Coefficients Of Characteristic Polynomial Research Articles

Robust Controller Design Ensuring the Desired Aperiodic Stability Degree of a Control System with Affine Uncertainty

This paper considers a system whose characteristic polynomial coefficients are linear combinations of the interval parameters of a plant forming a parametric polytope. A linear robust controller is parametrically designed to place a dominant pole of the system within the desired interval of the negative real semi-axis and ensure an aperiodic transient in the system. The parametric design procedure involves a low-order controller with dependent and free parameters: the former serve to place the dominant pole within the desired interval on the complex plane whereas the latter to shift the other poles to some localization regions beyond a given bound (to the left of the dominant pole to satisfy the pole dominance principle). To evaluate the dependent parameters of the controller, the originals of the interval bounds of the dominant pole are determined for the plant’s parametric polytope based on a corresponding theorem (see below). The free parameters of the controller are chosen using the robust vertex or edge D-partition method, depending on the boundary edge branches of the localization regions of the free poles. A numerical example of the parametric design procedure is provided: a PID controller is built to ensure an acceptable aperiodic transient time in a load-lifting mechanism with interval values of cable length and load weight.

Fitting of Coupled Potential Energy Surfaces via Discovery of Companion Matrices by Machine Intelligence.

Fitting coupled potential energy surfaces is a critical step in simulating electronically nonadiabatic chemical reactions and energy transfer processes. Analytic representation of coupled potential energy surfaces enables one to perform detailed dynamics calculations. Traditionally, fitting is performed in a diabatic representation to avoid fitting the cuspidal ridges of coupled adiabatic potential energy surfaces at conical intersection seams. In this work, we provide an alternative approach by carrying out fitting in the adiabatic representation using a modified version of the Frobenius companion matrices, whose usage was first proposed by Opalka and Domcke. Their work involved minimizing the errors in fits of the characteristic polynomial coefficients (CPCs) and diagonalizing the resulting companion matrix, whose eigenvalues are adiabatic potential energies. We show, however, that this may lead to complex eigenvalues and spurious discontinuities. To alleviate this problem, we provide a new procedure for the automatic discovery of CPCs and the diagonalization of a companion matrix by using a special neural network architecture. The method effectively allows analytic representation of global coupled adiabatic potential energy surfaces and their gradients with only adiabatic energy input and without experience-based selection of a diabatization scheme. We demonstrate that the new procedure, called the companion matrix neural network (CMNN), is successful by showing applications to LiH, H3, phenol, and thiophenol.

An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements

Consider a finite collection of affine hyperplanes in mathbb R^d. The hyperplanes dissect mathbb R^d into finitely many polyhedral chambers. For a point xin mathbb R^d and a chamber P the metric projection of x onto P is the unique point yin P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by text {dim}(x,P). We prove that for every given kin {0,ldots , d}, the number of chambers P for which text {dim}(x,P) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010)].

Extending a conjecture of Graham and Lovász on the distance characteristic polynomial

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊n2⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.

On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg+(D)−A(D) and Q(D)=Deg+(D)+A(D), where A(D) represents the adjacency matrix and Deg+(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations.

Coefficients of Randic and Sombor characteristic polynomials of some graph types

Let GG be a graph. The energy of GG is defined as the summation of absolute values of the eigenvalues of the adjacency matrix of GG. It is possible to study several types of graph energy originating from defining various adjacency matrices defined by correspondingly different types of graph invariants. The first step is computing the characteristic polynomial of the defined adjacency matrix of GG for obtaining the corresponding energy of GG. In this paper, formulae for the coefficients of the characteristic polynomials of both the Randic and the Sombor adjacency matrices of path graph PnPn , cycle graph CnCn are presented. Moreover, we obtain the five coefficients of the characteristic polynomials of both Randic and Sombor adjacency matrices of a special type of 3−regular graph RnRn.

Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$ . We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$ , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$ , and one of the formulas in the case $$n=5$$ . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$ . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

On the coefficients of skew Laplacian characteristic polynomial of digraphs

Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see text] edges. Let [Formula: see text] be the digraph obtained by orienting the edges of [Formula: see text] arbitrarily. The digraph [Formula: see text] is called an orientation of [Formula: see text] or oriented graph corresponding to [Formula: see text]. The skew Laplacian matrix of the digraph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] is the skew matrix and [Formula: see text] is the diagonal matrix with [Formula: see text]th diagonal entry [Formula: see text]. In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of [Formula: see text]. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determinations of the directed path and directed cycle.

Hybrid dual-loop control method for dead-time second-order unstable inverse response plants with a case study on CSTR

Abstract Unstable processes are hard to stabilize as they contain one or more positive poles which result in unbounded dynamic behavior. The occurrence of the delay and positive zeros often creates more difficulty in controlling such unstable plants. Most control strategies meant for first-order unstable processes fail to stabilize and control the higher-order unstable processes. Hence, a new dual-loop hybrid control structure is suggested for dead-time unbounded second-order processes with positive/negative zeros. Inner-loop has a stabilizing PID controller. The PID controller parameters are derived by comparing the numerator and characteristic polynomial coefficients in the transfer function for internal-loop servo action. A fractional-order internal model controller (FOIMC) is used in the external-loop. Methods for the selection of outer-loop tuning parameter and fractional-order are also discussed. By comparing the simulation results with a contemporary single-loop scheme, the usefulness of the suggested scheme is proved. The suggested scheme is capable of minimizing the overshoot and improving the overall dynamic performance. Finally, the usefulness of the suggested scheme is also demonstrated by a case study on temperature control of a continuous stirred tank reactor (CSTR) during a first-order irrevocable exothermic reaction.

Applications of the Harary-Sachs theorem for hypergraphs

The Harary-Sachs theorem for k-uniform hypergraphs equates the codegree-d coefficient of the adjacency characteristic polynomial of a uniform hypergraph with a weighted sum of subgraph counts over certain multi-hypergraphs with d edges. We begin by showing that the classical Harary-Sachs theorem for graphs is indeed a special case of this general theorem. To this end we apply the generalized Harary-Sachs theorem to the leading coefficients of the characteristic polynomial of various hypergraphs. In particular, we provide explicit and asymptotic formulas for the contribution of the k-uniform simplex to the codegree-d coefficient. Moreover, we provide an explicit formula for the leading terms of the characteristic polynomial of a 3-uniform hypergraph and further show how this can be used to determine the complete spectrum of a hypergraph. We conclude with a conjecture concerning the multiplicity of the zero-eigenvalue of a hypergraph.

Correction*

Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coefficients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

COLLECTION OF MECHANISMS FOR 10 LINKS KINEMATIC CHAINS OF GROUP G

In this paper a new, easy, reliable, and efficient method to detect isomorphism and a catalogue of fixed link with its corresponding equivalent links in the distinct mechanisms of kinematic chains of Group-G, has been presented. It is helpful to the new researchers and designers to choose the best mechanism to perform the desired task at the conceptual stage of design. The proposed method is presented by comparing the structural invariants ‘sum of the absolute values of the characteristic polynomial coefficients’ [SCPC] and ‘maximum absolute value of the characteristic polynomial coefficient’ [MCPC] of [JJ] matrices. These invariants are used to detect isomorphism in the mechanism kinematic chain having simple joints. The method is explained with the help of examples of planar kinematic chain.

State feedback control for stabilization of PMSM ‐based servo‐drive with parametric uncertainty using interval analysis

For a class of multivariable uncertain dynamic systems, the parametric uncertainties are belonging to a closed interval with lower and upper boundaries a priori known. Thereby, these systems can be analyzed based on interval structures and interval matrices. In this article, a fully interval analysis–based method is developed and applied to the state feedback control of permanent magnet synchronous motor (PMSM) in order to design a stabilized servo-drive. Indeed, the parametric uncertainties are investigated in state space representation, which represents a simplified and effective way to analyze the robust stability for the interval system. Firstly, a robust state feedback control technique, dedicated to an uncertain system is introduced. For that, the robust controllability test is performed for the interval system by using the linear independency condition of column interval vectors. It is proven that there is a direct correlation between the controllability test and the existence of a robust modal P-regulator for the correction of the uncertain system. It is also shown that it invariably relies on each input's controllability indices and thus their effect on the uncertain system's state variables. In order to ensure stability in a closed loop, the modal P-regulator is designed with possibility of incorporation of an integral action. The modified modal PI regulator has the ability to reject disturbance and guarantee zero-steady-state error for step inputs. In fact, the stability is achieved by placing all coefficients of the system characteristic polynomial within assigned intervals based on Kharitonov's Theorem. The technique provides a matrix gain with interval coefficients for the stabilizing regulator. Finally, the developed approach is applied to position control of linearized model of the PMSM-based servo-drive, presenting parametrical uncertainties. To demonstrate the efficiency of the proposed method, a numerical and graphical comparison of conventional LQR and pole placement, state feedback controllers for the PMSM servo-drive with the robust interval controller is provided. In order to verify the feasibility of the whole proposed technique, calculations and simulations are performed by using Matlab/Intlab toolbox. Real-time simulation is also investigated using Lab-View Compact-RIO.

On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension n. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev–LeVerrier algorithm based on the Cayley–Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph

A directed cyclic graph is a directed graph that has at least one directed cycle graph, that is a cycle graph in which all edges are oriented, so that the direction passes through each vertex once, except the end of vertex. The directed unicyclic tadpole graph is the graph created by concatenating a cycle and a path with an edge from any vertex of to a pendant of for integers m ≥ 3 and n ≥ 1. Antiadjacency matrix is a directed graph representation matrix based on whether or not there is a relation between one vertex and the others. This paper gives the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph. To find the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph we need to do the grouping of induced subgraphs into acyclic and cyclic and verify with related theorems. After that, the characteristic polynomial is factorized and the roots are calculatedto find its eigenvalues. The coefficients of the characteristic polynomial consist of three distinct values and the eigenvalues are divided into odd case and even case.

Quantum matrix algebras of BMW type: Structure of the characteristic subalgebra

A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a ‘quantum’ matrix M possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are naturally derived in a more general framework of the QM-algebras. In this work we consider a family of Birman–Murakami–Wenzl (BMW) type QM-algebras. These algebras are defined with the use of R-matrix representations of the BMW algebras. Particular series of such algebras include orthogonal and symplectic types RTT- and RE-algebras, as well as their super-partners.For a family of BMW type QM-algebras, we investigate the structure of their ‘characteristic subalgebras’ — the subalgebras where the coefficients of characteristic polynomials take values. We define three sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. We also define an associative ⋆-product for the matrix M of generators of the QM-algebra which is a proper generalization of the classical matrix multiplication. We determine the set of all matrix ‘descendants’ of the quantum matrix M, and prove the ⋆-commutativity of this set in the BMW type.

Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph

This research discussed the characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The entries of the antiadjacency matrix of a directed graph represent the presence or the absence of a directed arc from one vertex to the others. If A is the adjacency matrix of a graph , then the antiadjacency matrix B of graph is B = J − A, where J is the square matrix with all entries equal to one. The general form of characteristic polynomial coefficients of the antiadjacency matrix of directed unicyclic flower vase graph can be obtained by calculating the sum of determinants of the antiadjacency matrices of all induced cyclic and acyclic subgraphs, while the eigenvalues were obtained by using polynomial factorization and Horner’s method. This paper gives the characteristic polynomial coefficients and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The characteristic polynomial can be considered as a function that depends on the number of vertices.

A novel two-polynomial criteria for higher-order systems stability boundaries detection and control

There are many methods for identifying the stability of complex dynamic systems. Routh and Hurwitz’s criterion is one of the earliest and commonly used analytical tools analysing the stability of dynamic systems. However, it requires tedious and lengthy derivations of all components of the Routh array to solve the stability problem. Therefore, it is not a simple method to define analytically, stability boundaries for the coefficients of the system characteristic equation.The proposed brand-new criterion is an effective alternaztive technique in identifying stabilityhigher-order linear time-invariant dynamic system that binds the coefficients of the system characteristic polynomial at the stability boundaries by means of an additional single constantk. It defines the necessary and sufficient conditions for the absolute stability of higher-order dynamic systems. It also allows the analysing of the system’s precise marginal stabilityor marginal instability condition when the roots are relocated on imaginary jω-axis of s-plane. The criterion proposed bythe authors, in contrast to Routh criteria, simplifies the identification of maximum and minimum stability limits for any coefficient of the higher-order characteristic equation significantly. The derived in the paperstability boundary formulas for the polynomial coefficients are successfully used for the proportional integral derivative (PID) controller with single or multiple gains selections in closed-loop control system.

Graphs that are cospectral for the distance Laplacian

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.

Robustness research of interval dynamic systems by algebraic method

The paper considers robust stability study of continuous and discrete interval dynamic systems by algebraic method. The original robustness results obtained for continuous and discrete linear interval dynamic systems within the algebraic direction of robustness stability are presented. The author formulated and proved the basic theorem on the robustness of linear continuous dynamic system with interval elements of the right-hand part matrix, which is determined through the separate angular coefficients of characteristic polynomial of the system. The basic theorem is proved on the basis of a lemma on the separative coefficients of the characteristic polynomial obtained by optimization methods of nonlinear programming on multiple interval elements of the system matrix. Their possible values can be the upper or lower limits of the corresponding interval or zero. A clarification note to the basic theorem for continuous systems is formulated. The idea lies in the need for a complete set of four angular polynomials for the robustness stability of the system, excluding multiple cases of the characteristic polynomial, when the set of Kharitonov polynomials degenerates and will consist of the less required four different polynomials. The theorem is obtained on the necessary and sufficient conditions of robustness stability for the polyhedron of interval matrices. A discrete analogue of the Kharitonov theorem is obtained for discrete systems. The algorithm of robustness stability determination for discrete interval dynamic systems is presented. Comparative characteristics of the results obtained in the works of well-known authors having studied the algebraic trend of robust stability problem are considered. They show the distinctive feature of this method, which consists in consideration of interval matrices of general type. The validity of the method is tested on the known counterexamples to Bialas’s theorem, as well as the other researchers studying robustness problems of interval dynamic systems.