The paper is devoted to variable order estimation process when measurements are obtained in two different ways: directly and by lossy network. Since the problem of fractional order estimation is highly nonlinear, dual estimation algorithm based on Unscented Fractional Order Kalman filter has been used. In dual estimation process, state variable and order estimation have been divided into two sub-processes. For estimation state variables and variable fractional order, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system realized as an analog circuit.

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The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical scheme for a variable order derivatives based on matrix approach. Using this approach, the identity of the switching scheme and considered definition is derived. The switching scheme can be used as an interpretation of this type of definition. Paper presents also numerical examples for introduced methods. Finally, the idea and results of analog (electrical) realization of the switching fractional order integrator (of orders 0.5 and 1) are presented and compared with numerical approach. *

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In this paper, a switching strategy for recursive fractional variable-order derivative is proposed. This strategy can be interpreted as an explanation of order switching mechanism for this particular type of derivative. Additionally, important properties of variable fractional order derivatives, required for prove the main result, are introduced both in a difference equation and a matrix form. Duality between the recursive and standard variable-order derivative is detailed derived. Based on the switching scheme, an analog realization of the recursive variable-order derivative definition is presented. Experimental results obtained for the analog realization are compared to the numerical results.

This paper presents a method for including initial conditions in recursive constant and variable fractional-order derivatives. The initial conditions are assumed to be in the form of a time-constant function. The numerical scheme for solving fractional-order differential equations, given in the matrix form, is presented as well. Results obtained with the proposed numerical algorithm are compared with the analog realization of a fractional-order inertial system. For the constant-order case, analog models were built based on domino ladder approximations for 0.5 and 0.25 orders. For the variable-order case, the analog model was built based on the reductive-switching structure. Comparison with the physical systems shows the ability of the proposed method to describe the behavior of real fractional-order systems with initial conditions.

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