Persistence of Excitation in Uniformly Embedded Reproducing Kernel Hilbert (RKH) Spaces

Guo, Jia; Paruchuri, Sai Tej; Kurdila, Andrew J.

doi:10.23919/acc45564.2020.9147851

Cited by 18 publications

(43 citation statements)

References 19 publications

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“…This paper extends the results in the recent papers [23], [41], [24], [25], [42] in several fundamental ways. The first result states sufficient conditions that guarantee the PE condition for finite-dimensional RKHS over a manifold M that is defined in terms of a finite number of kernel basis functions.…”

Section: B Summary Of New Resultssupporting

confidence: 86%

“…Recent articles on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) extend the notion of partial PE to cases where the unknown function appearing in ordinary differential equations (ODEs) belongs to an infinite-dimensional RKHS. [23], [24], [25] The PE condition is difficult, and sometimes impossible, to verify a priori in practical applications. To overcome this limitation, authors have studied simpler sufficient conditions that ensure PE.…”

Section: A Pe Conditions and Convergencementioning

confidence: 99%

“…Recent articles on adaptive estimation in RKHS provide a framework for adaptive estimation of dynamic systems whose states evolve in more generic spaces, including embedded manifolds. [23], [41] The corresponding PE conditions are given in [24], [25]. Guo et al study the rate of convergence of the finite-dimensional approximations of reproducing kernel Hilbert spaces defined over manifolds in [42].…”

Section: A Pe Conditions and Convergencementioning

confidence: 99%

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Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Paruchuri

Guo

Kurdila

2023

IEEE Trans. Automat. Contr.

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The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example.

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Section: B Summary Of New Resultssupporting

confidence: 86%

Section: A Pe Conditions and Convergencementioning

confidence: 99%

Section: A Pe Conditions and Convergencementioning

confidence: 99%

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Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Paruchuri

Guo

Kurdila

2023

IEEE Trans. Automat. Contr.

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“…The analysis of convergence of parameters (ie, functions in our case) is notoriously long, so in this short paper we merely outline the proof in a special case. The full and lengthy details (for general P and Hurwitz A) are given in [31].…”

Section: The Rkh Embedding Methodsmentioning

confidence: 99%

“…In the case at hand, Equation 16 follows from the fact that f (x(•)) ∈ F, a family of uniformly equi-continuous functions. The lengthy details of the proof can be found in [31].…”

Section: The Rkh Embedding Methodsmentioning

confidence: 99%

Persistence of Excitation in Reproducing Kernel Hilbert Spaces, Positive Limit Sets, and Smooth Manifolds

Kurdila¹,

Guo²,

Paruchuri³

et al. 2019

Preprint

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This paper studies the relationship between the positive limit sets of continuous semiflows and the newly introduced definition of persistently excited (PE) sets and associated subspaces of reproducing kernel Hilbert (RKH) spaces. It is shown that if the RKH space contains a rich collection of cut-off functions, persistently excited sets are contained as subsets of the positive limit set of the semiflow. The paper demonstrates how the new PE condition can be used to guarantee convergence of function estimates in the RKH space embedding method for adaptive estimation. In particular, the paper is applied to uncertain ODE systems with positive limit sets given by certain types of smooth manifolds, and it establishes convergence of adaptive function estimates over the manifolds.

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Kernel center adaptation in the reproducing kernel Hilbert space embedding method

Paruchuri

Guo

Kurdila

2022

Adaptive Control & Signal

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SummaryThe performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state‐space. In this article, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems by exploiting the fact that the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self‐organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.

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Persistence of Excitation in Uniformly Embedded Reproducing Kernel Hilbert (RKH) Spaces

Cited by 18 publications

References 19 publications

Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Persistence of Excitation in Reproducing Kernel Hilbert Spaces, Positive Limit Sets, and Smooth Manifolds

Kernel center adaptation in the reproducing kernel Hilbert space embedding method

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