Adaptive Control Using Fixed Point Transformations for Nonlinear Integer and Fractional Order Dynamic Systems

Abstract

In this paper an automatic learning based adaptive approach elaborated for the control of nonlinear dynamic systems is analyzed from various points of view concerning the imprecision and incompleteness of the available system model used by the controller. The proposed approach is compared to the most elaborated classical techniques using Lyapunov functions and dynamic models that are exact in their mathematical form but imprecise in their parameters, yield globally and asymptotically stable solutions but do not allow the presence of permanent external perturbations. It is shown that the novel control allows both numerical imprecision and enduring external disturbances unknown by the controller, but generally cannot guarantee global stability. It is also shown that its simple structure makes it a prospective candidate for the control of fractional order dynamic phenomena in which the conventional techniques based on the application of integer order time-derivatives of quadratic Lyapunov functions have great difficulties. The possible implementation of the proposed method is mathematically tackled and expounded from various backgrounds as the application of Cauchy Sequences in complete, linear, normed metric (Banach) spaces, and the use of coupled differential equations that may also be obtained from simple quadratic Lyapunov functions the decreasing nature of which generally can be guaranteed only within bounded regions. The operation of the proposed method is illustrated by simulation examples made for integer and fractional order dynamical systems as examples.KeywordsLyapunov FunctionAdaptive ControlFractional OrderFractional DerivativeInput Multiple OutputThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.