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Nonlinear observability based on the concept of indistinguishability of states was introduced by Hermann and Krener in 1977. In this approach, the observability map needs to be tested for injectivity. In general, observability is difficult to verify for nonlinear systems. The well-known observability rank condition, which is based on the inverse function theorem, only makes a sufficient statement about a version of local observability.
The entries of the observability map consist of time derivatives of the output along the system’s dynamics. These derivatives can be expressed as Lie derivatives. For linear time-invariant systems, the number of output time derivatives required for a observability test is bounded with regard to the system’s dimension due to the Cayley–Hamilton theorem. In this case, the observability rank condition is both sufficient and necessary. Recently, it has been shown the the observability of polynomial systems can be decided with a finite number of steps using Hilbert's basis theorem. In this paper, the methodology developed for polynomial systems will be extended to rational systems.
