TABLE OF CONTENTS The invertibility of dynamical systems was first widely investigated in mechanics, in order to find the forces that cause the observable behavior of mechanical systems [Santilli (1978)]. In the general case, this meaning of the term leads to the concept of left invertibility. A system is said to be left invertible if a unique control function exists and can be found for the given system model, initial state, and output function.
So, on one hand, the left invertibility condition for a dynamical control system is the condition for uniqueness of the control function that provides the desired output behavior [Zadeh and Desoer (1963)]. On the other hand, right invertibility is the necessary condition for the existence of a control function such that the output behavior is an arbitrarily assigned smooth function [Brockett and Mesarovic (1965); Porter (1970)].
From a theoretical viewpoint, the desired input-controlled output map can be provided by a controller in the form of a serial system of the reference model and the right inverse system. Control of nonlinear systems through the use of their inverse dynamics is a topic that has received much attention [Boychuk (1966); Silverman (1969); Porter (1970); Popov and Krutko (1979); Petrov and Krutko (1980); Singh (1980); Slotine and Li (1991)]. The application of the inversion method in the discrete-time nonlinear control system synthesis problem was discussed in [Kotta (1995)]. Note that the well known input-output linearization technique for nonlinear systems is...
