Richard Rebarber

Internal model based tracking and disturbance rejection for stable well-posed systems

In this paper we solve the tracking and disturbance rejection problem for infinite-dimensional linear systems, with reference and disturbance signals that are finite superpositions of sinusoids. We explore two approaches, both based on the internal model principle. In the first approach, we use a low gain controller, and here our results are a partial extension of results by Hamalainen and Pohjolainen. In their papers, the plant is required to have an exponentially stable transfer function in the Callier-Desoer algebra, while in this paper we only require the plant to be well-posed and exponentially stable. These conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable. Our second approach concerns the case when the second component of the plant transfer function (from control input to tracking error) is positive. In this case, we identify a very simple stabilizing controller which is again an internal model, but which does not require low gain. We apply our results to two problems involving systems modeled by partial differential equations: the problem of rejecting external noise in a model for structure/acoustics interactions, and a similar problem for two coupled beams.

Stability of infinite-dimensional sampled-data systems

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space X and the control space U are Hilbert spaces, the system is of the form x(t) = Ax(t) + Bu(t), where A is the generator of a strongly continuous semigroup on X, and the continuous time feedback is u(t) = Fx(t). The answer to the above question is known to be yes if X and U are finite-dimensional spaces. In the infinite-dimensional case, if F is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is yes, if B is a bounded operator from U into X. Moreover, if B is unbounded, we show that the answer yes remains correct, provided that the semigroup generated by A is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.

Global asymptotic stability of density dependent integral population projection models

Many stage-structured density dependent populations with a continuum of stages can be naturally modeled using nonlinear integral projection models. In this paper, we study a trichotomy of global stability result for a class of density dependent systems which include a Platte thistle model. Specifically, we identify those systems parameters for which zero is globally asymptotically stable, parameters for which there is a positive asymptotically stable equilibrium, and parameters for which there is no asymptotically stable equilibrium.

Generalized Sampled-Data Stabilization of Well-Posed Linear Infinite-Dimensional Systems

We consider well-posed linear infinite-dimensional systems, the outputs of which are sampled in a generalized sense using a suitable weighting function. Under certain natural assump- tions on the system, the weighting function, and the sampling period, we show that there exists a generalized hold function such that unity sampled-data feedback renders the closed-loop system exponentially stable (in the state-space sense) as well as L 2 -stable (in the input-output sense). To illustrate our main result, we describe an application to a structurally damped Euler-Bernoulli beam.

Feedback control systems analysis of density dependent population dynamics

Abstract We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.

Choice of density-dependent seedling recruitment function affects predicted transient dynamics: a case study with Platte thistle

Modelers have to make choices about which functional forms to use for representing model components, such as the relationship between the state of individuals and their vital rates. Even though these choices significantly influence model predictions, this type of structural uncertainty has been largely ignored in theoretical ecology. In this paper, we use integral projection models (IPMs) for Platte thistle as a case study to illustrate that the choice of functional form characterizing density dependence in seedling recruitment has important implications for predicting transient dynamics (short-term population dynamics following disturbances). In one case, the seedling recruitment function is modeled as a power function, and in the other case, we derive density dependence in seedling recruitment from biological first principles. We chose parameter values for the recruitment functions such that both IPMs predicted identical equilibrium population densities and both recruitment functions fit the empirical recruitment data sufficiently well. We find that the recovery from a transient attenuation, and the magnitude of transient amplification, can vary tremendously depending on which function is used to model density-dependent seedling recruitment. When we loosen the restriction of having identical equilibrium densities, model predictions not only differ in the short term but also in the long term. We derive some mathematical properties of the IPMs to explain why the short-term differences occur.

ROBUST POPULATION MANAGEMENT UNDER UNCERTAINTY FOR STRUCTURED POPULATION MODELS

Structured population models are increasingly used in decision making, but typically have many entries that are unknown or highly uncertain. We present an approach for the systematic analysis of the effect of uncertainties on long-term population growth or decay. Many decisions for threatened and endangered species are made with poor or no information. We can still make decisions under these circumstances in a manner that is highly defensible, even without making assumptions about the distribution of uncertainty, or limiting ourselves to discussions of single, infinitesimally small changes in the parameters. Suppose that the model (determined by the data) for the population in question predicts long-term growth. Our goal is to determine how uncertain the data can be before the model loses this property. Some uncertainties will maintain long-term growth, and some will lead to long-term decay. The uncertainties are typically structured, and can be described by several parameters. We show how to determine which parameters maintain long-term growth. We illustrate the advantages of the method by applying it to a Peregrine Falcon population. The U.S. Fish and Wildlife Service recently decided to allow minimal harvesting of Peregrine Falcons after their recent removal from the Endangered Species List. Based on published demographic rates, we find that an asymptotic growth rate lambda > 1 is guaranteed with 5% harvest rate up to 3% error in adult survival if no two-year-olds breed, and up to 11% error if all two-year-olds breed. If a population growth rate of 3% or greater is desired, the acceptable error in adult survival decreases to between 1% and 6% depending of the proportion of two-year-olds that breed. These results clearly show the interactions between uncertainties in different parameters, and suggest that a harvest decision at this stage may be premature without solid data on adult survival and the frequency of breeding by young adults.

Parameterizing the growth-decline boundary for uncertain population projection models

We consider discrete time linear population models of the form n(t+1)=An(t) where A is a population projection matrix or integral projection operator, and n(t) represents a structured population at time t. It is well known that the asymptotic growth or decay rate of n(t) is determined by the leading eigenvalue of A. In practice, population models have substantial parameter uncertainty, and it might be difficult to quantify the effect of this uncertainty on the leading eigenvalue. For a large class of matrices and integral operators A, we give sufficient conditions for an eigenvalue to be the leading eigenvalue. By preselecting the leading eigenvalue to be equal to 1, this allows us to easily identify, which combination of parameters, within the confines of their uncertainty, lead to asymptotic growth, and which lead to asymptotic decay. We then apply these results to the analysis of uncertainty in both a matrix model and an integral model for a population of thistles. We show these results can be generalized to any preselected leading eigenvalue.

Feedback invariance of SISO infinite-dimensional systems

We consider a linear single-input single-output system on a Hilbert space X, with infinitesimal generator A, bounded control element b, and bounded observation element c. We address the problem of finding the largest feedback invariant subspace of X that is in the space c⊥ perpendicular to c. If b is not in c⊥, we show this subspace is c⊥. If b is in c⊥, a number of situations may occur, depending on the relationship between b and c.

A Sampled-Data Servomechanism for Stable Well-Posed Systems

In this technical note, an approximate tracking and disturbance rejection problem is solved for the class of exponentially stable well-posed infinite-dimensional systems by invoking a simple sampled-data low-gain controller (suggested by the internal model principle). The reference signals are finite sums of sinusoids and the disturbance signals are asymptotic to finite sums of sinusoids.