Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics

Abstract

This dissertation introduces the notion of a dynamic inverse of a nonlinear map. The dynamic inverse is used in the construction of nonlinear dynamical system, called a dynamic inverter, that asymptotically solves inverse problems with time-varying vector-valued solutions. Dynamic inversion generalizes and extends many previous results on the inversion of maps using continuous-time dynamic systems. By posing the dynamic inverse itself as the solution to an inverse problem, we show how one may solve for a dynamic inverse dynamically while simultaneously using the dynamic inverse solution to solve for the time-varying root of interest. Dynamic inversion is a continuous-time dynamic computational paradigm that may be incorporated into controllers in order to continuously provide estimates of time-varying parameters necessary for control. This allows nonlinear control systems to be posed entirely in continuous-time, replacing discrete root-finding algorithms as well as discrete algorithms for matrix inversion with integration. Example applications include solving for the intersection of time-varying polynomials, inversion of nonlinear control systems, regular and generalized inversion of fixed and time-varying matrices, polar decomposition of fixed and time-varying matrices, output tracking of implicitly defined reference trajectories, end-effector tracking control for robotic manipulators, and causal approximate output tracking for nonlinear nonminimum-phase systems. For the problem of output tracking for nonminimum-phase systems, an internal equilibrium manifold is introduced. This manifold is intrinsic to the class of nonlinear nonminimum-phase systems studied. Approximate output tracking is achieved by constructing a controller that makes a neighborhood of the internal equilibrium manifold attractive and invariant. Dynamic inversion is incorporated into the controller to provide a continuous estimate of the manifold location. This estimate is incorporated into the tracking control law. We demonstrate, by application to the tracking problem for the inverted pendulum on a cart, that the resulting internal equilibrium controller significantly outperforms a linear quadratic regulator, where the linearization of the internal equilibrium controller is made identical to the linear quadratic regulator. We also apply internal equilibrium control to the problem of causing a nonlinear, nonholonomic model of a bicycle to track a time-parameterized trajectory in the ground plane while retaining balance.