Abstract
Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is further demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including (a) a nonlinear system with a known linear embedding, (b) a variety of Hamiltonian systems, and (c) a high-dimensional double-gyre model for ocean mixing.
Highlights
In contrast to linear systems, a generally applicable and scalable framework for the control of nonlinear systems remains an engineering grand challenge
We examine the effect of an error εψ(x) in the representation of a Koopman eigenfunction, φ(x) := φ(x) + εψ(x), on its closed-loop dynamics based on (33) and provide an upper bound for the error
We note that the full-state model and reduced-order models (ROMs) obtained from the implicit formulation are identical, as all discovered eigenpairs fall below the threshold (in (f)) and only unique pairs are selected to construct either model
Summary
In contrast to linear systems, a generally applicable and scalable framework for the control of nonlinear systems remains an engineering grand challenge. There is no overarching framework for nonlinear control, that is generically applicable to a wide class of potentially high-dimensional systems, as exists for linear systems [37, 104]. Applicable frameworks, such as dynamic programming [15], Pontryagin’s maximum principle [92], and model-predictive control (MPC) [2, 26], often suffer from the curse of dimensionality and require considerably computational effort, e.g. solving adjoint equations, when applied to nonlinear systems, which can be mitigated to some degree by combining these with low-dimensional, linear representations. Koopman operator theory has recently emerged as a leading framework to obtain linear representations of nonlinear dynamical systems from data [80]. We build on the existing EDMD and reformulate the Koopman-based control problem in eigenfunction coordinates and provide strategies to identify lightly damped eigenfunctions from data, that can be
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