Summary
Carleman linearization is used to transform a polynomial control system with output, defined on n-dimensional space, into a linear or bilinear system evolving in the space of infinite sequences. Such a system is described by infinite matrices with special properties. Linear observability of the original system is studied. It means that all coordinate functions can be expressed as linear combinations of functions from the observation space. It is shown that this property is equivalent to a rank condition involving matrices that appear in the Carleman linearization. This condition is equivalent to observability of the first n coordinates of the linearized system.
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Mozyrska, D., Bartosiewicz, Z. (2008). Carleman Linearization of Linearly Observable Polynomial Systems. In: Sarychev, A., Shiryaev, A., Guerra, M., Grossinho, M.d.R. (eds) Mathematical Control Theory and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69532-5_17
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