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.

Summary In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (ΔE,ΔA) such that (E+ΔE,A+ΔA) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that QTE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.

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 structure-preserving perturbations are considered.

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by Eẋ = Ax+Bu and y = Cx+Du, minimize the Frobenius norm of (∆ E , ∆ A , ∆ B , ∆ C , ∆ D ) such that (E + ∆ E , A + ∆ A , B + ∆ B , C + ∆ C , D + ∆ D ) is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices (E, A, B, C, D) describing the system, and (iii) does not make any assumption on the original given system. generate energy: the system (1.1) is called passive if there exists a nonnegative scalar valued function V, called the storage function, such that V(0) = 0 and the dissipation inequalityholds for all admissible u, t 0 , and t 1 ≥ t 0 ; see for example [2,27]. The energy is defined via the inner product of the input and output vectors u(t) and y(t) of the system hence these vectors need to be of the same length.Given (E, A, B, C, D), the goal of this paper is to findWe will refer to this problem as the nearest PR system problem. We will also consider the case of nearest standard PR systems when E = I n imposing ∆ E = 0. Since passivity and positive realness are equivalent for LTI systems, the distance to positive realness has direct applications in passive model approximations. For example, when a real-world problem is approximated by a model (1.1), the passivity of the physical system may not be preserved, that is, the approximation process (for example, finite element or finite difference models, linearization, or model order reduction) makes the passive system nonpassive. The nonpassive system has to be approximated by a nearby passive system by perturbing E, A, B, C, and D.Several algorithms tackle this problem using the spectral properties of the related Hamiltonian/skew-Hamiltonian matrices or pencils for the input systems that are asymptotically stable, controllable, observable and almost passive; see [19,37,50,48] and the references therein. The algorithms in [19] and [37] impose additional assumptions on the input system, namely E = I n and D + D T nonsingular, and are restricted to perturbations of the matrix C only. These algorithms are based on the displacement of the imaginary eigenvalues of the related Hamiltonian matrix. The methods in [50] and [48] can deal with general systems (i.e., when E is ...

Abstract. We derive a formula for the backward error of a complex number λ 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, * -even, and * -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.