Non-causal Fliess operators and their shuffle algebra - Gray, W. S.; Wang, Y. (International journal of control, 2008)
Fliess operators as a class of non-linear operators have been well studied in several respects. They have a well developed realization theory and convenient representations in terms of directed infinite products of exponential Lie series. Their interconnection as subsystems has been studied, as has their relationship to rational systems. They find applications in such diverse areas as discretization methods for controls systems, optimal control, neural network analysis, and the numerical solution of stochastic differential equations. One issue concerning Fliess operators, however, that has received little attention is their possible generalization to the non-causal case. Examples of such operators appear implicitly in the literature addressing Hilbert adjoints of causal non-linear operators and system inversion for the purpose of output tracking. But a general, systematic treatment of the subject has not appeared. In this paper, a non-causal extension of a Fliess operator is developed with the primary focus being on local convergence, continuity, the associated shuffle algebra, and computing adjoint operators.