Local nested transverse feedback linearization

This is a post-peer-review, pre-copyedit version of an article published in Mathematics of Control, Signals, and Systems. The final authenticated version is available online at: http://dx.doi.org/https://doi.org/10.1007/s00498-015-0149-yWe study two local feedback equivalence problems for a nonlinear control-affine system with two nested, controlled-invariant, embedded submanifolds in its state space. The first, less restrictive, result gives necessary and sufficient conditions for the dynamics of the system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable. This normal form facilitates designing controllers that locally stabilize the smaller set relative to the larger set. The second, more restrictive, result additionally imposes that the transversal dynamics to the larger set be linear and controllable. This result can simplify designing controllers to locally stabilize the larger submanifold. This is illustrated by sufficient conditions under which these normal forms can be used to locally solve a nested set stabilization problem.Funder 1,This research was partially supported by the Natural Sciences and Engineering Research Council of Canada (N.S.E.R.C.) ||. Funder 2, the University of Waterloo

Coordinated path following: A nested invariant sets approach

In this thesis we study a coordinated path following problem for multi-agent systems. Each agent is modelled by a smooth, nonlinear, autonomous, deterministic control-affine ordinary differential equation. Coordinated path following involves designing feedback controllers that make each agent's output approach and traverse a pre-assigned path while simultaneously coordinating its motion with the other agents. Coordinated motion along paths includes tasks like maintaining formations, traversing paths at a common speed and more general tasks like making the positions of some agents obey functional constraints that depend on the states of other agents. The coordinated path following problem is viewed as a nested set stabilization problem. In the nested set stabilization approach, stabilization of the larger set corresponds to driving the agents to their assigned paths. This set, under suitable assumptions, is an embedded, controlled invariant, product submanifold and is called the multi-agent path following manifold. Stabilization of the nested set, contained in the multi-agent path following manifold, corresponds to meeting the coordination specification. Under appropriate assumptions, this set is also an embedded controlled invariant submanifold which we call the coordination set. Our approach to locally solving nested set stabilization problems is based on feedback equivalence of control systems. We propose and solve two local feedback equivalence problems for nested invariant sets. The first, less restrictive, solution gives necessary and sufficient conditions for the dynamics of a system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable. This normal form facilitates designing controllers that locally stabilize the coordination set relative to the multi-agent path following manifold. The second, more restrictive, result additionally imposes that the transversal dynamics to the larger submanifold be linear and controllable. This result can simplify designing controllers to locally stabilize the multi-agent path following manifold. We propose sufficient conditions under which these normal forms can be used to locally solve the nested set stabilization problem. To illustrate these ideas we consider a coordinated path following problem for a multi-agent system of dynamic unicycles. The multi-agent path following manifold is characterized for arbitrary paths. We show that each unicycle is feedback equivalent, in a neighbourhood of its assigned path, to a system whose transversal and tangential dynamics to the path following manifold are both double integrators. We provide sufficient conditions under which the coordination set is nonempty. The effectiveness of the proposed approach is demonstrated experimentally on two robots

Passivity-based stabilization of non-compact sets

Abstract—We investigate the stabilization of closed sets for passive nonlinear systems which are contained in the zero level set of the storage function. I