Koopman-Based Reduced Order Controller Design for Quadrotors

A novel Koopman-based reduced order controller for quadrotor stabilization and trajectory tracking is derived, implemented and tested. Controller design exploits Koopman spectral theory for which an infinite dimensional Koopman-based linear dynamics model, embedding the dynamics of the underlying nonlinear system, may be represented by a finite subset of the eigenfunctions of the Koopman operator. The proposed controller exploits the fact that, for Hamiltonian systems, the Hamiltonian function is always an eigenfunction of the Koopman operator. Using this approach, a hierarchical controller is derived that is composed of a PID outer loop position controller and a state dependent Riccati equation Koopman-Hamiltonian inner loop attitude controller. A key advantage of this approach is that it achieves multivariable attitude control through the use of a single scalar Hamiltonian function, thus, drastically simplifying controller design. Moreover, by minimizing an energy function, the optimal controller guarantees optimal trajectory tracking. Results show that the designed controller is able to stabilize the quadrotor dynamics and to track a helix trajectory even in the presence of wind gust disturbances, drag effects, and first order motor dynamics.