Koopman-LQR Controller for Quadrotor UAVs from Data

TL;DR

This work uses the method of extended dynamic mode decomposition (EDMD) to identify the Koopman operator from data and demonstrates that the identified model can be stabilized and controlled by designing a controller using the linear quadratic regulator (LQR).

Abstract

Quadrotor systems are common and beneficial for many fields, but their intricate behavior often makes it challenging to design effective and optimal control strategies. Some traditional approaches to nonlinear control often rely on local linearizations or complex nonlinear models, which can be inaccurate or computationally expensive. We present a data-driven approach to identify the dynamics of a given quadrotor system using Koopman operator theory. Koopman theory offers a framework for representing nonlinear dynamics as linear operators acting on observable functions of the state space. This allows to approximate nonlinear systems with globally linear models in a higher dimensional space, which can be analyzed and controlled using standard linear optimal control techniques. We leverage the method of extended dynamic mode decomposition (EDMD) to identify Koopman operator from data with total least squares. We demonstrate that the identified model can be stabilized and controllable by designing a controller using linear quadratic regulator (LQR).

Figures (3)

• Figure 1: Illustration of the Koopman Operator: The Koopman Operator ($\mathcal{K}_t$) maps state-space into a lifted linear space governed by ${\xi}$. Green panels represent linear space (convex search space for control design), while red panels represent nonlinear state-space (non-convex search space).
• Figure 2: The spectrum analysis of the discovered system without control (right) with LQR stabilization (left).
• Figure 3: Illustration of the performance comparison between Nominal PID Control (blue), Koopman Control (black), and the Reference trajectory (red) for a quadrotor. The plots depict the state trajectories over a 200-step prediction horizon, covering the states of the quadrotor including position ($x$, $y$, $z$), velocity ($\dot{x}$, $\dot{y}$, $\dot{z}$), Euler angles ($\phi$, $\theta$, $\psi$), and angular velocities ($\omega_x$, $\omega_y$, $\omega_z$).

Theorems & Definitions (6)

• Definition 1: Koopman Operator
• Definition 2: Regression using DMD
• Remark 1
• Remark 2
• Remark 3
• Remark 4