Some applications of randomized algorithms for control system design

Patel, Vijay V.; Deodhare, Girish; Viswanath, Tharini

doi:10.1016/s0005-1098(02)00112-7

Cited by 17 publications

(39 citation statements)

References 13 publications

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“…Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature [17,21]. …”

supporting

confidence: 79%

“…Obviously, any element in U (cl(D)) and its inverse are analytic on cl(D). The following problem is the well-known "Champagne Problem" [6,8,9,17,20,21] of simultaneous stabilization. It illustrates the difficulty in the simultaneous stabilization of three systems in general.…”

Section: Solution To the Generalized Champagne Problemmentioning

confidence: 99%

“…We give a complete answer to the open problem proposed by Patel et al [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established.…”

Section: Introductionmentioning

confidence: 99%

“…

Abstract:The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis [1,11,13,18,22] and Blondel's technique [5,6,8].We give a complete answer to the open problem proposed by Patel et al [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established.

…”

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confidence: 99%

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Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

et al. 2007

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Abstract:The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis [1,11,13,18,22] and Blondel's technique [5,6,8].We give a complete answer to the open problem proposed by Patel et al. [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature [17,21].

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supporting

confidence: 79%

Section: Solution To the Generalized Champagne Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

et al. 2007

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Abstract:The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis [1,11,13,18,22] and Blondel's technique [5,6,8].We give a complete answer to the open problem proposed by Patel et al. [20,21], which automatically includes the solution to the original "Champagne Problem" [6,8,9,17,20,21]. Based on the recent development in automated inequality-type theorem proving [31,32,33,35,36], a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature [17,21].

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“…The best previously known upper bound was δ * < 0.99999712, obtained from Batra's [6] estimate t * ≥ A 2 ≥ 0.0012. (The frequently cited [10,11,25] upper bound δ * < 0.9999800002 seems to come from a computational error using the lower bound A 2 ≥ 10 −5 given in [8]. )…”

Section: The Belgian Chocolate Problemmentioning

confidence: 99%

Gol’dberg’s constants

Bergweiler

Erëmenko

2013

JAMA

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Abstract. We study two extremal problems of geometric function theory introduced by A. A. Goldberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them and make a conjecture. Goldberg's problems have important applications to control theory. IntroductionGoldberg [16] studied a class of extremal problems for meromorphic functions. Let F 0 be the set of all holomorphic functions f defined in the rings {z : ρ(f ) < |z| < 1}, omitting 0 and 1, and such that the indices of the curve f ({z : |z| = ρ(f )}) with respect to 0 and 1 are non-zero and distinct.Let F 1 ⊂ F 0 be the subclass consisting of functions meromorphic in the unit disk U. Functions in F 1 can be described as meromorphic functions in U with the property that the numbers of preimages of 0, 1 and ∞, counted with multiplicities, are all finite and pairwise distinct.Let F 2 , F 3 , F 4 be the subclasses of F 1 consisting of functions holomorphic in the unit disk, rational functions and polynomials, respectively. For f in any of these classes F j , 1 ≤ j ≤ 4, we define ρ(f ) as ρ(f ) = sup{|z| : f (z) ∈ {0, 1, ∞}}. Goldberg's constants areGoldberg credits the problem of minimizing ρ(f ) to E. A. Gorin. He proved thatand showed that there exist extremal functions for A 0 and A 2 , but extremal functions for A 1 , A 3 or A 4 do not exist. He also proved the estimates A 0 < 0.0091 and 0.0000038 < A 2 < 0.0319.

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Global optimization in stabilizing controller design

Chang

Sahinidis

2006

J Glob Optim

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Some applications of randomized algorithms for control system design

Cited by 17 publications

References 13 publications

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

Gol’dberg’s constants

Global optimization in stabilizing controller design

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