Some stability theorems for polygons of polynomials

@article{Pujara1992SomeST,
  title={Some stability theorems for polygons of polynomials},
  author={L. R. Pujara and Naresh R. Shanbhag},
  journal={IEEE Transactions on Automatic Control},
  year={1992},
  volume={37},
  pages={1845-1849}
}
Some necessary and sufficient conditions for a polynomial in a polygon of polynomials to vanish on the imaginary axis are obtained. These theorems generalize the segment lemma of H. Chapellat and S.P. Bhattachartta (see ibid., vol.34, p.448-50, 1989) to a polygon of polynomials. > 

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References

SHOWING 1-10 OF 13 REFERENCES
On the partition of unstable standard polytopes of polynomials
  • L. PujaraN. Shanbhag
  • Mathematics
    [1991] Proceedings of the 30th IEEE Conference on Decision and Control
  • 1991
The set of all polynomials vanishing on the imaginary axis in a polytope of polynomials is determined. This constitutes a significant step towards finding the boundary of stability in a
Fast stability checking for the convex combination of stable polynomials
TLDR
In this algorithm, the major computations involved are those of solving for the positive real roots of two polynomials with degree less than or equal to n/2 for each vertex, and the burden of the combinatoric explosion of the number of edges is greatly reduced.
An alternative proof of Kharitonov's theorem
An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in
On the Stability of Uncertain Polynomials with Dependent Coefficients
  • L. Pujara
  • Mathematics
    1989 American Control Conference
  • 1989
The main objective of this paper is to give a sufficient condition for reducing the conservatism of the stablility-bounds for a family of polynomials with dependent coefficients including nonlinear
Graphical stability robustness tests for linear time-invariant systems: generalizations of Kharitonov's stability theorem
The authors derive two graphical tests for the U-Hurwitz stability of convex polyhedra of polynomials. The first is a Nyquist-type test, which has been extended to arbitrary connected sets of
A Generalization of Kharitonov's Four Polynomial Concept for Robust Stability Problems with Linearly Dependent Coefficient Perturbations
  • B. Barmish
  • Mathematics
    1988 American Control Conference
  • 1988
From a systems-theoretic point of view, Kharitonov's seminal theorem on stability of interval polynomials suffers from two fundamental limitations: First, the theorem only applies to polynomials with
Robustness analysis for uncertain dynamical systems with structured perturbations
  • A. TesiA. Vicino
  • Mathematics
    Proceedings of the 27th IEEE Conference on Decision and Control
  • 1988
The authors propose two approaches for robust pole location analysis of dynamical systems with structured parametric uncertainties. The first approach, based on geometric considerations, solves the
Frequency Domain Conditions for the Robust Stability of Linear and Nonlinear Dynamical Systems
TLDR
This paper presents general frequency domain criteria for the robust stability of systems with parametric uncertainities and of LTI systems operating under possibly nonlinear passive feedback.
A Polynomial time Algorithm for Checking the Robust Stability of a Polytope of Polynomials
TLDR
A efficient algorithm to check the robust stability of a polytope of polynomials with a zero exclusion condition at each frequency is proposed and it is shown that such a condition has to be checked at only a finite number of frequencies.
...
...