Cyber-Natural Control Systems

Abstract

The thesis explores complex dynamics at the interface of natural systems and control systems. It is divided into three main parts. The first part considers adaptive stabilisation and destabilisation for continuous-time systems. Here a Byrnes-Willems high-gain adaptive controller is revisited for classes of positive systems that arise in a context of population dynamics. We show that the convergent adaptive, positive high-gain, Byrnes-Willems controller results in convergence of the gain to a stabilising gain — so that the adaptive controller learns to stabilise. We then develop a convergent, adaptive destabilising controller for the same class of systems but with adaptive negative high gain. In this case, the resulting convergent gain is destabilising — so that the adaptive controller learns to destabilise. We then show that when the two adaptation algorithms are combined, then the resulting convergent gain is the critical gain — so that a bifurcation between stable and unstable behaviour is learnt. In the second part, we develop similar results for discrete-time systems that arise in a context of population protection matrix models. We construct adaptive stabilising and destabilising controllers which converge to stabilising and destabilising controllers, respectively. When the stabilising and destabilising adaptation algorithms are combined, we find more complicated outcomes than in the continuous-time case — a so called arms race ensues. Sometimes the arms race converges, and a concord is reached. But other times, the arms race is divergent. These results are explored in a number of simulations for $1$, $2$ and $3$-dimensional systems. In the third part we consider the competitive exclusion principle for two populations in competition. For certain ranges of the species interaction parameters and their respective carrying capacities competitive exclusion occurs — either one or the other species persists and the other dies out. We show that if the carrying capacities are not constant, but instead adapt to the prevailing population abundances — with relatively more of one species causing its own carry capacity to decrease (degrade) and the others to increase (renew) — then competitive co-existence is reached. All the results are illustrated with numerous examples throughout.