Finite Time Lyapunov Analysis in Optimal Control

University Of California-Irvine, Irvine CA

Investigators

Abstract

The research aims to develop methodology for optimally controlled systems that evolve on two or more disparate timescales. At the heart of the approach is the use of finite-time Lyapunov exponents and vectors associated with the linearized dynamics along trajectories of the Hamiltonian system that, with the appropriate boundary conditions, augments the original dynamics with the first-order necessary conditions for optimality and governs the evolution of the optimally controlled system. The timescales are diagnosed from the spectrum of exponents. Geometric structure is identified by using Lyapunov vectors to construct the appropriate partially hyperbolic splitting of the tangent bundle to the phase space for the Hamiltonian system and to formulate conditions satisfied by phase points on key manifolds. The capability of calculating points on key manifolds enables a solution approximation method in which the hyper-sensitivity due to the disparate timescales is suppressed. The finite-time Lyapunov exponents and vectors also provide information that aids the development of implementable near-optimal control laws. Deliverables include analysis methods and tools, application to and validation of the methodology to problems in flight mechanics and robotics, documentation of research results, and engineering student education. Optimal control of machines, structures, and vehicles is of increasing importance in this era of demands for both high performance and efficient operation. Yet obtaining solutions to optimal control problems and especially developing near-optimal implementable control laws is often difficult. A common source of this difficulty is hyper-sensitivity which is problematic for many numerical solution approaches, yet it implies a geometric structure in the solution space, knowledge of which can lead to simplified solution approaches, greater understanding, and implementable near-optimal control laws. If successful, the project results will facilitate the implementation of high performance, efficient controllers for engineered systems. The methodology is expected to benefit other application areas, such as weather prediction and chemical kinetics, where understanding timescales and manifold structure has value.