MAINTAINING AN OPTIMAL STEADY STATE IN THE PRESENCE OF PERSISTENT DISTURBANCES.
AuthorXABA, BUSA ABRAHAM.
KeywordsEquilibrium -- Mathematical models.
AdvisorVincent, Thomas L.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractThe central goal of this dissertation is to develop a simple but powerful theory to handle a problem which arises in management situations where an optimally exploited, system at steady state is subjected to a set of continuous, persistent and unpredictable disturbances emanating from the system's environment. Such disturbances drive the system out of steady state. The question that arises in such a situation is whether there exists any additional control which can be imposed on the disturbed system in order to drive it back to the steady state and maintain it there for all future time? We show in this dissertation that such a control is possible provided bounds for the disturbances are known. We develop the additional control using concepts from reachability and the so-called Liapunov's "second method". We further develop some theory concerning certain problems which arise in generating the boundary of the reachable set, ∂R(•) using the controllability maximum principle. In generating ∂R(•) several boundary controls may be used to generate different parts of ∂R(•). We show that all the parts of ∂R(•) are polygonally connected. We also show that for a second-order system if an equilibrium point under constant control is hyperbolic and lies on ∂R(•), it is asymptotically stable. Further, in persistently disturbing a system, it is desirable to have some idea about the boundedness of the disturbed system. If the system is bounded then a boundary can be generated using controllability maximum principle. We give some theory and discussion on how to test such boundedness for linear, quasilinear and some cases of nonlinear systems. The last two chapters of this dissertation show how the theory is applied to a second-order system; in particular to a second-order grazing system.
Degree ProgramApplied Mathematics