Abstract
A new approach is developed to the analysis and synthesis of non-linear discrete control systems x[ k + 1] = f ( k, x [ k], u [ k]), x ϵ R n , u ϵ R m first proposed in /1/ for continuous non-linear systems. The underlying idea of the approach is to redefine the addition of state and control vectors and multiplication of vectors by scalars in such a way that the system becomes linear in the new linear space. As an application, a description is given of a class of non-linear control systems which are isomorphic to their linear approximations, and explicit formulae for this isomorphism are presented. This makes it possible to construct a control with prescribed dynamic characteristics for the linear approximation system, using the well-developed theory of the linear case; this control is then converted via the isomorphism into a control for the non-linear system, generating the required closed-loop dynamics of the system, by introducing linear feedback that compensates for the non-linearity of the open-loop system.
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