The Initialization Response of Multi-Term Linear Fractional-Order Systems With Constant History Functions

In this paper, the distinction between an operator’s historical initial condition function, the consequential initialization function of the operator, and the resulting initialization response of an entire system, is discussed. The single term and two-term differential equation results with constant history functions from earlier studies are reviewed. A three-term linear fractional-order differential equation with constant history function is studied next. This system is solved by using the proper Laplace transforms for the fractional-order derivatives. The paper then presents the initialization responses for multi-term linear fractional-order systems with commensurate orders that have had arbitrarily-long constant displacements in negative time. Results for short-times and for long-times are provided. These results are obtained by using the proper Laplace transform for the fractional-order derivatives. Using the results of this paper, the initialization response of any linear, commensurate-order, fractional-order system, with arbitrarily-long constant displacements in negative time can be determined.