| NSF Org: |
CMMI Division of Civil, Mechanical, and Manufacturing Innovation |
| Recipient: |
|
| Initial Amendment Date: | July 14, 2014 |
| Latest Amendment Date: | July 14, 2014 |
| Award Number: | 1436856 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Atul Kelkar
CMMI Division of Civil, Mechanical, and Manufacturing Innovation ENG Directorate for Engineering |
| Start Date: | September 1, 2014 |
| End Date: | August 31, 2017 (Estimated) |
| Total Intended Award Amount: | $186,855.00 |
| Total Awarded Amount to Date: | $186,855.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
800 WEST CAMPBELL RD. RICHARDSON TX US 75080-3021 (972)883-2313 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
TX US 75080-3021 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | Dynamics, Control and System D |
| Primary Program Source: |
|
| Program Reference Code(s): |
|
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.041 |
ABSTRACT
Special Topics in Dynamical Systems: A New Mathematical Framework for the Design of Switching and Continuous Control Strategies
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Nonsmooth dynamical systems provide simple mechanisms for known smooth phenomena. For example, switching between systems of linear differential equations is capable to produce limit cycles, while a linear system along is not able to model cycling behavior. Furthermore, nonsmooth dynamical systems exhibit phenomena that cannot be modeled in the context of smooth systems, e.g. lack of the derivative at an equilibrium is required for the equilibrium to be finite-time stable. That is why nonsmooth dynamical systems have lately become an efficient tool of control theory with applications ranging from robotics and power electronics to neuroscience and climate modeling.
This project proposed a new framework to control the dynamics of nonsmooth dynamical systems building upon the methods of bifurcation theory. For nonsmooth systems of relay type (when one or another subsystem is activated upon touching a respective sensor), the distance between the sensors turned out to be a fruitful choice for the bifurcation parameter. However, such a bifurcation cannot be studied by the methods of the theory of discontinuous (Filippov) system as the switching manifold of relay systems is itself discontinuous. The project offered a general result to ensure that such a bifurcation leads to cycling dynamics. The most direct applications of this result are in control of dc-dc converters and anti-lock braking systems. The latter application was studied in greater details and effectiveness of the proposed control strategy compared to the precessors was documented. The second group of results concerns finite-time stable equilibria and limit cycles. For equilibria, the project discovered a normal form whose finite-time dynamics is classified against coefficients. This result can be applied to study bifurcations in finite-time integrators used to control robotic locomotion. We offered a simple route to finite-time stable limit cycles where a cycle that just touches the discontinuity threshold converts to a finite-time limit cycle under a suitable class of perturbations. The third part of the project offers bifurcation theory for nonsmooth differential equations with continuous right-hand-sides with non-uniqueness of solutions. By introducing a normal form system of differential equations with Holder continuous right-hand-sides the result of this part classifies the occurrence of terminal attractors and terminal repellers against the parameters of the normal form. The final forth group of the results concerns those bifurcations in nonsmooth mechanical systems that are due to regularization of absolutely elastic constraints. It was discovered that such a perturbation can preserve stability of periodic orbits or lead to multi-stability according to the geometry of the respective sweeping process that the project used to account for possible plastic ingredients in the regularized constraints.
Last Modified: 09/11/2017
Modified by: Oleg Makarenkov
Please report errors in award information by writing to: awardsearch@nsf.gov.
