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Achievability of linear systems up to bisimulation

08 July 2011

PhD ceremony: Mr. H.G. Vinjamoor, 16.15 uur, Doopsgezinde kerk, Oude Boteringestraat 33, Groningen

Dissertation: Achievability of linear systems up to bisimulation

Promotor(s): prof. A.J. van der Schaft

Faculty: Mathematics and Natural Sciences

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 \ref{nsp} 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 \emph{canonical}. The conditions presented in this theorem are checkable using standard algorithms from linear geometric control. We also presented 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 too 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 was to investigate 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 \ref{nsp} to the nonlinear case leads us to Theorem \ref{mres}. This theorem shows that for a certain class of systems, the nonlinear canonical controller is indeed still canonical. However, canonical now means that one has to look at the set of systems obtained by restricting the state space of the canonical controller to full simulation relations of the desired system by the plant. At the end of Chapter \ref{nlin} we also showed examples where the canonical controller construction conveys no information about the existence of a controller. This entails the need to look for approaches which do not depend on the canonical controller.

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