# Non-Statistical Method for Validation the Time Characteristics of Digital Control Systems with a Cyclic Processing Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

- parallel processing of multiple tasks on same computing resource;
- significant change in the amount of information at the input and output of a component (for example, when a component compresses information);
- heterogeneity of data in a digital control system, in contrast to the information in streaming systems; it means that each element of data (bit) has its own value and can be processed according to a proper algorithm.

- comparing the network calculus computed delays with delays obtained by the statistical methods;
- calculation of the flow envelope and service curves from experimental data.

## 2. The Structure of a Typical Industrial Automation Control System

- Equipment embedded controllers and low-level data gateways—level G (1);
- Data storage and data processing servers—level S (2);
- Human-machine interface components—level Z (3).

**Definition**

**1.**

## 3. Application of Network Calculus to Evaluate Control System Time Characteristics

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 4. Network Calculus Main Curves Estimation Approaches

#### 4.1. Calculation of the Flow Envelope Based on Experimental Data

#### 4.2. Maximum and Minimum Service Curves Calculation Approaches Based on the Experimental Data

**Proposition**

**1.**

**Proof.**

## 5. I&C System Model and CS Time Characteristics Validation

## 6. Network Calculus Method Application Verification for System’s Time Characteristics Estimation

#### 6.1. Reference Data and Verification Procedure

#### 6.2. Comparing the Network Calculus and Statistical Results

- delay ($D$) calculated by the network calculus (5) using experimental maximum and minimum service curves;
- maximum measured delay in the sample (${D}_{x}$);
- ratio of $D/{D}_{x}$;
- dependence of the maximum calculated delay on the sample size ($L)$ and distribution.

#### 6.3. Comparison of Network Calculus and Statistical Calculation Results

## 7. Practical Example of Calculating the Delay for a Real I&C System

## 8. Results and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Dependence of the maximum delay estimate on the sample size for minimum service curve ${\beta}^{\prime}$, $N$ is a sample size. The symbol * means the data is measured in a real computational environment; the data without * are generated with a random number generator.

**Figure 5.**Dependence of the maximum delay estimation on the sample size for the maximum service curve ${\mathsf{\gamma}}^{\prime},N\mathrm{is}\mathrm{a}\mathrm{sample}\mathrm{size}$. The symbol * means the data is measured in a real computational environment; the data without * are generated with a random number generator.

**Figure 6.**The ratio of the measured and calculated delay for the Rayleigh distribution with parameters $\mu =0,\sigma =300B$ depending on the magnitude of a single outlier in $\mathsf{\sigma}$ for several fixed sets of sample lengths.

**Figure 7.**Experimental Network Calculus curves for a quasi-normal distributed sample with $\mu =500,\mathsf{\sigma}=100$; size of $\mathrm{L}=10$. Arrow 1 points to the maximum delay for the service curve (7), arrow 2 points to the maximum delay estimation for the service curve (11).

**Figure 8.**An example of two cumulative flows (flow 1, flow 2) and envelopes (${\mathsf{\alpha}}_{1},{\mathsf{\alpha}}_{2}$) for them. Flow 1 has outliers in the data (point 4 on the X-axis), and flow 2 has no significant outliers.

**Figure 9.**The empirical probability density function of the cycle time of the Z component. The solid line shows the smoothed approximation of the distribution.

Parameter | Value |
---|---|

${D}_{x}$ | 0.37 |

$p\left({D}_{x}\right)$ | $~1$ |

${D}_{}$ | 5.1 |

$p\left({D}_{}\right)$ | $~1$ |

$D/{D}_{x}$ | 4.9 |

${D}^{\prime}$ | 0.32 |

$p\left({D}^{\prime}\right)$ | 0.87 |

${D}^{\prime}/{D}_{x}$ | 0.3 |

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Promyslov, V.; Semenkov, K.
Non-Statistical Method for Validation the Time Characteristics of Digital Control Systems with a Cyclic Processing Algorithm. *Mathematics* **2021**, *9*, 1732.
https://doi.org/10.3390/math9151732

**AMA Style**

Promyslov V, Semenkov K.
Non-Statistical Method for Validation the Time Characteristics of Digital Control Systems with a Cyclic Processing Algorithm. *Mathematics*. 2021; 9(15):1732.
https://doi.org/10.3390/math9151732

**Chicago/Turabian Style**

Promyslov, Vitaly, and Kirill Semenkov.
2021. "Non-Statistical Method for Validation the Time Characteristics of Digital Control Systems with a Cyclic Processing Algorithm" *Mathematics* 9, no. 15: 1732.
https://doi.org/10.3390/math9151732