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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

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**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