Abstract
Let Pfaffian system ω define an intrinsically nonlinear control system on manifold M that is invariant under the free, regular action of a Lie group G. The problem of identifying and constructing static feedback linearizable G-quotients of ω was solved in De Doná et al. (2016). Building on these results, the present paper proves that the trajectories of ω can often be expressed as the composition of the trajectories of a static feedback linearizable quotient control system, ω/G, on quotient manifold M/G, and those of a separate control system, γ G, evolving on a principal G-bundle over a jet space. Furthermore, we point out that ω may not only have a static feedback linearizable quotient, ω/G but additionally, γ G itself may possess a static feedback linearizable reduction as well. This enables one to express the trajectories of an intrinsically nonlinear control system as the composition of the trajectories of static feedback linearizable control systems, thereby providing a geometric criterion for the explicit integrability of intrinsically nonlinear systems. Moreover, special integrability properties arise when G is solvable. Examples are presented in which the above phenomena are explicitly demonstrated. An important aspect of the examples is that they gather evidence for the conjecture that our sufficient conditions for explicit integrability are also necessary.
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Notes
We will say that (2) is not explicitly integrable; see Definition 1.
The term “cascade static feedback linearizable” will be elucidated in the next section.
The notion of “reduction” here is explained in Section 2.1.
4Sometimes “partial prolongation” is used as a noun as it is here.But sometimes, it is used as a verb as in “a partial prolongationβ 〈1〉has been executed to produce β 〈0,1〉.”
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Vassiliou, P.J. Cascade Linearization of Invariant Control Systems. J Dyn Control Syst 24, 593–623 (2018). https://doi.org/10.1007/s10883-017-9389-0
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DOI: https://doi.org/10.1007/s10883-017-9389-0
Keywords
- Lie symmetry
- Contact geometry
- Static feedback linearization
- Dynamic feedback linearization
- Trajectory decomposition
- Integrability