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

Abstract

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.

Robust matrix 𝒟U‐stability analysis

“…where the set PðaÞ is a set of m HPD matrices P k ðaÞ defined by (35). From (44), it can be deduced that S n ðAÞM U ðAðaÞ; PðaÞ; G U ÞSðAÞ50…”

“…Derivation of robustness bounds related to non-connected regions was a bit handled for unions of disjoint disks and unstructured uncertainty in Reference [33]. It was treated in a more general manner in References [3,34,35] through a quadratic approach.…”

Transient response shaping in H/sub ∞/ control by loose eigenstructure assignment technique

“…Lyapunov region is a subclass of the 2nd-order a2-region [6]. A Lyapunov region which is convex and symmetric with respect to the real axis is described as a LMI[3] or EMI [5] region.…”