Robustness Margin in Linear Time Invariant Fractional Order Systems

2010 ◽  
Vol 43 (21) ◽  
pp. 198-203 ◽  
Author(s):  
Kamran Akbari Moornani ◽  
Mohammad Haeri
Keyword(s):  
Fractional Order ◽  
Linear Time ◽  
Fractional Order Systems ◽  
Linear Time Invariant ◽  
Time Invariant

scholarly journals Optimal Control Problem Investigation for Linear Time-Invariant Systems of Fractional Order with Lumped Parameters Described by Equations with Riemann-Liouville Derivative

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
V. A. Kubyshkin ◽  
S. S. Postnov
Keyword(s):  
Fractional Order ◽  
Caputo Derivative ◽  
Linear Time ◽  
Fractional Order Systems ◽  
Linear Time Invariant ◽  
Liouville Derivative ◽  
Lumped Parameters ◽  
Linear Time Invariant Systems ◽  
Time Invariant ◽  
Problem Setting

This paper studies two optimal control problems for linear time-invariant systems of fractional order with lumped parameters whose dynamics is described by equations which contain Riemann-Liouville derivative. The first problem is to find control with minimal norm and the second one is to find control with minimal control time at given restriction for control norm. The problem setting with nonlocal initial conditions is considered which differs from other known settings for integer-order systems and fractional-order systems described in terms of equations with Caputo derivative. Admissible controls are allowed to belong to the class of functions which arep-integrable on half segment. The basic investigation approach is the moment method. The correctness and solvability of moment problem are validated for considered problem setting for the system of arbitrary dimension. It is shown that corresponding conditions are analogous to those derived for systems which are described in terms of equations with Caputo derivative. For several particular cases of one- and two-dimensional systems the posed problems are solved explicitly. The dependencies of basic values from derivative index and control time are analyzed. The comparison is performed of obtained results with known results for analogous integer-order systems and fractional-order systems which are described by equations with Caputo derivative.

scholarly journals Extending the Root-Locus Method to Fractional-Order Systems

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Farshad Merrikh-Bayat ◽  
Mahdi Afshar
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Linear Time ◽  
Rational Functions ◽  
Fractional Order Systems ◽  
Root Locus ◽  
Linear Time Invariant ◽  
Locus Method ◽  
Linear Time Invariant Systems ◽  
Time Invariant

The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variables. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

Geometrical design of fractional PDμ controllers for linear time-invariant fractional-order systems with time delay

2019 ◽  
Vol 233 (7) ◽  
pp. 815-829
Author(s):  
Adrián Josué Guel-Cortez ◽  
César-Fernando Méndez-Barrios ◽  
Emilio Jorge González-Galván ◽  
Gilberto Mejía-Rodríguez ◽  
Liliana Félix
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Linear Time ◽  
Simple Procedure ◽  
Geometric Approach ◽  
Fractional Order Systems ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Single Output ◽  
Time Invariant

This article presents a simple procedure that allows a practical design of fractional –[Formula: see text] controllers for single-input single-output linear time-invariant fractional-order systems subject to a constant time delay. The methodology is based on a geometric approach, which provides practical guidelines to design stabilizing and non-fragile PDμ controllers. The simplicity of the proposed approach is illustrated by considering several numerical examples encountered in the control literature. Moreover, with the aim of showing the performance of the PDμ over a classical PD controller, both controllers were implemented at the end of the article in an experimental test-bench consisting a teleoperated robotic system.

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