Branch and bound algorithm with applications to robust stability

Ravanbod, Laleh; Noll, Dominikus; Apkarian, Pierre

doi:10.1007/s10898-016-0424-6

Cited by 7 publications

(2 citation statements)

References 26 publications

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“…Differential equations with interval coefficients arise also when stability problems are examined. We naturally encounter with the robust (or interval) stability concept when we study real‐life problems in control theory . A system is called robust (interval absolutely stable), if it is absolutely stable for each matrix, the elements of which lie in the specified intervals .…”

Section: Introductionmentioning

confidence: 99%

On differential equations with interval coefficients

Gasılov

Amrahov

2019

Math Methods in App Sciences

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In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept.Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case. KEYWORDSbunch of functions, interval differential equations, linear differential equations MSC CLASSIFICATION34A12; 65L05; 03E75; 68W25

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Section: Introductionmentioning

confidence: 99%

On differential equations with interval coefficients

Gasılov

Amrahov

2019

Math Methods in App Sciences

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“…1 Introduction Nonsingularity of a matrix is well known to be an important property in many applications of linear algebra. Let us mention an important area of systems stability studied for example for linear time-invariant dynamical systems with parameters having uncertain values [19]. Within these systems questions like distance to instability or minimum stability degrees are solved.…”

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confidence: 99%

Regularity Radius: Properties, Approximation and a Not a Priori Exponential Algorithm

Hartman

Hladík

2018

ELA

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The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are applied. Improvements of both approaches have been recently shown by Hartman and Hladík (doi:10.1007/978-3-319-31769-4 9) utilizing relaxation of computation to semidefinite programming. An estimation of the regularity radius using either of mentioned ways is usually applied to general matrices considering none or just weak assumptions about the original matrix. Surprisingly less explored area is represented by utilization of properties of special class of matrices as well as utilization of classical algorithms extended to be used to compute the considered radius. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values. At first, checking finiteness of this characteristic is shown to be a polynomial problem along with determining a maximal bound on number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. Considering situations where known bounds are not enough, a new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several corresponding results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the radius of regularity.

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Optimization-Based Control Design Techniques and Tools

Apkarian

Noll

2019

Encyclopedia of Systems and Control

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Branch and bound algorithm with applications to robust stability

Cited by 7 publications

References 26 publications

On differential equations with interval coefficients

On differential equations with interval coefficients

Regularity Radius: Properties, Approximation and a Not a Priori Exponential Algorithm

Optimization-Based Control Design Techniques and Tools

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