An overview of recent results on the parametric approach to robust stability

Abstract

This pa er discusses recent results relating to the parametric approact to robust stability. The general problem of robust stability is defined and a review of Kharitonov e and edge theorems, zero exclusion results, interval matrix staBity, maximal perturbation bounds, multilinear and nonlinear perturbations, time delay systems and numerical/graphical approaches is presented. Some open research directions are indicated, concluding remarks on the future of the approach are given and an extensive list of references is included.

Note on Schur stability of three‐dimensional interval matrices

“…Nevertheless, Theorem 3 is true if $$$ n=2 $$$; this follows from the results of Reference 14. Sufficient conditions for Hurwitz stability of interval matrices can be found in the surveys 1,3,5,15 …”

“…A large number of works are devoted to the study of Hurwitz and Schur stability of a family of interval polynomials and interval matrices. These questions have been presented at different times in various reviews [1][2][3][4][5] and monographs. [6][7][8] We note some key results in the context of this work.…”

“…Consider a polynomial p(𝜆, q) = p(𝜆) = q 0 + q 1 𝜆 + • • • + q n 𝜆 n (1) with real coefficients q j ∈ R, j = 0, 1, … , n. The polynomial (1) is called Hurwitz stable (Schur stable) if all its roots lie in the open left half-plane (open unit disc) of the complex plane. For a linear homogeneous time-invariant differential (difference) equation, a necessary and sufficient condition for its asymptotic stability is the condition of Hurwitz (Schur) stability of its characteristic polynomial.…”

The Dual Characterization of Structured and Skewed Structured Singular Values

“…The main aim of the work by Kharitonov is to determine the stability robustness with a finite number of conditions. This approach also aims to study the problems in control when real parametric uncertainties consisting of real-valued uncertain parameters are involved, for a more details see, e.g., [5][6][7].…”