#
Further Remarks on Irrational Systems and Their Applications^{ †}

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**.**The Branch point (BP) or point of accumulation is defined as the point with the smallest magnitude for which a function is multivalued. Another definition would be: a branch point is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.

**Definition**

**3**

**.**An irrational system is a multi-valued transfer function $G(s)$ with one or more terms raised to the power $\alpha \in \mathbb{Q}$.

#### Origins and Connection with Fractional Calculus

- The network should contain only linear lumped elements. For instance, viscous dampers, springs, capacitors, or inductors.
- All initial conditions should be equal to zero.
- Elements in the network should have equal impedance value. For example, the tree-like network shown in Figure 1 contains only two linear operators ${\mathcal{L}}_{1}$ and ${\mathcal{L}}_{2}$, which have the same value throughout all the layers of the network.
- The network is one-dimensional and infinite.

**Hypothesis**

**1.**

**Hypothesis**

**2.**

## 3. Stability Analysis

**Theorem**

**1**

**.**A given multivalued transfer function is stable if and only if it has no pole in ${\mathbb{C}}_{+}$ and no branch points in ${\mathbb{C}}_{-}$. Here, ${\mathbb{C}}_{+}$ and ${\mathbb{C}}_{-}$ stand for the closed right half plane (RHP) and the open RHP of the first Riemann sheet in the complex plane, respectively.

**Example**

**1.**

**Example**

**2.**

## 4. Control Design

#### PD${}^{\mu}$ Control

**Remark**

**1.**

## 5. Applications

#### 5.1. Control of IS

#### 5.2. Bessel

#### 5.3. First Order IS

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

IS | Irrational system |

PD | Proportional derivative |

PI | Proportional integral |

BP | Branch point |

PID | Proportional integral derivative |

## Appendix A. Example 1

## Appendix B. Example 2

## References

- Guel-Cortez, A.J. Modeling and Control of Fractional-Order Systems. The Linear Systems Case. Ph.D. Thesis, CIEP-UASLP, San Luis, Mexico, 2018. [Google Scholar]
- Sen, M.; Hollkamp, J.P.; Semperlotti, F.; Goodwine, B. Implicit and fractional-derivative operators in infinite networks of integer-order components. Chaos Solitons Fractals
**2018**, 114, 186–192. [Google Scholar] [CrossRef] - Guel-Cortez, A.J.; Sen, M.; Goodwine, B. Closed form time response of an infinite tree of mechanical components described by an irrational transfer function. In Proceedings of the 2019 American Control Conference (ACC), Philadelphia, PA, USA, 10–12 July 2019; pp. 5828–5833. [Google Scholar]
- Guel-Cortez, A.J.; Kim, E. A Fractional-Order Model of the Cardiac Function. In Proceedings of the 13th Chaotic Modeling and Simulation International Conference, Florence, Italy, 9–12 June 2020; Springer: Berlin/Heidelberg, Germany, 2020; pp. 273–285. [Google Scholar]
- Leyden, K.; Sen, M.; Goodwine, B. Large and infinite mass–spring–damper networks. J. Dyn. Syst. Meas. Control
**2019**, 141, 061005. [Google Scholar] [CrossRef] - Ni, X.; Goodwine, B. Frequency Response and Transfer Functions of Large Self-similar Networks. arXiv
**2020**, arXiv:2010.11015. [Google Scholar] [CrossRef] - Mayes, J.; Sen, M. Approximation of potential-driven flow dynamics in large-scale self-similar tree networks. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2011**, 467, 2810–2824. [Google Scholar] [CrossRef] [Green Version] - Merrikh-Bayat, F.; Karimi-Ghartemani, M. On the essential instabilities caused by fractional-order transfer functions. Math. Probl. Eng.
**2008**, 2008, 419046. [Google Scholar] [CrossRef] - Guel-Cortez, A.J.; Sen, M.; Goodwine, B. Fractional- PD
^{μ}Controllers for Irrational Systems. In Proceedings of the 2019 International Conference on Control, Decision and Information Technologies, Paris, France, 23–26 April 2019. [Google Scholar] - Guel-Cortez, A.J.; Méndez-Barrios, C.F.; Kim, E.j.; Sen, M. Fractional-order controllers for irrational systems. IET Control Theory Appl.
**2021**, 15, 965–977. [Google Scholar] [CrossRef] - Cohen, H. Complex Analysis with Applications in Science and Engineering; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Harvey, C. Conformal mapping on Riemann Surfaces; Dover Publications, Inc.: New York, NY, USA, 1980. [Google Scholar]
- Needham, T. Complex Visual Analysis; Oxford University Press, Inc.: New York, NY, USA, 1997. [Google Scholar]
- Capoccia, M. Development and characterization of the arterial W indkessel and its role during left ventricular assist device assistance. Artif. Organs
**2015**, 39, E138–E153. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Piovesan, D.; Pierobon, A.; DiZio, P.; Lackner, J.R. Measuring multi-joint stiffness during single movements: Numerical validation of a novel time-frequency approach. PLoS ONE
**2012**, 7, e33086. [Google Scholar] [CrossRef] [Green Version] - Ni, X.; Goodwine, B. Frequency Response of Transmission Lines with Unevenly Distributed Properties with Application to Railway Safety Monitoring. arXiv
**2020**, arXiv:2012.09247. [Google Scholar] - Leyden, K.; Goodwine, B. Using fractional-order differential equations for health monitoring of a system of cooperating robots. In Proceedings of the 2016 IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, 16–21 May 2016; pp. 366–371. [Google Scholar]
- Leyden, K.; Sen, M.; Goodwine, B. Models from an implicit operator describing a large mass-spring-damper network. IFAC-PapersOnLine
**2018**, 51, 831–836. [Google Scholar] [CrossRef] - Shukla, A.; Prajapati, J. On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl.
**2007**, 336, 797–811. [Google Scholar] [CrossRef] - Sabatier, J.; Farges, C.; Tartaglione, V. Some alternative solutions to fractional models for modelling power law type long memory behaviors. Mathematics
**2020**, 8, 196. [Google Scholar] [CrossRef] [Green Version] - Sabatier, J. Some Proposals for a Renewal in the Field of Fractional behavior Analysis and Modelling. In Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA’21); Springer: Berlin/Heidelberg, Germany, 2022; pp. 1–25. [Google Scholar]
- Ramos-Avila, D.; Rodrıguez, C.; Hernández-Carrillo, J.; Guel-Cortez, A.; Sen, M.; Méndez-Barrios, C.; González-Galván, E.; Goodwine, B. Experiments with PD-controlled robots in ring formation. In Proceedings of the XXI Congreso Mexicano de Robótica–COMRob, Ciudad de Manzanillo, Colima, Mexico, 13–15 November 2019. [Google Scholar]
- Gryazina, E.N. The D-decomposition theory. Autom. Remote Control
**2004**, 65, 1872–1884. [Google Scholar] [CrossRef] - Hernández-Díez, J.E.; Méndez-Barrios, C.F.; Mondié, S.; Niculescu, S.I.; González-Galván, E.J. Proportional-delayed controllers design for LTI-systems: A geometric approach. Int. J. Control
**2018**, 91, 907–925. [Google Scholar] [CrossRef] - Barrios, C.F.M. Low-Order Controllers for Time-Delay Systems: An Analytical Approach. Ph.D. Thesis, Université Paris Sud-Paris XI, Bures-sur-Yvette, France, 2011. [Google Scholar]
- Abate, J.; Whitt, W. A unified framework for numerically inverting Laplace transforms. INFORMS J. Comput.
**2006**, 18, 408–421. [Google Scholar] [CrossRef] - Moslehi, L.; Ansari, A. Some remarks on inverse Laplace transforms involving conjugate branch points with applications. UPB Sci. Bull. Ser. A-Appl. Math. Phys.
**2016**, 78, 107–118. [Google Scholar]

**Figure 1.**Tree-like network of N layers that can be described by an ISs transfer function. In the network, it is necessary to have ${\mathcal{L}}_{\mathcal{1}}$ and ${\mathcal{L}}_{\mathcal{2}}$ to be linear operators. Note that all end-points ${x}_{out}$ are in the same position. The movement is in one-dimension.

**Figure 2.**Examples of the application of Hypothesis 1 and 2 in realistic scenarios. (

**a**) Model reduction of the cardiovascular system by an electrical system using a fractance. (

**b**) Ladder network description of mobile robots described by mechanical elements and driven by PID controls [6].

**Figure 3.**Example of the D-composition method. The method maps the complex plane stability region to the controller parameters’ plane. In this case, the plane has not stability boundary at $s\to \infty $.

**Figure 4.**Stability analysis of system (11). (

**a**) Stability region (gray) of the closed-loop system with $\mu =0.3$. (

**b**) Time response for control gains inside different regions on the parameter’s plane.

**Figure 5.**Stability analysis of system (13). (

**a**) Stability region (gray) of the closed-loop system with $\mu =0.4$. (

**b**) Time response for control gains inside different regions on the parameter’s plane.

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**MDPI and ACS Style**

Guel-Cortez, A.-J.; Méndez-Barrios, C.-F.; Torres-García, D.; Félix, L.
Further Remarks on Irrational Systems and Their Applications. *Comput. Sci. Math. Forum* **2022**, *4*, 5.
https://doi.org/10.3390/cmsf2022004005

**AMA Style**

Guel-Cortez A-J, Méndez-Barrios C-F, Torres-García D, Félix L.
Further Remarks on Irrational Systems and Their Applications. *Computer Sciences & Mathematics Forum*. 2022; 4(1):5.
https://doi.org/10.3390/cmsf2022004005

**Chicago/Turabian Style**

Guel-Cortez, Adrián-Josué, César-Fernando Méndez-Barrios, Diego Torres-García, and Liliana Félix.
2022. "Further Remarks on Irrational Systems and Their Applications" *Computer Sciences & Mathematics Forum* 4, no. 1: 5.
https://doi.org/10.3390/cmsf2022004005