Explicit Solution of Bilinear Systems with and Without Time Delays by Lie Algebraic Approach and its Application to Controllability

Abstract

The purpose of this paper is to obtain the explicit solutions of bilinear systems with and without time-delay by Lie algebraic approach and to apply the results to controllability problems. First, an algorithm to construct the canonical basis of Lie algebra is presented. Using the canonical basis, explicit solution of bilinear system without time-delay is obtained in vector form for the case where the Lie algebra generated by the system matrices is solvable. Defining a set of new variables, the bilinear system equation with time-delay is expressed in a form without time-delay. The explicit solution of the bilinear system with time-delay is then obtained in a vector form by the the procedure developed for the case without time-delay. The controllability conditions of bilinear systems with and without time-delay for the case of independent linear and bilinear controls are directly obtained from the explicit solutions.