A new vertex result for robustness problems with interval matrix uncertainty (original) (raw)

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.

On Robust Stability of Multivariable Interval Control Systems

2005

This paper studies robustness of MIMO control systems with parametric uncertainties, and establishes a lower dimensional robust stability criterion. For control systems with interval transfer matrices, we identify the minimal testing set whose stability can guarantee the stability of the entire uncertain set. Our results improve the results in the literature, and provide a constructive solution to the robustness of a family of MIMO control systems.

On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty

SIAM Journal on Optimization, 2002

We present efficiently verifiable sufficient conditions for the validity of specific NPhard semi-infinite systems of Linear Matrix Inequalities (LMI's) arising from LMI's with uncertain data and demonstrate that these conditions are "tight" up to an absolute constant factor. In particular, we prove that given an n × n interval matrix Uρ = {A | |A ij − A * ij | ≤ ρC ij }, one can build a computable lower bound, accurate within the factor π 2 , on the supremum of those ρ for which all instances of Uρ share a common quadratic Lyapunov function. We then obtain a similar result for the problem of Quadratic Lyapunov Stability Synthesis. Finally, we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube.

Stability Analysis of the Interval Systems Based on Linear Matrix Inequalities

Springer Nature, 2019

Positive definiteness and Hurwitz stability of the interval systems are discussed. A linear matrix inequality representation is introduced to simplify the analysis of the interval system. First, it is shown that the interval matrix can be stable if it has 2 conditions. Afterward, they converted to linear matrix inequalities for simplifying the conditions solution. A Lyapunov function is introduced to prove the new representation based on linear matrix inequalities.

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.

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.

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 ≤ ≤∞ .

Semidefinite Programs with Interval Uncertainty: Reduced Vertex Results

nt.ntnu.no

In this paper, we derive a reduced vertex result for robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n×n, a direct vertex approach would require satisfaction of 2 n(m+1)(n+1)/2 vertex constraints: a huge number, even for small values of n and m. The conditions obtained here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 n−1 , thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty.