Coprime fractional representations and stability of non-linear feedback systems

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Abstract
Fractional representation methods for finite-dimensional linear time-invariant (FDLTI) systems have been developed in an algebraic framework of principal ideal domains. The concept of coprime fractional representations has been generalized to discrete-time non-linear systems by Hammer (1985, 1987), by viewing the Bezout identity as the basis of the definition of coprimeness. In this paper, we view the role of (right) coprime fractional representation in representing the set of all stable input-output pairs of a (possibly unstable) non-linear system. This notion of coprimeness is shown to be equivalent to that of FDLTI systems. Using coprime fractional representations of non-linear systems, we obtain a characterization of the stability of non-linear feedback systems. This characterization is similar to that obtained in the case of FDLTI systems. We also obtain a parameterization of all non-linear and time-varying stabilizing controllers for linear (possibly time-varying) plants which has the same form as the Youla parameterization for FDLTI systems.