State Space Formulas for a Suboptimal Rational Leech Problem I: Maximum Entropy Solution

Frazho, A. E.; Horst, S. ter; Kaashoek, M. A.

doi:10.1007/s00020-014-2147-8

Cited by 6 publications

(29 citation statements)

References 21 publications

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“…Recall, cf., [18,Theorem 3.61] or [6,Section 2], that the H ∞ -corona problem is solvable if and only if T G admits a right inverse. Here T G is the analytic Toeplitz operator…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Bezout-Corona Problem Revisited: Wiener Space Setting

Groenewald

Horst

Kaashoek

2015

Complex Anal. Oper. Theory

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The matrix-valued Bezout-corona problem G(z)X (z) = I m , |z| < 1, is studied in a Wiener space setting, that is, the given function G is an analytic matrix function on the unit disc whose Taylor coefficients are absolutely summable and the same is required for the solutions X . It turns out that all Wiener solutions can be described explicitly in terms of two matrices and a square analytic Wiener function Y satisfying det Y (z) = 0 for all |z| ≤ 1. It is also shown that some of the results hold in the H ∞ setting, but not all. In fact, if G is an H ∞ function, then Y is just an H 2 function. Nevertheless, in this case, using the two matrices and the function Y , all H 2 solutions to the Bezout-corona problem can be described explicitly in a form analogous to the one appearing in the Wiener setting. Communicated by Bernd Kirstein. M. A. Kaashoek gratefully thanks the mathematics department of North-West University, Potchefstroom campus, South Africa, for the generous support during his visit May 22-June 12, 2014. B S. ter Horst

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“…Recall, cf., [18,Theorem 3.61] or [6,Section 2], that the H ∞ -corona problem is solvable if and only if T G admits a right inverse. Here T G is the analytic Toeplitz operator…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Bezout-Corona Problem Revisited: Wiener Space Setting

Groenewald

Horst

Kaashoek

2015

Complex Anal. Oper. Theory

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“…The following ℋ ∞ problem, known as the Leech problem (see related works [6][7][8][9], is shown in this paper to be equivalent with the strong (J, J ′ )-lossless property of an RM. Problem 1.…”

Section: Introductionmentioning

confidence: 90%

“…For some given RMs G(s) and K(s), find an RM X(s) without poles in ℜ[s] ≥ 0 (ie, a stable RM) satisfying ||X(s)|| ∞ ≤ 1 and G(s) X(s) = K(s). (9) Note that ||X(s)|| ∞ ≤ 1 implies that X X * ≤ I in ℜ[s] ≥ 0, hence a necessary condition for Problem 1 is…”

Section: Introductionmentioning

confidence: 99%

Strongly (J,J′)‐lossless rational matrices and ℋ_∞ problem

Stefanovski

2018

Intl J Robust & Nonlinear

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We strengthen the existing definition of (J, J ′ )-lossless rational matrices (RMs) and find an algebraic characterization of the newly defined class of strongly (J, J ′ )-lossless RMs. The algebraic characterization is given for possibly improper RMs. A connection is presented to a rational ℋ ∞ problem, known as the Leech problem, which is elaborated on with necessary and sufficient conditions, given in terms of the strongly (J, J ′ )-lossless property, as an alternative of the Leech conditions, which can be expressed with positivity of a kernel. An algorithm for solving the Leech problem is given and illustrated by examples. KEYWORDS robustness, robust control, (J, J ′ )-lossless rational matrix, ℋ ∞ problem Int J Robust Nonlinear Control. 2018;28:4261-4286.wileyonlinelibrary.com/journal/rnc

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“…We state the following problem. The problem with excluded condition (1.2) is known in the literature as the Leech problem (see [21, p. 107], [17] and [18]) with some recent activity on getting explicit state-space formulas for a solution (see [27], [28], [10], and [11]). Therefore, Problem 1 could be called a two-sided rational Leech problem.…”

Section: Introductionmentioning

confidence: 99%

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

Stefanovski¹

2015

SIAM J. Control Optim.

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For given rational matrices V a, U a, V b , U b , we find necessary and sufficient conditions for existence of a stable rational matrix Φ satisfying Φ ∞ ≤ 1, V aΦ = U a, and ΦV b = U b . A condition is the positive semidefiniteness of a matrix, denoted by R. We present a parametrization of all problem solutions. A property of the proposed algorithm is, as a first step, to reduce the problem to a minimal realization, by an orthogonal transformation. Another property is the ability to transform the problem into one with constant matrices, by another orthogonal transformation. A problem motivation is the optimal H∞ control problem of descriptor systems. We show by an example that the existing numerical H∞ control optimization algorithms, which solve the problem of obtaining stabilizing controllers such that Φ ∞ ≤ γ, where Φ is the closed loop transfer matrix, and γ is slightly greater than the optimal one, compute controllers such that the closed loop system suffers from lack of stability robustness. Our proposed algorithm doesn't require γ to be greater than the optimal one, and the closed loop system has the property of stability robustness. Consequences of the singularity of matrix R, which property generically appears for the optimal γ, are (1) that Φ(jω) = γ for all real ω and all optimal controllers, and (2) if the measured output is single, or the control input is single, then the H∞ controller is unique. A numerical example is presented to illustrate the H∞ control optimization algorithm. Introduction.Consider rational matrices (rm's) V a (s) ∈ R(s) μ×m , U a (s) ∈ R(s) μ×r , V b (s) ∈ R(s) r×ρ , and U b (s) ∈ R(s) m×ρ , where by R(s) μ×m we denote the set of rm's with real coefficients, of dimension μ × m, which are possibly improper. We state the following problem.Problem 1. Find a proper stable rm Φ(s) ∈ R(s) m×r such that Φ ∞ ≤ 1 and the following conditions are satisfied:

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State Space Formulas for a Suboptimal Rational Leech Problem I: Maximum Entropy Solution

Cited by 6 publications

References 21 publications

The Bezout-Corona Problem Revisited: Wiener Space Setting

The Bezout-Corona Problem Revisited: Wiener Space Setting

Strongly (J,J′)‐lossless rational matrices and ℋ_∞ problem

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

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State Space Formulas for a Suboptimal Rational Leech Problem I: Maximum Entropy Solution

Cited by 6 publications

References 21 publications

The Bezout-Corona Problem Revisited: Wiener Space Setting

The Bezout-Corona Problem Revisited: Wiener Space Setting

Strongly (J,J′)‐lossless rational matrices and ℋ∞ problem

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

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Product

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Strongly (J,J′)‐lossless rational matrices and ℋ_∞ problem