Sturm Sequence Research Articles

One-dimensional Fibonacci tilings

We use the -operator to construct a family of Fibonacci tilings of the unit interval consisting of short and long elementary intervals with the ratio of the lengths equal to the golden section . We prove that the tilings satisfy a recurrence relation similar to the relation for the Fibonacci numbers. The ends of the elementary intervals in the tilings form a sequence of points whose derivatives are sequences similar to the sequence . We compute the direct and inverse renormalizations for the sequences . We establish a connection between our tilings and the Sturm sequence, and give some applications of the tilings in the theory of numbers.

A minimum-time control strategy for torque tracking in permanent magnet AC motor drives

A minimum-time torque control strategy for permanent-magnet AC motor drives is presented. The proposed technique neither requires the solution of a HJB-type equation, which would be practically unfeasible, nor uses Pontryagin's maximum principle. Instead, the solution is obtained by an ad hoc procedure based on the computation of reachability and controllability sets. In principle, the optimal control strategy can be carried out by iteratively solving a fourth-degree polynomial equation. For its efficient implementation, an algorithm based on Sturm sequences is suggested. The sequence of online operations required by the algorithm for a given tolerance on the optimal time is illustrated. The method has been tested on a laboratory prototype. Experimental results show the effectiveness of the technique.

Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Multiple delay Rössler system—Bifurcation and chaos control

Multiple delayed Rössler system is analyzed from the view point of stability and chaos control. Usually these systems occur in active sensing problems where a signal is transmitted and received at a later time. Analytical and numerical results are obtained from the basic characteristic equation, using the Routh–Hurwitz criterion and Sturm sequences. The bifurcation pattern as the delay increases is displayed in detail, finally leading to chaos. In the second half we analyze the structure of the unstable periodic orbits and construct the controller which gets back the system to periodic state. Quantitative measure of the accuracy of the computation is obtained through the use of conditional Lyapunov exponent. At this point a Galerkin projection technique is used, which sets up a system of ODE in place of the delayed system, and makes the computation much simpler. Importance of this analysis is due to the role of the delay terms in the generation of the attractor, various bifurcation scenario, along with their control.

Modified Sturm Sequence Property for Damped Systems

This technical note presents a method of checking the number of complex eigenvalues in some interested regions or the multiplicity of some complex eigenvalues for nonproportionally damped system. A Schur-Cohn matrix is constructed from the coeffi- cients of the characteristic polynomial for the damped system, and LDL T factorized using some standard numerical algorithms. By observing signs of the diagonal elements of the above diagonal matrix D, we can determine the number of complex eigenvalues in some interested regions or the multiplicity of some complex eigenvalues, which is very similar to the well-known Sturm sequence property for undamped systems. To verify the applicability of the proposed method, two numerical examples are considered.

The predicates of the Apollonius diagram: Algorithmic analysis and implementation

We study the predicates involved in an efficient dynamic algorithm for computing the Apollonius diagram in the plane, also known as the additively weighted Voronoi diagram. We present a complete algorithmic analysis of these predicates, some of which are reduced to simpler and more easily computed primitives. This gives rise to an exact and efficient implementation of the algorithm, that handles all special cases. Among our tools we distinguish an inversion transformation and an infinitesimal perturbation for handling degeneracies. The implementation of the predicates requires certain algebraic operations. In studying the latter, we aim at minimizing the algebraic degree of the predicates and the number of arithmetic operations; this twofold optimization corresponds to reducing bit complexity. The proposed algorithms are based on static Sturm sequences. Multivariate resultants provide a deeper understanding of the predicates and are compared against our methods. We expect that our algebraic techniques are sufficiently powerful and general to be applied to a number of analogous geometric problems on curved objects. Their efficiency, and that of the overall implementation, are illustrated by a series of numerical experiments. Our approach can be immediately extended to the incremental construction of abstract Voronoi diagrams for various classes of objects.

A MINIMUM-TIME CONTROL STRATEGY FOR TORQUE TRACKING IN PERMANENT MAGNET AC MOTOR DRIVES

A minimum-time torque control strategy for permanent-magnet ac motor drives is presented. The proposed solution is obtained by an ad-hoc procedure based on the computation of reachability and controllability sets which in turn requires solving iteratively a fourth-degree polynomial equation. For its efficient implementation an algorithm based on Sturm sequences is proposed. The algorithm has been implemented on a laboratory prototype and the experimental results, showing the effectiveness of the proposed technique, are included in the paper.

Automatic Symmetrization and Energy Estimates Using Local Operators for Partial Differential Equations

We develop a method for automatically symmetrizing Petrowsky well-posed Cauchy problems for constant coefficient linear partial differential equations. The method is rooted in the Sturm sequence technique for establishing the location of the roots of a complex polynomial and can be automated using standard symbolic computation tools. In the special case of homogeneous strictly hyperbolic scalar equations, we show that the resulting estimates are strong enough to control all principal order derivatives and thus can be used in place of the Leray energies. We also illustrate the method by applying it to various problems of mixed type.

The Godunov-inverse iteration: A fast and accurate solution to the symmetric tridiagonal eigenvalue problem

We present a new algorithm for computing eigenvectors of real symmetric tridiagonal matrices based on Godunov's two-sided Sturm sequence method and inverse iteration, which we call the Godunov-inverse iteration. We use eigenvector approximations computed recursively from two-sided Sturm sequences as starting vectors in inverse iteration, replacing any nonnumeric elements of these approximate eigenvectors with uniform random numbers. We use the left-hand bounds of the smallest machine presentable eigenvalue intervals found by the bisection method as inverse iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one or two iterations, producing accurate residuals.

Applications of Sturm sequences to bifurcation analysis of delay differential equation models

This paper formalizes a method used by several others in the analysis of biological models involving delay differential equations. In such a model, the characteristic equation about a steady state is transcendental. This paper shows that the analysis of the bifurcation due to the introduction of the delay term can be reduced to finding whether a related polynomial equation has simple positive real roots. After this result has been established, we utilize Sturm sequences to determine whether a polynomial equation has positive real roots. This work has extended the stability results found in previous papers and provides a novel theorem about stability switches for low degree characteristic equations.

On the computation of an arrangement of quadrics in 3D

In this paper, we study a sweeping algorithm for computing the arrangement of a set of quadrics in R 3 . We define a “trapezoidal” decomposition in the sweeping plane, and we study the evolution of this subdivision during the sweep. A key point of this algorithm is the manipulation of algebraic numbers. In this perspective, we put a large emphasis on the use of algebraic tools, needed to compute the arrangement, including Sturm sequences and Rational Univariate Representation of the roots of a multivariate polynomial system.

The Use of Eigenvalues for Finding Equilibrium Probabilities of Certain Markovian Two-Dimensional Queueing Problems

A number of papers have appeared recently using eigenvalues for solving steady-state queueing problems. In this paper, we analyze Markovian systems with two state variables, the level X1 and the phase X2, X1 ≤ 0, 0 ≤ X2 ≤ N. Except for some boundary levels, the rates of the events are independent of the level, and no event can change X1, X2, or X1 + X2 by more than 1. In this case, the eigenvectors are essentially Sturm sequences, which implies that all eigenvalues are real. The properties of the Sturm sequences allow us to design an extension of the binary search to find all eigenvalues. As it turns out, once the interval containing an eigenvalue is narrowed down sufficiently, it is preferable to use Newton's method. A computational-complexity study indicates that the resulting algorithm should be signicantly faster than matrix-iterative methods. Two numerical examples are discussed involving servers with breakdowns, and in both cases, our method yields highly accurate results. Tentative reasons why this is to be expected are provided.

Exact computation of the medial axis of a polyhedron

We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding neighboring junctions along the seam curves. To speed up the computation, we have designed specialized algorithms for fast computation with algebraic curves and surfaces. These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

Eigenvalue-counting methods for non-proportionally damped systems

This paper presents numerical methods of counting the number of eigenvalues for non-proportionally damped system in some interested regions on the complex plane. Most of the eigenvalue analysis methods for proportionally damped systems use the well-known Sturm sequence property to check the missed eigenvalues when only a set of the lowest modes is used. However, in the case of the non-proportionally damped systems such as the soil–structure interaction system, the structural control system and composite structures, no counterpart of the Sturm sequence property for undamped systems has been established yet. In this study, a numerical method based on argument principle is explained with emphasis on the discretization of the contour and a new method based on Gleyse’s theorem is proposed. To verify the applicability of the methods, two numerical examples are considered.

An iterative system equivalent reduction expansion process for extraction of high frequency response from reduced order finite element model

An iterative system equivalent reduction expansion process (SEREP) is proposed for extraction of high frequency response from a reduced-order model (ROM) under frequency band-limited excitation. Various model order reduction methods are discussed. To alleviate the present drawback of the SEREP, which requires computation of full system modal matrix, an iterative method based on Sturm sequence check is proposed. The method uses eigenvalue separation properties on the excitation frequency band to identify the optimal number of the eigenpairs required to capture the accurate response. The basic steps for numerical implementation are given. Numerical results are presented to validate the predicted response. Comparison is also made to explain the reasons for the instability of the ROM based on dynamic condensation to capture the high frequency dynamics. Case studies are carried out to demonstrate the effect of the chosen frequency band for response extraction. Also, it is shown that accurate results are obtained while using the proposed method when the chosen frequency band encompass the excitation frequency band.

A test for robust Hurwitz stability of convex combinations of complex polynomials

In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1− λ) p 0( s)+ λp 1( s), where λ∈[0,1] and p 0( s) and p 1( s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2 n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.

A TABULAR ALGORITHM FOR TESTING THE ROBUST HURWITZ STABILITY OF CONVEX COMBINATIONS OF COMPLEX POLYNOMIALS

In this paper we present a tabular algorithm for testing the Hurwitz property of a segment of complex polynomials. The algorithm consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real polynomials in the interval (0, 1). The presented algorithm is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.

Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications

Many queueing problems lead to tridiagonal lambda-matrices containing polynomials that have, except for the diagonal, non-negative coefficients. This paper deals with the question, addressed in the literature only for special cases, whether the eigenvalues corresponding to such lambda-matrices are real. In most cases, they are, as the theorems of this paper show, but sometimes, complex eigenvalues occur. Our results are derived by using Sturm sequences. In addition to simplifying the proofs of our theorems, Sturm sequences are also valuable to verify whether or not a given interval contains eigenvalues.

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagrange's formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.

Piecewise algebraic curve

A piecewise algebraic curve is defined by a bivariate spline function. Using the techniques of the B- net form of bivariate splines function, discriminant sequence of polynomial (cf. Yang Lu et al. (Sci. China Ser. E 39(6) (1996) 628) and Yang Lu et al. (Nonlinear Algebraic Equation System and Automated Theorem Proving, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1996)) and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence, we determine the number of real intersection points of two piecewise algebraic curves whose common points are finite. A lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field.