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The four illustrative examples demonstrated the application of the values set concept and the zero exclusion condition for the families of fractional order polynomials with multilinear uncertainty structure, polynomial uncertainty structure, general uncertainty structure, and for the family of the fractional order retarded quasi-polynomials.

The obtained results showed the effectivity of the method for robust stability analysis of fractional order polynomials with various complex uncertainty structures. The potential directions for future research can be seen in robust stability analysis of e.

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The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS One. Published online Jun Xiaosong Hu, Editor. Author information Article notes Copyright and License information Disclaimer. Competing Interests: The authors have declared that no competing interests exist.

Received Feb 2; Accepted Jun This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order quasi- polynomials with complicated uncertainty structure. Introduction Fractional order control represents promising and attractive research topic, which has been widely studied recently.

Standard classification for integer order systems is [ 33 ], [ 65 ], [ 66 ]: Independent uncertainty structure called interval one for Q in the shape of a box Affine linear uncertainty structure called polytopic one for Q in the shape of a polytope Multilinear uncertainty structure Polynomial polynomic uncertainty structure General uncertainty structure On top of that, so-called single parameter uncertainty is a special case, which can be seen as the simplest one despite the structure itself can be formally affine linear or even more complicated.

Value sets and zero exclusion condition As mentioned above, the complicated structures of uncertainty suffer from the lack of suitable techniques for robust stability analysis. Open in a separate window. Fig 1. Value sets of the family of fractional order polynomials with multilinear uncertainty structure Eq Fig 4.

## Robust Control Design A Polynomial Approach

Value sets of the family of fractional order retarded quasi-polynomials Eq Illustrative examples In order to show the practical applicability of the graphical approach to robust stability analysis discussed hereinbefore, four illustrative examples with families of fractional order quasi- polynomials are presented in this Section. Fig 2. Value sets of the family of fractional order polynomials with polynomial uncertainty structure Eq Fig 3. Value sets of the family of fractional order polynomials with general uncertainty structure Eq Conclusion This article was focused on a graphical approach to robust stability investigation for families of fractional order polynomials or even quasi-polynomials with complicated uncertainty structure.

Data Availability All relevant data are within the paper. References 1. Oldham KB, Spanier J. New York—London: Academic Press; Miller KS, Ross B. Fractional Differential Equations. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation. Mathematical Problems in Engineering. Some applications of fractional order calculus. Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; Dordrecht, Netherlands: Springer; Magin RL.

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Fractional Calculus in Bioengineering. High Tatras, Slovakia; Magin RL, Ovadia M. Modeling the cardiac tissue electrode interface using fractional calculus. Journal of Vibration and Control. Perdikaris P, Karniadakis GE. Annals of Biomedical Engineering. On the Fractional Order Model of Viscoelasticity. Mechanics of Time-Dependent Materials.

Heymans N. Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state. Fractional-order chaotic systems. Palma de Mallorca, Spain; — Control and switching synchronization of fractional order chaotic systems using active control technique. Journal of Advanced Research. Li C, Chen G. Chaos in the fractional order Chen system and its control. Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana—Baleanu derivatives with fractional order.

Electronic realization of the fractional-order systems. Acta Montanistica Slovaca. Active and Passive Electronic Components. Fractional-order modeling and State-of-Charge estimation for ultracapacitors. Journal of Power Sources. Fractional order control of a hexapod robot.

Nonlinear Dynamics. Experimental signal analysis of robot impacts in a fractional calculus perspective. Ostalczyk P, Stolarski M. Asian Journal of Control. London, UK: Springer; Das S, Pan I. Abstract: The polynomial discrete-time systems are the type of systems where the dynamics of the systems are described in polynomial forms. This system is classified as an important class of nonlinear systems due to the fact that many nonlinear systems can be modelled as, transformed into, or approximated by polynomial systems.

The focus of this thesis is to address the problem of controller design for polynomial discrete-time systems. The main reason for focusing on this area is because the controller design for such polynomial discrete-time systems is categorised as a difficult problem. This is due to the fact that the relation between the Lyapunov matrix and the controller matrix is not jointly convex when the parameter-dependent or state-dependent Lyapunov function is under consideration. Therefore the problem cannot possibly be solved via semidefinite programming SDP. In light of the aforementioned problem, we establish novel methodologies of designing controllers for stabilising the systems both with and without H-infinity performance and for the systems with and without uncertainty.

Two types of uncertainty are considered in this research work; 1. Polytopic uncertainty, and 2. Norm-bounded uncertainty. A novel methodology for designing a filter for the polynomial discrete-time systems is also developed.

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We show that through our proposed methodologies, a less conservative design procedure can be rendered for the controller synthesis and filter design. In particular, a so-called integrator method is proposed in this research work where an integrator is incorporated into the controller and filter structures. Quite in the spirit of the Polynomial Toolbox the coefficients are assumed to be polynomial functions of the uncertain parameter and can be naturally passed to the macro stabint which returns the margins within which the family thus described remains stable.

Interval Polynomials Another important class of uncertain systems is described by interval polynomials with independent uncertainties in the coefficients.