Fast Methods for Estimating the Distance to Uncontrollability

Gu, Ming; Mengi, Emre; Overton, Michael L.; Xia, Jianlin; Zhu, Jingge

doi:10.1137/05063060x

Cited by 51 publications

(50 citation statements)

References 7 publications

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“…The smallest real eigenvalue can then be computed efficiently by using a robust real eigenvalue searching strategy (cf. Section 3.3 in [11]). …”

Section: Legendre-galerkin Methodsmentioning

confidence: 99%

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

An¹,

Shen²

2014

JMS

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Abstract. An important step in estimating the index of refraction of electromagnetic scattering problems is to compute the associated transmission eigenvalue problem. We develop in this paper efficient and accurate spectral methods for computing the transmission eigenvalues associated to the electromagnetic scattering problems. We present ample numerical results to show that our methods are very effective for computing transmission eigenvalues (particularly for computing the smallest eigenvalue), and together with the linear sampling method, provide an efficient way to estimate the index of refraction of a non-absorbing inhomogeneous medium.

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“…The smallest real eigenvalue can then be computed efficiently by using a robust real eigenvalue searching strategy (cf. Section 3.3 in [11]). …”

Section: Legendre-galerkin Methodsmentioning

confidence: 99%

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

An¹,

Shen²

2014

JMS

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“…In the context of linear control problems, the first polynomial time algorithm for computing the distance to uncontrollability was proposed by Gu 2000 [10]. Later, Gu published a faster method in 2007 [11]. These methods by Gu are based on bisection.…”

Section: Previous Workmentioning

confidence: 99%

Solving Overdetermined Eigenvalue Problems

Das¹,

Neumaier²

2013

SIAM J. Sci. Comput.

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Abstract. We propose a new interpretation of the generalized overdetermined eigenvalue problem (A − λB) v ≈ 0 for two m × n (m > n) matrices A and B, its stability analysis, and an efficient algorithm for solving it. Usually, the matrix pencil {A − λB} does not have any rank deficient member. Therefore we aim to compute λ for which A − λB is as close as possible to rank deficient; i.e., we search for λ that locally minimize the smallest singular value over the matrix pencil {A − λB}. Practically, the proposed algorithm requires O(mn 2 ) operations for computing all the eigenpairs. We also describe a method to compute practical starting eigenpairs. The effectiveness of the new approach is demonstrated with numerical experiments. A MATLAB based implementation of the proposed algorithm can be found at:

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“…With the stratification theory, the quantitative results presented in [22,44] and additional results like distance to uncontrollability [34,46] are complemented with new qualitative information. In the following, we step-by-step illustrate the procedure to obtain the bundle stratification of the controllability pencil S C (λ) of size 4 × 5, which (A 0 , B 0 ) is part of.…”

Section: F B(a C) Covers B( a C)mentioning

confidence: 99%

“…For a controllable pair (A, B), the distance to uncontrollability [48] where σ min (X) denotes the smallest singular value of X ∈ C n×(n+m) [19]. Using the Matlab implementation [47] of the methods presented in [34,46], the distance to uncontrollability can be computed where τ (A, B) is bounded within an interval (l, u] with any desired accuracy tol ≥ u − l. For the above system the computed distance to uncontrollability is within (3.0323e−2, 3.0332e−2], where tol = 10…”

Section: Codimensionmentioning

confidence: 99%

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Stratification of Controllability and Observability Pairs—Theory and Use in Applications

Elmroth¹,

Johansson²,

Kågström³

2009

SIAM J. Matrix Anal. & Appl.

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Abstract. Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.Key words. Stratification, matrix pairs, controllability, observability, robustness, Kronecker structures, orbit, bundle, closure hierarchy, cover relations, StratiGraph.1. Introduction. Computing the canonical structure of a linear time-invariant (LTI) system,ẋ(t) = Ax(t) + Bu(t) with states x(t) and inputs u(t), is an ill-posed problem, i.e., small changes in the input data matrices A and B may drastically change the computed canonical structure of the associated system pencil A − λI B (e.g., see [13]). Besides knowing the canonical structure, it is equally important to be able to identify nearby canonical structures in order to explain the behavior and possibly determining the robustness of a state-space system under small perturbations. For example, a state-space system which is found to be controllable may be very close to an uncontrollable one, and can therefore by only a small change in some data, e.g., due to round-off or measurement errors, become uncontrollable. If the LTI system considered and all nearby systems in a given neighborhood are controllable, the system is called robustly controllable (e.g., see [46]).The qualitative information about nearby linear systems is revealed by the theory of stratification for the corresponding system pencil. A stratification shows which canonical structures are near to each other (in the sense of small perturbations) and their relation to other structures, i.e., the theory reveals the closure hierarchy of orbits and bundles of canonical structures. A cover relation guarantees that two canonical structures are nearest neighbours in the closure hierarchy.For square matrices, Arnold [1] examined nearby structures by small perturbations using versal deformations. For matrix pencils, Elmroth and Kågström [23] first investigated the set of 2-by-3 matrix pencils and later extended the theory, in collaboration with Edelman, to general matrices and matrix pencils [17,18]. In line of this work, the theory has further been developed in [21], and for matrix pairs in [20,42]. Several other people have worked on the theory of stratifications and similar topics, and we refer...

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Fast Methods for Estimating the Distance to Uncontrollability

Cited by 51 publications

References 7 publications

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

Solving Overdetermined Eigenvalue Problems

Stratification of Controllability and Observability Pairs—Theory and Use in Applications

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