Adaptive controller for systems of fractional dynamics based on robust fixed point transformations

In this paper a discrete time approximation of Caputo´s fractional order derivatives is used for modeling the dynamic behavior of hypothetical fractional order systems the appropriate responses of which that can directly be manipulated by some physical agents are some fractional order time-derivatives of their state variables. A possible generalization of the concept of “initial conditions” of the integer order systems is proposed as “preceding history” for fractional order ones. It is shown that the number of the independent data characteristic to the “preceding history” can be made independent of the order of derivation. It is shown that the discrete time approximation proposed makes it possible to interpret the order of derivation in a higher range than in the case of the original integral form of Caputo´s definition. By providing a simple analysis of the so obtained time-sequences it is shown that by manipulating the order of differentiation in this model both dissipative and unstable behavior can be modeled. The dissipative case corresponds to the presence of unmodeled internal degrees of freedom that are dynamically coupled to the directly controlled ones but cannot directly be controlled. The unstable case seems to be appropriate for modeling the behavior of systems coupled to some directly unmodeled exciting environment. For this purpose very simple mathematical estimations can be applied. The paradigm controlled is a fractional order Φ⁶ type Van der Pol oscillator that already obtained certain attention in the literature. It is shown that the simple fixed point transformations based adaptive control elaborated for integer order systems can be applied without any modification for fractional order ones.