On approximating the nearest Ω‐stable matrix

Choudhary, Nisha; Gillis, Nicolas; Sharma, Punit

doi:10.1002/nla.2282

Cited by 8 publications

(14 citation statements)

References 18 publications

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“…Our approach to tackle the SOF and SSF problems can be extended to find feedbacks that make the system Ω-stable. A matrix is said to be Ω-stable if its eigenvalues belong to the set Ω ⊂ C. In [7], a parametrization of Ω-stable matrices was obtained in terms of DH matrices that satisfy additional LMI constraints, where Ω is the intersection of specific regions in the complex plane; namely conic sectors, vertical strips, and disks. For example, one may want the real parts of the eigenvalues of A − BKC to be strictly smaller than some given negative value so that A − BK is robustly stable (see also Remark 3 below).…”

Section: Extension To ω-Stabilizationmentioning

confidence: 99%

“…Using these parametrizations, the results obtained in Sections 3 and 4 can be directly extended to obtain a characterization of static stabilizing feedbacks that guarantee Ω-stability in terms of DH matrices. The corresponding optimization problems (5.7) and (5.9) would be subject to additional LMIs on the variables J, R and Q depending on the Ω region; see [7,.…”

Section: Extension To ω-Stabilizationmentioning

confidence: 99%

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Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Gillis

Sharma

2021

Linear Algebra and its Applications

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In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to tackle this problem. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library. IntroductionConsider a continuous linear-time invariant (LTI) system in the forṁwhere, for all t ∈ R, x(t) ∈ R n is the state space, u(t) ∈ R m is the control input, A ∈ R n,n , B ∈ R n,m , and C ∈ R p,n . The notion of stabilizing the system pair (A, B) using feedback controllers is a fundamental one, and is referred to as the static-state feedback (SSF) problem. It requires to find K ∈ R m,n such that A − BK is stable, that is, all eigenvalues of the matrix A − BK are in the left half of the complex plane and those on the imaginary axis are semisimple; see for example [4,2]. The first goal of this paper is to solve the SSF problem; this can be divided into two parts: 1) Feasibility. Check the existence of a feedback matrix K such that A − BK is stable. IIT Delhi.2) Optimization. If the problem is feasible, minimize the norm of the feedback matrix, that is,where · is a given norm such as the ℓ 2 norm or the Frobenius norm.The second goal is to consider the analogous problem for system triplets (A, B, C), referred to as the static-output feedback (SOF) problem. The SOF problem requires to find K ∈ R m,p such that A − BKC is stable; see [27] for a survey on the SOF problem. This decision problem is believed to be NP-hard as no polynomial-time algorithm is known. Moreover, if extra constraints are imposed on the entries of the static controller, then this decision problem is NP-hard [19,6]. As a consequence, the minimal-norm SOF problem, for which the norm of K is minimized, is hard as well. For more discussion on the hardness of this problem, we refer to the recent paper by Peretz [21] and the references therein. The solution of the SOF problem is important for systems which models structural dynamics, and naturally needs a static feedback that can be built into the structure [25,31,32,22,21]. It was shown that optimal SOFs may achieve similar performance as optimal dynamic feedbacks. Contribution and outline of the paperRecently, in [11], a parametrization of the set of all stable matrices was obtained in terms of dissipative Hamiltonian (DH) systems. DH systems are special cases of port-Hamiltonian systems, which recently have received a lot attention in energy based modeling; see for example [13,23,24], and also [12,5,18] for ...

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Section: Extension To ω-Stabilizationmentioning

confidence: 99%

Section: Extension To ω-Stabilizationmentioning

confidence: 99%

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

Gillis

Sharma

2021

Linear Algebra and its Applications

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“…Moreover, in the literature interest has been given also to more exotic choices. For instance, in [9] this problem is considered for the case where Ω is a region generated by the intersection of lines and circles.…”

Section: Introductionmentioning

confidence: 99%

“…-Curtis, Mitchell and Overton [10], improving on a previous method by Lewis and Overton [33], use a BFGS method for non-smooth problems, applying it directly to the largest real part among all eigenvalues. -Choudhary, Gillis and Sharma [9,16] use a reformulation using dissipative Hamiltonian systems, paired with various optimization methods, including semidefinite programming.…”

Section: Introductionmentioning

confidence: 99%

Nearest $$\varOmega $$-stable matrix via Riemannian optimization

Noferini

Poloni

2021

Numer. Math.

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We study the problem of finding the nearest $$\varOmega $$ Ω -stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $$\varOmega $$ Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

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“…The authors of [4] use the projected gradient descent method to solve such a problem. In paper [1], a block coordinate descent method for this optimization problem (1.1) is proposed. The nearest stable matrix pair problems have been considered in [2].…”

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confidence: 99%