowns and ∂X/∂h = T^-1((∂T/∂h)-(∂W/∂h)) is a sensitivity function with respect to variation of system components h. The algorithm (PSME) divides the mathematical input model into two partitions and  uses  only (n-1) processors to find the vector of unknowns for original system x = (x₁,x₂,…,x_n)^T and in parallel using (n-1) processors to find the vector of unknowns for similar system (x’)^t = d^tT^-1 = (x₁’,x₂’,…x_n’)^T by using Net-Processors, where d is a constant vector. Finally, sensitivity function (with respect to variation of component ∂X/∂h_i = (x_i×x_i’) can be calculated in parallel by multiplication unknowns X_i×X_i’, where i = 0,1,…n-1. Results: The running time t is reduced to O(t/n-1) and, the performance of (PSME) was increased by 30-40%. Conclusion: Hence, used (PSME) algorithm reduced the time to calculate sensitivity function for a large dimension control system and the performance was increased.