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Real and Complex Stability Radii of Polynomial Matrices
"... In this paper, analytic expressions are derived for the complex and real stability radii of nonmonic polynomial matrices with respect to an arbitrary stability region of the complex plane. Numerical issues for computing these radii for di#erent perturbation structures are also considered with appli ..."
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In this paper, analytic expressions are derived for the complex and real stability radii of nonmonic polynomial matrices with respect to an arbitrary stability region of the complex plane. Numerical issues for computing these radii for di#erent perturbation structures are also considered with application to robust stability of Hurwitz and Schur polynomial matrices. 1
Robustness Analysis for Systems with Ellipsoidal Uncertainty
 International Journal of Robust and Nonlinear Control
, 1998
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Formally reviewed communication A note on a nearest polynomial with a given root
"... In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coef ..."
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In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coefficients of the polynomial and the root are real numbers), and in the realcomplex case (the coefficients of the polynomial are real numbers and the root is a complex number). We derive new formulas for computing such polynomials.
The Rank One Mixed µ Problem and "KharitonovType" Analysis
 Manuscript in Preparation
"... The general mixed problem has been shown to be NP hard, so that the exact solution of the general problem is computationally intractable, except for small problems. In this paper we consider not the general problem, but a particular special case of this problem, the rank one mixed problem. We show ..."
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The general mixed problem has been shown to be NP hard, so that the exact solution of the general problem is computationally intractable, except for small problems. In this paper we consider not the general problem, but a particular special case of this problem, the rank one mixed problem. We show that for this case the mixed problem is equivalent to its upper bound (which is convex), and it can in fact be computed easily (and exactly). This special case is shown to be equivalent to the so called "affine parameter variation" problem (for a polynomial with perturbed coefficients) which has been examined in detail in the literature, and for which several celebrated "Kharitonovtype" results have been proven. 1 Introduction It is now known that the general mixed problem is NP hard, and this strongly suggests that the exact solution of the general problem is computationally intractable, except for small problems [3]. In this paper we consider not the general problem, but a particular ...
Stability Bounds for Higher Order Linear Dynamical Systems
 in Fourteenth International Symposium of Mathematical Theory of Networks and Systems
, 2000
"... This paper derives analytic expressions for the real stability radius of polynomial matrices with respect to an arbitrary region in the complex plane. We are also discussing numerical issues for computing these radii for different perturbation structures, with application to robust stability of Hurw ..."
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This paper derives analytic expressions for the real stability radius of polynomial matrices with respect to an arbitrary region in the complex plane. We are also discussing numerical issues for computing these radii for different perturbation structures, with application to robust stability of Hurwitz and Schur polynomial matrices. 1
273 Calculation of stability margins and design of robust controllers for multilinear systems
, 2000
"... Consider a linear timeinvariant system with uncertain parameters. One can study the performance of the uncertain system by investigating the effects of uncertainties on the characteristic equation of the system. Different methods have already been presented for the calculation of the bounds of allo ..."
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Consider a linear timeinvariant system with uncertain parameters. One can study the performance of the uncertain system by investigating the effects of uncertainties on the characteristic equation of the system. Different methods have already been presented for the calculation of the bounds of allowable parameter variations and the design of robust controllers for such systems. In a few recent papers ([1], [2]), the Minimum Distance Function (MDF) method is developed to address the above problems. The cases where the coefficients of the characteristic polynomials are linearly dependent on uncertain parameters, have already been investigated. In this paper, the more general and the more realistic case of polynomials with multilinearlydependent coefficients is investigated. The MDF method is generalised to this case and a new algorithm for the calculation of allowable parameter perturbations is presented. The algorithm requires the numerical solution of a system of equations in each fr...
Solving Factorable Programs with Applications to Cluster Analysis,
, 2005
"... Despite recent advances in optimization research and computing technology, deriving global optimal solutions to nonconvex optimization problems remains a daunting task. Existing approaches for solving such formidable problems are typically heuristic in nature, often leading to significantly subopti ..."
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Despite recent advances in optimization research and computing technology, deriving global optimal solutions to nonconvex optimization problems remains a daunting task. Existing approaches for solving such formidable problems are typically heuristic in nature, often leading to significantly suboptimal solutions. This motivates the need to develop a framework for optimally solving a broad class of nonconvex programming problems, which yet retains sufficient flexibility to exploit inherent special structures. Toward this end, we focus in this dissertation on a variety of applications that occur in practice as instances of polynomial programming problems or more general nonconvex factorable programs, and we employ a central theme based on the ReformulationLinearization Technique (RLT) to design theoretically convergent and practically effective and robust solution methodologies. We begin our discussion in this dissertation by providing a basis for developing efficient solution methodologies for solving the class of nonconvex factorable programming problems. Recognizing the ability of the RLT to solve polynomial programs to (global) optimality, the basic idea is to solve the given nonconvex program
Abstract Efficient Algorithms for Computing the Nearest Polynomial with Constrained Roots ∗
"... Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem finding the minimal perturbations to the coefficients to move a root to a given locus, such as a single point, the real or imaginary axis, the unit circle, or the right half plane. We measure ..."
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Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem finding the minimal perturbations to the coefficients to move a root to a given locus, such as a single point, the real or imaginary axis, the unit circle, or the right half plane. We measure minimality in both the Euclidean distance to the coefficient vector and maximal coefficientwise change in absolute value (infinity norm), either with entirely real or with complex coefficients. If the locus is a piecewise parametric curve, we can give efficient, i.e., polynomial time algorithms for the Euclidean norm; for the infinity norm we present an efficient algorithm when a root of the minimally perturbed polynomial is constrained to a single point. In terms of robust control, we are able to compute the radius of stability in the Euclidean norm for a wide range of convex open domains of the complex plane. 1
A note on a nearest polynomial with a given root
"... In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coef ..."
Abstract
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In this paper, we consider the problem of a nearest polynomial with a given root in the complex field (the coefficients of the polynomial and the root are complex numbers). We are interested in the existence and the uniqueness of such polynomials. Then we study the problem in the real case (the coefficients of the polynomial and the root are real numbers), and in the realcomplex case (the coefficients of the polynomial are real numbers and the root is a complex number). We derive new formulas for computing such polynomials.