Root-locus approach to the stability analysis of interval matrices

A new approach, called the root-locus approach, is developed for the stability analysis of perturbed matrices. The basic idea of this approach is to ensure that the root loci of a continuously perturbed matrix remain in the open-left-half complex s-plane. Based on this approach, new techniques are presented to analyse the stability of interval matrices. Examples are given to demonstrate the merit of the proposed theorems.     

On stability of interval matrices

New sufficient, and sometimes necessary and sufficient conditions, are obtained for Schur- and Hurwitz-stability of interval matrices by relying on the concept of connective stability and M-matrices. The necessity part is broadened to include interval matrices with mixed signs of the off-diagonal elements, provided the sign patterns follow that of the Morishima matrix. The obtained results are extended to cover convex combinations of interval matrices     

Sufficient conditions for the asymptotic stability of interval matrices

Jiang (1987) gave a sufficient condition for the asymptotic stability of interval matrices. We shall show here that his results can be derived in a trivial way using the Lyapunov equation. An extension of his results and a similar result for discrete systems is stated.     

On sufficient conditions for the stability of interval matrices

We investigate the Hurwitz and Schur stability of interval matrices using Lyapunov's second method and interval analysis techniques. Previous results require a check of the definiteness of 2^{n(n − 1)/2} concerns of a certain interval matrix. The present results require a check of the definiteness of only 2^{n − 1} corners.     

On sufficient conditions for the stability of interval matrices

Using Gershgorin's diagonal dominance theorem, new sufficient conditions for the stability of interval matrices are obtained. The new conditions remove a restriction in Heinen's (1983) analysis and are applicable to a wider class of problems than previously possible.     

Simultaneous Schur stability of interval matrices

Interval matrix structures are ubiquitous in nature and engineering. Ordinarily, in an uncertain system there is associated with a set of coupled interval matrices, a basic issue of exploring its asymptotic stability. Here we introduce the notion of simultaneous Schur stability by linking the concepts of the majorant and the joint spectral radius, and prove the asymptotic stability of a set of interval matrices governed by simultaneous Schur stability. The present result may lead to the stability analysis of discrete dynamical interval systems.     

An improved method for determining the stability of interval matrices

In this paper, the problem of the stability of interval matrices has been tackled using the properties of real stability radius. Based on these, a necessary and sufficient condition has been developed for the Hurwitz (Schur) stability of an interval matrix. An algorithm has been suggested on the basis of the above results to determine the stability of such a system. This work provides an alternative tool to that proposed by Wang et al. and it has been claimed, on the basis of comparative results, that it is a more efficient method in terms of both the computation time and the number of matrices to be checked.     

Simplified sufficient conditions for the asymptotic stability of interval matrices

It will be shown that the sufficient conditions for the asymptotic stability of interval matrices proved in Mansour (1988) can be further simplified by considering only a part of the extreme matrices.     

An algorithm for checking stability of symmetric interval matrices

A branch-and-bound algorithm for checking Hurwitz and Schur stability of symmetric interval matrices is proposed. The algorithm in a finite number of steps either verifies stability or finds a symmetric matrix which is not stable. It can also be used for checking positive definiteness of asymmetric interval matrices     

Necessary and sufficient conditions for stability of time-varying discrete interval matrices

Sufficient conditions for the stability of time-varying discrete interval systems are presented. The spectral radius of the maximum absolute matrix of all interval matrices is employed as opposed to utilizing the matrix norm or Lyapunov theorem. For special cases, our result becomes a necessary and sufficient condition for stability. Furthermore, the proposed conditions are inclusive of the previous results of stability of time-invariant interval systems.     

Sufficient conditions on stability of interval matrices:connections and new results

The stability properties of interval matrices are discussed. Sufficient conditions are given for these properties based upon Gershgorin's theorem and its extension, and they are shown to improve previous results. The derivation helps clarify relations between certain previously available results and shows they can be derived using Gershgorin's theorem. Furthermore, the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular M-matrices     

Convergence property of interval matrices and interval polynomials

In association with robust control-system design and analysis, the Hurwitz property of interval matrices and interval polynomials has recently been actively investigated. However, its discrete counterpart, the convergence property, has seemingly not been much discussed. In this paper, this property is studied in comparison with the Hurwitz counterpart. Some conditions under which interval matrices or interval polynomials are convergent are derived.     

A Kharitonov-like theorem for interval polynomial matrices

As an extension of Kharitonov's theorem, robust stability of interval polynomial matrices is studied. Here a polynomial matrix is said to be stable if its determinant has all roots with negative real parts. The present paper shows that the robust stability of interval polynomial matrices is equivalent to that of the subclasses where each row (column) has only one element that involves Kharitonov edge polynomials and all the other elements take on one of the four Kharitonov vertex polynomials.     

Sufficient condition for the positive definiteness of symmetric interval matrices

A sufficient condition for the positive definiteness of symmetric interval matrices is obtained.     

A measure for Ill-conditioning of matrices in interval arithmetic

The new condition numberV(A) is proposed. This condition number measures ill-conditioning in interval arithmetic. The reliability of the condition numberV(A) has been proved. It is shown that Bauer's minimum condition number [2] and the condition numberV(A) are essentially equivalent although different approaches were used to derive them.     

Eigenvalues of tridiagonal symmetric interval matrices

An algorithm for the determination of the eigenvalues of tridiagonal symmetric interval matrices is presented. The intervals of the eigenvalues are not overestimated, but exactly calculated. The algorithm is an efficient one, because it only needs twice as many operations as the Sturm algorithm, which is used for real matrices     

Simple criteria for stability interval polynomials

Some new simple sufficient conditions are presented for the stability and complex instability of interval polynomials.     

A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices

A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices is obtained, and a useful application is shown.     

Necessary and sufficient conditions for the Hurwitz and Schurstability of interval matrices

Establishes a set of new sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors use these results to establish necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. The authors relate the above results to the existence of quadratic Lyapunov functions for linear time-invariant systems with interval-valued coefficient matrices. Using the above results, the authors develop an algorithm to determine the Hurwitz and the Schur stability properties of interval matrices. The authors demonstrate the applicability of their results by means of two specific examples     

Counter-example to a recent result on the Schur stability of interval matrices by C.-l. Jiang

Jiang (1988) stated that the Schur stability of an interval matrix A _I is equivalent to that of the vertices of A _I . We point out, via a counter-example, that this result is incorrect.