Stabilization of a class of slow–fast control systems at non-hyperbolic points

Jardón-Kojakhmetov, Hildeberto; Scherpen, Jacquelien M. A.; Puerto-Flores, Dunstano del

doi:10.1016/j.automatica.2018.10.008

Cited by 20 publications

(20 citation statements)

References 58 publications

(141 reference statements)

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“…To overcome this, we shall show that it is possible to "inject" a normally hyperbolic behavior to the fast variable through the backstepping algorithm [12]. • A general treatment of a more general class of SFSs around a fold point shall be presented in [6]. Our main contribution is as follows.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

Jardón-Kojakhmetov

Scherpen

Puerto-Flores

2017

2017 American Control Conference (ACC)

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Non-hyperbolic points of slow-fast systems (also known as singularly perturbed ordinary differential equations) are responsible for many interesting behavior such as relaxation oscillations, canards, mixed-mode oscillations, etc. Recently, the authors have proposed a control strategy to stabilize nonhyperbolic points of planar slow-fast systems. Such strategy is based on geometric desingularization, which is a well suited technique to analyze the dynamics of slow-fast systems near non-hyperbolic points. This technique transforms the singular perturbation problem to an equivalent regular perturbation problem. This papers treats the nonlinear adaptive stabilization problem of slow-fast systems. The novelty is that the point to be stabilized is non-hyperbolic. The controller is designed by combining geometric desingularization and Lyapunov based techniques. Through the action of the controller, we basically inject a normally hyperbolic behavior to the fast variable. Our results are exemplified on the van der Pol oscillator.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

Jardón-Kojakhmetov

Scherpen

Puerto-Flores

2017

2017 American Control Conference (ACC)

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“…We now examine the concept of a folded critical manifold, determine the conditions needed to prove its existence, and then apply these ideas to our specific critical manifold S c . The discussion of geometric singular perturbation theory presented in [36,11] covers fold points of critical manifolds in some detail.…”

Section: First Of All We Notice Thatmentioning

confidence: 99%

Traveling wave solutions for a cancer stem cell invasion model

Mayer¹,

Stachura²

2021

DCDS-B

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“…Perturbation theory is a well studied area and has a rich background in linear operator theory and the controls field [26]- [28], and it has seen renewed interest in recent years [29]- [31]. It is commonly applied in the context of wellseparated time scales.…”

Section: B Overviewmentioning

confidence: 99%

“…Proof. Consider the equation forẋ n (t) in (31). Performing an MBAM approximation by taking the limit r n → 0 gives thaṫ x n = 0.…”

Section: A Balanced Truncation From Mbammentioning

confidence: 99%

Model Boundary Approximation Method as a Unifying Framework for Balanced Truncation and Singular Perturbation Approximation

Paré

Grimsman

Wilson

et al. 2019

IEEE Trans. Automat. Contr.

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We show that two widely accepted model reduction techniques, Balanced Truncation and Balanced Singular Perturbation Approximation, can be derived as limiting approximations of a carefully constructed parameterization of Linear Time Invariant (LTI) systems by employing the Model Boundary Approximation Method (MBAM), a recent development in the Physics literature [1]. This unifying framework of these popular model reduction techniques shows that Balanced Truncation and Balanced Singular Perturbation Approximation each correspond to a particular boundary point on a manifold, the "model manifold," which is associated with the specific choice of model parameterization and initial condition, and is embedded in a sample space of measured outputs, which can be chosen arbitrarily, provided that the number of samples exceeds the number of parameters. We also show that MBAM provides a novel way to interpolate between Balanced Truncation and Balanced Singular Perturbation Approximation, by exploring the set of approximations on the boundary of the manifold between the elements that correspond to the two model reduction techniques; this allows for alternative approximations of a given system to be found that may be better under certain conditions. The work herein suggests similar types of approximations may be obtainable in topologically similar places (i.e. on certain boundaries) on the model manifold of nonlinear systems if analogous parameterizations can be achieved, therefore extending these widely accepted model reduction techniques to nonlinear systems.

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Stabilization of a class of slow–fast control systems at non-hyperbolic points

Cited by 20 publications

References 58 publications

Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

Nonlinear adaptive stabilization of a class of planar slow-fast systems at a non-hyperbolic point

Traveling wave solutions for a cancer stem cell invasion model

Model Boundary Approximation Method as a Unifying Framework for Balanced Truncation and Singular Perturbation Approximation

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