Superstable Control Systems

Hard Problems in Linear Control Theory: Possible Approaches to Solution

“…We then readily obtain the radius of maximal robustness, which is the maximal value of the uncertainty span γ that allows for robust superstabilization. Indeed, if the matrix A 0 c (K) of the nominal closed-loop system is superstable for some K, then, in accordance with (63), the superstability of the perturbed system is retained for all…”

“…At the same time, to check if a matrix is superstable is not a serious problem, since the corresponding conditions are formulated directly in terms of the entries of a matrix rather than its eigenvalues. Sufficient conditions of stability formulated in terms of inequalities on the entries of the system matrix, as well as the properties of such matrices have been discussed in the earlier literature, e.g., see [47,[64][65][66][67]; however, it is only recently that the notion of superstability has been applied to design, see [61,[68][69][70]. We also note that a similar superstability condition can be formulated for SISO systems (in discrete time); we however put this issue aside.…”

Superstabilization of positive linear electrical circuit by state-feedbacks

“…From Gershgorin theorem and (6) it follows that the state variables u i (t), i = 1, …, n of positive electrical circuits have not overshoots. See also [17][18][19]. □…”

“…Stability of continuous-time and discrete-time linear systems with inverse state matrices has been analyzed in [15] and positive stable minimal realization of fractional linear systems in [16]. Superstability and superstabilization of dynamical systems have been considered in [17][18][19].…”

Stability and Performance of Complex Systems Affected by Parametric Uncertainty

“…The second approach to solving "hard" problems in robust stability relates to the notion of superstability (Polyak and Shcherbakov 2002). The matrix A of system (1) (and the system itself) is said to be superstable, if its entries a ij ; i; j D 1; : : : ; n, satisfy the relations…”