Koopman Operator for Nonlinear Flight Dynamics

Koopman Operator for Nonlinear Flight Dynamics

Dynamical systems representing vehicle flight are inherently nonlinear. Currently there are no generalised frameworks for explicit characterisation and solution of such systems. Conversely, linear systems are well understood, and many efficient algorithms are available for explicit characterisation and prediction.

Koopman operator theory presents a framework for constructing finite linear approximations to nonlinear systems, by projecting the nonlinear dynamics onto a Hilbert space constructed of Koopman operator eigenfunctions. This work summarises the theory underpinning the Galerkin method of constructing Koopman linear approximations of known dynamics using polynomial basis functions. Details of a system-agnostic program for computing the Koopman linear matrix is presented.

Linear approximations are constructed for two nonlinear systems: a simple Duffing oscillator, and a three degree-of-freedom glider trajectory. The ability of linear systems to closely follow the time-integrated nonlinear solution is demonstrated. Furthermore, expansions of the lifted linear model to incorporate variable system parameter, to incorporate bilinear time-varying inputs necessary for optimal control calculations, as well as methods to avoid the inherent linear system limitations are explored.

This work demonstrates feasibility and merit of the lifted linear system approach for analysis of vehicle dynamics and trajectory problems.