Problem Of Root Clustering Research Articles

A Derivative Free Fourth-Order Optimal Scheme for Applied Science Problems

We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more flexibility for obtaining new members, and is two-point, cost-effective, more stable and yields better numerical results. The derivation of our scheme is based on the weight function technique. The convergence order is studied in three main theorems. We have demonstrated the applicability of our methods on four numerical problems. Out of them, two are real-life cases, while the third one is a root clustering problem and the fourth one is an academic problem. The obtained numerical results illustrate preferable outcomes as compared to the existing ones in terms of absolute residual errors, CPU timing, approximated zeros and absolute error difference between two consecutive iterations.

A conformal mapping approach to a root-clustering problem

This paper presents a new approach for matrix root-clustering in sophisticated and multiply-connected regions of the complex plane. The parametric sweeping method and a concept of the closed forbidden region covered by a set of modified three-parametrical Cassini regions are used. A conformal mapping approach was applied to formulate the main results of the paper. An application of the developed method to the problem of matrix root-clustering in a multiply connected region is shown for illustration.

Method of forbidden regions in the dynamic system matrices root clustering problem

New criteria for root-clustering of the dynamic system matrices in multiply connected regions were presented. A concept of the forbidden region was introduced, and a generalization of the Lyapunov-Jury theorem onto the multiply connected regions was proposed. The necessary and sufficient conditions for root-clustering outside the forbidden region were established as a sequence of matrix inequalities. Construction of the conditions for root-clustering outside the frequency bands and robust root-clustering in multiply connected region was illustrated by way of examples.

A Sweeping Method for Matrix Root Clustering

Abstract This paper presents a new criteria for matrix root-clustering problem. It provides generalizations of the existing theorems onto multiply-connected and non-convex regions of the complex plane.

極の領域を指定した拡張<i>H</i><sub>∞</sub>制御のLMI解法の導出とその応用

Abstract As well known, extended H ∞ control accepts unstable weighting functions which are included in a generalized plant, so it has wider availability than standard H ∞ control. On the other hand, the root-clustering problem for H ∞ control has been considered by Chilali and Gahinet and the solution is derived. But it is not so easy to apply the solution to extended H ∞ control, because the formulation of the root-clustering controllers is discordant with that of extended H ∞ controllers. Therefore, we can not simply combine the root-clustering conditions with extended H ∞ control. In this paper, we propose a LMI based design method for this control problem by using modified formulation of controllers of the root-clustering problem derived by Chilali and Gahinet. These types of control problems appear in design problems of a vibration isolation controller which attenuates vibrations forced by narrow-band frequency disturbances. A controller for a simple vibration isolation experimental system is designed by the proposed method and performance of a designed controller is verified by experimental results.

EXTENDED H∞ CONTROL WITH POLE PLACEMENT CONSTRAINTS VIA LMI APPROACH AND ITS APPLICATION

As well known, extended H∞ control accepts unstable weighting functions which are included in a generalized plant, so it has wider availability than standard H∞ control. On the other hand, the root-clustering problem for H∞ control has been considered by Chilali and Gahinet and the solution is derived. But it is not so easy to apply the solution to extended H∞ control, because the formulation of the root-clustering controllers is discordant with that of extended H∞ controllers. Therefore, we can not simply combine the root-clustering conditions with extended H∞ control. In this paper, we propose a LMI based design method for this control problem by using modified formulation of controllers of the root-clustering problem derived by Chilali and Gahinet. These types of control problems appear in design problems of a vibration isolation controller which attenuates vibrations forced by narrow-band frequency disturbances. A controller for a simple vibration isolation experimental system is designed by the proposed method and performance of a designed controller is verified by experimental results.

Common Lyapunov function approach to matrix root clustering

This paper proposes an alternative route to the matrix root clustering problem, which enables to use a result on the common Lyapunov function for solving the problem efficiently. A necessary and sufficient condition is obtained in terms of the existence of a common positive definite solution to a set of Lyapunov inequalities for eigenvalues of a matrix to lie in a prescribed subregion of the complex plane. Applications to root clustering in sector regions are shown for illustration.

A test for root-clustering transformability

Consider the problem of root clustering: given a square matrix A with spectrum \sigma(A) , for what region S in the complex plane is it possible to state a criterion (necessary and sufficient conditions) so that \sigma(A) \in S ? Recently it has been shown that one subclass Ω of S satisfies a certain transformability condition. In this note we test transformability via polynomial global nonnegativity.

A root-clustering theorem

We consider the problem of root clustering of a real matrix in an algebraic region of the complex plane. It is shown that a criterion (previously obtained) based on an n^{2} \times n^{2} matrix reduces to a criterion based on an \frac{1}{2}n(n-1) \times \frac{1}{2}n(n-1) matrix. This is done for any transformable region and thus generalizes earlier results.

A general theory for matrix root-clustering in subregions of the complex plane

We consider the general problem of root-clustering of a matrix in the complex plane: Let A \in C^{n \times n} and S \subset C . Find the largest class of S and an algebraic criterion which is necessary and sufficient for \lambda_{i}[A] \in S, i=1,2,..., n . We introduce two types of regions which constitute the largest class of S known to date. The criterion is presented both for open regions and closed ones. The results are used to define a design methodology for control systems. Moreover, all classical results are shown to be special cases of the present theory.

Inner-minor transformation

A general inner-minor transformation is presented. This transformation is based on a permutation matrix which preserves the determinants values of the corresponding inners and minors. Furthermore, for the root clustering problem, this permutation matrix takes a special simple form which preserves a certain feature in the innerwise matrix. For the sake of unification, it is shown in an example that the inner form is the unifying form.

Symmetric and innerwise matrices for the root-clustering and root-distribution of a polynomial

The general problem of root-clustering and root-distribution of a polynomial in a certain region Γ in the complex plane has been investigated in this paper. The region Γ is general and includes all the previously investigated regions. For the root-clustering problem, it is shown that by using a certain transformation, the necessary and sufficient condition can be represented either in terms of positive definite (p.d.) or positive innerwise (p.i.) matrices. The entries in these matrices are rational functions of the coefficients of the polynomial. The connection between p.d. and p.i. matrices is established in terms of matrix multiplication. The investigations of this paper are quite general in that all the previously known criteria for root-clustering can be obtained from the general formulation. Furthermore, this study shows that the “Inner” matrix representation offers unification, simplification and a systematic way for studying the mentioned problems.