Gradient flow approximation for stabilization and motion planning in dynamic environments

• Applicant: Professor Dr. Christian Pötzsche, since 4/2020
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 403230946

Motion planning for nonlinear control systems is one of the most important problems of mathematical control theory because of its theoretical challenges and many practical applications. Typically, this problem involves the construction of controls that ensure the motion of a system to a target. Often, it is important that such controls also stabilize the system. Considerable efforts have been made to the solution of stabilization and motion planning problems for static target and obstacles. However, many applied problems are related to dynamic environments, where the locations of targets and obstacles change with time, e.g, when an autonomous vehicle should follow a moving target avoiding collisions with other moving objects. From a theoretical point of view, the complexity of control design for such tasks increases significantly, and many classical methods developed for static environments are no longer applicable. To the best of our knowledge, most publications in this area deal with specific classes of systems, and the explicit control design for general classes of nonlinear systems remains an open task. The main goal of the proposed project is to develop a general framework for stabilization and motion planning of nonholonomic systems governed by driftless control-affine systems. Unlike fully actuated holonomic systems, the number of controls in such systems can be significantly smaller than the number of state variables, so that an arbitrary path is not necessary feasible for a nonholonomic system due to motion constraints. The main idea of this project is to guarantee that the motion of the system steers along the approximated gradient flow of a certain time-varying potential function, for example, Lyapunov functions for stabilization and navigation functions for obstacle avoidance. Depending on the approximation method, two types of controls will be constructed: gradient-based and gradient-free. Gradient-based controls may explicitly depend on the derivatives of a potential function and can be used in situations where its analytical expression is known, for example, when complete information on the target and obstacles is available. Although gradient-based controllers serve as well-established tools for the stabilization and motion planning, in a variety of practical scenarios an explicit analytical expression of the potential function is partially or completely unknown; the gradient cannot hence be computed explicitly. This issue is of high relevance, for example, when the target trajectory is unknown, or in extremum seeking problems. For such cases, it is planned to construct gradient-free controls, which, nevertheless, ensure the gradient-like behavior of the system. The main expected results of this project are aimed to contribute to mathematical control theory and nonlinear dynamics by novel control design approaches, and by a rigorous asymptotic stability analysis of the corresponding closed-loop non-autonomous systems.

• DFG Programme: Research Grants
• International Connection: Austria
• Ehemalige Antragstellerin: Dr. Victoria Grushkovskaya, until 4/2020