An Algorithm for Computing the Distance to Instability

He, Chunyang; Watson, G. A.

doi:10.1137/s0895479897314838

Cited by 39 publications

(28 citation statements)

References 11 publications

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“…However, the situation is less bleak in practice; numerical results reveal that the method often converges to the correct solution for the initial values proposed in [19]. As discussed in [32], there is always the possibility to check whether H .˛ / has purely imaginary eigenvalues, but this global optimality certificate may become too expensive for large-scale systems. As discussed in [21], the described algorithm can be extended in a rather straightforward manner to H 1 norm computations, even for the general case of descriptor systems.…”

Section: The Implicit Determinant Methodsmentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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This chapter is concerned with distance problems for linear timeinvariant differential and differential-algebraic equations. Such problems can be formulated as distance problems for matrices and pencils. In the first part, we discuss characterizations of the distance of a regular matrix pencil to the set of singular matrix pencils. The second part focuses on the distance of a stable matrix or pencil to the set of unstable matrices or pencils. We present a survey of numerical procedures to compute or estimate these distances by taking into account some of the historical developments as well as the state of the art. IntroductionConsider a linear time-invariant differential-algebraic equation (DAE)with coefficient matrices E; A 2 R n n , a sufficiently smooth inhomogeneity f W OE0; 1/ ! R n , and an initial state vector x 0 2 R n . When E is nonsingular, (20.1) is turned into a system of ordinary differential equations by simply multiplying with E 1 on both sides. The more interesting case of singular E arises, for example, when imposing algebraic constraints on the state vector x.t/ 2 R n . Equations of this type play an important role in a variety of applications, including electrical circuits [54,55] and multi-body systems [48]. The analysis and numerical solution of more general DAEs (which also take into account time-variant coefficients, nonlinearities and time delays) is a central part of Volker Mehrmann's work, as witnessed by the monograph [43] and by the several other chapters of this book

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Section: The Implicit Determinant Methodsmentioning

confidence: 99%

Distance Problems for Linear Dynamical Systems

Kreßner

Voigt

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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“…Aiming at a large sparse matrix A, the use of inverse iteration in Step 2 is proposed and analyzed in [16]. Nevertheless, the need for computing all the eigenvalues of H( ) in Step 1 leaves Algorithm 2 inaccessible for larger matrices.…”

Section: Other Methodsmentioning

confidence: 99%

Finding the Distance to Instability of a Large Sparse Matrix

Kreßner¹

2006

2006 IEEE Conference on Computer-Aided Control Systems Design

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Abstract-The distance to instability of a matrix A is a robust measure for the stability of the corresponding dynamical systeṁ x = Ax, known to be far more reliable than checking the eigenvalues of A. In this paper, a new algorithm for computing such a distance is sketched. Built on existing approaches, its computationally most expensive part involves a usually modest number of shift-and-invert Arnoldi iterations. This makes it possible to address large sparse matrices, such as those arising from discretized partial differential equations.

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“…[6], [9] and [17]). The real stability radius also has been well-studied, the most notable example of which is [14], where it was shown that This formula indicates that computing the real stability radius is somewhat difficult but computationally tractable.…”

Section: Stability Of Continuous-time Linear Systems: a Ppc Specimentioning

confidence: 99%

“…[4], [12] and [16]) and continuous-time (cf. [2], [5], [6], [7], [10], [11], [14] and [17]) also has played a central role in providing a coherent and meaningful formulation of complexity and fragility in the context of linear systems. The exact way in which the aforementioned work has influenced this work will be made clear throughout the rest of the paper.…”

Section: Introductionmentioning

confidence: 99%