Abstract
The terms observability and reachability stem from the very beginning of modern control theory for linear, time-invariant systems. These are properties of the system, they are independent of the choice of solutions. This picture changes, if we turn to the nonlinear, time-variant or even time-invariant case. We present a short review of observability and reachability investigations for the latter class of systems based on functional analysis and differential geometry.
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Notes
- 1.
A fiber of \(\mathcal{X} \rightarrow \mathbb{R}\) over \(t\) is the set of points \(\left (t,x\right ) \in \mathcal{X}\).
- 2.
A foliation partitions a manifold into submanifolds called leaves.
- 3.
We use the abbreviation \(\mathrm{d}h =\mathrm{ d}h_{1},\ldots,\mathrm{ d}h_{l}\). etc. to keep the notation as simple as possible.
- 4.
f ∂ x is a shortcut for \(\sum _{i=1}^{n}{f}^{i}\partial _{x_{i}}\).
- 5.
∂ u is a shortcut for \(\partial _{u_{1}},\ldots,\partial _{u_{l}}\), \(\partial _{u}\left (g\right )\) is a shortcut for \(\partial _{u}\left (g_{1}\right ),\ldots,\partial _{u}\left (g_{l}\right )\).
- 6.
We use also the abbreviation \(\left [f_{e,u},\partial _{u}\right ] = \left [f_{e,u},\partial _{u_{1}}\right ],\ldots,\left [f_{e,u},\partial _{u_{m}}\right ]\).
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Acknowledgements
The first author want to thank ACCM (Austrian Center of Competence in Mechatronics) for the partial support of this contribution. Markus Schöberl is an APART fellowship holder of the Austrian Academy of Sciences.
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Schlacher, K., Schöberl, M. (2014). Observability and Reachability, a Geometric Point of View. In: Belyaev, A., Irschik, H., Krommer, M. (eds) Mechanics and Model-Based Control of Advanced Engineering Systems. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1571-8_29
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DOI: https://doi.org/10.1007/978-3-7091-1571-8_29
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