Averaging and control of nonlinear systems

Abstract

This dissertation investigates three principal areas regarding the dynamics and control of nonlinear systems: averaging theory, controllability of mechanical systems, and control of underactuated nonlinear systems. The most effective stabilizing controllers for underactuated nonlinear systems are time-periodic, which leads to the study of averaging theory for understanding the nonlinear effect generated by resonant oscillatory inputs. The research on averaging theory generalizes averaging theory to arbitrary order by synthesizing series expansion methods for nonlinear time-varying vector fields and their flows with nonlinear Floquet theory. It is shown that classical averaging theory is the application of perturbation methods in conjunction with nonlinear Floquet theory. Many known properties and consequences of averaging theory are placed within a single framework. The generalized averaging theory is merged with controllability analysis of underactuated nonlinear systems to derive exponentially stabilizing controllers. Although small-time local controllability (STLC) is easily demonstrated for driftless systems via the Lie algebra rank condition, STLC for systems with drift is more complicated. Furthermore, there exists a variety of techniques and canonical forms for determining STLC. This thesis exploits notions of geometric homogeneity to show that STLC results for a large class of mechanical systems with drift can be recovered by considering a class of nonlinear dynamical systems satisfying certain homogeneity conditions. These theorems generalize the controllability results for simple mechanical control systems found in Lewis and Murray [85]. Most nonlinear controllability results for classes of mechanical systems may be obtained using these methods. The stabilizing controllers derived using the generalized averaging theory and STLC analysis can be used to stabilize both systems with and without drift. Furthermore, they result in a set of tunable gains and oscillatory parameters for modification and improvement of the feedback strategy. The procedure can not only derive known controllers from the literature, but can also be used to improve them. Examples demonstrate the diversity of controllers constructed using the generalized averaging theory. This dissertation concludes with a chapter devoted to biomimetic and biomechanical locomotive control systems that have been stabilized using the generalized averaging theory and the controller construction procedure. The locomotive control systems roll, wriggle, swim, and walk, demonstrating the universal nature of the control strategy proposed.