Feedback Control Approaches to Uncertain Nonlinear Structured Population Models

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

Like engineering systems, population systems often involve nonlinear feedback loops, parameter uncertainty, and external (often persistent) disturbances. For instance, in plant populations, seedling recruitment is often a nonlinear feedback of the structured population. As another example, populations coupled via dispersal are naturally described by a closed-loop feedback system. This project will develop techniques for the analysis and control of a class of nonlinear structured population models with uncertainties and disturbances. Any ecological system is subject to a wide range of natural disturbances and parameters uncertainties, and in population management it is critical to do a robustness analysis of the effect of these disturbances and uncertainties on the population. This research project aims to obtain new mathematical results about stability analysis, tracking controllers, and observers and observer-based controllers, and robustness for these systems. Optimal control, where a cost function is to be minimized, has also been used frequently in ecology. Proportional-integral-derivative controllers are feedbacks of observed measurements, and are known to have good robustness properties. This work introduces ideas from robust feedback control theory into management of plant and animal populations, with a particular emphasis on endangered plants. The results of this research project will have potential impact in environmental, economic, and social policy areas. The concepts of robustness of feedback loops, small gains, absolute stability, and input-to-state stability are well suited to modeling and analysis of uncertain, structured, nonlinear population dynamics. Feedback control can be applied robustly without knowledge of the probability distributions of the uncertainties and disturbances, so that it can be applied to some models for which a stochastic analysis is not possible. We study the class of Lure systems, which can be described as a feedback interconnection between a linear system and a nonlinearity. We consider these systems in either discrete time or continuous time, and with either a finite dimensional state space or an infinite dimensional state space. We include uncertainties and disturbances in the model. We propose to develop mathematics that will extend systems theory techniques and concepts to Lure systems that describe structured populations. A key feature is the interplay between the linear part and nonlinear part, and will require a combination of classical engineering input-output techniques and mathematical state space analysis. Our key goals are: to determine the asymptotic stability structure of the uncontrolled continuous time system; to study the effect of a tracking controllers, which are feedback controllers that uses an observation of the system (instead of the whole state) as their input; and to construct and analyze an observer and observer-based controller for Lure systems. An observer is a dynamical system that takes the observation of the original system as its input, and asymptotically reconstructs the whole state of the original system. We will apply these results and techniques to population management problems.