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.