# Investigating Data-Driven Systems as Digital Twins: Numerical Behavior of Ho–Kalman Method for Order Estimation

## Abstract

**:**

## 1. Introduction

## 2. Framework

- Data are aggregated for various cases (i.e., different materials)
- Ho–Kalman algorithm is applied to estimate the order of the system
- Plain estimation techniques are applied (i.e., mean least squares) to retrieve the transfer function(s)

- Sensors are used to detect the model that should be applied
- The controller is designed (i.e., Proportional–Integral–Derivative)

- Sensors are used to measure input and output of the system
- (optional) An observer is used to estimate the state (inner variables) of the system
- The control signal is generated and control is applied (these may be two different steps depending on the implementation)

## 3. Method

#### Comparison to Other Methods and Correlation to Information

## 4. Numerical Behaviour and Applicability

#### 4.1. Simple Numerical Examples

#### 4.2. Performance on Systems of Higher Order

#### 4.3. Non-Homogeneous Systems

## 5. Summary and Future Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Akaike information criterion (AIC) values as a function of the system order (for system of Equation (3)).

**Figure 4.**The rank of the responses matrices as a function of the responses matrix dimensions for the case of a first order system.

**Figure 5.**The rank of the responses matrices as a function of the responses matrix dimensions and the proximity of the poles in the case of a second order system. Noise is present.

**Figure 6.**The rank of the responses matrices as a function of the responses matrix dimensions, for the case of Equation 6, where ${f}_{0}\left[n\right]=sin\left[n\right]$ and the upper limit of the sum ${N}_{0}$ varies. The values of ${N}_{0}$ are shown within the plots.

**Figure 7.**The rank of the responses matrices as a function of the responses matrix dimensions, for the case of Equation 6, where ${f}_{0}\left[n\right]={e}^{-n}$ and the upper limit of the sum ${N}_{0}$ varies. The values of ${N}_{0}$ are shown above the plots.

Order of System | Determinant of Co-variance Matrix |
---|---|

2 | $0.00106278$ |

3 | $1.96022\times {10}^{-12}$ |

4 | $-5.85109\times {10}^{-26}$ |

5 | $-4.11625\times {10}^{-40}$ |

6 | $6.85315\times {10}^{-55}$ |

7 | $1.71849\times {10}^{-68}$ |

8 | $1.78185\times {10}^{-82}$ |

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**MDPI and ACS Style**

Papacharalampopoulos, A. Investigating Data-Driven Systems as Digital Twins: Numerical Behavior of Ho–Kalman Method for Order Estimation. *Processes* **2020**, *8*, 431.
https://doi.org/10.3390/pr8040431

**AMA Style**

Papacharalampopoulos A. Investigating Data-Driven Systems as Digital Twins: Numerical Behavior of Ho–Kalman Method for Order Estimation. *Processes*. 2020; 8(4):431.
https://doi.org/10.3390/pr8040431

**Chicago/Turabian Style**

Papacharalampopoulos, Alexios. 2020. "Investigating Data-Driven Systems as Digital Twins: Numerical Behavior of Ho–Kalman Method for Order Estimation" *Processes* 8, no. 4: 431.
https://doi.org/10.3390/pr8040431