Koopman Control Factorization: Data-Driven Convex Controller Design for a Class of Nonlinear Systems
Taha Ondogan, Ran Jing, Andrew P. Sabelhaus, Roberto Tron
TL;DR
The paper tackles the difficulty of designing convex controllers for nonlinear systems within the Koopman framework, where inputs typically break global linearity. It introduces the Koopman Control Factorization (KCF), which preserves linearity of the control input in the lifted space by using a linear controller on nonlinear state features, yielding a closed-loop Koopman operator $\tilde{K}$ that is bilinear in system and controller coefficients via $\\tilde{K} = K_{xx} + K_{xu}(I \otimes K_u) H$. A data-driven identification pipeline based on Motor Babbling estimates $K_x$, $S$, and $H$, after which a semidefinite program computes a stabilizing $K_u$ with a Lyapunov guarantee. The method is demonstrated on inverted pendulum examples, showing fast, verifiable stability without receding-horizon optimization and broad applicability to nonlinear control-affine systems. Overall, KCF provides a practical, offline, data-driven route to convex control synthesis with stability guarantees for a class of nonlinear dynamics.
Abstract
Although Koopman operators provide a global linearization for autonomous dynamical systems, nonautonomous systems are not globally linear in the inputs. State (or output) feedback controller design therefore remains nonconvex in typical formulations, even with approximations via bilinear control-affine terms. We address this gap by introducing the Koopman Control Factorization, a novel parameterization of control-affine dynamical systems combined with a feedback controller defined as a linear combination of nonlinear measurements. With this choice, the Koopman operator of the closed-loop system is a bilinear combination of the coefficients in two matrices: one representing the system, and the other the controller. We propose a set of sufficient conditions such that the factorization holds. Then, we present an algorithm that calculates the feedback matrix via semi-definite programming, producing a Lyapunov-stable closed-loop system with convex optimization. We evaluate the proposed controllers on two canonical examples of control-affine nonlinear systems (inverted pendulums), and show that our factorization and controller successfully stabilize both under properly-chosen basis functions. This manuscript introduces a broadly generalizable control synthesis method for stabilization of nonlinear systems that is quick-to-compute, verifiably stable, data-driven, and does not rely on approximations.