# On the Linear–Quadratic–Gaussian Control Strategy for Fractional-Order Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. Controller Design

## 4. Output Variance

## 5. Simulation Results

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Realization of the output (

**left**) and control signal (

**right**) for $\lambda =0$ and $\Delta \alpha =0$.

**Figure 6.**Realization of the output (

**left**) and control (

**right**) signal for $\lambda =0.1$ and $\Delta \alpha =0.35$.

**Figure 7.**Realization of the output (

**left**) and control (

**right**) signal for $\lambda =0.5$ and $\Delta \alpha =0.66$.

**Figure 8.**High performance control (minimal output variance—reflected by 2${\sigma}_{y}$) against corresponding $\lambda $ for various $\Delta \alpha $ values (

**left**), and the $\Delta \alpha $ as a function of $\lambda $ penalty factor.

**Figure 12.**Realization of the output (

**left**) and control (

**right**) signal for $\lambda =0$ and $\Delta \alpha =0$.

**Figure 13.**Realization of the output (

**left**) and control (

**right**) signal for $\lambda =0.1$ and $\Delta \alpha =0.04$.

**Figure 14.**Realization of the output (

**left**) and control (

**right**) signal for $\lambda =0.5$ and $\Delta \alpha =-0.06$.

**Figure 15.**High performance control (minimal output variance—reflected by 2${\sigma}_{y}$) against corresponding $\lambda $ for various $\Delta \alpha $ values (

**left**), and the $\Delta \alpha $ as a function of $\lambda $ penalty factor (

**right**).

$\mathit{\lambda}$ | $\mathbf{\Delta}\mathit{\alpha}$ | $2{\mathit{\sigma}}_{\mathit{y}}$ |
---|---|---|

0 | 0 | 0.199 |

0.01 | 0.05 | 0.374 |

0.02 | 0.10 | 0.472 |

0.05 | 0.20 | 0.628 |

0.1 | 0.33 | 0.877 |

0.2 | 0.50 | 1.131 |

0.3 | 0.55 | 1.271 |

0.5 | 0.66 | 1.428 |

$\mathit{\lambda}$ | $\mathbf{\Delta}\mathit{\alpha}$ | $2{\mathit{\sigma}}_{\mathit{y}}$ |
---|---|---|

0 | 0 | 0.282 |

0.01 | 0.07 | 0.288 |

0.02 | 0.08 | 0.291 |

0.05 | 0.07 | 0.299 |

0.1 | 0.04 | 0.317 |

0.2 | 0 | 0.351 |

0.3 | −0.03 | 0.385 |

0.5 | −0.06 | 0.452 |

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**MDPI and ACS Style**

Bialic, G.; Stanisławski, R.
On the Linear–Quadratic–Gaussian Control Strategy for Fractional-Order Systems. *Fractal Fract.* **2022**, *6*, 248.
https://doi.org/10.3390/fractalfract6050248

**AMA Style**

Bialic G, Stanisławski R.
On the Linear–Quadratic–Gaussian Control Strategy for Fractional-Order Systems. *Fractal and Fractional*. 2022; 6(5):248.
https://doi.org/10.3390/fractalfract6050248

**Chicago/Turabian Style**

Bialic, Grzegorz, and Rafał Stanisławski.
2022. "On the Linear–Quadratic–Gaussian Control Strategy for Fractional-Order Systems" *Fractal and Fractional* 6, no. 5: 248.
https://doi.org/10.3390/fractalfract6050248