Comparison between two approximation methods of state space fractional systems

A large number of real dynamic systems are better characterized using a non-integer order dynamic model based on fractional order differential equations. Nevertheless, their analytical solutions are rather complicated. In this paper we present two alternative methods of simulation of the fractional order dynamical system in state space. We particularly present a comparison between two approximation methods. The first one is based on the differentiation operator approximation and thus uses the usual form of the state space representation D(α)(x)=Ax+BuD(α)(x)=Ax+Bu. The second method is based on the approximation of the integration operator and uses, in this case, the new representation w=AI(α)(w)+Buw=AI(α)(w)+Bu. The comparison between these two methods is made on the basis of initial and steady state approximation state and output errors which are characterized in this present paper. Numerical examples are also given to support this comparison.