A link between H∞, control of a discrete-time nonlinear system and its linearization

This paper studies relations between nonlinear H_∞, control of a discrete-time nonlinear system and H_∞ control of the linearized system. It is shown that an input-output discrete-time nonlinear system has a local L₂-gain γ > 0, with internal stability if its linearized system is asymptotically stable and has an L2-gain strictly less than γ. Using this foundamental result, it is further proved that the existence of a local solution to the problem of disturbance attenuation with internal stability for nonlinear systems is guaranteed by the solution of the corresponding H_∞ control problem for the linearized systems. All the results are established directly via the simple and intuitive linearization technique which provides, in contrast to the existing approach presented in [7], an appealing alternative to the discrete-time nonlinear H_∞ control problem.