Distance Problems for Linear Dynamical Systems

Kreßner, Daniel; Voigt, Matthias

doi:10.1007/978-3-319-15260-8_20

Cited by 11 publications

(7 citation statements)

References 53 publications

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“…As a result, the optimization problem SR is not convex, and it may have multiple local minima. This is a typical property of all 2-norm/F -norm, complex/real SR problems, as well as most other minimum distance problems [25].…”

Section: Definition 1 (Sparse Stability Radius)mentioning

confidence: 78%

On the real stability radius of sparse systems

Katewa

Pasqualetti

2020

Automatica

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In this paper, we study robust stability of sparse LTI systems using the stability radius (SR) as a robustness measure. We consider real perturbations with an arbitrary and pre-specified sparsity pattern of the system matrix and measure their size using the Frobenius norm. We formulate the SR problem as an equality-constrained minimization problem. Using the Lagrangian method for optimization, we characterize the optimality conditions of the SR problem, thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system. Further, we use the Sylvester equation based parametrization to develop a penalty based gradient/Newton descent algorithm which converges to the local minima of the optimization problem. Finally, we illustrate how our framework provides structural insights into the robust stability of sparse networks.

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Section: Definition 1 (Sparse Stability Radius)mentioning

confidence: 78%

On the real stability radius of sparse systems

Katewa

Pasqualetti

2020

Automatica

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“…Previous to that this problem was posed for linear matrix pencils in Byers and Nichols (1993) and followed up in Byers et al (1998). The nearest singular matrix polynomial relates to the stability of polynomial eigenvalue problems, linear time invariant systems and differential-algebraic equations studied subsequently in (Kressner and Voigt, 2015;Guglielmi et al, 2017). For non-linear matrix polynomials/pencils, previous works rely on embedding a non-linear (degree greater than 1) matrix polynomial into a linear matrix polynomial of much higher order.…”

Section: Previous Researchmentioning

confidence: 99%

Computing lower rank approximations of matrix polynomials

Giesbrecht

Haraldson

Labahn

2020

Journal of Symbolic Computation

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Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest matrix polynomial that is algebraically singular with a prescribed lower bound on the dimension given in a previous paper by the authors. In this paper we prove that such lower rank matrices at minimal distance always exist, satisfy regularity conditions, and are all isolated and surrounded by a basin of attraction of non-minimal solutions. In addition, we present an iterative algorithm which, on given input sufficiently close to a rank-at-most matrix, produces that matrix. The algorithm is efficient and is proven to converge quadratically given a sufficiently good starting point. An implementation demonstrates the effectiveness and numerical robustness of our algorithm in practice.

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“…is a challenging open problem [2,18] and only in few special cases computational methods have been developed, see [2,23]. The reason that this problem is difficult arises from the fact that the nearest singular pencil may be one which has a higher dimensional singular block in its Kronecker canonical form (KCF), see [5], and it is not clear a priori which block structure is the one which is closest.…”

Section: The Computation Of the Distances D(a E) And D A (A E)mentioning

confidence: 99%

“…18) where K is defined in(2.8) and η = Re ∆, K .All examples have dimension 3 and we use µ ℓ = e iℓπ/2 , ℓ = 1, . .…”

mentioning

confidence: 99%

On the Nearest Singular Matrix Pencil

Guglielmi¹,

Lubich²,

Mehrmann³

2017

SIAM J. Matrix Anal. & Appl.

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Given a regular matrix pencil A + µE, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A + ∆A + µ(E + ∆E), one approach for general singular pencils and another one such that A + ∆A and E + ∆E have a common left/right null vector. For the latter case the nearest singular pencil is shown to differ from the original pencil by rank-one matrices ∆A and ∆E. In both cases we consider also the situation where only A is perturbed. The nearest singular pencil is approached by a two-level iteration, where a gradient flow is driven to an equilibrium point in the inner iteration and the outer level uses a fast iteration for the distance parameter. This approach extends also to structured matrices A and E.

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Distance Problems for Linear Dynamical Systems

Cited by 11 publications

References 53 publications

On the real stability radius of sparse systems

On the real stability radius of sparse systems

Computing lower rank approximations of matrix polynomials

On the Nearest Singular Matrix Pencil

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