# Extended Hierarchical Fuzzy Interpreted Petri Net

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Refinement of Places and Transitions

#### 1.2. Reduction

#### 1.3. Formal Description of the Hierarchical PNs

#### 1.4. Hierarchical Nets in Industrial Standards

#### 1.5. Methodology of Dealing with Complex Systems

#### 1.6. Recent Studies

#### 1.7. Paper Scope and Organization

- (a)
- Concept of a macroplace that can have several input, output and input-output places;
- (b)
- Functionality of macroplace instances;
- (c)
- Formal definition of HFIPN;
- (d)
- Concept and a definition of a hierarchy graph displaying a hierarchical structure and allowing a quick access to all subnets in an implementation version;
- (e)
- Formal algebraic representation of HFIPN;
- (f)
- Conversion of HFIPN to its flat version;
- (g)
- Formal way to sum any two subnets.

#### 1.8. Comparison to Similar Solutions

## 2. The Formal Basis and the Concept of FIPN

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. New HFIPN Concept and Its Definitions

#### 3.1. The Concept of HFIPN

#### 3.2. Formal Description of HFIPN

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

#### 3.3. The Hierarchy in HFIPN

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

#### 3.4. The Algebraic Representation

**Definition**

**13.**

- $\ast {\mathit{M}}_{\mathit{i}}$ is the vector of the length $1\times \ast {a}_{i}$, holding the current marking of places, where $\ast {a}_{i}=card(\ast {P}_{i})$;
- $\ast {\mathit{M}}_{\mathit{i}}^{\prime}$ is the vector of the length $1\times \ast {a}_{i}$, holding the new marking of places;
- $\ast {\mathit{U}}_{\mathit{i}}$ is the vector of the length $1\times \ast {b}_{i}$, in which the given coefficient is equal to one if it corresponds to the enabled transition by the marking $\ast {\mathit{M}}_{\mathit{i}}$ in the subnet, where $\ast {b}_{i}=card(\ast {T}_{i}$);
- $\Delta \ast {\Theta}_{i}$ is the vector of the length $1\times \ast {b}_{i}$, in which the coefficient $\Delta {\theta}_{j}$ ($1\le j\le \ast {b}_{i}$) describes the increment in the degree to which the condition corresponding to the transition ${t}_{j}$ (${t}_{j}\in \ast {T}_{i}$) is satisfied;
- $\ast {\mathit{K}}_{\mathbf{i}}$ is the vector of the length $1\times \ast {a}_{i}$, holding the places capacity.

#### 3.5. HFIPN-SML Tool

- (a)
- the creation of a net graph,
- (b)
- the use of a hierarchical structure including macroplace instances,
- (c)
- the automatic code generation in Structured Text (ST) language for PLC controllers based on a non-hierarchical graph,
- (d)
- the hierarchy graph presentation,
- (e)
- displaying all vectors and matrices from algebraic representation,
- (f)
- automatic and step simulations of a net operation,
- (g)
- monitoring the operation of a program generated based on a non-hierarchical net.

## 4. Exemplary Application

## 5. Conclusions

- (a)
- Concept of a macroplace that can have several input, output and input-output places;
- (b)
- Functionality of a macroplace instance;
- (c)
- Formal definition of HFIPN;
- (d)
- Concept and a definition of a hierarchy graph displaying a hierarchical structure and allowing a quick access to all subnets in an implementation version;
- (e)
- Formal algebraic representation of HFIPN;
- (f)
- Conversion of HFIPN to its flat version;
- (g)
- Formal description of the combination of any two subnets in the hierarchical net.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**HFIPN graph with the macroplace having two input places, two output places and one input-output place.

**Figure 6.**The net with two macroplaces of the same type, in which input and output variables are not shared.

**Figure 7.**The application of two concepts of macroplace instances: (

**a**) shared variables, (

**b**) unshared variables.

**Figure 9.**The hierarchy graph for the subnet from Figure 8.

**Figure 11.**The scheme of the system to mix three components: two liquids and soluble bricks. Based on [64].

**Figure 15.**The module checking whether the liquids ${L}_{1}$ and ${L}_{2}$ are transferred to the mixer at a similar speed.

HFIPN | GrafcetSFC | SIPN | Other Fuzzy PNs | |
---|---|---|---|---|

hierarchy/modularization | ✓ | ✓ | ✓ | ✗ |

analogue signals | ✓ | ✗ | ✗ | ✓ |

software tool support | ✓ | ✓ | ✓ | ✗ |

modeling of resources using a net structure | ✓ | ✗ | ✗ | ✗ |

automatic executable code generation | ✓ /✗ | ✓ | ✓ | ✗ |

automatic investigation of properties | ✓ /✗ | ✓ /✗ | ✓ | ✗ |

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Markiewicz, M.; Gniewek, L.; Warchoł, D.
Extended Hierarchical Fuzzy Interpreted Petri Net. *Sensors* **2021**, *21*, 8433.
https://doi.org/10.3390/s21248433

**AMA Style**

Markiewicz M, Gniewek L, Warchoł D.
Extended Hierarchical Fuzzy Interpreted Petri Net. *Sensors*. 2021; 21(24):8433.
https://doi.org/10.3390/s21248433

**Chicago/Turabian Style**

Markiewicz, Michał, Lesław Gniewek, and Dawid Warchoł.
2021. "Extended Hierarchical Fuzzy Interpreted Petri Net" *Sensors* 21, no. 24: 8433.
https://doi.org/10.3390/s21248433