# Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fractional Order Calculus

#### 2.2. Fractional-Order PID (FOPID) Controller

## 3. Problem Formulation

## 4. Design of FOPID Controller

#### 4.1. Description of Fractional Order Controller Based on Bode Envelope

- $x={K}_{a}+{K}_{b}{\omega}_{gc}^{-\alpha}cos\left(\right)open="("\; close=")">\frac{\pi}{2}\alpha $
- $y=-{K}_{b}{\omega}_{gc}^{-\alpha}sin\left(\right)open="("\; close=")">\frac{\pi}{2}\alpha $

## 5. Brief Description of Reduction Method for HOCIP

#### 5.1. Procedure for Denominator

**Condition**

**1.**

**Condition**

**2.**

#### 5.2. Procedure for Numerator

## 6. Sine-Cosine Algorithm (SCA)

## 7. Test System

#### 7.1. Improved Routh–Pade Approximation Method

#### 7.1.1. Calculation of Denominator

#### 7.1.2. Calculation of Numerator

#### 7.2. Design of FOPID Controller

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FOPID | fractional-order proportional-integral-derivative |

PM | phase margin |

MPs | Markov parameters |

TMs | time moments |

ROCIP | reduced-order continuous interval plant |

HOCIP | high-order continuous interval plant |

SCA | sine-cosine algorithm |

SISO | single-input-single-output |

MIMO | multi-input-multi-output |

PID | proportional-integral-derivative |

IOPID | integer-order proportional-integral-derivative |

AVR | automatic voltage regulator |

MOR | model order reduction |

ISE | integral-square-error |

PSO | particle swarm optimization |

RL | Riemann–Liouville |

GL | Grunwald–Letnikov |

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**Figure 7.**Sine and cosine functions within the range of [−2, 2] allow a way to move around (inside the space among them) or beyond (outside the space among them) the target [26].

**Figure 8.**Step response of system ${G}_{3}\left(s\right)$ and model ${\stackrel{\u2322}{G}}_{1}\left(s\right)$.

${T}_{1,1}={B}_{n}$ | ${T}_{1,2}={B}_{n-2}$ | ${T}_{1,3}={B}_{n-4}$ | … |

${T}_{2,1}={B}_{n-1}$ | ${T}_{2,2}={B}_{n-3}$ | … | |

${T}_{3,1}$ | ${T}_{3,2}$ | ||

⋮ | ⋮ | ⋱ | |

${T}_{n,1}$ | ${T}_{n,2}$ | ||

${T}_{n+1,1}$ |

**Table 2.**Routh array for denominator polynomial of (39).

${s}^{3}$ | $\left(\right)open="["\; close="]">7,8$ | $\left(\right)open="["\; close="]">90,91$ |

${s}^{2}$ | $\left(\right)$ | $\left(\right)$ |

${s}^{1}$ | $\left(\right)$ | |

${s}^{0}$ | $\left(\right)$ |

Algorithms | Parameters | Values |
---|---|---|

PSO | Cognitive parameter, ${c}_{1}$ Social parameter, ${c}_{2}$ Inertia weight, w | 2 2 $0.5$ |

NMS | Refection coefficient, $\alpha $ Contraction coefficient, $\beta $ Expansion coefficient, $\gamma $ Shrinking coefficient, $\delta $ | 1 $1/2$ 2 $1/2$ |

LJ | Contraction factor, $\gamma $ | $0.95$ |

SCA | No parameter | − |

Algorithms | ${\mathit{\varphi}}_{\mathit{p}\mathit{c}}$ (deg) | ${\mathit{\omega}}_{\mathit{g}\mathit{c}}$ (rad/sec) | ${\mathit{K}}_{\mathit{a}}$ | ${\mathit{K}}_{\mathit{b}}$ | ${\mathit{K}}_{\mathit{c}}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|---|

SCA | $87.32$ | $54.7$ | $324.565$ | $658.081$ | $204.214$ | $0.616$ | $0.992$ |

PSO | $75.83$ | $43.2$ | $245.496$ | $335.474$ | $443.675$ | $0.457$ | $0.845$ |

NMS | $58.57$ | $28.3$ | $832.765$ | $147.164$ | $379.852$ | $0.329$ | $0.693$ |

LJ | $31.54$ | $33.7$ | $473.359$ | $259.876$ | $156.952$ | 478 | $0.573$ |

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**MDPI and ACS Style**

Bokam, J.K.; Patnana, N.; Varshney, T.; Singh, V.P.
Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability. *Algorithms* **2020**, *13*, 317.
https://doi.org/10.3390/a13120317

**AMA Style**

Bokam JK, Patnana N, Varshney T, Singh VP.
Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability. *Algorithms*. 2020; 13(12):317.
https://doi.org/10.3390/a13120317

**Chicago/Turabian Style**

Bokam, Jagadish Kumar, Naresh Patnana, Tarun Varshney, and Vinay Pratap Singh.
2020. "Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability" *Algorithms* 13, no. 12: 317.
https://doi.org/10.3390/a13120317