Geometric pseudospectral method for spatial integration of dynamical systems

Moulla, Redha; Lefèvre, Laurent; Maschke, Bernhard

doi:10.1080/13873954.2010.537524

Cited by 14 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Construction of the defect constraints ( ). For , refer to (12) and 13 For configuration , refer to (26) and 27, and define the other part of the defect constraints ( = 1, . .…”

Section: Transcription Of the Oac On So(3) Into An Nlp Problemmentioning

confidence: 99%

“…Among the direct methods, the collocation method, also known as a pseudo-spectral method, is widely used in rigid body attitude optimization (including UAV, satellite, space shuttle, launch vehicle, and supersonic aircraft) by virtue of its high accuracy and spectral (or exponential) convergence rate [1]. In the context of a Lie group, to our knowledge, Moulla et al [12] are the first to propose the concept of the "geometric pseudo-spectral method." They suggest a polynomial pseudo-spectral method that preserves the geometric structure of port-Hamiltonian systems, phenomenological laws, and conservation laws without introducing any uncontrolled numerical dissipation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

Xiang

2015

Mathematical Problems in Engineering

View full text Add to dashboard Cite

The collocation method is extended to the special orthogonal group SO(3) with application to optimal attitude control (OAC) of a rigid body. A left-invariant rigid-body attitude dynamical model on SO(3) is established. For the left invariance of the attitude configuration equation in body-fixed frame, a geometrically exact numerical method on SO(3), referred to as the geometric collocation method, is proposed by deriving the equivalent Lie algebra equation inso(3)of the left-invariant configuration equation. When compared with the general Gauss pseudo-spectral method, the explicit RKMK, and Lie group variational integrator having the same order and stepsize in numerical tests for evolving a free-floating rigid-body attitude dynamics, the proposed method is higher in accuracy, time performance, and structural conservativeness. In addition, the numerical method is applied to solve a constrained OAC problem on SO(3). The optimal control problem is transcribed into a nonlinear programming problem, in which the equivalent Lie algebra equation is being considered as the defect constraints instead of the configuration equation. The transcription method is coordinate-free and does not need chart switching or special handling of singularities. More importantly, with the numerical advantage of the geometric collocation method, the proposed OAC method may generate satisfying convergence rate.

show abstract

“…Construction of the defect constraints ( ). For , refer to (12) and 13 For configuration , refer to (26) and 27, and define the other part of the defect constraints ( = 1, . .…”

Section: Transcription Of the Oac On So(3) Into An Nlp Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

Xiang

2015

Mathematical Problems in Engineering

View full text Add to dashboard Cite

show abstract

“…It can be shown that the control synthesis and design for the distributed parameter systems have been broadly studied in the literature [6][7][8][9]. On the one hand, a very natural approach for control synthesis and design is to spatially discretize by approximating equations or solutions of the original PDEs using finite difference method, finite volume or Galerkin's methods [10][11][12]. The c 2015 Vietnam Academy of Science & Technology goal is to obtain a set of ordinary differential equations (ODEs) for which the nonlinear control strategies specially developed for the finite dimensional systems [13][14][15] can be applied.…”

Section: Introductionmentioning

confidence: 99%

“…Let us cite for example, [16] for predictive control of transport reaction processes, [17][18][19], with robust control of parabolic PDE systems using classical Lyapunov based approach and [20] for passivity based control of a reduced port controlled Hamiltonian model for the shallow water equations. On the other hand, spectral methods (such as proper orthogonal decomposition [21] or Hammerstein modeling approach [22], symmetry groups and invariance conditions [23,24], geometric pseudo-spectral method [11] and energy based discretization [12] provide powerful tools to handle the dynamics described by PDEs directly. All these allow reducing the dimensionality of the system before synthesizing the feedback controllers.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Control of Temperature Profile of Unstable Heat Conduction Systems: A Port Hamiltonian Approach

Phan¹,

Hoang

2016

JCC

View full text Add to dashboard Cite

This paper focuses on boundary control of distributed parameter systems (also called infinite dimensional systems). More precisely, a passivity based approach for the stabilization of temperature profile inside a well-insulated bar with heat conduction in a one-dimensional described by parabolic partial differential equations (PDEs) is developed. This approach is motivated by an appropriate model reduction schema using the finite difference approximation method. On this basis, it allows to discretize and then, write the original parabolic PDEs into a Port Hamiltonian (PH) representation. From this, the boundary control input is therefore synthesized using passive tools to stabilize the temperature at a desired reference profile asymptotically. The infinite dimensional nature of the original distributed parameter system in the PH framework is also discussed. Numerical simulations illustrate the application of the developments.

show abstract

“…There have been several publications on structure preserving spatial discretizations of lossless pH systems [8], [9], [10], [11]. The spatial discretization of a purely diffusive system has been discussed in [12].…”

mentioning

confidence: 99%

Structure preserving spatial discretization of 1D convection-diffusion port-Hamiltonian systems

Voß

Weiland

2011

IEEE Conference on Decision and Control and European Control Conference

View full text Add to dashboard Cite

Convection-diffusion is a physical phenomenon that appears in a multitude of dynamical systems, e.g. vibrating string with damping or chemical and thermal systems. This paper focuses on a structure preserving spatial discretization scheme of a general dynamical system with convection and diffusion in the port-Hamiltonian framework. The preservation of the port-Hamiltonian structure ensures that specific properties, such as passivity, of the infinite dimensional system are preserved. I. INTRODUCTIONSystems with spatial and temporal dynamics occur in many engineering applications where they are typically modeled by partial differential equations or as state space models over infinite dimensional state or phase spaces. The control of infinite dimensional systems is generally troublesome due to the fact that an infinite number of states needs to be controlled through a finite number of control variables. Currently, one can distinguish two common approaches for the control of infinite dimensional systems. The first one, late lumping [1], [2], amounts to designing an infinite dimensional control law which renders the infinite dimensional system stable while achieving some desired performance. The second approach, early lumping [3], amounts to first spatially discretizing the infinite dimensional system and, subsequently, designing a finite dimensional control law based on the dynamics of the spatially discretized system. All control design approaches based on finite element models belong to this category. This paper considers the first step in the early lumping control approach, namely the spatial discretization of a 1D convective and diffusive infinite dimensional port-Hamiltonian (pH) system [4]. There are compelling reasons to model convective and diffusive phenomena in infinite dimensional system in the port-Hamiltonian framework. The most important one is that pH systems have structural properties, such as passivity, that make them extremely suitable for control design, see [5], [6]. These properties are ensured by the special mathematical structure of a pH system and are typically lost by arbitrary spatial discretization schemes.The problem of structure preservation in discretization schemes of pH systems is of paramount importance. Indeed, from the general perspective of modeling, simulation, model approximation, and control system design, there is a need for specialized discretization schemes that preserve the pH structure in spatial discretizations. Classical spatial T. Voß and S. Weiland are with

show abstract

Geometric pseudospectral method for spatial integration of dynamical systems

Cited by 14 publications

References 21 publications

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

Geometric Collocation Method on SO(3) with Application to Optimal Attitude Control of a 3D Rotating Rigid Body

Nonlinear Control of Temperature Profile of Unstable Heat Conduction Systems: A Port Hamiltonian Approach

Structure preserving spatial discretization of 1D convection-diffusion port-Hamiltonian systems

Contact Info

Product

Resources

About