The 21st Century Systems: An Updated Vision of Continuous-Time Fractional Models

This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is shown that such systems can have integer and fractional components. From the integer component we deduce the stability. The fractional order component is always stable. The initial-condition problem is analyzed and it is verified that it depends on the structure of the system. For a correct definition and backward compatibility with classic systems, suitable fractional derivatives are also introduced. The Grünwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated.