# Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Representation

**Remark**

**1.**

## 3. Model Order Reduction

## 4. Controllability and Observability Gramians for Discrete-Time Fractional-Order Systems

#### 4.1. Gramians in the Time Domain

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

#### 4.2. Gramians in the Frequency Domain

**Lemma**

**2.**

**Proof.**

#### 4.3. Frequency Weighted Gramians

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Remark**

**3.**

## 5. Simulation Examples

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**left**) Frequency responses for full- and reduced-order models and (

**right**) approximation errors.

**Figure 2.**(

**left**) Step responses for full- and reduced-order models and (

**right**) approximation errors.

**Figure 3.**(

**left**) Frequency responses for full- and reduced-order models and (

**right**) approximation errors.

**Figure 4.**(

**left**) Impulse responses for full- and reduced-order models and (

**right**) approximation errors.

**Figure 5.**(

**left**) Frequency responses for full- and reduced-order models and (

**right**) approximation errors.

$\mathbf{DCE}$ | ${\mathit{MSE}}_{\mathit{\omega}}$ | ${\mathcal{H}}_{\mathbf{\infty}}$ | ${\mathit{MSE}}_{\mathit{t}}$ | |
---|---|---|---|---|

BT | $22.2\times {10}^{-3}$ | $4.58\times {10}^{-4}$ | $53.0\times {10}^{-3}$ | $4.62\times {10}^{-4}$ |

FLBT | $1.88\times {10}^{-5}$ | $2.68\times {10}^{-10}$ | $88.8\times {10}^{-3}$ | $6.37\times {10}^{-6}$ |

TLBT | $28.1\times {10}^{-3}$ | $7.33\times {10}^{-4}$ | $54.9\times {10}^{-3}$ | $7.40\times {10}^{-4}$ |

FW | $4.52\times {10}^{-5}$ | $1.42\times {10}^{-9}$ | $83.7\times {10}^{-3}$ | $5.60\times {10}^{-6}$ |

${\mathit{MSE}}_{\mathit{\omega}}$ | ${\mathcal{H}}_{\mathbf{\infty}}$ | ${\mathit{MSE}}_{\mathit{t}}$ | |
---|---|---|---|

BT | $2.54\times {10}^{-4}$ | $0.053$ | $9.21\times {10}^{-8}$ |

FLBT | $3.46\times {10}^{-5}$ | $0.538$ | $3.23\times {10}^{-8}$ |

TLBT | $3.68\times {10}^{-5}$ | $0.385$ | $5.05\times {10}^{-14}$ |

FW | $4.31\times {10}^{-5}$ | $0.539$ | $2.24\times {10}^{-8}$ |

$\mathit{DCE}$ | ${\mathit{MSE}}_{\mathit{\omega}}$ | ${\mathcal{H}}_{\mathbf{\infty}}$ | |
---|---|---|---|

BT | $5.538$ | $29.93$ | $5.538$ |

FLBT | $4.94\times {10}^{-4}$ | $1.90\times {10}^{-7}$ | $11.12$ |

FW | $6.30\times {10}^{-4}$ | $3.14\times {10}^{-7}$ | $11.35$ |

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**MDPI and ACS Style**

Rydel, M.; Stanisławski, R.; Latawiec, K.J.
Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems. *Symmetry* **2019**, *11*, 258.
https://doi.org/10.3390/sym11020258

**AMA Style**

Rydel M, Stanisławski R, Latawiec KJ.
Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems. *Symmetry*. 2019; 11(2):258.
https://doi.org/10.3390/sym11020258

**Chicago/Turabian Style**

Rydel, Marek, Rafał Stanisławski, and Krzysztof J. Latawiec.
2019. "Balanced Truncation Model Order Reduction in Limited Frequency and Time Intervals for Discrete-Time Commensurate Fractional-Order Systems" *Symmetry* 11, no. 2: 258.
https://doi.org/10.3390/sym11020258