Nonlinear control design for nonholonomic and underactuated systems

This dissertation contains new results on the design and analysis of nonholonomic and underactuated systems. The interest in these systems is motivated both by practical needs and technical challenges. On the one hand, there are numerous examples of nonholonomic systems, of which some have substantial engineering interest. The control construction for these systems is an important part of the system design. On the other hand, nonholonomic systems have velocity constraints. These systems are fundamental nonlinear in the sense that they cannot be controlled by smooth time-invariant state feedback controllers. Therefore, nonlinear control algorithms need to be developed for these dynamic systems.; There are mainly two canonical forms for nonholonomic systems, the chained form and the power form. Since a group of systems are equivalent to one canonical form through transformation, the control law for a canonical form can be applied to this group of systems. A discontinuous, time-invariant controller is constructed for the n dimensional power form system. A recursive algorithm is presented which uses invariant manifolds to construct a generated 3-dimensional system in power form. Utilizing the controller for this generated system, we develop a control law for the original n dimensional system and the closed loop system achieves exponential convergence.; A common problem associated with discontinuous, time-invariant stabilizing controllers for nonholonomic systems is addressed. A common characteristic of all these discontinuous controllers is that control inputs may become excessively large, especially for initial conditions close to certain singular manifold which includes the origin. We deal with this problem by first dividing the state space into “good” and “bad” regions. A bounded controller is developed for n dimensional chained form systems based on this. The trajectories of the closed-loop system converge to the origin exponentially with the control effort bounded by a priori given value.; For the underactuated systems, we consider attitude stabilization and tracking problems under two independent control torques. We provide a stabilizing feedback control law for the kinematic system subject to input constraints. The proposed control law forces all closed-loop trajectories into a region of the state space where the control inputs are small and bounded. The control law is subsequently extended to solve the case of attitude tracking of an underactuated spacecraft using two controls. As a special case we also provide a feedback control to track a specified direction in inertial space. All proposed control laws achieve asymptotic stability with exponential convergence.; Finally, the bounded angular velocity controller is provided for the dynamic attitude control problem using gas jet actuators or momentum wheel actuators. The control inputs are torques applied to the spacecraft. Again the state space is divided into “good” and “bad” region. Convergent controller is designed in the “bad” region to drive the closed loop trajectories to the “good” region exponentially. The “good” region is invariant under the feedback controller and the closed loop trajectories are rendered exponentially convergent.