David S. Gilliam

Output regulation for linear distributed parameter systems

This work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator, in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system. In particular it is shown that the full state feedback and error feedback regulator problems are solvable, under the standard assumptions of stabilizability and detectability, if and only if a pair of regulator equations is solvable. For linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in Francis (1997) and Isidori and Byrnes (1990). We also provide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer function. Examples are given of periodic tracking, set point control, and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system.

Regular Linear Systems Governed by a Boundary Controlled Heat Equation

AbstractIn this paper we consider a class of distributed parameter systems governed by the heat equation on bounded domains in

Systems and control in the twenty-first century

\mathbb{R}

Static and dynamic controllers for boundary controlled distributed parameter systems

n. We consider two types of boundary inputs (actuators) and two types of boundary outputs (sensors). Allowing for any possible pairing of these, we consider a totality of four possible arrangements of our system. The first type of input (control) is through the Neumann boundary condition on a part of the boundary, together with a homogenous Neumann boundary condition on the remaining part of the boundary. For this type of input, the input space is infinite-dimensional. The second type of input (with a finite-dimensional input space) is obtained by imposing constant normal derivatives on each element of a finite partition of the boundary. The first type of output (observation) is given by evaluation (trace) of the state of the system on a part of the boundary, so that the output space is infinite-dimensional. For the second type of output (with a finite-dimensional output space), we again consider a partition of the boundary of the spatial domain (which can be different from the one considered for the inputs) and each output channel contains the average of the values of the state of the plant on one element of this partition. Our main result is that any possible combination of the aforementioned inputs and outputs provides a regular linear system.

The delta method for analytic functions of random operators with application to functional data

The problem of discrete observation for the heat equation is considered. It is shown that under a wide variety of conditions it is possible to recover the initial data for the heat equation from measurements that are discrete in time and space. The conditions under which the initial data is recoverable depend on the geometry of the underlying domain and consequently on the spectrum of the associated eigenvalue problem.

Asymptotic properties of root locus for distributed parameter systems

State space method for inverse spectral problems, D. Alpay and I. Gohberg new developments in the theory of positive systems, B.D.O. Anderson modelling methodology for elastomer dynamics, H.T. Banks and N. Lybeck numerical methods for linear control systems, D. Boley and B.N. Datta notes on stochastic processes on manifolds, R. Brockett on duality between filtering and interpolation, C.I. Byrnes and A. Lindquist controlling nonlinear systems by flatness, M. Fliess et al how set-values maps pop up in control theory, H. Frankowska circuit simulation techniques based on Lanczos-type algorithms, R.W. Freund dynamical systems approach to target motion perception and ocular motion control, B.K. Ghosh et al the Jacobi method - a tool for computation and control, U. Helmke and K. Huper ellipsoidal calculus for estimation and feedback control, A.B. Kurzhanski control and stabilization of interactive structures, I. Lasiecka risk sensitive Markov decision processes, S.I. Marcus et al on inverse spectral problems and pole-zero assignment, Y.M. Ram inverse eigenvalue problems for multivariable linear systems, J. Rosenthal and X.A. Wang recursive designs and feedback passivation, Rodolphe Sepulchre et al ergodic algorithms on special Euclidean groups for ATR, A. Srivastava et al some recent results on the maximum principle of optimal control theory, H.J. Sussmann nonlinear input-output stability and stabilization, A.R. Teel repetitive control systems - old and new ideas, G. Weiss fitting data sequences to linear systems, Jan C. Willems fighter aircraft control challenges and technology transition, K.A. Wise.

Boundary feedback design for nonlinear distributed parameter systems

This paper is about the state feedback regulator problem for infinite-dimensional linear systems. The plant, assumed to be an exponentially stable regular linear system, is driven by a linear (possibly infinite-dimensional) exosystem via a disturbance signal. The exosystem has its spectrum in the closed right half-plane and also generates the reference signal for the plant output. The regulator problem is to design a controller that, while guaranteeing the stability of the closed-loop system without the exosystem, drives the tracking error to zero. A particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller. Under suitable assumptions, we show that the latter problem is solvable if and only if a pair of algebraic equations, called the regulator equations, is solvable. We derive conditions, in terms of the transfer function of the plant and eigenvalues of the exosystem, for the solvability of the regulator equations. Three examples illustrating the theory are presented.

Stability of certain distributed parameter systems by low dimensional controllers: a root locus approach

In this paper the authors describe a systematic methodology for solving certain problems of output regulation for abstract boundary control systems using dynamic and static controllers. For the special systems considered in this work, the controllers are designed using the composite zero dynamics system obtained from the plant and exosystem by constraining the error (the difference between the measured output and signal to be tracked) to be zero. Under our assumptions the proof of the main result is very simple. On the other hand, in applications this result is quite easy to apply and provides a very simple design procedure for a wide range of problems that can otherwise be difficult to solve.