Constrained feedback control of imperfectly known singularly perturbed non-linear systems

Robust stability analysis of Linsker-Type Hebbian learning multi-time scale neural networks under parametric uncertainties

“…The goal of the present paper is to study stability and robustness properties of Linsker-type Hebbian learning mod- eled by a system of competitive differential equations, from a rigorous analytic standpoint and to apply results of the theory of nonlinear uncertain singularly perturbed systems [4]. The networks under study model the nonlinear dynamics of both fast and slow states under consideration of nonlinear uncertainties.…”

“…The Linsker-type Hebbian learning multi-time scale neural network is a two time-scale system consisting of two coupled subsystems with the following structure [4] where…”

“…These perturbations can lead to location errors of equilibria, to instabilities, and even to spurious states. Therefore, a rigorous understanding of the qualitative robustness properties of neural systems with respect to parameter variations on both a fast or slow time scale becomes imperative.The goal of the present paper is to study stability and robustness properties of Linsker-type Hebbian learning modAnke Meyer-Baese is with the eled by a system of competitive differential equations, from a rigorous analytic standpoint and to apply results of the theory of nonlinear uncertain singularly perturbed systems [4]. The networks under study model the nonlinear dynamics of both fast and slow states under consideration of nonlinear uncertainties.…”

Stabilizing Feedback Controls for a Class of Singularly Perturbed, Nonlinear, Uncertain, Time-Delay Systems

“…This particular problem can be addressed utilizing singular perturbation theory (for more details, see (Kokotovic et al, 1986) or, for a differentialgeometric approach, (Isidori, 1989)). There has been much research, over the past decades, on control of uncertain singularly perturbed systems using a deterministic approach; for example, (Binning and Goodall, 1999;Binning and Goodall, 2000;Corless, 1991;Corless et al, 1993;Corless et al, 1990;Corless and Ryan, 1991;Garofalo and Leitmann, 1990), and (Leitmann et al, 1986), to name but a few. In addition, time-delays, which can have a significant effect on the dynamic behaviour of a system, is a phenomenon that has been investigated by a number of researchers in recent times; in particular, aspects of stability analysis using a deterministic approach.…”

“…One component of a controller assures desired behaviour of the slow dynamics, whilst a second component yields the desired stability properties for the fast dynamics in the presence of the active slow controller. The stability analysis is similar to that used in (Binning and Goodall, 1999) and (Binning and Goodall, 2000) but, here, the full-order system is a nonlinear delay system of the retarded-type, containing both discrete and distributed delays. Utilizing memoryless feedback controllers, together with a deterministic methodology based on Lyapunov theory and Lyapunov-Krasovskiȋ functionals, some stability criteria are proposed that will ensure the desired stability property for the prescribed class of singularly perturbed delay systems, provided the singular perturbation parameter is small enough.…”