Stability Region for Fractional-Order Linear System with Saturating Control

AbstractIn this paper we investigate the stability of the equilibrium solution of the νth order linear system of difference equations $(\Delta _{a + \nu - 1}^\nu y)(t) = \Lambda y(t + \nu - 1);t \in \mathbb{N}_a ,a \in \mathbb{R},and\Lambda \in \mathbb{R}^{p \times p} ,$ subject to the initial condition $y(a + \nu - 1) = y - 1,$, where 0 < ν < 1 and y−1 ∈ ℝp.

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Stability Analysis of a Fractional-Order Linear System Described by the Caputo–Fabrizio Derivative

Mathematics ◽

10.3390/math7020200 ◽

2019 ◽

Vol 7 (2) ◽

pp. 200 ◽

Cited By ~ 6

Author(s):

Hong Li ◽

Jun Cheng ◽

Hou-Biao Li ◽

Shou-Ming Zhong

Keyword(s):

Stability Analysis ◽

Linear System ◽

Fractional Order ◽

Stability Domain ◽

Order System ◽

Stability Conditions ◽

Order Linear ◽

Dynamic Matrix ◽

Sufficient Stability ◽

Sufficient Stability Conditions

In this paper, stability analysis of a fractional-order linear system described by the Caputo–Fabrizio (CF) derivative is studied. In order to solve the problem, character equation of the system is defined at first by using the Laplace transform. Then, some simple necessary and sufficient stability conditions and sufficient stability conditions are given which will be the basis of doing research of a fractional-order system with a CF derivative. In addition, the difference of stability domain between two linear systems described by two different fractional derivatives is also studied. Our results permit researchers to check the stability by judging the locations in the complex plane of the dynamic matrix eigenvalues of the state space.

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Robust stability of a class of uncertain fractional order linear systems with pure delay

Archives of Control Sciences ◽

10.1515/acsc-2015-0011 ◽

2015 ◽

Vol 25 (2) ◽

pp. 177-187 ◽

Cited By ~ 2

Author(s):

Mikołaj Busłowicz ◽

Andrzej Ruszewski

Keyword(s):

Linear Systems ◽

Fractional Order ◽

Robust Stability ◽

Complex Plane ◽

Stability Region ◽

The State ◽

Interval Matrix ◽

State Matrix ◽

Order Linear ◽

Pure Delay

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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