These notes accompany four lectures, giving an introduction to new developments in, and tools for problems in nonlinear control. Roughly speaking, after the successful development, starting in the 1960s, of methods from linear algebra, complex analysis and functional analysis for solving linear control problems, the 1970s and 1980s saw the emergence of differential geometric tools that were to mimic that success for nonlinear systems. In the past 30 years this theory has matured, and now connects with many other branches of mathematics. The focus of these notes is the role of algebraic combinatorics for both illuminating structures and providing computational tools for nonlinear systems. On the control side, we focus on problems connected with controllability, although the combinatorial tools obviously have just as much use for other control problems, including e.g. pathplanning, realization theory, and observability.

The use of \coordinates of the rst kind" provides an alternative to Sussmann's exponential product expansion of the Chen Fliess series. The chronological formalism together with Hall bases for the free Lie algebra allow one to give explicit recursive descriptions of the iterated integral coecients (i.e., the coordinates of the rst kind) occurring in the logarithm of the Chen Fliess series. Of interest are the interplay of left and right chronological powers, and the disappearance of the factorials when working with chronological products. 1 Introduction The Chen Fliess series describes the solutions of nonlinear (analytic) systems that are linear (or ane if setting u a0 1) in the control _ x = X a2Z u a f a (x): (1) Arguably one may consider this (exponential) Lie series as the noncommutative analogue (with dynamic scalars) of the Taylor series, which is an essential tool of analysis when working with commutative, static scalars. The Chen Fliess series in its original form ...

The chronological formalism, in particular, exponential product expansions and combinatorial features of Viennot-Hall bases are shown to lead to streamlined proofs of conditions for controllability and optimality. The focus is on those high-order conditions for smalltime local controllability that originally were derived in the 1980s. The key features are adapted ViennotHall bases and Lazard elimination tailored to the speci c conditions, which together rene the construction of Sussmann's exponential product expansion. 1 Introduction Nonlinear control systems commonly feature singular states where standard linearization techniques fail completely. Such states are distinguished as singular points of the distributions spanned by the vector elds that de- ne the system. The best known example may be the angular velocity equations of a controlled rigid body _ !(t) = f 0 (!(t)) + m X k=1 u i (t)f i (!(t)) (1) whose uncontrolled drift vector eld, taking the form f 0 (!) = c 1 ! 2 ...

This paper reports progress in the analysis of interconnections of nonlinear systems, employing the chronological formalism. A fundamental observation is the close analogy between feeding outputs of one system back as inputs to another system and the process of Lazard elimination which is at the root of Hall-Viennot bases and chronological products. Possible applications of the algebraic description of interconnections of systems include static and dynamic output feedback, and formal inversions of systems which are of interest for tracking problems. Our description in terms of iterated integral functionals is most readily applicable in the case of nilpotent systems, especially strictly triangular homogeneous systems. 1 Introduction One of the distinguishing characteristics of controlled dynamical systems is their predestination to be interconnected: This is particularly prominent when working with input-output descriptions of control systems. But state-space descriptions are just as ...

Since the articles on nonlinear controllability of the early 1970s by Jurdjevic and Sussmann, Lobry, Brockett, Hermes and Haynes and others, one common thread of the dierential geometric approach has been the desired conceptual separation of the time-varying features from the xed geometrical structures. Integral manifolds are intrinsically related to Lie-brackets of vector elds, and they determine accessibility. Iterated integral functionals of the controls carry a natural chronological algebra structure, and they carry the information that distinguishes controllability from accessibility { in the case of nonlinear systems. This article surveys some of the progress made in the study of nonlinear controllability since the 1970s, leading to a modern reformulation of the concept of obstructions to controllabilty (traditionally misnamed bad brackets) which takes advantage of newly found formulas for the logarithm of the Chen-Fliess series, i.e. using coordinates of the rst kind, and co...

this report, I survey and summarize the literature on nonlinear controllability. Since most of the results are based on Lie series representations of systems, I devote the early portion of the paper to this topic without reference to controllability. In particular, the Chen-Fliess series is developed, and then represented in terms of an exponential Lie series. Next several types of controllability are dened, the most important of these being the accessibility property (AP) and small-time local controllability (STLC). A necessary and sucient condition for a system to have the accessibility property is given, followed by several necessary conditions and sucient conditions for STLC. Finally, various extensions of these results are discussed. I have written with the GE 489 student in mind, hence the discussion is somewhat tutorial. I have sought to introduce concepts through intuitive development and to illustrate results with examples. At most, I have in some places included proof synopses for important results. This being said, the nature of the topic does require an understanding of some more abstract concepts, hence I do incorporate some higher mathematics in order to make the denitions and results precise, as well as to provide intuition. The paper assumes an exposure to dierential geometry on the part of the reader, although some basic denitions and results are reviewed. A mathematical appendix containing denitions and results perhaps unfamiliar to the GE 489 student is included. Furthermore, I have endeavored to unify the notation of the literature, and a list of notation also appears as an appendix. Finally, a detailed bibliography is given including abstracts where possible and my comments where appropriate. 2 Problem Formulation

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new su#cient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [1].