An Introduction to Aspects of Geometric Control Theory

Abstract

There are many texts on linear control theory, and a number of introductions to nonlinear control theory and in particular its differential geometric formulation, which is important for this book. We do not pretend here to give a comprehensive introduction to this subject; we just touch on a few aspects that are important to our major topics. We mention the books by Jurdjevic [1997], Isidori [1995], Nijmeijer and van der Schaft [1990], Sontag [1990], and Brockett [1970b]. The first three deal with the differential-geometric approach, Sontag’s book gives a mathematical treatment of both linear theory and various parts of the nonlinear theory, while Brockett’s book gives an approach to linear theory. We refer the reader to these books for a detailed treatment of nonlinear control theory as well as for a more exhaustive list of references. We also mention the important articles of Sussmann [1987], Sussmann and Jurdjevic [1972], Hermann and Krener [1977], and Brockett [1970a,b]. Two recent books that are relevant are van der Schaft [2000] and Ortega, Loria, Nicklasson and Sira-Ramirez [1998]. A useful recent book on optimal control is Liberzon [2012]. There are, of course, many other good sources as well, and we shall refer to them as needed.