Geometric Methods in Analysis and Control of Implicit Differential Systems

Abstract

In this article, we discuss the application of differential geometric
methods in analyzing the structure and designing control laws for
implicit differential systems or differential algebraic equation (DAE) systems.
While there have been several efforts toward numerical and quantitative
analysis of DAE problems, the theoretical contributions especially
in the case of nonlinear systems are scarce. We discuss two popular
techniques from differential geometric control theory and bring out their
merits in addressing implicit differential systems. In the first section, we
review the theory of noninteracting control via input–output decoupling
and its application in analyzing the intrinsic structure of DAE control
problems. In particular, we focus on addressing the problem of well-posedness
of DAE systems as well as feedback control design through a
regularization process which allows one to solve the DAE by expressing
the constraint variable as a dynamically dependent endogenous
function of the states, inputs, and their derivatives. Further, extensions
of these techniques to stochastic differential algebraic equations have
been presented. In the second section, we review the theory of differential
flatness and its applicability to feedback control design for a class of
DAE systems. Here, the DAE system is expressed as a Cartan field on a
manifold of jets of infinite order, and necessary and sufficient conditions
for its equivalence to a linear, controllable system have been derived in
order to design globally stabilizing nonlinear feedback laws. Examples
from constrained mechanics have been presented in order to demonstrate
the practical applicability of these methods.