Punit Sharma

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

Abstract In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix A is stable if and only if it can be written as A = ( J − R ) Q , where J = − J T , R ⪰ 0 and Q ≻ 0 (that is, R is positive semidefinite and Q is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables ( J , R , Q ) : (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require O ( n 3 ) operations per iteration, where n is the size of A . We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.

Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical systems. One of the major advantages of the DH formulation is that system properties are encoded in an algebraic way. For instance, the algebraic structure of DH systems guarantees that the system is automatically stable. In this paper the question is discussed when a linear constant coefficient DH system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well understood. In this paper, explicit formulas for this distance under structure-preserving perturbations are determined. It is also shown (via numerical examples) that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbations, since the distance to instability can be much larger when struc...

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

We derive a formula for the backward error of a complex number

Reconstruction of aggregation tree in spite of faulty nodes in wireless sensor networks

\lambda

Finding the Nearest Positive-Real System

when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian,

Structured eigenvalue backward errors of matrix pencils and polynomials with palindromic structures

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Computing the nearest stable matrix pairs: Computing nearest stable matrix pairs

-even, and

A semi-analytical approach for the positive semidefinite Procrustes problem

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Structured eigenvalue backward errors of matrix polynomials

-odd. Numerical experiments suggest that in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.

Stability radii for real linear Hamiltonian systems with perturbed dissipation

Recent advances in wireless sensor networks (WSNs) have led to many new promissing applications. However data communication between nodes consumes a large portion of the total energy of WSNs. Consequently efficient data aggregation technique can help greatly to reduce power consumption. Data aggregation has emerged as a basic approach in WSNs in order to reduce the number of transmissions of sensor nodes over aggregation tree and hence minimizing the overall power consumption in the network. If a sensor node fails during data aggregation then the aggregation tree is disconnected. Hence the WSNs rely on in-network aggregation for efficiency but a single faulty node can severely influence the outcome by contributing an arbitrary partial aggregate value. In this paper we have presented a distributed algorithm that reconstruct the aggregation tree from the initial aggregation tree excluding the faulty sensor node. This is a synchronous model, completed in several rounds. Our proposed scheme can handle multiple number of faulty nodes as well.