Calculate Sensitivity Function Using Parallel Algorithm

Abstract

Problem statement: To calculate sensitivity functions for a large dimension control system using one processor, it takes huge time to find the unknowns vectors for a linear system, which represents the mathematical model of the physical control system. This study is an attempt to solve the same problem in parallel to reduce the time factor needed and increase the efficiency. 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^-1 W, 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 parallel algorithm divided 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 was reduced to O(t/n-1) and, The Performance of parallel algorithm was increased by 40-55%. Conclusion: Used parallel algorithm reduced the time to calculate sensitivity function for a large dimension control system and the performance was increased.