# An Efficient Chicken Search Optimization Algorithm for the Optimal Design of Fuzzy Controllers

^{*}

## Abstract

**:**

## 1. Introduction

`M`aximum Power Point Tracking (MTTP) in photovoltaic system. Finally, in [15] Yaseen et al. presented a Hybrid Invasive Weed Optimization Algorithm with Chicken Swarm Optimization Algorithm to solve Global Optimization Problems; in [16], Zarlis et al. presented a framework of training Adaptive Neuro-Fuzzy Inference System (ANFIS) using Chicken Swarm Optimization for solving classification problems, and in [17] Liang et al. presented a fast Synthetic Aperture Radar (SAR) image segmentation method based on an improved Chicken Swarm Optimization Algorithm. All these previous works allow us to observe that the CSO algorithm shows efficiency in various fields of study.

## 2. Chicken Search Optimization Algorithm

Algorithm 1 General Steps of the CSO Algorithm |

1. Initialize a population of N chickens and define the relation parameters; |

2. Evaluate the N chickens’ fitness values, t=0; |

3. While (t < Max_Generation) |

4. If (t % G = = 0) |

5. Rank the chickens’ fitness values and establish a hierarchal order in the swarm; |

6. Divide the swarm into different groups, and determine the relationship between the chicks and mother hens in a group; |

7. End if; |

8. For i = 1; N |

9. If (i == rooster) |

10. Update its solution/location using Equation (3); |

11. End if; |

12. If (i == hen) |

13. Update its solution/location using Equation (5); |

14. End if; |

15. If (i == chick) |

16. Update its solution/location using Equation (7); |

17. End if; |

18. Evaluate the new solution; |

19. If the new solution is better its previous one; update it; |

20. End For; |

21. End While; |

## 3. Control Problems

#### 3.1. Water Tank Controller

#### 3.2. Inverted Pendulum Controller

#### 3.3. Autonomous Mobile Robot Controller

- $q={\left(x,y,\theta \right)}^{\mathrm{T}}$ is the vector of the configuration coordinates,
- $v={\left(v,\right)}^{\mathrm{T}}$ is the vector of velocities,
- $\mathsf{\tau}=\left({\mathsf{\tau}}_{1,}{\mathsf{\tau}}_{2}\right)$ is the vector of torques applied to the wheels of the robot where ${\mathsf{\tau}}_{1,}$ and ${\mathsf{\tau}}_{2}$ denote the torques of the right and left wheel,
- $P\in {R}^{2}$ is the uniformly bounded disturbance vector,
- $M\left(q\right)\in {R}^{2X2}$ is the positive-definite inertia matrix,
- $C\left(q,\dot{q}\right)\vartheta $ is the vector of centripetal and Coriolis forces, and
- $D\in {R}^{2X2}$ is a diagonal positive-definite damping matrix.

## 4. Proposed Structure of Type-1 Fuzzy Logic Systems for Control

#### 4.1. Fuzzy Logic System

#### 4.2. Fuzzy Logic Controller

#### 4.3. Proposed Structure of the Type-1 FLS

#### 4.3.1. Water Tank Controller

#### 4.3.2. Inverted Pendulum Controller

#### 4.3.3. Autonomous Mobile Robot Controller

_{v}(error in the linear velocity), and e

_{w}(error in the angular velocity). Each input has three MFs of which in the extremes are trapezoidal MFs and in the middle a triangular MF, for each input. Regarding the outputs, it has two, called T1 (Torque 1) and T1 (Torque 2) with three triangular MFs, respectively. Figure 16 illustrates the Fuzzy Inference System (FIS) (a) and the inferences rules (b). Also, Figure 17 shows the FLC model for this third control problem.

## 5. Simulations Results

**2.77 × 10**, with the noise pulse generator is

^{+3}**3.00 × 10**and with the noise band-limited is

^{+3}**3.38 × 10**. Thus, errors remain similar even when a high level of noise is added; this is possible due to the exploitation ability that CSO algorithm presents. In the case of the third control problem the best average in RMSE is with the noise of band-limited perturbation.

^{+3}^{−1}compared to the average without perturbation is 2.29 × 10

^{0}; similarly, for the third control problem the average value is 6.99 × 10

^{−2}with band-limited perturbation more stable that the value of 1.79 × 10

^{−1}with pulse generator perturbation and the value of 1.35 × 10

^{−1}without perturbation in the model.

## 6. Comparative Analysis and Discussion of Results

**8.22 × 10**and for BCO algorithm is

^{−4}**5.50 × 10**.

^{−2}## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**General idea for the Chicken Search Optimization (CSO) algorithm to solve control problems.

**Figure 3.**Points of membership functions of (

**a**) Input 1 and Input 2 for the water tank problem controller, (

**b**) Input 1 and Input 2 for the autonomous mobile robot problem controller.

**Figure 8.**Second problem of control. (

**a**) Representation in the model, and (

**b**) graphic idea for the inverted pendulum controller.

**Figure 12.**Proposed structure or the first Fuzzy Inference System (FIS) Control Problem. (

**a**) Input and output of the FLS, and (

**b**) inference rules for the Water Tank Controller.

**Figure 13.**FLC model for execution of simulation by CSO algorithm of the Water Tank Controller. (

**a**) Pulse generator perturbation, and (

**b**) Band-limited perturbation.

**Figure 14.**Proposed structure of the second FIS Control Problem. (

**a**) Input and output of the FLS, and (

**b**) inference rules for the Inverted Pendulum Controller.

**Figure 15.**FLC model for the simulation by CSO algorithm of the Inverted Pendulum Controller. (

**a**) Pulse generator perturbation, and (

**b**) Band-Limited perturbation.

**Figure 16.**Proposed structure of the third FIS Control Problem. (

**a**) Input and output of the FLS, and (

**b**) inference rules for the Autonomous Mobile Robot Controller.

**Figure 17.**FLC model to execution of simulation by CSO algorithm of the AMR Controller. (

**a**) Pulse generator perturbation, and (

**b**) Band-Limited perturbation.

**Figure 18.**Convergence of Best Results. (

**a**) Water Tank Controller, (

**b**) Inverted Pendulum Controller and (

**c**) Autonomous Mobile Robot Controller.

**Figure 24.**Comparative results of best (

**a**) MSE and (

**b**) RMSE for the Autonomous Mobile Robot Controller.

**Figure 25.**Comparative of trajectories for the Autonomous Mobile Robot Controller. (

**a**) Initial FIS, (

**b**) CSO algorithm without perturbation, (

**c**) CSO algorithm with Pulse Generator Perturbation, and (

**d**) CSO algorithm with Band-Limited Perturbation.

Control Problem | Input | Output | Total Size of Vector | ||
---|---|---|---|---|---|

Total | Type of MFs | Total | Type of MFs | ||

Water Tank Controller | 2 | 3—Triangular in each Input | 1 | 5–Triangular in each output | 33 |

Inverted Pendulum Controller | 4 | 2—Trapezoidal in each Input | Sugeno-Takagi (Function) | 32 | |

Autonomous Mobile Robot Controller | 2 | Trapezoidal—Triangular—Trapezoidal | 2 | 3—Triangular in each Output | 40 |

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

N (Population) | 30 |

RN | 0.15 * N |

HN | 0.7 * N |

MN | 0.15 * N |

G | 10 |

FL | [0.5, 0.9] |

Generations | 40 |

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

N (Population) | 20 |

RN | 0.15 * N |

HN | 0.7 * N |

MN | 0.15 * N |

G | 10 |

FL | [0.5, 0.9] |

Generations | 15 |

Control Problem | Performance Index | Types of Experiment | ||
---|---|---|---|---|

Not AP | Type-1 Perturbation | Type-2 Perturbation | ||

Water Tank Controller | ITAE | 2.77 × 10^{+5} | 2.94 × 10^{+5} | 3.05 × 10^{+5} |

ITSE | 7.26 × 10^{+5} | 7.80 × 10^{+5} | 8.27 × 10^{+5} | |

IAE | 1.08 × 10^{+3} | 1.15 × 10^{+3} | 1.23 × 10^{+3} | |

ISE | 2.77 × 10^{+3} | 3.00 × 10^{+3} | 3.38 × 10^{+3} | |

RMSE | 2.43 × 10^{−1} | 6.86 × 10^{−1} | 7.27 × 10^{−1} | |

Inverted Pendulum Controller | ITAE | 2.23 × 10^{+2} | 2.23 × 10^{+2} | 2.84 × 10^{+2} |

ITSE | 5.59 × 10^{+2} | 5.58 × 10^{+2} | 6.07 × 10^{+2} | |

IAE | 2.78 × 10^{+1} | 2.77 × 10^{+1} | 3.12 × 10^{+1} | |

ISE | 6.93 × 10^{+1} | 6.94 × 10^{+1} | 7.02 × 10^{+1} | |

RMSE | 1.48 × 10^{0} | 1.48 × 10^{0} | 8.76 × 10^{−1} | |

Autonomous Mobile Robot Controller | ITAE | 3.37 × 10^{+2} | 3.41 × 10^{+2} | 3.41 × 10^{+2} |

ITSE | 1.38 × 10^{+2} | 1.42 × 10^{+2} | 1.41 × 10^{+2} | |

IAE | 1.44 × 10^{+1} | 1.46 × 10^{+1} | 1.46 × 10^{+1} | |

ISE | 6.07 × 10^{0} | 6.28 × 10^{0} | 6.21 × 10^{0} | |

RMSE | 2.06 × 10^{−1} | 2.14 × 10^{−1} | 2.03 × 10^{−1} |

Control Problem | Performance Index | Types of Experiment | ||
---|---|---|---|---|

Not AP | Type-1 Perturbation | Type-2 Perturbation | ||

Water Tank Controller | BEST | 8.22 × 10^{−4} | 2.34 × 10^{−1} | 2.19 × 10^{−1} |

WORST | 8.17 × 10^{−2} | 9.25 × 10^{−1} | 7.69 × 10^{−1} | |

AVERAGE | 3.61 × 10^{−2} | 4.21 × 10^{−1} | 5.05 × 10^{−1} | |

STANDARD DEVIATION (σ) | 2.35 × 10^{−2} | 1.86 × 10^{−1} | 1.09 × 10^{−1} | |

Inverted Pendulum Controller | BEST | 4.67 × 10^{−1} | 4.80 × 10^{−1} | 5.79 × 10^{−1} |

WORST | 5.98 × 10^{0} | 5.69 × 10^{0} | 1.77 × 10^{0} | |

AVERAGE | 2.29 × 10^{0} | 2.35 × 10^{0} | 8.59 × 10^{−1} | |

STANDARD DEVIATION (σ) | 1.84 × 10^{0} | 1.79 × 10^{0} | 3.66 × 10^{−1} | |

Autonomous Mobile Robot Controller | BEST | 3.87 × 10^{−5} | 7.56 × 10^{−3} | 3.81 × 10^{−4} |

WORST | 1.44 × 10^{0} | 2.16 × 10^{0} | 3.48 × 10^{−1} | |

AVERAGE | 1.35 × 10^{−1} | 1.79 × 10^{−1} | 6.99 × 10^{−2} | |

STANDARD DEVIATION (σ) | 2.63 × 10^{−1} | 4.18 × 10^{−1} | 8.63 × 10^{−2} |

Method | Control Problem | Minimum | Maximum | Average RMSE | Population | Iterations (BCO)—Generations (CSO) |
---|---|---|---|---|---|---|

Proposed CSO Algorithm | Water Tank Controller | 8.22 × 10^{−4} | 8.17 × 10^{−2} | 2.43 × 10^{−1} | 30 | 40 |

BCO Algorithm [65] | Water Tank Controller | 5.50 × 10^{−2} | 1.47 × 10^{−1} | 5.60 × 10^{−1} | 50 | 30 |

Proposed CSO Algorithm | Autonomous Mobile Robot Controller | 3.87 × 10^{−5} | 1.44 × 10^{0} | 2.06 × 10^{−1} | 20 | 15 |

BCO Algorithm [65] | Autonomous Mobile Robot Controller | 2.00 × 10^{−3} | 1.40 × 10^{+1} | 2.26 × 10^{+1} | 50 | 30 |

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**MDPI and ACS Style**

Amador-Angulo, L.; Castillo, O.; Peraza, C.; Ochoa, P.
An Efficient Chicken Search Optimization Algorithm for the Optimal Design of Fuzzy Controllers. *Axioms* **2021**, *10*, 30.
https://doi.org/10.3390/axioms10010030

**AMA Style**

Amador-Angulo L, Castillo O, Peraza C, Ochoa P.
An Efficient Chicken Search Optimization Algorithm for the Optimal Design of Fuzzy Controllers. *Axioms*. 2021; 10(1):30.
https://doi.org/10.3390/axioms10010030

**Chicago/Turabian Style**

Amador-Angulo, Leticia, Oscar Castillo, Cinthia Peraza, and Patricia Ochoa.
2021. "An Efficient Chicken Search Optimization Algorithm for the Optimal Design of Fuzzy Controllers" *Axioms* 10, no. 1: 30.
https://doi.org/10.3390/axioms10010030