V.I. Shubov

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

Well-posedness for a One Dimensional Nonlinear Beam

\mathbb{R}

Zero dynamics modeling and boundary feedback design for parabolic 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.

Zero dynamics boundary control for regulation of the Kuramoto-Sivashinsky equation

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.

The regulator equations for retarded delay differential equations

In this note we present well-posedness results for a class of nonlinear beam models with linear damping and general external time dependent forcing in the Sobolev space H-2. Our efforts here are motivated by the eventual goal of developing computational methodologies for the control of smart material composites undergoing large deformations and/or deformations that fall within the regime of nonlinear stress-strain laws. It is known [2] in engineering applications (especially with curved and twisted structures) that large deflections can occur even when strain levels remain relatively low. On the other hand in composites and certain types of elastomers one encounters a nonlinear stress-strain relationship even in the case of small deformations [2], [8]. The equations considered here include the important class of nonlinear but monotone stress-strain laws encountered in the derivation of the transverse bending equation for a beam (cf. [10], [11]). The external applied forces (or controls f = Bu) are precisely the class that arise in current versions of “smart material” structures when beams or plates are loaded with piezoceramic actuators and sensors (e.g. see [3], [4] and the references therein).

Example of output regulation for a system with unbounded inputs and outputs

Abstract This paper is concerned with the effect of boundary feedback control on the dynamics of a class of nonlinear distributed parameter systems governed by convective reaction diffusion equations acting on bounded domains inn-dimensional Euclidean space.

Interior point control of a heat equation using zero dynamics design

In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular for a scalar proportional output feedback trajectories of the closed loop system converge to those of the zero dynamics system as the gain tends to infinity. The main contribution of this work is a simple modeling and design mechanism for solving certain tracking and disturbance rejection problems. This mechanism provides an alternative to the geometric approach involving regulator equations. Our approach is based on the fact that for certain boundary control systems it is very easy to model the systems zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation. As special cases we describe dynamic and static controllers from the associated zero dynamics system for set-point and harmonic tracking. In order to provide a simple roadmap to using this methodology we present numerical examples for a one dimensional heat equation.