scholarly journals Robust Stability of Fractional-Order Linear Time-Invariant Systems: Parametric versus Unstructured Uncertainty Models

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Radek Matušů ◽  
Bilal Şenol ◽  
Libor Pekař
Keyword(s):  
Fractional Order ◽  
Robust Stability ◽  
Linear Time ◽  
Parametric Uncertainty ◽  
Uncertainty Modeling ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Robust Stability Analysis ◽  
Uncertainty Models ◽  
Time Invariant

The main aim of this paper is to present and compare three approaches to uncertainty modeling and robust stability analysis for fractional-order (FO) linear time-invariant (LTI) single-input single-output (SISO) uncertain systems. The investigated objects are described either via FO models with parametric uncertainty, by means of FO unstructured multiplicative uncertainty models, or through FO unstructured additive uncertainty models, while the unstructured models are constructed on the basis of appropriate selection of a nominal plant and a weight function. Robust stability investigation for systems with parametric uncertainty uses the combination of plotting the value sets and application of the zero exclusion condition. For the case of systems with unstructured uncertainty, the graphical interpretation of the utilized robust stability test is based mainly on the envelopes of the Nyquist diagrams. The theoretical foundations are followed by two extensive, illustrative examples where the plant models are created; the robust stability of feedback control loops is analyzed, and obtained results are discussed.

Robust Stability Analysis of Distributed-Order Linear Time-Invariant Systems With Uncertain Order Weight Functions and Uncertain Dynamic Matrices

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Hamed Taghavian ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Weight Function ◽  
Robust Stability ◽  
Linear Time ◽  
Upper And Lower Bounds ◽  
Weight Functions ◽  
Order Linear ◽  
Linear Time Invariant ◽  
Robust Stability Analysis ◽  
Time Invariant ◽  
Distributed Order

Bounded-input bounded-output (BIBO) stability of distributed-order linear time-invariant (LTI) systems with uncertain order weight functions and uncertain dynamic matrices is investigated in this paper. The order weight function in these uncertain systems is assumed to be totally unknown lying between two known positive bounds. First, some properties of stability boundaries of fractional distributed-order systems with respect to location of eigenvalues of dynamic matrix are proved. Then, on the basis of these properties, it is shown that the stability boundary of distributed-order systems with the aforementioned uncertain order weight functions is located in a certain region on the complex plane defined by the upper and lower bounds of the order weight function. Thereby, sufficient conditions are obtained to ensure robust stability in distributed-order LTI systems with uncertain order weight functions and uncertain dynamic matrices. Numerical examples are presented to verify the obtained results.

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.

scholarly journals Value-Set-Based Approach to Robust Stability Analysis for Ellipsoidal Families of Fractional-Order Polynomials with Complicated Uncertainty Structure

2019 ◽  
Vol 9 (24) ◽  
pp. 5451 ◽  
Author(s):  
Radek Matušů ◽  
Bilal Şenol ◽  
Libor Pekař
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Robust Stability ◽  
Linear Time ◽  
Complex Structure ◽  
Parametric Uncertainty ◽  
Value Set ◽  
Uncertainty Structure ◽  
Robust Stability Analysis ◽  
Exclusion Condition

This paper presents the application of a value-set-based graphical approach to robust stability analysis for the ellipsoidal families of fractional-order polynomials with a complex structure of parametric uncertainty. More specifically, the article focuses on the families of fractional-order linear time-invariant polynomials with affine linear, multilinear, polynomic, and general uncertainty structure, combined with the uncertainty bounding set in the shape of an ellipsoid. The robust stability of these families is investigated using the zero exclusion condition, supported by the numerical computation and visualization of the value sets. Four illustrative examples are elaborated, including the comparison with the families of fractional-order polynomials having the standard box-shaped uncertainty bounding set, in order to demonstrate the applicability of this method.

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