  It has been recognized that using time-varying initialization functions to
solve the initial value problem of fractional-order systems (FOS) is both
complex and essential in defining the dynamical behavior of the states of FOSs.
In this paper, we investigate the use of the initialization functions for the
purpose of estimating unknown parameters of linear non-commensurate FOSs. In
particular, we propose a novel "pre-initial" process that describes the dynamic
characteristic of FOSs before the initial state and consists of designing an
appropriate time-varying initialization function that ensures accurate
convergence of the estimates of the unknown parameters. To do so, we propose an
estimation technique that consists of two steps: (i) to design of practical
initialization function that is output-dependent and which is employed; (ii) to
solve the joint estimation problem of both parameters and fractional
differentiation orders (FDOs). A convergence proof has been presented. The
performance of the proposed method is illustrated through different numerical
examples. Potential applications of the algorithm to joint estimation of
parameters and FDOs of the fractional arterial Windkessel and neurovascular
models are also presented using both synthetic and real data. The added value
of the proposed "pre-initial" process to solve the studied estimation problem
is shown through different simulation tests that investigate the sensitivity of
estimation results using different time-varying initialization functions.
