Mathematical Problems in Engineering
Volume 2005 (2005), Issue 4, Pages 465-475
doi:10.1155/MPE.2005.465
On the optimal control of affine nonlinear systems
Statistics and Applied Mathematics Institute of the Romanian Academy, P.O. Box 1-24, Bucharest 010145, Romania
Received 20 January 2004
Copyright © 2005 M. Popescu and A. Dumitrache. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The minimization control problem of quadratic functionals for the
class of affine nonlinear systems with the hypothesis of nilpotent
associated Lie algebra is analyzed. The optimal control
corresponding to the first-, second-, and third-order nilpotent
operators is determined. In this paper, we have considered the
minimum fuel problem for the multi-input nilpotent control and for
a scalar input bilinear system for such systems. For the
multi-input system, usually an analytic closed-form solution for
the optimal control ui∗(t) is not possible and it is
necessary to use numerical integration for the set of m
nonlinear coupled second-order differential equations. The optimal
control of bilinear systems is obtained by considering the Lie
algebra generated by the system matrices. It should be noted that
we have obtained an open-loop control depending on the initial
value of the state x0.