Researchers often change a physical process by manipulating an external parameter to get the response they desire. It is hard to create a controller that can continuously change the external parameter to get the desired results, so often controllers have only a limited amount of options. This limited option system is called a switching control. For example, in an anti-lock braking system, the controller can either open or close valves completely to relieve pressure in the brakes rather than by degrees. While the mathematics of each of these stages is simple, the design of interplay between the two valves is a challenging mathematical problem. This project provides the knowledge that engineers will use when designing switching control strategy that move the dynamics toward a required robust regime. Not only will this project improve designs for simple systems like anti-lock braking systems, but also in complex systems like a robot walking on two legs. When a robot walks on two legs, it must be able to change its center of mass, alternate leg and knee motion and account for the nearly infinite number of repeated collisions the feet have with the ground. In addition to implications in the robotics and automotive industry, the theory developed in this project is relevant in fluid mechanics (wind and river flow, where the control is continuous, but nonsmooth), ecology (grazing management), and medicine (intermittent therapies, such as hormone therapies) where, as in robotics, the systems are complex. This project offers training for mathematics students interested in collaboration with researchers in other fields.
The project will investigate the intrinsically nonsmooth (i.e. new) aspects of the dynamics of differential equations with nondifferentiable right-hand terms. The current theory is incapable to predict the emergence (bifurcation) of complex oscillations, if the points of discontinuity of the right-hand side (switchings) form discontinuous manifolds, if the trajectories feature infinite number of discontinuities (collisions) in finite time (chattering), if a trajectory develops into a funnel of curves over time (i.e. makes long-term prediction impossible), or if one smooths the discontinuities of the model (i.e. introduces the slow-fast dynamics). These four mechanisms describe the four objectives of the project because they are the four central routes for intrinsically nonsmooth attractive regimes to occur in nonsmooth control applications. To achieve the goal, such chapters of the field as dimension reduction, normal forms and bifurcation theory will be significantly advanced.
