Baramov, L. and Kimura, H.
Nonlinear J-lossless conjugation and factorization.
International Journal of Robust and Nonlinear Control, 6, .
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A new definition of nonlinear local J-lossless factorization is introduced, which plays a crucial role in nonlinear $H^\infty$ control theory. Sufficient (and in two special cases also necessary) conditions for the existence of this factorization and state-space formulae of the factor systems are given here. The main tools for the J-lossless factorization are the local right and left J-lossless conjugations, introduced in this paper. The former corresponds to the standard linear J-lossless conjugation, while the latter has no counterpart in the linear theory where it is completely dual to the former one and hence conceptually redundant. In the nonlinear case, however, this duality is much weaker and therefore the left J-lossless conjugation is essential for solving the local J-lossless factorization for unstable systems. This factorization requires a transformation of the given system to a special form and solving two independent Hamilton-Jacobi partial differential equations. Solutions of the two Hamilton-Jacobi equations have to satisfy a simple coupling condition.
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