Intersection Problem Research Articles

Closed‐loop identification of the time‐varying dynamics of variable‐speed wind turbines

AbstractThe trend with offshore wind turbines is to increase the rotor diameter as much as possible to decrease the costs per kWh. The increasing dimensions have led to the relative increase in the loads on the wind turbine structure. Because of the increasing rotor size and the spatial load variations along the blade, it is necessary to react to turbulence in a more detailed way: each blade separately and at several separate radial distances. This combined with the strong nonlinear behavior of wind turbines motivates the need for accurate linear parameter‐varying (LPV) models for which advanced control synthesis techniques exist within the robust control framework. In this paper we present a closed‐loop LPV identification algorithm that uses dedicated scheduling sequences to identify the rotational dynamics of a wind turbine. We assume that the system undergoes the same time variation several times, which makes it possible to use time‐invariant identification methods as the input and the output data are chosen from the same point in the variation of the system. We use time‐invariant techniques to identify a number of extended observability matrices and state sequences that are inherent to subspace identification identified in a different state basis. We show that by formulating an intersection problem all states can be reconstructed in a general state basis from which the system matrices can be estimated. The novel algorithm is applied on a wind turbine model operating in closed loop. Copyright © 2008 John Wiley & Sons, Ltd.

Classification and resolution of critical cases in Grandine and Klein's topology determination using a perturbation method

We classify and resolve all critical cases in the topology determination method proposed in [Grandine, T.A., Klein IV, F.W., 1997. A new approach to the surface intersection problem. Computer Aided Geometric Design 14 (2), 111–134]. Their algorithm for finding the intersection of two parametric surfaces has two steps: determining the topology of the intersection curves and using that information to find the curves themselves. The essence of the first step is to decide whether the boundary points and the turning points are at the start or the end of a contour. However, there are several cases in which the decision criteria proposed by Grandine and Klein are not applicable. We classify all these cases, which include the tangential intersection of two surfaces, the tangential intersection of the contour with the boundary of the domain of a surface, and the vanishing of ( u ″ ( τ ) , v ″ ( τ ) ) in the interior of the ( u , v ) domain. Then we resolve all these cases using a perturbation method, which is based on the fact that transversality is a stable and generic property of an intersection. We also present a classical method for resolution, which uses higher-order derivatives.

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.

On the Intersection of $ {\BBZ }_{2} {\BBZ }_{4}$-Additive Perfect Codes

The intersection problem for Z2Z4-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z2Z4-additive codes C1 and C2 of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these Z2Z4-additive codes C1 and C2, the abelian group structure of the intersection codes C1 cap C2 is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given.

Alternation and redundancy analysis of the intersection problem

The intersection of sorted arrays problem has applications in search engines such as Google. Previous work has proposed and compared deterministic algorithms for this problem, in an adaptive analysis based on the encoding size of a certificate of the result (cost analysis). We define the alternation analysis , based on the nondeterministic complexity of an instance. In this analysis we prove that there is a deterministic algorithm asymptotically performing as well as any randomized algorithm in the comparison model. We define the redundancy analysis , based on a measure of the internal redundancy of the instance. In this analysis we prove that any algorithm optimal in the redundancy analysis is optimal in the alternation analysis, but that there is a randomized algorithm which performs strictly better than any deterministic algorithm in the comparison model. Finally, we describe how these results can be extended beyond the comparison model.

Alternating Projections on Manifolds

We prove that if two smooth manifolds intersect transversally, then the method of alternating projections converges locally at a linear rate. We bound the speed of convergence in terms of the angle between the manifolds, which in turn we relate to the modulus of metric regularity for the intersection problem, a natural measure of conditioning. We discuss a variety of problem classes where the projections are computationally tractable, and we illustrate the method numerically on a problem of finding a low-rank solution of a matrix equation.

Closed-loop LPV identification of the time-varying dynamics of a variable-speed wind turbine

In this paper we present a closed-loop LPV identification algorithm that uses a periodic scheduling sequence to identify the rotational dynamics of a wind turbine. In the algorithm we assume that the system undergoes the same time variation several times, which make it possible to use time-invariant identification methods since the input and output data are chosen from the same point in the variation of the system. We use closed-loop time-invariant techniques to identify a number of extended observability matrices and state sequences that are, inherent to subspace identification, identified in a different state basis. We show that by formulating an intersection problem all the states can be reconstructed in a general state basis from which the system matrices can be estimated. The novel algorithm is applied on a wind turbine model operating in closed loop.

Geodetic intersection on the ellipsoid

Through each of two known points on the ellipsoid a geodesic is passing in a known azimuth. We solve the problem of intersection of the two geodesics. The solution for the latitude is obtained as a closed formula for the sphere plus a small correction, of the order of the eccentricity of the ellipsoid, which is determined by numerical integration. The solution is iterative. Once the latitude is obtained, the longitude is determined without iteration.

Scalarizations for adaptively solving multi-objective optimization problems

In this paper several parameter dependent scalarization approaches for solving nonlinear multi-objective optimization problems are discussed. It is shown that they can be considered as special cases of a scalarization problem by Pascoletti and Serafini (or a modification of this problem). Based on these connections theoretical results as well as a new algorithm for adaptively controlling the choice of the parameters for generating almost equidistant approximations of the efficient set, lately developed for the Pascoletti-Serafini scalarization, can be applied to these problems. For instance for such well-known scalarizations as the ε-constraint or the normal boundary intersection problem algorithms for adaptively generating high quality approximations are derived.

Mixing symbolic and numerical approaches for the surface‐to‐surface intersection problem

AbstractThis paper shows how the efficiency of the current methodologies applied to the surface‐to‐surface intersection problem can be improved by combining an algebraic/symbolic framework with efficient and robust numerical techniques. The algebraic/symbolic framework is used to translate the computation of resultants, subresultants, discriminants, etc. to one or several generalized eigenvalue problems and SVD computations. The framework requires only the values of the involved polynomials at some set of points, and it will guide the numerical computations, providing thus a certificate of the topological correctness of the output. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied.

Intersection of a ruled surface with a free-form surface

This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics.

Analysis of validated error bounds of surface-to-surface intersection

In this paper, we address the problem of computing the error bounds of surface-to-surface intersection and propose a novel procedure to reduce them. We formulate the surface-to-surface intersection problem as solving a system of ordinary differential equations and using the validated ODE solver we compute the validated a priori enclosures of an intersection curve, in which the existence and the uniqueness of a solution are guaranteed. Then we use straight line enclosures to reduce the size of the a priori enclosures. These reduced enclosures are again enclosed by bounding curves, which can be used as the reduced error bounds of the intersection curve. We demonstrate our method with tangential and transversal intersections.

A two-steps algorithm for approximating real roots of a polynomial in Bernstein basis

The surface/curve intersection problem, through the resultants process results in a high degree ( n ≥ 100 ) polynomial equation on [ 0 , 1 ] in the Bernstein basis. The knowledge of multiplicities of the roots is critical for the topological coherence of the results. In this aim, we propose an original two-steps algorithm based on successive differentiations which separates any root (even multiple) and guarantees that the assumptions of Newton global convergence theorem are satisfied. The complexity is ϑ ( n 4 ) but the algorithm can easily be parallelized. Experimental results show its efficiency when facing ill-conditioned polynomials.

The intersection problem for [formula omitted

For every v ≡ 5 ( mod 6 ) there exists a pairwise balanced design (PBD) of order v with exactly one block of size 5 and the rest of size 3. We will refer to such a PBD as a PBD ( 5 * , 3 ) . A flower in a PBD ( 5 * , 3 ) is the set of all blocks containing a given point. If ( S , B ) is a PBD ( 5 * , 3 ) and F is a flower, we will write F * to indicate that F contains the block of size 5. The intersection problem for PBD ( 5 * , 3 ) s is the determination of all pairs ( v , k ) such that there exists a pair of PBD ( 5 * , 3 ) s ( S , B 1 ) and ( S , B 2 ) of order v containing the same block b of size 5 such that | ( B 1 ⧹ b ) ∩ ( B 2 ⧹ b ) | = k . The flower intersection problem for PBD ( 5 * , 3 ) s is the determination of all pairs ( v , k ) such that there exists a pair of PBD ( 5 * , 3 ) s ( S , B 1 ) and ( S , B 2 ) of order v having a common flower F * such that | ( B 1 ⧹ F * ) ∩ ( B 2 ⧹ F * ) | = k . In this paper we give a complete solution of both problems.

Fully adaptive finite element based tomography using tetrahedral dual-meshing for fluorescence enhanced optical imaging in tissue

We have developed fluorescence enhanced optical tomography based upon fully adaptive finite element method (FEM) using tetrahedral dual-meshing wherein one of the two meshes discretizes the forward variables and the other discretizes the unknown parameters to be estimated. We used the 8-subtetrahedron subdivision scheme to create the nested dualmesh in which each are independently refined. However, two tetrahedrons from the two different meshes pose an intersection problem that needs to be resolved in order to find the common regions that the forward variables (the fluorescent diffuse photon fluence fields) and the parameter estimates (the fluorescent absorption coefficients) can be mutually assigned. Using an efficient intersection algorithm in the nested tetrahedral environments previously developed by the authors, we demonstrate fully adaptive tomography using a posteriori error estimates. Performing the iterative reconstructions using the simulated boundary measurement data, we demonstrate that small fluorescent targets embedded in the breast simulating phantom in point illumination/detection geometry can be resolved at reasonable computational cost.

Resultant Optimization of the Three-dimensional Intersection Problem

Resultant approach is here employed to optimize the dimensionless space angles to solve in a closed form the over-determined three-dimensional intersection problem. The advantages of the resultant optimization approach are the non-requirement of the approximate initial starting values, non iterative and does not rely on linearization during its operation, save for the nonlinear variance-covariance/error propagation to generate the weight matrix. Resultant method, a branch of abstract algebra, is employed to compute the combinatorial scatters, which are then optimized to offer a closed form solution. Using the test network Stuttgart Central as an example, it is demonstrated that the resultant optimization approach can be applied as an alternative approach to conventional methods such as least squares for point positioning within the over-determined intersection framework, especially when the approximate starting values for linearization and iterative approaches are not known as may happen in Photogrammetry, Machine Vision or in Robotics.

A Novel Approach to Fast Eliminating the Characteristic Curves Crossing Problem of Relay Coordination in Radial Subtransmission Networks

A novel approach is presented to solve the coordination problem of overcurrent relays in radial subtransmission networks resulting from the intersection of characteristic curves. Based on the ANSI/IEEE standard characteristic curve equation of digital overcurrent relays, a slope adjustment equation for a selected point on the characteristic curve is derived and then used as the core of a dedicated software program developed to compute the corresponding relay setting with different tap value in order that, passed through the selected point, another curve with steeper slope is chosen and thus able to escape from the crossed position. Finally, with the assistance of the graphical capability of commercial package OneLiner in drawing the upstream and downstream relay coordination curves after adjustment, the curves intersection problem in protection coordination can be readily eliminated. In addition, an actual case of a radial subtransmission network with six buses in Taiwan Power Company is simulated to validate the feasibility of the approach.

On the Intersection Problem for Steiner Triple Systems of Different Orders

A Steiner triple system of order v, or STS(v), is a pair (V, **) with V a set of v points and ** a set of 3-subsets of V called blocks or triples, such that every pair of distinct elements of V occurs in exactly one triple. The intersection problem for STS is to determine the possible numbers of blocks common to two Steiner triple systems STS(u), (U, **), and STS(v), (V, **), with U⊆V. The case where U=V was solved by Lindner and Rosa in 1975. Here, we let U⊂V and completely solve this question for v−u=2,4 and for v≥2u−3.

Space-efficient geometric divide-and-conquer algorithms

We develop a number of space-efficient tools including an approach to simulate divide-and-conquer space-efficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multi-dimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divide-and-conquer. Specifically, we present a deterministic algorithm running in O ( n log n ) time using O ( 1 ) extra memory given inputs of size n for the closest pair problem and a randomized solution running in O ( n log n ) expected time and using O ( 1 ) extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O ( n log n + k ) time using O ( 1 ) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections.