Bilinearization, Reachability, and Optimal Control of Control-Affine Nonlinear Systems: A Koopman Spectral Approach

Abstract

This article considers the problem of bilinearization and optimal control of a control-affine nonlinear system by projecting the system dynamics onto the Koopman eigenspace. Although there are linearization techniques like Carleman linearization for embedding a finite-dimensional nonlinear system into an infinite-dimensional space, they depend on the analytic property of the vector fields and work only on polynomial space. The proposed method utilizes the Koopman canonical transform, specifically the Koopman eigenfunctions of the drift vector field, to transform the dynamics into a bilinear system under certain assumptions. While the bilinearization is exact, if there exists a Koopman-invariant finite-dimensional subspace for the drift vector field, sometimes this condition is too conservative. An approximate approach is to minimize an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}^2$</tex-math></inline-formula> norm on the truncated state-space. The approximation can be carried out from time-series data without explicit knowledge of the drift vector field. Controllability of the bilinear system is analyzed using the Myhill semigroup method and Lie algebraic structures. Pontryagin’s principle is applied to the bilinear system to yield a two-point boundary-value problem for the optimal control design. A single shooting method solves the boundary value problem in order to determine the control signal. Alternatively, a gradient-based method is also outlined to find the optimal control which exploits the bilinear structure. Several examples of control-affine nonlinear systems numerically illustrate the bilinearization and optimal control design, assuming a cost function quadratic in the states and control input.