Coefficients Of Characteristic Polynomial Research Articles

A State Space Method for Modal Identification of Mechanical Systems from Time Domain Responses

A new state space method is presented for modal identification of a mechanical system from its time domain impulse or initial condition responses. A key step in this method is the identification of the characteristic polynomial coefficients of an adjoint system. Once these coefficients are determined, a canonical state space realization of the adjoint system and the system′s modal parameters are formulated straightforwardly. This method is conceptually and mathematically simple and is easy to be implemented. Detailed mathematical treatments are demonstrated and numerical examples are provided to illustrate the use and effectiveness of the method.

계수의 특성비에 대한 선형계의 파라미터적 감도해석(I): 일반적인 경우

The characteristic ratio assignment (CRA) method〔1〕 is new polynomial approach which allows to directly address the transient responses such as overshoot and speed of response time in time domain specifications. The method is based on the relationships between time response and characteristic ratios(<TEX>$\alpha_i$</TEX> ) and generalized time constant (T), which are defined in terms of coefficients of characteristic polynomial. However, even though the CRA can apply to developing a linear controller that meets good transient responses, there are still some fundamental questions to be explored. For the purpose of this, we have analyzed several sensitivities of a linear system with respect to the changes of coefficients itself and <TEX>$\alpha_i$</TEX> of denominator polynomial. They are (i) the unnormalized root sensitivity : to determine how the poles change as <TEX>$\alpha_i$</TEX> changes, and (ii) the function sensitivity to determine the sensitivity of step response to the change of o, and to analyze the sensitivity of frequency response as o, changes. As an other important result, it is shown that, under any fixed T and coefficient of the lowest order of s in denominator, the step response is dominantly affected merely by <TEX>$\alpha_1, alpha_2 and alpha_3$</TEX> regardless of the order of denominator higher than 4. This means that the rest of the<TEX>$\alpha_i$</TEX> s have little effect on the step response. These results provide some useful insight and background theory when we select <TEX>$\alpha_i$</TEX> and T to compose a reference model, and in particular when we design a low order controllers such as PID controller.

Directional stability radius: a stability analysis tool for uncertain polynomial systems

Coefficients of characteristic polynomials for stable parametrically uncertain systems are allowed to perturb to some extent for stability. Stability radius is a useful tool to assess the allowance of the stability for the systems. To enhance its usefulness, we modify stability radius so that it takes into account of given restricted perturbations, which we call directional stability radius. For an application, we show shifted-Hurwitz stability conditions and a stability analysis method for interval polynomial systems using the directional stability radii.

Max algebra and the linear assignment problem

Max-algebra, where the classical arithmetic operations of addition and multiplication are replaced by a⊕b:=max(a, b) and a⊗b:=a+b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems and the eigenvalue problem. The max-algebraic permanent of a matrix A corresponds to the maximum value of the classical linear assignment problem with cost matrix A. The analogue of van der Waerden's conjecture in max-algebra is proved. Moreover the role of the linear assignment problem in max-algebra is elaborated, in particular with respect to the uniqueness of solutions of linear equation systems, regularity of matrices and the minimal-dimensional realisation of discrete event systems. Further, the eigenvalue problem in max-algebra is discussed. It is intimately related to the best principal submatrix problem which is finally investigated: Given an integer k, 1≤k≤n, find a (k×k) principal submatrix of the given (n×n) matrix which yields among all principal submatrices of the same size the maximum (minimum) value for an assignment. For k=1,2,...,n, the maximum assignment problem values of the principal (k×k) submatrices are the coefficients of the max-algebraic characteristic polynomial of the matrix for A. This problem can be used to model job rotations.

On the coefficients of the Laplacian characteristic polynomial of trees

Let the Laplacian characteristic polynomial of an n-vertex tree T be of the form ?(T, ?) = ?nk=0(-l)n-k ck(T)?k . Then, as well known, C0(T) = 0 and c1 (T) = n. If T differs from the star (Sn) and the path (Pn), which requires n ? 5, then c2(Sn} &lt; C2(T) &lt; C2(Pn) and c3(Sn) &lt; c3(T) &lt; c3(Pn). If n = 4, then c3(Sn) = c3(Pn).

Four Coefficients of Characteristic Polynomials of Coxeter Transformations

Four Coefficients of Characteristic Polynomials of Coxeter Transformations

Using interval arithmetic for robust state feedback design

A problem of a modal P-regulator synthesis for a linear multivariable dynamical system with uncertain (interval) parameters in state space is considered. The designed regulator has to place all coefficients of the system characteristic polynomial within assigned intervals. We have developed the approach proposed earlier in Dugarova and Smagina (Avtomat. i Telemech. 11 (1990) 176) and proved a direct correlation between interval system controllability and existence of robust modal P-regulator.

Non-Alternant Hydrocarbon Polycycles: Signs of Unusual Reactivity

A connection is developed between typical characteristic polynomial coefficient sign patterns and Δ, the difference between the numbers of π-bonding and π-antibonding molecular orbitals for hydrocarbons. A link is revealed between Δ and dominant Sachs' subgraphs for the non-zero characteristic polynomial coefficient that is closest to the end of the polynomial. Through the intermediacy of the appropriate Sachs' subgraphs, the connection between a given Lewis structure and Δ is clarified. Exploiting the Sachs' subgraphs, one can redesign non-alternant hydrocarbon structures to alter π-system stability and, therefore, the kinetic stability of interesting molecules. A PM3 test of this methodology is provided in the accompanying paper.

On the Coefficients of the Max-Algebraic Characteristic Polynomial and Equation

No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices is NP-complete. (2) The task of finding the lowest order finite term for a {0, -∞} matrix can be converted to the assignment problem. (3) The task of finding the max-algebraic characteristic equation of a {0, -∞} matrix can be converted to that of finding the conventional characteristic equation for a {0, 1} matrix and thus it is solvable in polynomial time.

Introduction to Coefficient Diagram Method

We introduce a linear controller design method, called the Coefficient Diagram Method (CDM), which is a kind of model matching approach. In this paper, we have especially focused on the main role and characteristics of specific design parameters γi, and τ, which are defined by only coefficients of characteristic polynomial. The relationships between these parameters and the system response as well as the stability are investigated systematically. Based on the relations, we have proposed a prototype for target transfer functions of type I by which one is able to fit easily the desired transient response. It is ascertained that the CDM has some strong points in cases where fixed lower-order controllers wish to be designed.

A NEW PRONY-BASED CIRCULAR-HYPERBOLIC DECOMPOSITION

We introduce a new closed-form decomposition technique for estimating the model parameters of an evenly sampled signal known to be composed of circular and hyperbolic sine and cosine functions in the presence of Gaussian white noise. The techniqe is closely related to Prony's method and hereditary algorithms that fit complex exponential functions to evenly sampled data. The circular and hyperbolic sine and cosine functions are obtained by adding constraints that limit the form of the characteristic polynomial coefficients. It avoids the leakage effects associated with the discrete fourier transform (DFT) for circular sine and cosine functions. When the signal contains frequency components that are not rational multiples of each other, the proposed decomposition yields amplitude and phase parameters that are more accurate than those obtained with the DFT in moderate levels of noise. First, we review Prony's method and one hereditary algorithm (the complex exponential algorithm). Then, we detail three implementation procedures of the new technique. The first is a two-stage least-squares approach. The second utilises a novel concept of noise reduction which is attributed to Pisarenko. The last provides additional means of noise reduction through a covariance formulation that avoids zero-lag terms. Experimental and numerical examples of the application of the circular-hyperbolic decomposition (CHD) are given.

Some Preliminary Results on Robust Reflection Coefficients Placement

A new version of pole placement controller design, called robust reflection coefficients placement, is proposed for discrete-time systems. Instead of a single stable point a stable simplex must be preselected in the closed loop characteristic polynomials coefficients space. A constructive procedure for generating simplexis inside the "nice stability region" is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedure is quite straightforward: an appropriate stable point has to be chosen and then the edges of the stable simplex will be generated by a linear Schur invariant transformation. The procedure of robust controller design makes use of a stability measure defined as the minimal distance between a preselected stable simplex and vertices of an uncertain interval plant.

Characteristic polynomials of linear polyacenes and their subspectrality

Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal’s triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP’s. Thus it has been shown that the entire eigenspectrum of ann-ring LP is included in that of (2n + 1)-ring LP. Correspondence between the eigenspectra of linear chains and LP’s is brought out by a recently developed vertex-alternation and squaring algorithm.

On the robust control via reflection coefficients

Abstract A robust version of the modal controller design for discrete-time systems is introduced. Instead of a single stable point a stable simplex must be preselected in the closed loop characteristic polynomials coefficients space. A constructive procedure for generating simplexis inside the “nice stability region” is given starting from the unit hypercube of reflection coefficients of monic polynomials. This procedure is quite straightforward because, first, n edges are generated by a linear Schur invariant transformation and, second, the most critical edge is determined by the difference of reflection coefficients numbers. The procedure of robust controller design makes use of a stability measure defined as the minimal distance between a preselected stable simplex and vertices of an uncertain polytopic plant.

A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs

A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs

A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs

Characteristic polynomial coefficients of three classes of graph, namely Ln + n(p), Cn + n(p) and K 1,n-1 + n(p), which are known to have reciprocal pairs of eigenvalues, have been shown to be generated by simple manipulation of the Pascal's triangle.

Strong stabilization over polytopes

The problem considered consists of finding a stable controller assigning a closed-loop characteristic polynomial over a stability domain specified as a polytopic region of the space of characteristic polynomial coefficients. The controller is required to be stable with respect to the same polytopic stability region specified for the closed loop system.

Automorphism and similarity groups of forms determined by the characteristic polynomial

The paper is concerned with forms of degree higher than two. M.Oryan and D.B.Shapiro [OS] proved that if A is a central simple algebra over a field K, ’tr‘ denotes its reduced trace and d ≥ 3, then every similarity of the trace form tr(Xd ) of degree d on A is standard. Now let A be a commutative and étale K— algebra of degree n ≥ 3. The coefficients of the generic characteristic polynomial χ A determine the norm form N A of degree n, the trace form of degree d, where d ≥ 3, and the intermediate forms M A,t of degree t, 2 ≤ t ≤ n - 1. We prove that all similarities of the intermediate forms are standard. The main result of the paper is the complete description of the similarity and the automorphism groups of the norm, trace and intermediate forms.

A century of phugoid approximations

Abstract It is stated in most flight dynamics texts that the phugoid approximations provide poor estimates while the short period approximations are accurate. The survey in this paper reveals that there are at least five approximations to the phugoid. The extent of departure of each of these from the exact value is determined for a fairly extensive database representing various aircraft in different flight conditions. It is found that most of them are inadequate in predicting the phugoid characteristics accurately. Nonetheless, two approximations to the phugoid frequency that seem to have remained unnoticed are seen to be exemplary. On the other hand, no worthy approximation exists for the phugoid damping. With this background, a fresh approximation for the phugoid mode is put forth herein. It is derived by equating the coefficients of the product of the short period equation (which has been shown to be very accurate) and the phugoid equation (as yet unknown) to the coefficients of the fourth-order characteristic polynomial. The new approximation is shown to be accurate.

Criterion of high-codimensional bifurcations with several pairs of purely imaginary eigenvalues

This paper presents a criterion for high-codimensional bifurcations with several pairs of purely imaginary eigenvalues in terms of the properties of coefficients of characteristic polynomials instead of those of eigenvalues. It uses the Routh-Hurwitz determinants and is convenient in the symbolic analysis of complicated systems.