A note on checking regularity of interval matrices (original) (raw)
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
Studies in Computational Intelligence, 2020
It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well.
Improved bounds for the spectrum of interval matrices
IET Control Theory & Applications, 2013
This study presents new sufficient conditions for Hurwitz and Schur stability of interval matrices. Tight bounds for the spectrum of interval matrices are estimated using computationally simple optimisation problems. The conservativeness is reduced further by application of ordinary similarity transformation. A necessary and sufficient vertex based criterion for the stability of a subclass of interval systems in continuous and discrete-time cases is also proposed. This enables the spectra for this class of interval systems to be determined exactly. A selection of various examples adopted from existing literature is used to demonstrate the utility of the proposed criteria.
On stability of interval matrices
IEEE Transactions on Automatic Control, 1994
Lai and H. Robbins, "Consistency and asymptotic efficiency of slope estimates in stochastic schemes." Z Wahrscheinlichkeirs theorie y e w .
Diagonal stability of interval matrices and applications
Linear Algebra and its Applications, 2010
P is a positive definite diagonal matrix and the notation "≺ 0" means negative definite. The first part of the paper • provides SDS p and HDS p criteria, • presents methods for finding the positive definite diagonal matrix requested by the definition of SDS p and HDS p , • analyzes the robustness of SDS p and HDS p and • explores the connection with the Schur and Hurwitz stability of A I . The second part shows that the SDS p or HDS p of A I is equivalent to the following properties of a discrete-or continuous-time dynamical interval system whose motion is described by A I : • the existence of a strong Lyapunov function defined by the p-norm and • the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system's trajectories.
Sufficient conditions for the stability of interval matrices
International Journal of Control, 1993
In this paper, the global asymptotic stability of interval systems with multiple unknown time-varying delays is considered. Some criteria are derived to guarantee the global asymptotic stability of such systems. A numerical example is provided to illustrate the use of our main results.
New sufficient conditions for robust stability analysis of interval matrices
Systems & Control Letters, 2012
This letter presents new sufficient conditions for robust Hurwitz stability of interval matrices. The proposed conditions are based on two approaches: (i) finding a common Lyapunov matrix for the interval family and (ii) converting the robust stability problem into a robust non-singularity problem using Kronecker operations. The main contribution of the letter is to derive accurate and computationally simple optimal estimates of the robustness margin and spectral bound of general interval matrices. The evaluation of the condition relies on the solutions of linear matrix inequalities (LMIs) and eigenvalue problems, both of which are solved very efficiently. The improvements gained by using the proposed conditions are demonstrated through application to previous examples in the literature.
Interval matrix norms induced by point matrix norms are introduced in the space of interval matrices. It is shown that evaluating the interval matrix norm induced by a point matrix norm · p is exponential (probably NP-hard) for p = 2 and requires computation of only one point matrix norm for p ∈ {1, ∞, (1, ∞), F }. . 4 Above: logo of interval computations and related areas (depiction of the solution set of the system [2, 4]x 1 + [−2, 1]x 2 = [−2, 2], [−1, 2]x 1 + [2, 4]x 2 = [−2, 2] (Barth and Nuding [3])).
Positive definiteness and stability of parametric interval matrices
arXiv: Numerical Analysis, 2017
We investigate positive definiteness, Hurwitz stability and Schur stability of parametric interval matrices. We give a verifiable sufficient condition for positive definiteness of parametric interval matrices with non-linear dependencies. We also give several sufficient and necessary conditions for stability of symmetric parametric interval matrices with affine-linear dependencies. The presented results extend the results on positive definiteness and stability of interval matrices. In addition, we provide a formula for the radius of stability of symmetric parametric interval matrices.
Lecture Notes in Computer Science, 2004
We consider the interval iteration [x]kH = [A][X]k+ [b]with p(I[A]I) ::; 1 where I[A]I denotes the absolute value of the given interval matrix [A]. If I[A]I is irreducible we derive a necessary and sufficient criterion für the existence of the limit [x]*= [x]*([x]o)of each sequence ([xt) of interval iterates. In this way we generalize a well-known theorem of O. Mayer [6] on the above-mentioned iteration, and we are able to enclose solutions of certain singular systems (I-A
Stability and Set-Invariance Testing for Interval Systems
2008
Many works dealing with the stability analysis of interval systems developed criteria based on matrices that majorize (in a certain sense) the interval matrices describing the system dynamics. Besides this already classical employment, we prove that the majorant matrices also contain valuable information for the study of the exponentially decreasing sets, invariant with respect to the trajectories of the interval systems. The interval systems are considered with both discrete-and continuous-time dynamics. The invariant sets are characterized by arbitrary shapes, defined in terms of Hőlder vector p-norms, 1 p ≤ ≤ ∞ . Our results cover two types of interval systems, namely described by interval matrices of general form and by some particular classes of interval matrices. For the general case, we formulate necessary and sufficient conditions, when the shape of the invariant sets is defined by the norms 1, p = ∞ , and sufficient conditions, when the shape is defined by the norms 1 p < <∞ . For the particular cases, we provide necessary and sufficient conditions for all norms 1 p ≤ ≤∞ .