1,189 research outputs found
Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
In this work, we explore finite-dimensional linear representations of
nonlinear dynamical systems by restricting the Koopman operator to an invariant
subspace. The Koopman operator is an infinite-dimensional linear operator that
evolves observable functions of the state-space of a dynamical system [Koopman
1931, PNAS]. Dominant terms in the Koopman expansion are typically computed
using dynamic mode decomposition (DMD). DMD uses linear measurements of the
state variables, and it has recently been shown that this may be too
restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing
nonlinear observable functions to form an invariant subspace where it is
possible to obtain linear models, especially those that are useful for control,
is an open challenge.
Here, we investigate the choice of observable functions for Koopman analysis
that enable the use of optimal linear control techniques on nonlinear problems.
First, to include a cost on the state of the system, as in linear quadratic
regulator (LQR) control, it is helpful to include these states in the
observable subspace, as in DMD. However, we find that this is only possible
when there is a single isolated fixed point, as systems with multiple fixed
points or more complicated attractors are not globally topologically conjugate
to a finite-dimensional linear system, and cannot be represented by a
finite-dimensional linear Koopman subspace that includes the state. We then
present a data-driven strategy to identify relevant observable functions for
Koopman analysis using a new algorithm to determine terms in a dynamical system
by sparse regression of the data in a nonlinear function space [Brunton et al.
2015, arxiv]; we show how this algorithm is related to DMD. Finally, we
demonstrate how to design optimal control laws for nonlinear systems using
techniques from linear optimal control on Koopman invariant subspaces.Comment: 20 pages, 5 figures, 2 code
Dynamic mode decomposition with control
We develop a new method which extends Dynamic Mode Decomposition (DMD) to
incorporate the effect of control to extract low-order models from
high-dimensional, complex systems. DMD finds spatial-temporal coherent modes,
connects local-linear analysis to nonlinear operator theory, and provides an
equation-free architecture which is compatible with compressive sensing. In
actuated systems, DMD is incapable of producing an input-output model;
moreover, the dynamics and the modes will be corrupted by external forcing. Our
new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all
of the advantages of DMD and provides the additional innovation of being able
to disambiguate between the underlying dynamics and the effects of actuation,
resulting in accurate input-output models. The method is data-driven in that it
does not require knowledge of the underlying governing equations, only
snapshots of state and actuation data from historical, experimental, or
black-box simulations. We demonstrate the method on high-dimensional dynamical
systems, including a model with relevance to the analysis of infectious disease
data with mass vaccination (actuation).Comment: 10 pages, 4 figure
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