2,618 research outputs found
On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems
A recently-derived alternative method for computing the relative gain array
(RGA) for singular and/or non-square systems has been proposed that provably
guarantees unit invariance. This property is not offered by the conventional
method that uses the Moore-Penrose (MP) pseudoinverse. In this paper we note
that the absence of the scale-invariance property by the conventional MP-RGA
does not necessarily imply a practical disadvantage in real-world applications.
In other words, while it is true that performance of a controller should not
depend on the choice of units on its input and output variables, this does not
{\em necessarily} imply that the resulting MP-RGA measures of component
interaction lead to different controller-design input-output pairings. In this
paper we consider the application of the MP-RGA to a realistic system (a Sakai
fractional distillation system) to assess whether or not the choice of unit,
which in this case relates to temperature, affects the choice of input-output
pairings determined by the resulting RGA matrix. Our results show that it does,
thus confirming that unit-sensitivity of the MP-RGA undermines its rigorous use
for MIMO controller design
Matrix-free polynomial preconditioning of saddle point systems using the hyper-power method
This study explores the integration of the hyper-power sequence, a method
commonly employed for approximating the Moore-Penrose inverse, to enhance the
effectiveness of an existing preconditioner. The approach is closely related to
polynomial preconditioning based on Neumann series. We commence with a
state-of-the-art matrix-free preconditioner designed for the saddle point
system derived from isogeometric structure-preserving discretization of the
Stokes equations. Our results demonstrate that incorporating multiple
iterations of the hyper-power method enhances the effectiveness of the
preconditioner, leading to a substantial reduction in both iteration counts and
overall solution time for simulating Stokes flow within a 3D lid-driven cavity.
Through a comprehensive analysis, we assess the stability, accuracy, and
numerical cost associated with the proposed scheme
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Linear Approximation to Optimal Control Allocation for Rocket Nozzles with Elliptical Constraints
In this paper we present a straightforward technique for assessing and realizing the maximum control moment effectiveness for a launch vehicle with multiple constrained rocket nozzles, where elliptical deflection limits in gimbal axes are expressed as an ensemble of independent quadratic constraints. A direct method of determining an approximating ellipsoid that inscribes the set of attainable angular accelerations is derived. In the case of a parameterized linear generalized inverse, the geometry of the attainable set is computationally expensive to obtain but can be approximated to a high degree of accuracy with the proposed method. A linear inverse can then be optimized to maximize the volume of the true attainable set by maximizing the volume of the approximating ellipsoid. The use of a linear inverse does not preclude the use of linear methods for stability analysis and control design, preferred in practice for assessing the stability characteristics of the inertial and servoelastic coupling appearing in large boosters. The present techniques are demonstrated via application to the control allocation scheme for a concept heavy-lift launch vehicle
A simple preconditioner for a discontinuous Galerkin method for the Stokes problem
In this paper we construct Discontinuous Galerkin approximations of the
Stokes problem where the velocity field is H(div)-conforming. This implies that
the velocity solution is divergence-free in the whole domain. This property can
be exploited to design a simple and effective preconditioner for the final
linear system.Comment: 27 pages, 4 figure
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