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## Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model

### Abstract

Abstract The stability problems of fractional discrete-time linear scalar systems described by the new model are considered. Using the classical D-partition method, the necessary and sufficient conditions for practical stability and asymptotic stability are given. The considerations are il-lustrated by numerical examples.
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## Stability regions of closed loop system with time delay inertial plant of fractional order and fractional order PI controller

### Abstract

The paper presents the stability problem of control systems composed of a fractional-order PI controller and a inertial plant of a fractional order with time delay. Simple and efficient computational method for determining stability regions in the controller and plant parameters space is given. Knowledge of these regions permits tuning of the fractional-order PI controller. The method proposed is based on the classical D-partition method.
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## Robust stability of a class of uncertain fractional order linear systems with pure delay

### Abstract

Abstract The paper considers the robust stability problem of uncertain continuous-time fractional order linear systems with pure delay in the following two cases: a) the state matrix is a linear convex combination of two known constant matrices, b) the state matrix is an interval matrix. It is shown that the system is robustly stable if and only if all the eigenvalues of the state matrix multiplied by delay in power equal to fractional order are located in the open stability region in the complex plane. Parametric description of boundary of this region is derived. In the case a) the necessary and sufficient computational condition for robust stability is established. This condition is given in terms of eigenvalue-loci of the state matrix, fractional order and time delay. In the case b) the method for determining the rectangle with sides parallel to the axes of the complex plane in which all the eigenvalues of interval matrix are located is given and the sufficient condition for robust stability is proposed. This condition is satisfied if the rectangle multiplied by delay in power equal to fractional order lie in the stability region. The considerations are illustrated by numerical examples.
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