Matthias Kawski

Stabilization of nonlinear systems in the plane

Abstract It is shown that every small-time locally controllable system in the plane can (locally) be asymptotically stabilized by employing locally Holder continuous feedback laws, as essentially was conjectured by E. Sontag. An explicit algorithm for the construction of such feedback laws is given.

Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory

In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled differential equations, exponentials of vector fields, and Lie brackets. We show by means of an example —the computation of control variations that give rise to the Legendre-Clebsch condition— how a good choice of formalism, based on expanding diffeomorphisms as products of exponentials, can simplify the calculations. We then describe the algebraic structure underlying the formal part of these calculations, showing that it is based on the theory of formal power series, Lie series, the Chen series —introduced in control theory by M. Fliess— and the formula for the dual basis of a Poincare-Birkhoff-Witt basis arising from a generalized Hall basis of a free Lie algebra.

GEOMETRIC HOMOGENEITY AND STABILIZATION

Abstract We present a consistent, geometric notion of homogeneity, for vector fields (differential equations and control systems), functions, differential forms and endomorphisms. The fundamental observation is that homogeneity is an intrinsic, geometric property. Accordingly, a coordinate-free characterization is given, and shown to yield the desired/expected results. This also addresses: “When do there exist local coordinates such that in these coordinates a vector field is homogeneous in the traditional sense?” While no simple algebraic test is known, the coordinate-free approach still allows one to exploit the geometric homogeneity properties without having to transform the objects into homogenizing local coordinates. We clarify which objects to declare homogeneous of degree zero, tracing the persisting confusion to a common mix-up of vector fields and endomorphisms.

Control variations with an increasing number of switchings

1. Introduction. The purpose of this paper is to introduce new families of control variations and exhibit how they lead to high-order conditions for controllability which cannot be obtained by the usual methods. Also, we explain why the underlying phenomenon is likely to be very important for the synthesis of (time-optimal) feedback. Suppose X(x) and Y (x) are real analytic vectorfields on R n with X(0) = 0. They give rise to the single-input affine control system

Needle Variations that Cannot be Summed

This article analyzes sets of higher order tangent vectors to reachable sets of analytic control systems (affine in the control). Both small-time local output controllability and small-time local controllability about a nonstationary reference trajectory are considered. In a series of purposefully constructed examples it is shown that the cones generated by needle variations may fail to be convex. The examples demonstrate that the usual technical condition that needle variations must be movable is essential to guarantee desirable convexity properties. Moreover, new doubts are cast on the structural stability of controllability properties, as apparently higher order perturbations can reverse the (lack of) controllability of lower order nilpotent approximating systems, thereby providing new insights about the ultimate question of whether controllability is finitely determined.

Chronological Algebras: Combinatorics and Control

This article investigates the geometric and algebraic foundations of exponential product expansions in nonlinear control A survey of historic developments in geometric control theory on one side and algebraic combinatorics on the other side exhibits parallel developments and demonstrates how respective ndings translate into powerful tools on the other side Chronological algebras are shown to provide the fundamental structure that uni es geometric algebraic and combinatorial theories

Homogeneous Feedback Stabilization

We address a special case of the general problem of finding continuous feedback laws (that lead to unique solutions) and asymptotically stabilize the origin of (in general) nonlinear smooth systems of the form

Stability and nilpotent approximations

\dot x = f(x) + ug(x)with x \in {R^n},u \in R