Performance robustness bounds for linear systems to guarantee root-clustering in second order subregions of the complex plane

Bakker, W.H.; Luo, J. S.; Johnson, A.

doi:10.1109/cdc.1993.325860

Cited by 10 publications

(6 citation statements)

References 8 publications

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“…In a diOE erent manner, this topic was also studied by Zhou and Khargonekar (1987). The ® rst extensions to robust performance analysis were achieved by choosing D as a sector (see Luo et al (1993) and and references therein) but a more signi® cant generalization to a large set of regions, namely the simple O-regions, is proposed by Yedavalli (1993 b) and later improved by Bakker et al (1993), Chouaib (1994), Chouaib and Pradin (1995 b) and . The following results constitute a step towards extending the previous Lyapunov bounds to the case of non-connected regions.…”

Section: Robust D-stability Boundsmentioning

confidence: 98%

“…In this paper, we propose an alternative means of deriving robustness bounds relevant to non-connected regions by considering any union of any possibly disjoint O-subregions. Although this work is based on an LMI condition, it is an extension of the various results proposed in the Lyapunov approach, from the nominal view point (Gutman and Jury 1981) and from the robust viewpoint (Bakker et al 1993, Yedavailli 1993b, Chouaib 1994, Chouaib and Pradin 1995. The paper is organized as follows: in } 2, an NSC for complex matrix root clustering in such a region is given.…”

Section: Introductionmentioning

confidence: 98%

“…To be more precise, the bound is either a bound on the 2-norm of E for the case where the structure of E is unknown (unstructured uncertainty) or a bound on the maximal perturbation in the entries of the uncertain matrix for the case where E linearly depends on parameter variations (structured parametric uncertainty). If those bounds are not exceeded and if A is already D-stable, then the eigenvalues of A ‡ E keep lying in D. The bounds, which are called robustness bounds or robust D-stability bounds, are deduced from the various NSCs for nominal matrix root clustering mentioned above, derived from the Lyapunov approach (Patel and Toda 1980, Sezer and S Ï iljak 1989, Yedavailli 1993b, Luo et al 1993, Bakker et al 1993, Chouaib and Pradin 1995, from the logarithmic approach (only available for unstructured uncertainty (Wang and Lin 1992, Chouaib and Pradin 1994 or from the LMI approach (Bachelier and Pradin 1998 b). Also, especially for structured uncertainty, the reader can also consult the results proposed by Luo et al (1996) as well as the good work achieved by Yedavalli (1993 a) and improved by Gardiner (1997).…”

Section: Introductionmentioning

confidence: 99%

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Robust matrix root-clustering analysis in a union of ?-subregions

Bachelier¹,

Pradin²

2001

International Journal of Systems Science

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This paper addresses the research of robust D-stability bounds. A matrix is D-stable when its eigenvalues lie in a speciWed region D of the complex plane. Such a property, which is easily testable for some regions, is of practical interest for linear systems analysis in terms of pole location. Ensuring the state matrix D-stability can guarantee some performances on the transient response of this system, but the matrix D-stability is not testable any longer when this matrix is subject to an additive uncertainty. A bound on the uncertainty domain can be computed. This bound is established either on the 2-norm of the uncertainty matrix in the case of an unstructured uncertainty or on the maximal perturbation in the entries of the matrix in the case of a structured uncertainty. It guarantees that the D-stability of the uncertain matrix is preserved when the nominal matrix is already D-stable. Most of the results in this area of work are relevant to connected regions. In this work, because of a linear matrix inequality approach, D is a union of possibly disjoint O-subregions.

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Section: Robust D-stability Boundsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Robust matrix root-clustering analysis in a union of ?-subregions

Bachelier¹,

Pradin²

2001

International Journal of Systems Science

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“…Those results were later improved in References [22][23][24]. They were also extended to classes of regions that are usually connected (often even convex) and symmetric with respect to the real axis (let us just say 'symmetric') [25][26][27][28][29][30]. These bounds are deduced from necessary and sufficient conditions for nominal matrix root-clustering based, for example, on generalized Lyapunov equations [31], on the notion of logarithmic norms [32] or on LMI frame-work [13,12].…”

Section: Introductionmentioning

confidence: 99%

Robust matrix 𝒟_U‐stability analysis

Bachelier¹,

Mehdi²

2003

Intl J Robust & Nonlinear

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SUMMARYIn this paper, the problem of robust matrix root-clustering is addressed. The studied matrices are subject to both polytopic and unstructured uncertainties. An original point is the large choice of clustering regions enabled by the proposed approach since these regions can be unions of possibly disjoint and nonsymmetric subregions of the complex plane. The precise purpose is, considering a specified polytope, to determine the greatest robustness bound on the unstructured uncertainty such that robust matrix rootclustering is ensured. To reduce conservatism in the derivation of the bound, the reasoning relies on a framework based upon parameter-dependent Lyapunov functions. The bound value is computed by solving an LMI problem.

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“…To be more precise, the bound is either a bound on the 2-norm of E for the case where the structure of E is unknown (unstructured uncertainty) or a bound on the maximal perturbation in the entries of the uncertain matrix for the case where E linearly depends on parameter vari-ations (structured parametric uncertainty). If those bounds are not exceeded and if A is already D-stable, then the eigenvalues of A ‡ E keep lying in D. The bounds, which are called robustness bounds or robust D-stability bounds, are deduced from the various NSCs for nominal matrix root clustering mentioned above, derived from the Lyapunov approach (Patel and Toda 1980, Sezer and S Ï iljak 1989, Yedavailli 1993b, Luo et al 1993, Bakker et al 1993, Chouaib and Pradin 1995, from the logarithmic approach (only available for unstructured uncertainty (Wang and Lin 1992, Chouaib and Pradin 1994 or from the LMI approach (Bachelier and Pradin 1998 b). Also, especially for structured uncertainty, the reader can also consult the results proposed by Luo et al (1996) as well as the good work achieved by Yedavalli (1993 a) and improved by Gardiner (1997).…”

Section: Introductionmentioning

confidence: 99%