Computations with half-range Chebyshev polynomials

Orel, B.; Perne, Andrej

doi:10.1016/j.cam.2011.10.006

Cited by 6 publications

(16 citation statements)

References 10 publications

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“…where D is the differention matrix defined in [24], R is the radiation matrix defined in [13], corresponding to the radiation force generated by the velocity over the control horizon I.…”

Section: ) Optimal Trajectory Calculationmentioning

confidence: 99%

Adaptive Control of a Wave Energy Converter

Davidson

Genest

Ringwood

2018

IEEE Trans. Sustain. Energy

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Energy maximising controllers (EMCs), for wave energy converters (WECs), based on linear models, are attractive in terms of simplicity and computation. However, such (Cummins equation) models are normally built around the still water level as an equilibrium point and assume small movement, leading to poor model validity for realistic WEC motions, especially for the large amplitude motions obtained by a well controlled WEC. The method proposed here is to use an adaptive algorithm to estimate the control model online, whereby system identification techniques are employed to identify a linear model that is most representative of the actual controlled WEC behaviour. Using exponential forgetting, the linear model can be regularly adapted to remain representative in changing operational conditions. To that end, this paper presents a novel adaptive controller based on a receding horizon pseudospectral formulation. A case study is presented, providing an preliminary evaluation of the adaptive receding horizon pseudospectral control (ARHPC), using simulations based on both linear hydrodynamic modelling and computational fluid dynamics (CFD). The parameter adaptation is shown to behave as expected in both different simulation environments. The ARHPC is observed to perform well in irregular sea states, absorbing more power than its nonadaptive counterpart. The optimal trajectory calculated by the adaptive model is seen to have a smaller motion and power takeoff (PTO) forces, compared to those calculated by the nonadaptive linear control model, due to the increased amount of hydrodynamic resistance estimated by the adaptive model, as identified from the nonlinear viscous CFD simulation.

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“…where D is the differention matrix defined in [24], R is the radiation matrix defined in [13], corresponding to the radiation force generated by the velocity over the control horizon I.…”

Section: ) Optimal Trajectory Calculationmentioning

confidence: 99%

Adaptive Control of a Wave Energy Converter

Davidson

Genest

Ringwood

2018

IEEE Trans. Sustain. Energy

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“…Each constraints is expressed as a residual terms R that needs to be minimized by the employed optimization algorithm. As an example, the dynamical equation of the WEC is described in terms of residuals at the collocation nodes in (11).…”

Section: Pseudospectral Optimal Controlmentioning

confidence: 99%

“…(11) where D represents the differentiation matrix [11], and P the convolution matrix defined in [8]. In the same way, the energy absorption, i.e.…”

Section: Pseudospectral Optimal Controlmentioning

confidence: 99%

“…The obtained control problem designed to maximized the cost function (12), while insuring the nullity of the residuals at each collocation points t k , such as (11) and under inequality constraints, can be solved using standard quadratic problem solving algorithms, such as active-set or interior-point methods. The obtained solution of the derived control problem will determined each projection vectors of the state and control variables.…”

Section: Pseudospectral Optimal Controlmentioning

confidence: 99%

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Receding horizon pseudospectral optimal control for wave energy conversion

Genest

Ringwood

2016

2016 UKACC 11th International Conference on Control (CONTROL)

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Abstract-The present study introduces a real-time control algorithm for applications involving energy maximisation and subject to technological limitations. Development of Wave energy converters constitutes an actual topic of research, in their designs and fluid-interactions, and particularly on their controllability. Immersed WECs are subject to fluid-interaction forces, generating unusual solicitations specific to hydrodynamic equation, such as radiation force or excitation forces.Various control strategies were recently developed and more advance control algorithms are currently applied on WECS, such as Model Predictive Control [1], or Pseudospectral optimal control [2]. Such control strategies, capable of maximizing the energy production while insuring the respect of path constraints, are more realistic and applicable in real conditions involving technological limitation aspects. This paper presents a receding horizon type control, based on pseudospectral approach in the control problem resolution. Dealing with irregular waves on a fixed control horizon, the presented control need to work with nonperiodic functions, implying the change of the basis functions involved in the description of the state and control variables, namely the halfrange Chebyshev Fourier functions.Application of the receding horizon control is presented for a generic WEC, and compared with a standard MPC algorithm.

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“…Explicit evaluation of moments such as these is important in general for further study of the associated orthogonal polynomials and related quantities, where one does not have an explicit representation for the polynomials or the recurrence coefficients in terms of special functions. See for example, [8] and [11] where they make use of the half-range Chebyshev moments in a numerical study applied to the efficient computation of Fourier approximations. See also [9] where computations performed with half range Legendre moments are applied to radiative transfer in spherical geometry.…”

Section: Introductionmentioning

confidence: 99%

Ultraspherical moments on a set of disjoint intervals

AlSabi

Griffin

2019

Indagationes Mathematicae

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Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.

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Computations with half-range Chebyshev polynomials

Cited by 6 publications

References 10 publications

Adaptive Control of a Wave Energy Converter

Adaptive Control of a Wave Energy Converter

Receding horizon pseudospectral optimal control for wave energy conversion

Ultraspherical moments on a set of disjoint intervals

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