Controllability of linear systems, differential geometry curves in grassmannians and generalized grassmannians, and riccati equations

Abstract

We study the linear system ⨰=Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators eΛtB. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ℝn. We then view γ(t)=Im(eΛtB) as a curve in appropriate Grassmannian and see that, in local coordinates, γ is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on γ that are equivalent to the controllability of the system. To get quantitiative results, we lift γ to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ℝn). Explicit and simple expressions concerning the geometry of Γ are computed in terms of the Lie algebra of GL(ℝn), and these are related to the controllability of the system.