	Control Group

University of Cambridge > Department of Engineering > Control Group > Publications > Publication

H-infinity control of spatially distributed systems

Reinschke J.

1999

Abstract

This dissertation is concerned with the design of robust controllers for linear, time-invariant, spatially distributed systems. An input-output view is taken in which spatially distributed systems, that is, the plant to be controlled as well as the controller, have input and output signals that may depend on spatial variables as well as time. Such input and output signals take values in (possibly) infinite-dimensional Hilbert spaces at any point in time. In the frequency domain a linear, time-invariant, spatially distributed system can be described by a parameter-dependent integral operator, the parameter being the complex frequency s. The kernel of this integral operator is the Laplace-transformed Green's function of the system. In this dissertation, sensors and actuators are viewed as part of the controller rather than as part of the plant. Consequently, optimal controller design means finding optimal locations and shapes of the spatial distribution functions of the sensors and actuators as well as an optimal (lumped) controller transfer matrix connecting the optimally placed and shaped (spatial distribution functions of the) sensors and actuators. It turns out that many problems typically associated with the feedback control of spatially distributed systems can be treated in a systematic and integrated manner. The control-theoretic framework outlined in this dissertation puts particular emphasis on robust stability of feedback loops of spatially distributed systems, where system uncertainty is measured in the gap-metric. The notion of a finite-dimensional, distributed, linear, time-invariant (FDDLTI) system is introduced, and various fundamental results like computation of normalized coprime factorizations are shown to be capable of generalization from the standard, lumped-parameter case. The largest part of this dissertation proposes specific numerical methods that together form a comprehensive robust controller design methodology for spatially distributed plants. Given a description of the spatially distributed plant to be controlled, in terms of partial differential equations or the plant's Green's function, the idea is first to compute an approximate, FDDLTI system that models the low-frequency dynamics of the infinite-dimensional plant well. The finite-dimensional approximation error, i.e., the mismatch between the infinite-dimensional plant model and its finite-dimensional approximation, is accounted for by associating an uncertainty with the FDDLTI plant model, typically additive uncertainty if the plant is open-loop stable, and coprime factor uncertainty if the plant is open-loop unstable. The FDDLTI plant model plus the uncertainty associated with it are used for the actual controller synthesis. Depending on whether the aim of the controller design is to stabilize an open-loop unstable, spatially distributed plant, or to achieve some additional performance criterion, controller synthesis methods are proposed that are similar to well-known H-infinity controller synthesis methods for lumped-parameter systems: coprime factor synthesis in the former case, and mu-synthesis in the latter. Almost all methods proposed in this dissertation, from finite-dimensional plant approximation to controller computation and validation, are illustrated by numerical examples. Key words: Spatially distributed systems, distributed-parameter systems, gap-metric, robust stabilization, mu-synthesis, H-infinity control, optimal sensor and actuator placement

BibTex Entry

@PhdThesis{,: author = {Reinschke J.},; title = {H-infinity control of spatially distributed systems },; year = {1999}
}



© 2004 Control Group, Department of Engineering, University of Cambridge www-control-admin@eng.cam.ac.uk