# Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background and Preliminaries

#### 2.1. Stability Analysis of Fractional Order Systems According to Minimum Angle Root Placement

#### 2.2. Brief Introduction of PSO Algorithms

## 3. Methodology for Robust Stabilization of FOPID Control System

## 4. Illustrative Design Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

- This study demonstrated a v-domain design scheme that is straightforward for optimal robust stabilization of fractional order control systems and the development of computer-aided-design tools.
- The PSO algorithm can find optimal FOPID controller coefficients that lead to minimum angle roots of interval systems placed on the desired angle line within the stability region of the first Riemann sheet. Thus, the proposed approach can ensure the stability of FOPID control systems in uncertainty ranges of plant parameters. The angle of this line is configured by a target angle specification.
- Target angle specifications can improve the robust control performance of FOPID control systems. We observed that the target angle specifications could be utilized to reduce the sensitivity of the time response performances of control systems to the variation of plant parameters within the uncertainty ranges. Thus, the method can achieve optimal stabilization of FOPID control system designs.
- The proposed stabilization scheme may have implications for variable time-delay systems such as the control and robust stabilization problems of the networked control systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**The block diagram describes the implication of the particle swarm optimization (PSO) algorithm for optimal robust stabilization. SAE: squared angle error.

**Figure 7.**Step response of the stabilized FOPID control system for nominal plant function ${G}_{m}(s)$; (

**a**) FOPID controller and (

**b**) PI controller.

**Figure 9.**Changes of SAEs (

**a**) for ${\varphi}_{T1}=\frac{4\pi}{60}$, (

**b**) for ${\varphi}_{T2}=\frac{3\pi}{40}$ and (

**c**) for ${\varphi}_{T3}=\frac{7\pi}{80}$.

**Figure 10.**Minimum angle root placement in the first Riemann sheet (

**a**) for ${\varphi}_{T1}=\frac{4\pi}{60}$, (

**b**) for ${\varphi}_{T2}=\frac{3\pi}{40}$ and (

**c**) for ${\varphi}_{T3}=\frac{7\pi}{80}$.

**Figure 11.**Step responses of stabilized FOPID control systems for target angle specifications (

**a**) for ${\varphi}_{T1}=\frac{4\pi}{60}$, (

**b**) for ${\varphi}_{T2}=\frac{3\pi}{40}$ and (

**c**) for ${\varphi}_{T3}=\frac{7\pi}{80}$.

**Figure 12.**Step responses of 16 vertex plant functions for target angle specifications (

**a**) for ${\varphi}_{T1}=\frac{4\pi}{60}$, (

**b**) for ${\varphi}_{T2}=\frac{3\pi}{40}$ and (

**c**) for ${\varphi}_{T3}=\frac{7\pi}{80}$.

**Figure 13.**Step disturbance responses of FOPID controller designs for the target angle specifications ${\varphi}_{T1}$,${\varphi}_{T2}$ and ${\varphi}_{T3}$; (

**a**) a full view from simulation and (

**b**) a close view of disturbance responses.

Vertex Plant Forms | Vertex Plant Functions |
---|---|

${G}_{1}(s)=\frac{\underset{\_}{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{1}(s)=\frac{0.6}{1.6{s}^{0.6}+1.3}{e}^{-0.3s}$ |

${G}_{2}(s)=\frac{\underset{\_}{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{2}(s)=\frac{0.6}{1.6{s}^{0.6}+1.3}{e}^{-0.8s}$ |

${G}_{3}(s)=\frac{\underset{\_}{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{3}(s)=\frac{0.6}{2.1{s}^{0.6}+1.3}{e}^{-0.3s}$ |

${G}_{4}(s)=\frac{\underset{\_}{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{4}(s)=\frac{0.6}{2.1{s}^{0.6}+1.3}{e}^{-0.8s}$ |

${G}_{5}(s)=\frac{\underset{\_}{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{5}(s)=\frac{0.6}{1.6{s}^{0.6}+1.7}{e}^{-0.3s}$ |

${G}_{6}(s)=\frac{\underset{\_}{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{6}(s)=\frac{0.6}{1.6{s}^{0.6}+1.7}{e}^{-0.8s}$ |

${G}_{7}(s)=\frac{\underset{\_}{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{7}(s)=\frac{0.6}{2.1{s}^{0.6}+1.7}{e}^{-0.3s}$ |

${G}_{8}(s)=\frac{\underset{\_}{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{8}(s)=\frac{0.6}{2.1{s}^{0.6}+1.7}{e}^{-0.8s}$ |

${G}_{9}(s)=\frac{\overline{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{9}(s)=\frac{0.9}{1.6{s}^{0.6}+1.3}{e}^{-0.3s}$ |

${G}_{10}(s)=\frac{\overline{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{10}(s)=\frac{0.9}{1.6{s}^{0.6}+1.3}{e}^{-0.8s}$ |

${G}_{11}(s)=\frac{\overline{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{11}(s)=\frac{0.9}{2.1{s}^{0.6}+1.3}{e}^{-0.3s}$ |

${G}_{12}(s)=\frac{\overline{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\underset{\_}{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{12}(s)=\frac{0.9}{2.1{s}^{0.6}+1.3}{e}^{-0.8s}$ |

${G}_{13}(s)=\frac{\overline{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{13}(s)=\frac{0.9}{1.6{s}^{0.6}+1.7}{e}^{-0.3s}$ |

${G}_{14}(s)=\frac{\overline{{a}_{0}}}{\underset{\_}{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{14}(s)=\frac{0.9}{1.6{s}^{0.6}+1.7}{e}^{-0.8s}$ |

${G}_{15}(s)=\frac{\overline{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\underset{\_}{L}s}$ | ${G}_{15}(s)=\frac{0.9}{2.1{s}^{0.6}+1.7}{e}^{-0.3s}$ |

${G}_{16}(s)=\frac{\overline{{a}_{0}}}{\overline{{b}_{1}}{s}^{0.6}+\overline{{b}_{0}}}{e}^{-\overline{L}s}$ | ${G}_{16}(s)=\frac{0.9}{2.1{s}^{0.6}+1.7}{e}^{-0.8s}$ |

**Table 2.**Mean squared error (MSE) values of step responses for the FOPID control of 16 plant functions and the nominal plant function ${G}_{m}$.

Plants | ${\mathit{G}}_{\mathit{m}}$ | ${\mathit{G}}_{\mathbf{1}}$ | ${\mathit{G}}_{\mathbf{2}}$ | ${\mathit{G}}_{\mathbf{3}}$ | ${\mathit{G}}_{\mathbf{4}}$ | ${\mathit{G}}_{\mathbf{5}}$ | ${\mathit{G}}_{\mathbf{6}}$ | ${\mathit{G}}_{\mathbf{7}}$ | ${\mathit{G}}_{\mathbf{8}}$ |

MSEs | 0.0062 | 0.0044 | 0.0093 | 0.0049 | 0.0085 | 0.0052 | 0.0090 | 0.0057 | 0.0087 |

Plants | ${\mathit{G}}_{\mathbf{9}}$ | ${\mathit{G}}_{\mathbf{10}}$ | ${\mathit{G}}_{\mathbf{11}}$ | ${\mathit{G}}_{\mathbf{12}}$ | ${\mathit{G}}_{\mathbf{13}}$ | ${\mathit{G}}_{\mathbf{14}}$ | ${\mathit{G}}_{\mathbf{15}}$ | ${\mathit{G}}_{\mathbf{16}}$ | |

MSEs | 0.0036 | 0.0741 | 0.0037 | 0.0136 | 0.0039 | 0.0219 | 0.0040 | 0.0111 |

**Table 3.**Stabilizing FOPID controller designs that are obtained for ${\varphi}_{T1}$, ${\varphi}_{T2}$ and ${\varphi}_{T3}$.

Target Angles | SAE | ${\mathit{k}}_{\mathit{p}}$ | ${\mathit{k}}_{\mathit{i}}$ | ${\mathit{k}}_{\mathit{d}}$ | $\mathit{\lambda}$ | $\mathit{\mu}$ |
---|---|---|---|---|---|---|

${\varphi}_{T1}$ | 0.0043 | 3.3221 | 3.3578 | 0.7437 | 1.1502 | 0.7586 |

${\varphi}_{T2}$ | 0.0039 | 2.8017 | 2.3015 | 5.9713 | 0.7805 | 0.4722 |

${\varphi}_{T3}$ | 0.0033 | 6.1112 | 1.6694 | 3.7153 | 0.9317 | 0.7608 |

**Table 4.**MSE values of step responses for nominal plant functions ${G}_{m}$ and 7 plant functions (${G}_{1}$ to ${G}_{7}$ ).

Target Angles | ${\mathit{G}}_{\mathit{m}}$ | ${\mathit{G}}_{1}$ | ${\mathit{G}}_{2}$ | ${\mathit{G}}_{3}$ | ${\mathit{G}}_{4}$ | ${\mathit{G}}_{5}$ | ${\mathit{G}}_{6}$ | ${\mathit{G}}_{7}$ |
---|---|---|---|---|---|---|---|---|

${\varphi}_{T1}$ | 0.0043 | 0.0066 | 0.0078 | 0.0078 | 0.0092 | 0.0068 | 0.0078 | 0.0078 |

${\varphi}_{T2}$ | 0.0039 | 0.0086 | 0.0103 | 0.0090 | 0.0106 | 0.0100 | 0.0117 | 0.0103 |

${\varphi}_{T3}$ | 0.0033 | 0.0070 | 0.0084 | 0.0074 | 0.0088 | 0.0081 | 0.0095 | 0.0084 |

Target Angles | ${\mathit{G}}_{8}$ | ${\mathit{G}}_{9}$ | ${\mathit{G}}_{10}$ | ${\mathit{G}}_{11}$ | ${\mathit{G}}_{12}$ | ${\mathit{G}}_{13}$ | ${\mathit{G}}_{14}$ | ${\mathit{G}}_{15}$ | ${\mathit{G}}_{16}$ |
---|---|---|---|---|---|---|---|---|---|

${\varphi}_{T1}$ | 0.0090 | 0.0023 | 0.0041 | 0.0028 | 0.0047 | 0.0023 | 0.0040 | 0.0027 | 0.0046 |

${\varphi}_{T2}$ | 0.0120 | 0.0018 | 0.0215 | 0.0019 | 0.0061 | 0.0019 | 0.0199 | 0.0020 | 0.0062 |

${\varphi}_{T3}$ | 0.0098 | 0.0015 | 0.0124 | 0.0016 | 0.0049 | 0.0016 | 0.0122 | 0.0017 | 0.0050 |

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**MDPI and ACS Style**

Tufenkci, S.; Senol, B.; Matušů, R.; Alagoz, B.B. Optimal *V*-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems. *Fractal Fract.* **2021**, *5*, 3.
https://doi.org/10.3390/fractalfract5010003

**AMA Style**

Tufenkci S, Senol B, Matušů R, Alagoz BB. Optimal *V*-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems. *Fractal and Fractional*. 2021; 5(1):3.
https://doi.org/10.3390/fractalfract5010003

**Chicago/Turabian Style**

Tufenkci, Sevilay, Bilal Senol, Radek Matušů, and Baris Baykant Alagoz. 2021. "Optimal *V*-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems" *Fractal and Fractional* 5, no. 1: 3.
https://doi.org/10.3390/fractalfract5010003