Occupation Kernels and Densely Defined Liouville Operators for System Identification

Rosenfeld, Joel A.; Kamalapurkar, Rushikesh; Russo, Benjamin P.; Johnson, Taylor T.

doi:10.1109/cdc40024.2019.9029337

Cited by 16 publications

(20 citation statements)

References 15 publications

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“…Hence, as x 0 was arbitrary, every kernel may be approximated by an occupation kernel corresponding to a trajectory, and since kernels are dense in H, so are these occupation kernels. Finally, if H and H are spaces of real analytic functions, the dynamics must also be real analytic by the same proof found in [17]. Spaces of real analytic functions include the Gaussian RBF and the exponential dot product kernel space.…”

Section: And the Approximate Left Singular Vector Formentioning

confidence: 86%

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Singular Dynamic Mode Decompositions

Rosenfeld¹,

Kamalapurkar²

2021

Preprint

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This manuscript is aimed at addressing several long standing limitations of dynamic mode decompositions in the application of Koopman analysis. Principle among these limitations are the convergence of associated Dynamic Mode Decomposition algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in the general analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of two Dynamic Mode Decomposition algorithms that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.

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Section: And the Approximate Left Singular Vector Formentioning

confidence: 86%

“…Hence, there exist a function, Γ θ ∈ H, such that g, Γ θ H = T 0 g(θ(t))dt for all g ∈ H. The function Γ θ is called the occupation kernel in H corresponding to θ. These occupation kernels were first introduced in [17…”

Section: Reproducing Kernel Hilbert Spacesmentioning

confidence: 99%

Singular Dynamic Mode Decompositions

Rosenfeld¹,

Kamalapurkar²

2021

Preprint

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“…Recently, a collection of results surrounding the concept of occupation kernels have appeared, where system identification problems are addressed not through numerical differentiation, but rather through integration [4]. The integration approach is considerably less sensitive to signal noise, and can be incorporated in system identification routines naturally through reproducing kernel Hilbert spaces (RKHSs).…”

Section: Introductionmentioning

confidence: 99%

Control Occupation Kernel Regression for Nonlinear Control-Affine Systems

Abudia,

Channagiri,

Rosenfeld

et al. 2021

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This manuscript presents an algorithm for obtaining an approximation of nonlinear high order control affine dynamical systems, that leverages the controlled trajectories as the central unit of information. As the fundamental basis elements leveraged in approximation, higher order control occupation kernels represent iterated integration after multiplication by a given controller in a vector valued reproducing kernel Hilbert space. In a regularized regression setting, the unique optimizer for a particular optimization problem is expressed as a linear combination of these occupation kernels, which converts an infinite dimensional optimization problem to a finite dimensional optimization problem through the representer theorem. Interestingly, the vector valued structure of the Hilbert space allows for simultaneous approximation of the drift and control effectiveness components of the control affine system. Several experiments are performed to demonstrate the effectiveness of the approach.

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“…It was observed in [14] that the functional relationship between occupation measures and Liouville operators can be fruitfully exploited within the context of reproducing kernel Hilbert spaces, which gave rise to occupation kernels (cf. [14,13]).…”

Section: Introductionmentioning

confidence: 99%