# Optimization of Fuzzy Controller Using Galactic Swarm Optimization with Type-2 Fuzzy Dynamic Parameter Adjustment

^{*}

## Abstract

**:**

_{3}and c

_{4}parameters in the algorithm. In addition, the modification is used for the optimization of the fuzzy controller of an autonomous mobile robot. First, the galactic swarm optimization is tested for fuzzy controller optimization. Second, the GSO algorithm with the dynamic adjustment of parameters using T1 fuzzy systems is used for the optimization of the fuzzy controller of an autonomous mobile robot. Finally, the GSO algorithm with the dynamic adjustment of parameters using the IT2 fuzzy systems is applied to the optimization of the fuzzy controller. In the proposed approaches, perturbation (noise) was added to the plant in order to find out if our approach behaves well under perturbation to the autonomous mobile robot plant; additionally, we consider our ability to compare the results obtained with the approaches when no perturbation is considered.

## 1. Introduction

_{3}and c

_{4}parameters in the GSO algorithm [15]. In addition to a type-1 fuzzy system extension, we are also using type-2 fuzzy logic to give rise to an interval type-2 fuzzy system (IT2), and this proposal was used for the optimization of the membership functions of the controller of the autonomous mobile robot [16]. Experiments were carried out optimizing the fuzzy controller for the aforementioned study case, and the optimization was initially carried out with the original galactic swarm optimization, after which the T1 and IT2 fuzzy systems were used to perform the fuzzy controller optimization, in the same way that the original GSO algorithm was experimented with by adding noise to the plant and without noise in order to compare the proposed approach.

## 2. Galactic Swarm Optimization

- Start GSO.
- The population is divided into M subpopulations.$${X}_{i}\subset X:i=1,2,\dots ,M$$
- The population is initialized randomly.$${X}_{j}^{\left(i\right)}\in {X}_{i}:j=1,2,\dots ,N$$
- Begin Level 1.Begin PSO for each of the M subpopulations, and calculate the position and velocity of the particles.$${v}_{j}^{\left(i\right)}\leftarrow {W}_{1}{v}^{\left(i\right)}+{c}_{1}{r}_{1}\left({p}_{j}^{\left(i\right)}-{x}_{j}^{\left(i\right)}\right)+{c}_{2}{r}_{2}\left({g}^{\left(i\right)}-{x}_{j}^{\left(i\right)}\right)$$$${x}_{j}^{\left(i\right)}\leftarrow {x}_{j}^{\left(i\right)}+{v}_{j}^{\left(i\right)}$$End PSO.End Level 1.
- Begin Level 2.Initialization of the super swarm.$${Y}^{\left(i\right)}\in Y:i=1,2,\dots ,M$$Begin PSO.Calculate the position and velocity of the particles.$${v}^{\left(i\right)}\leftarrow {W}_{2}{v}^{\left(i\right)}+{c}_{3}{r}_{3}\left({p}^{\left(i\right)}-{Y}^{\left(i\right)}\right)+{c}_{4}{r}_{4}\left(g-{Y}^{\left(i\right)}\right)$$$${Y}^{\left(i\right)}\leftarrow {Y}^{\left(i\right)}+{v}^{\left(i\right)}$$End PSO.End Level 2.
- Return the best position g and fitness value f(g).
- End GSO.

## 3. Proposed Approach (FGSO)

_{3}and c

_{4}parameters as the iterations pass, which will be used for the optimization of the membership functions of the controller of the autonomous mobile robot. The proposed method can be found in Figure 1.

_{3}and c

_{4}parameters represent the output variables that likewise have 3 triangular membership functions. The design of these fuzzy systems is shown in Figure 3 and Figure 4.

_{3}and c

_{4}parameters are the iterations representing the output variables. The knowledge representation of the variables is shown in the following equations [28]:

_{1}, b

_{1}, c

_{1}, a

_{2}, b

_{2}and c

_{2}, where a

_{1}< a

_{2}, b

_{1}< b

_{2}and c

_{1}< c

_{2}is shown as follows [29]:

_{1}, a

_{2}, c

_{1}and c

_{2}are the parameters that represent the edges of the triangle, and the parameters b

_{1}and b

_{2}are the center for forming the IT2 triangular membership function, as shown in Figure 6 [30,31].

_{3}and c

_{4}parameters the output variables. The knowledge representation of the IT2 fuzzy system variables is presented in the following equations:

## 4. Autonomous Mobile Robot

^{T}is the coordinate vector, which describes the position of the robot, (v, w)

^{T}represents the linear and angular velocity vector, τ = (τ

_{1}, τ

_{2}) is the torque vector applied to the wheels of the robot, τ

_{1}and τ

_{2}represent the right and left wheels, $P\in {R}^{2}$ is the uniform disturbance vector, $M\left(q\right)\in {R}^{2{x}_{2}}$ is a symmetric and positive inertial matrix, $C\left(q,\dot{q}\right)v$ is the vector of the centripetal and Coriolis forces, and $D\in {R}^{2{x}_{2}}$ is a diagonal positive defined damping matrix [36,37].

#### Fuzzy Controller

_{1}(torque 1), the second output variable is τ

_{2}(torque 2); for the output variables, three triangular membership functions labeled N, Z and P are used. The design of the fuzzy controller of the autonomous mobile robot can be found in Figure 9 [4,33].

## 5. Experimental Results

_{3}and c

_{4}parameters used in the galactic swarm optimization.

#### Statistical Comparison

_{3}and c

_{4}parameters. Then a comparison is made with the PSO algorithm. All comparisons are made by experimenting with the autonomous mobile robot plant without adding noise or disturbances to the simulations and adding 5% noise in the simulations of the autonomous mobile robot plant.

_{0}is the null hypothesis; it establishes that the average of the proposals FGSO1 and FGSO1T2 (μ

_{1}) is greater than or equal to the average of the original GSO algorithm (μ

_{2}); furthermore, H

_{a}is the alternative hypothesis (claim) that establishes that the average of the proposals FGSO1 and FGSO1T2 (μ

_{1}) is less than the average of the original GSO algorithm (μ

_{2}).

_{0}and accepting H

_{a}; there is therefore enough evidence to affirm that the average of the proposal FGSO1 is lower than the average of the original GSO algorithm; there is a z value of 1.3656 for the second comparison, accepting H

_{0}, so that the average of the proposal FGSO1T2 is greater than or equal to the average of the original GSO algorithm.

_{0}is rejected and the H

_{a}is accepted; there is significant evidence to affirm that the average of the proposals FGSO1 and FGSO1T2 is lower than the average of the original GSO algorithm.

_{0}null hypothesis establishes that the average of the original GSO algorithm and the proposals FGSO1 and FGSO1T2 (μ

_{1}) is greater than or equal to the average of the original PSO algorithm, and the variants PSO + T1FS and PSO + IT2FS (μ

_{2}); furthermore, the H

_{a}alternative hypothesis (claim) establishes that the average of the original GSO algorithm and the proposals FGSO1 and FGSO1T2 (μ

_{1}) is less than the average of the original PSO algorithm, and the variants PSO + T1FS and PSO + IT2FS (μ

_{2}).

_{0}and accepting H

_{a}. There is therefore enough evidence to affirm that the average of the original GSO algorithm and the proposal FGSO1 is lower than the average of the original PSO algorithm and the variant PSO + T1FS. Finally, with a z value of −0.7116 for the comparison between the proposal FGSO1T2 and the variant PSO + IT2FS, H

_{0}is accepted; so the average of the proposal FGSO1T2 is greater than or equal to the average of the variant PSO + IT2FS.

_{0}is rejected and the H

_{a}is accepted; there is significant evidence to affirm the average of the original.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Composition of the individuals in GSO and the proposal fuzzy galactic swarm optimization type-1 (FGSO1) and fuzzy galactic swarm optimization interval type-2 (FGSO1T2).

**Figure 11.**Desired trajectory for the autonomous mobile robot [39].

Parameter | GSO | FGSO1 | FGSO1T2 |
---|---|---|---|

Population | 10 | 10 | 10 |

Subpopulation | 5 | 5 | 5 |

Iteration 1 | 15 | 15 | 15 |

Iteration 2 | 50 | 50 | 50 |

c_{1} and c_{2} | 2 | Dynamic | Dynamic |

c_{3} and c_{4} | 2 | Dynamic | Dynamic |

Autonomous Mobile Robot | |||
---|---|---|---|

MSE | GSO | FGSO1 | FGSO1T2 |

Best | $4.55\times {10}^{-6}$ | $1.92\times {10}^{-6}$ | $6.48\times {10}^{-4}$ |

Worst | $9.67\times {10}^{-2}$ | $9.15\times {10}^{-3}$ | $1.60\times {10}^{-1}$ |

Average | $1.14\times {10}^{-2}$ | $1.09\times {10}^{-3}$ | $2.19\times {10}^{-2}$ |

Standard Deviation | $2.25\times {10}^{-2}$ | $2.57\times {10}^{-3}$ | $3.56\times {10}^{-2}$ |

Autonomous Mobile Robot | |||
---|---|---|---|

MSE | GSO | FGSO1 | FGSO1T2 |

Best | $1.45\times {10}^{-1}$ | $2.14\times {10}^{-1}$ | $7.14\times {10}^{-4}$ |

Worst | $1.1565$ | $7.84\times {10}^{-1}$ | $9.37\times {10}^{-1}$ |

Average | $7.28\times {10}^{-1}$ | $4.70\times {10}^{-1}$ | $4.61\times {10}^{-1}$ |

Standard deviation | $2.68\times {10}^{-1}$ | $1.29\times {10}^{-1}$ | $2.22\times {10}^{-1}$ |

Parameter | Value |
---|---|

${H}_{0}$ | ${\mu}_{1}\ge {\mu}_{2}$ |

${H}_{a}$ | ${\mu}_{1}<{\mu}_{2}$ (Claim) |

Level of Significance | 95% |

A | 0.05 |

Critical Value | −1.645 |

Autonomous Mobile Robot | |||
---|---|---|---|

Methods | Average | Standard Deviation | z Value |

FGSO1 (μ_{1}) | $1.09\times {10}^{-3}$ | $2.57\times {10}^{-3}$ | −2.4936 |

GSO (μ_{2}) | $1.14\times {10}^{-2}$ | $2.25\times {10}^{-2}$ | |

FGSO1T2 (μ_{1}) | $2.19\times {10}^{-2}$ | $3.56\times {10}^{-2}$ | 1.3656 |

GSO (μ_{2}) | $1.14\times {10}^{-2}$ | $2.25\times {10}^{-2}$ |

Autonomous Mobile Robot | |||
---|---|---|---|

Methods | Average | Standard Deviation | z Value |

FGSO1 (μ_{1}) | 0.4700 | 0.1290 | −4.7511 |

GSO (μ_{2}) | 0.7280 | 0.2680 | |

FGSO1T2 (μ_{1}) | 0.4610 | 0.2220 | −4.2023 |

GSO (μ_{2}) | 0.7280 | 0.2680 |

Autonomous Mobile Robot | |||
---|---|---|---|

Methods | Average | Standard Deviation | z Value |

GSO (μ_{1}) | $1.14\times {10}^{-2}$ | $2.25\times {10}^{-2}$ | −4.4229 |

PSO (μ_{2}) | 2.4166 | 2.9784 | |

FGSO1 (μ_{1}) | $1.09\times {10}^{-3}$ | $2.57\times {10}^{-3}$ | −5.1972 |

PSO + T1FS (μ_{2}) | 0.3042 | 0.3194 | |

FGSO1T2 (μ_{1}) | $2.19\times {10}^{-2}$ | $3.56\times {10}^{-2}$ | −0.7116 |

PSO + IT2FS (μ_{2}) | $3.22\times {10}^{-2}$ | $7.11\times {10}^{-2}$ |

**Table 8.**Results for the z test between the proposal and PSO [4] with noise.

Autonomous Mobile Robot | |||
---|---|---|---|

Methods | Average | Standard Deviation | z Value |

GSO (μ_{1}) | 0.7280 | 0.2680 | −11.3293 |

PSO (μ_{2}) | 6.4302 | 2.7437 | |

FGSO1 (μ_{1}) | 0.4700 | 0.1290 | −29.3360 |

PSO + T1FS (μ_{2}) | 3.1108 | 0.4759 | |

FGSO1T2 (μ_{1}) | 0.4610 | 0.2220 | −26.0009 |

PSO + IT2FS (μ_{2}) | 2.3630 | 0.3335 |

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**MDPI and ACS Style**

Bernal, E.; Castillo, O.; Soria, J.; Valdez, F. Optimization of Fuzzy Controller Using Galactic Swarm Optimization with Type-2 Fuzzy Dynamic Parameter Adjustment. *Axioms* **2019**, *8*, 26.
https://doi.org/10.3390/axioms8010026

**AMA Style**

Bernal E, Castillo O, Soria J, Valdez F. Optimization of Fuzzy Controller Using Galactic Swarm Optimization with Type-2 Fuzzy Dynamic Parameter Adjustment. *Axioms*. 2019; 8(1):26.
https://doi.org/10.3390/axioms8010026

**Chicago/Turabian Style**

Bernal, Emer, Oscar Castillo, José Soria, and Fevrier Valdez. 2019. "Optimization of Fuzzy Controller Using Galactic Swarm Optimization with Type-2 Fuzzy Dynamic Parameter Adjustment" *Axioms* 8, no. 1: 26.
https://doi.org/10.3390/axioms8010026