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It has been recognized that using time-varying initialization functions tosolve the initial value problem of fractional-order systems (FOS) is bothcomplex and essential in defining the dynamical behavior of the states of FOSs.In this paper, we investigate the use of the initialization functions for thepurpose of estimating unknown parameters of linear non-commensurate FOSs. Inparticular, we propose a novel "pre-initial" process that describes the dynamiccharacteristic of FOSs before the initial state and consists of designing anappropriate time-varying initialization function that ensures accurateconvergence of the estimates of the unknown parameters. To do so, we propose anestimation technique that consists of two steps: (i) to design of practicalinitialization function that is output-dependent and which is employed; (ii) tosolve the joint estimation problem of both parameters and fractionaldifferentiation orders (FDOs). A convergence proof has been presented. Theperformance of the proposed method is illustrated through different numericalexamples. Potential applications of the algorithm to joint estimation ofparameters and FDOs of the fractional arterial Windkessel and neurovascularmodels are also presented using both synthetic and real data. The added valueof the proposed "pre-initial" process to solve the studied estimation problemis shown through different simulation tests that investigate the sensitivity ofestimation results using different time-varying initialization functions.

Contribution to the initialization of linear non-commensurate  fractional-order systems for the joint estimation of parameters and  fractional differentiation orders