Doctoral Dissertations

Date of Award

12-2006

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mechanical Engineering

Major Professor

VijaySekhar Chellaboina

Committee Members

William R. Hamel, Seddik M. Djouadi, Dongjun Lee

Abstract

Nonnegative systems are dynamical systems with nonnegative states for random non- negative initial conditions. A subclass of nonnegative systems are compartmental systems characterized by conservation laws.Nonnegative and compartmental systems have widespread applications in biological systems, medical systems, thermodynamic systems, network systems, economic systems, etc.

In this dissertation we have investigated the applications of nonnegative systems in biological and network systems. The specific focus in biological systems has been in the area of pharmacokinetics and we have addressed the consensus problem in network systems. We have specifically focused on the time-delay present in compartmental systems and have investigated the behavior of these systems in the presence of time- delay.

This dissertation describes necessary and sufficient conditions that guarantee mono- tonic decline of the drug concentration after drug administration to the patient has been discontinued. Results are presented for the cases when there is no delay in the transfer of the drugs between the body compartments and when there are multiple delays present in the system.

We developed an adaptive controller for uncertain linear nonnegative systems. The adaptive controller framework developed guarantees set-point regulation of the system in addition to guaranteeing nonnegativity of the control signal. We demonstrated the framework on a drug delivery model for general anesthesia.

Nonnegative and compartmental models are also widespread in agreement prob- lems in networks with directed graphs and switching topologies. We use compart- mental dynamical system models to characterize dynamic algorithms for linear and nonlinear networks of dynamic agents in the presence of inter-agent communication delays that possess a continuum of semistable equilibria, that is, protocol algorithms that guarantee convergence to Lyapunov stable equilibria. In addition, we show that the steady-state distribution of the dynamic network is uniform, leading to system state equipartitioning or consensus.

Finally, we incorporated the inertial effects into the dynamics of the multiagent system and have extended existing results in the literature to develop time-domain sufficient conditions in order to achieve consensus of the agents in the presence of time-delay.

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