Hierarchical control design for nonlinear trajectory planning and stabilization

• Applicant: Professor Dr. Alexander Zuyev
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 436433160

The proposal focuses on the development of hierarchical methods for essentially nonlinear control systems whose trajectories possess important features for the analysis at multiple time scales. As an important subclass of such systems, we will study dynamic and kinematic models of nonholonomic mechanical systems under controllability conditions with iterated Lie brackets. Three-layer hierarchical control algorithms will be developed in the case when the dynamics of the upper layer is generated by the gradient flow of a potential function. In these algorithms, the intermediary layer is governed by a discrete-time dynamical system, and the lower-layer dynamics (physical level) is ruled by a nonlinear control system with oscillating input functions. For more generality, we will consider discontinuous control functions and follow the concept of Carathéodory solutions. These ideas will be also extended for the stabilization of reference trajectories for non-autonomous control systems by using the separation of fast and slow dynamics under a suitable choice of frequency parameters. It is expected to derive new stability results by refining averaging methods for subsystems with fast variables and constructing Lyapunov functions for slow subsystems with disturbances. These theoretical results will be applied to nonlinear mathematical models in fluid dynamics and chemical engineering, such as finite dimensional approximations of the Euler and Navier-Stokes equations, preferential crystallization of enantiomers, and periodic non-isothermal reactions.

• DFG Programme: Research Grants