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Achievability of linear systems up to bisimulation. (English) Zbl 1246.93003
Groningen: Rijksuniversiteit Groningen, Wiskunde en Natuurwetenschappen (Diss.). vi, 76 p. (2011).

Summary: Given a plant and another system called the desired system. We address the following question: when can we construct a third system called the controller, such that when attached to the plant the combined system behaves like the desired system?

Towards this end the first main result of this thesis is Theorem 2.3.1. This uses an idea called the canonical controller. The canonical controller can be constructed from the equations of the plant and the desired system. Theorem 2.3.1 shows that for linear time invariant systems. if there at all exists an achieving controller then the canonical controller is itself one such controller: hence the adjective canonical. The conditions presented this theorem are checkable using standard algorithms from linear geometric control. We also present a solution to a variant of the above problem. namely. that of asymptotic achievability. Here the aim is to construct a controller such that when attached to the plant. The resulting combined system behaves like the desired system asymptotically. Here we have necessary and sufficient conditions which can be checked easily.

The canonical controller is in general not a feedback controller. Usually it imposes some state constraints on the plant. A natural question is the existence of feedback controllers. Feedback controllers accept the output of the plant as an input and produce a signal which then acts as the input to the plant and are easy to implement. Geometric conditions equivalent to the existence of a feedback controller have been presented. These conditions require us to ’list’ all controlled invariant subspaces of the canonical controller and then check if there is one with the desired properties. Extending the thought process behind Theorem 2.3.1 to the nonlinear case leads us to Theorem 3.2.4. This theorem shows that for a certain class of systems, the nonlinear canonical controller is indeed still canonicaL

MSC:
93-02Research monographs (systems and control)
93C05Linear control systems
93B10Canonical structure of systems
93B27Geometric methods in systems theory