On excessive Transverse Coordinates for Orbital Stabilization of Periodic Motions

Sætre, Christian Fredrik; Shiriaev, Anton S.

doi:10.1016/j.ifacol.2020.12.2212

Cited by 4 publications

(13 citation statements)

References 7 publications

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“…Since we are looking for a time-invariant feedback k(•), we require some mapping that allows us to recover the parameterizing variable s from the system's states within some neighborhood of the orbit, a so-called projection operator. Although such a mapping can always be constructed for periodic orbits (see, e.g., [14]), it needs to satisfy some additional requirements when the orbit begins and terminates at equilibrium points as we consider in this paper. Definition 4.…”

Section: Projection Operatorsmentioning

confidence: 99%

“…for some scalar function l(•) satisfying l(s, x) = O( x 2 ). The variables (11) may therefore be interpreted as to form an excessive set of so-called transverse coordinates inside T [14]. This means that the qualitative stability properties of the orbit may be assessed therein using the corresponding transverse linearization-the first-order approximation of their dynamics.…”

Section: Lemma 6 ([14]mentioning

confidence: 99%

“…Lemma 7 (Transverse linearization [11,14]). Inside the tube T , the first-order approximation of the dynamics of the coordinates (11) along the orbit O is described by the differential-algebraic equations:…”

Section: Lemma 6 ([14]mentioning

confidence: 99%

“…Indeed, it has long been known (see, e.g., [6]) that for non-trivial orbits, such as periodic ones, strict contraction in the directions transverse to it corresponds to the contraction of solutions towards the orbit itself; see also [7][8][9][10]. Moreover, this contraction can be determined from a specific linearization of the system dynamics along the nominal orbit [11], a so-called transverse linearization [12][13][14]. Since this contraction occurs on transverse hyperplanes, only the linearization of a set of transverse coordinates of dimension one less than the dimension of the state space needs to be stabilized; a fact which has been readily used to stabilize periodic orbits in UMSs (see, e.g., [12] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems

Sætre¹,

Shiriaev²

2021

Preprint

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The task of inducing, via continuous static state-feedback, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system, that is, to a smooth curve in its state-control space that is consistent with the system dynamics and which connects two stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver's orbit as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching and has a familiar feedforward plus feedback-like structure. The method is also complemented by synchronization functionbased arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numeric simulations of the non-prehensile manipulation task of a ball rolling between two points upon the "butterfly" robot demonstrates the efficacy of the full synthesis.

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Section: Projection Operatorsmentioning

confidence: 99%

Section: Lemma 6 ([14]mentioning

confidence: 99%

Section: Lemma 6 ([14]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems

Sætre¹,

Shiriaev²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…There exists several methods for designing orbitally stabilizing feedback for different classes of systems; see, for example [2][3][4][5][6][7][8][9][10]. However, with exception of a few methods which are either only applicable for a very limiting class of systems [11] or only ensure asymptotic orbital stability of some of the system's states [12], the existing methods in the literature are, to a large extent, both sensitive to-and highly dependent on the accuracy of the mathematical model.…”

Section: Introductionmentioning

confidence: 99%

Robust Orbital Stabilization: A Floquet Theory-based Approach

Sætre,

Shiriaev,

Freidovich

et al. 2020

Preprint

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The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal (disturbance-free) system, a robustifying feedback extensions is designed utilizing the sliding mode control (SMC) methodology. The main contribution of the paper is to provide a constructive procedure for designing the time-invariant switching function used in the SMC synthesis. More specifically, its zero-level set (the sliding manifold) is designed using a real Floquet-Lyapunov transformation as to locally correspond to an invariant subspace of the Monodromy matrix of a transverse linearization. This ensures asymptotic stability of the periodic orbit when the system is confined to the sliding manifold, despite any system uncertainties and external disturbances satisfying a matching condition. The challenging task of oscillation control of the underactuated Cart-Pendulum system subject to both matched-and unmatched disturbances/uncertainties demonstrates the efficacy of the proposed scheme.

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Robust orbital stabilization: A Floquet theory–based approach

Sætre¹,

Shiriaev

Freidovich

et al. 2021

Intl J Robust & Nonlinear

Self Cite

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The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal, disturbance‐free system, a robustifying feedback extension is designed utilizing the sliding‐mode control (SMC) methodology. The main contribution of the article is to provide a constructive procedure for designing the time‐invariant switching function used in the SMC synthesis. More specifically, its zero‐level set (the sliding manifold) is designed using a real Floquet–Lyapunov transformation to locally correspond to an invariant subspace of the Monodromy matrix of a transverse linearization. This ensures asymptotic stability of the periodic orbit when the system is confined to the sliding manifold, despite any system uncertainties and external disturbances satisfying a matching condition. The challenging task of oscillation control of the underactuated cart–pendulum system subject to both matched‐ and unmatched disturbances/uncertainties demonstrates the efficacy of the proposed scheme.

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On excessive Transverse Coordinates for Orbital Stabilization of Periodic Motions

Cited by 4 publications

References 7 publications

Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems

Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems

Robust Orbital Stabilization: A Floquet Theory-based Approach

Robust orbital stabilization: A Floquet theory–based approach

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