On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems
This addresses a bottleneck in control design for data-driven bilinear systems, but it is incremental as it focuses on limitations of a specific CLF class rather than a broad solution.
The paper tackles the challenge of designing control for high-dimensional bilinear systems derived from Koopman operators, showing that quadratic Control Lyapunov Functions (CLFs) are highly restrictive and often imply stabilizability only by constant control inputs. It establishes this via a QCQP formulation and proposes a convex relaxation for sufficient validity, with empirical evidence supporting the findings.
Koopman operator-based methods enable data-driven bilinear representations of unknown nonlinear control systems. Accurate representations often demand significantly higher dimensions than the original system, making control design challenging. Control Lyapunov Functions (CLFs) are widely used for controller synthesis, with quadratic CLF candidates being the most common due to their simplicity. Yet, we show that this class is highly restrictive, especially when the state dimension is large: under mild conditions, their existence implies stabilizability of the bilinear system by a constant input -- that is, the control remains fixed over time. We establish this result by formulating a quadratically constrained quadratic program (QCQP) that exactly characterizes valid CLFs. Since QCQPs are NP-hard, we propose a convex semidefinite relaxation that offers a sufficient validity condition. For single-input systems, we prove that a quadratic CLF requires constant control stabilizability, and empirically demonstrate that this extends to high-dimensional multi-input systems in many cases.