# Real-Time Metaheuristic Algorithm for Dynamic Fuzzification, De-Fuzzification and Fuzzy Reasoning Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- We evaluated how the type-2 FLS are more capable of performing under uncertain conditions and designed and developed a mechanism to integrate them into the proposed type-1 FLC;
- We designed and developed an adaptive metaheuristic FIM for type-1 FLS to overcome the problems that are currently faced when developing realistic fuzzy rules (as shown in Table 1, column 6);
- We designed and developed a fuzzification and de-fuzzification mechanism while integrating the features that were abstracted from the type-2 FLS into type-1 FLS;
- A real-time dynamic metaheuristic algorithm to automatically optimize all of the abovementioned processes related to dynamic fuzzification, de-fuzzification and fuzzy reasoning was designed and developed;
- To examine the performance of the proposed controller as a complex physical phenomenon, a four-wheeled independent-drive electric rover was designed and developed to regulate the wheel slip (under high-speed conditions on slippery roads).

## 2. Overall System Design

## 3. Implementation of the Dynamic Metaheuristic Fuzzy Logic Controller (TSK-PSO-FLC)

## 4. Implementation of the Static Fuzzy Logic Controller (Static FLC)

#### 4.1. Fuzzification Process of the Static Fuzzy Logic Controller

#### 4.2. Implementation of the Fuzzy Inference Mechanism of the Static Fuzzy Logic Controller

**or**$\tilde{B}$, then:

**and**$\tilde{B}$, then:

**then**$x$ is $\tilde{B}$, then:

**IF**$x$ $\tilde{A}$,

**THEN**$y$ is $\tilde{B}$; then, this is equivalent to the fuzzy relation $\tilde{R}$, where $\tilde{R}=\left(\left(\tilde{A}\times \tilde{B}\right){\displaystyle \cup}\left(\overline{\tilde{A}}\times Y\right)\right)$.

1.0 | 0.4 | 0.5 | and | 0.7 | 0.1 | ||||

$\tilde{R}$ | = | 0.3 | 0.0 | 0.7 | $\tilde{S}$ | = | 0.2 | 0.9 | |

0.6 | 0.8 | 0.2 | 0.8 | 0.4 |

V | W | ||||||||||||||

x | y | z | & | ∗ | |||||||||||

$\tilde{R}$ | = | a | 1.0 | 0.4 | 0.5 | $\tilde{S}$ | = | x | 0.7 | 0.1 | |||||

U | b | 0.3 | 0.0 | 0.7 | V | y | 0.2 | 0.9 | |||||||

c | 0.6 | 0.8 | 0.2 | z | 0.8 | 0.4 |

**IF**the (

**E**is

_{k}**NB**AND

**dE**is

_{k}**NB**)

**THEN**the

**U**is

_{k}**PB**

#### 4.3. De-Fuzzification Process of the Static Fuzzy Logic Controller

## 5. Dynamic Particle Swarm Optimization (PSO) Mechanism

#### Implementation of the Dynamic PSO Mechanism

_{1}+ a

_{2}≤ 4. [108]. The values ${r}_{1}$ and ${r}_{2}$ control the diversity of the optimal solutions, and originally they are uniformly distributed between the range of 0 and 1.

## 6. Implementation of the Dynamic Fuzzy Logic Controller (Dynamic FLC)

#### 6.1. Dynamic Fuzzification Process for the Fuzzy Antecedent and Consequent Dimensions with the Dynamic PSO Mechanism

#### 6.2. Dynamic Fuzzy Reasoning Process Optimized by the PSO

#### 6.3. Optimization of the Dynamic PSO Mechanism to Tune the Proposed Metaherustic FLC

## 7. Implementation of the Takagi–Sugeno–Kang (TSK) FLC

## 8. Modelling of Mechanical Dynamics of the Four-Wheeled Independent-Drive Electric Rover

## 9. Results and Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A2.**The straight road test on a slippery wet grass surface (captured when the speed of the rover was around 42.8 km/h.).

**Figure A3.**Top recorded speed while running on a gravel soil surface (as shown in Figure A8, the top recorded speed was around 90 km/h).

Physical Parameter | Amount with Units |
---|---|

Rover Width (W) | 0.415 m |

Rover Height (H) (Ground clearance) | 0.06 m |

Rover Length (L) | 0.465 m |

Diameter of a wheel | 0.13 m |

Weight of the rover body (M_{B}) | 3.288 kg |

Weight of a wheel (M_{Wh}) | 0.064 kg |

Total weight of the rover (M_{R}) | 5.066 kg |

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**Figure 3.**The experimental hardware setup of the developed static FLC [105].

**Figure 4.**The non-singleton fuzzy sets (E

_{k}, dE

_{k}and U (Speed)) of the proposed static FLC [105].

**Figure 6.**An interval-valued fuzzy set $\left(\tilde{A}\left(a\right)=\left[{\lambda}_{1},\text{}{\lambda}_{2}\right]\right)$.

**Figure 8.**The overall mechanism of the dynamic fuzzification process through the dynamic PSO mechanism.

**Figure 9.**The five consequent dimensions of the PSO mechanism for the dynamic fuzzy reasoning process.

**Figure 13.**The throttle signal is given to the controller to achieve the desired translational velocity. The neutral position of the steering angle level is 1.7 (to achieve a zero-yaw angle) [104].

**Figure 15.**The reference r.p.m. (refWs) and the actual r.p.m. of each wheel vs. time [104].

**Figure 16.**The actual three orthogonal acceleration directions vs. the time of the rover [104].

**Figure 17.**The actual yaw angle, pitch angle and roll angle vs. the time of the rover [104].

**Table 1.**A general summary of the current state of the research area that shows the sophisticated development of fuzzy input–output membership functions (shown in the 4th and 5th columns), especially when developing the FIM (the 6th column shows evidence of the development of a large number of fuzzy rules) to compensate for sophisticated physical phenomena.

Ref. | Year | Contribution | Number of Input Fuzzy Membership Functions | Number of Output Fuzzy Membership Functions | Number of Fuzzy Rules |
---|---|---|---|---|---|

[6] | 2021 | An Agent-Based Model-Driven Decision Support System for Assessment of Agricultural Vulnerability of Sugarcane Facing Climatic Change: Crop yield model (Scopus-Indexed) | 8 | 5 | 768 |

[7] | 2021 | New FMEA Risks Ranking Approach Utilizing Four Fuzzy Logic Systems (Scopus-Indexed) | 4 | 1 | 625 |

[8] | 2021 | Symptom Analysis Using Fuzzy Logic for Detection and Monitoring of COVID-19 Patients (Scopus-Indexed) | 6 | 1 | 512 |

[9] | 2021 | A Fuzzy Logic-Based Cost Modelling System for Recycling Carbon Fibre Reinforced Composites (Scopus-Indexed) | 5 | 1 | 243 |

[10] | 2021 | Enhanced Intelligent Closed Loop Direct Torque and Flux Control of Induction Motor for Standalone Photovoltaic Water Pumping System (Scopus-Indexed) | 3 | 1 | 180 |

[11] | 2021 | SAFEA application design on determining the optimal order quantity of chicken eggs based on fuzzy logic (Scopus-Indexed) | 3 | 1 | 144 |

[12] | 2021 | A fuzzy logic-based approach for evaluating forest ecosystem service provision and biodiversity applied to a case study landscape in Southern Germany (Scopus-Indexed) | 5 | 5 | 125 |

[13] | 2021 | A Fuzzy Logic Model for Early Warning of Algal Blooms in a Tidal-Influenced River (Scopus-Indexed) | 3 | 1 | 125 |

[14] | 2021 | Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation: A New Approach (Scopus-Indexed) | 3 | 1 | 125 |

[6] | 2021 | An Agent-Based Model-Driven Decision Support System for Assessment of Agricultural Vulnerability of Sugarcane Facing Climatic Change: Crop vulnerability model (Scopus-Indexed) | 5 | 3 | 120 |

[15] | 2021 | Inverter current control for reactive power compensation in solar grid system using Self-Tune Fuzzy Logic Controller (Scopus-Indexed) | 2 | 1 | 91 |

[16] | 2021 | A Fuzzy Logic Model for the Analysis of Ultrasonic Vibration Assisted Turning and Conventional Turning of Ti-Based Alloy (Scopus-Indexed) | 4 | 4 | 81 |

[17] | 2021 | Fuzzy Logic Based Synchronization Method for Solar Powered High Frequency On-Board Grid (Scopus-Indexed) | 2 | 1 | 81 |

[6] | 2021 | An Agent-Based Model-Driven Decision Support System for Assessment of Agricultural Vulnerability of Sugarcane Facing Climatic Change: Uncertain parameters model (Scopus-Indexed) | 5 | 3 | 72 |

[6] | 2021 | An Agent-Based Model-Driven Decision Support System for Assessment of Agricultural Vulnerability of Sugarcane Facing Climatic Change: Non-nutritional disorders model (Scopus-Indexed) | 5 | 3 | 72 |

[18] | 2021 | Prediction of gas velocity in two-phase flow using developed fuzzy logic system with differential evolution algorithm (Scopus-Indexed) | 3 | 1 | 64 |

[19] | 2021 | Comprehensive Knowledge-Driven AI System for Air Classification Process (Scopus-Indexed) | 5 | 3 | 55 |

[20] | 2021 | Overall fuzzy logic control strategy of direct driven PMSG wind turbine connected to grid (Scopus-Indexed) | 2 | 1 | 49 |

[21] | 2021 | Optimal Geno-Fuzzy Lateral Control of Powered Parachute Flying Vehicles (Scopus-Indexed) | 2 | 1 | 49 |

[22] | 2021 | Fuzzy Mathematics-Based Outer-Loop Control Method for Converter-Connected Distributed Generation and Storage Devices in Micro-Grids (Scopus-Indexed) | 2 | 1 | 49 |

[23] | 2021 | A Novel Fuzzy PI Control Method for Variable Frequency Brushless Synchronous Generators (Scopus-Indexed) | 2 | 1 | 49 |

[24] | 2021 | A Fuzzy Multi-Criteria Model for Municipal Waste Treatment Systems Evaluation including Energy Recovery: Workstation evaluation (Scopus-Indexed) | 2 | 1 | 49 |

[24] | 2021 | A Fuzzy Multi-Criteria Model for Municipal Waste Treatment Systems Evaluation including Energy Recovery: Treatment system evaluation (Scopus-Indexed) | 4 | 1 | 49 |

[25] | 2021 | A Temperature Control Method for Micro-accelerometer Chips Based on Genetic Algorithm and Fuzzy PID Control (Scopus-Indexed) | 2 | 1 | 49 |

[26] | 2022 | Induction Motor DTC Performance Improvement by Inserting Fuzzy Logic Controllers and Twelve-Sector Neural Network Switching Table (Scopus-Indexed) | 7 | 7 | 49 |

[27] | 2022 | Fuzzy Hysteresis Current Controller for Power Quality Enhancement in Renewable Energy Integrated Clusters (Scopus-Indexed) | 7 | 7 | 49 |

[28] | 2021 | Fuzzy Logic-Based Controller for Bipedal Robot (Scopus-Indexed) | 2 | 1 | 30 |

[29] | 2021 | A Swarm Intelligence Graph-Based Pathfinding Algorithm Based on Fuzzy Logic (SIGPAF): A Case Study on Unmanned Surface Vehicle Multi-Objective Path Planning (Scopus-Indexed) | 3 | 1 | 27 |

[30] | 2021 | Fuzzy Logic and Modified Butterfly Optimization with Efficient Fault Detection and Recovery Mechanisms for Secured Fault-Tolerant Routing in Wireless Sensor Networks (Scopus-Indexed) | 3 | 1 | 27 |

[31] | 2021 | Optimal Routing Protocol for Wireless Sensor Network Using Genetic Fuzzy Logic System (Scopus-Indexed) | 3 | 1 | 27 |

[32] | 2021 | GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles: Fuzzy system 1 (Scopus-Indexed) | 2 | 1 | 25 |

[33] | 2021 | Lifting and stabilizing of two-wheeled wheelchair system using interval type-2 fuzzy logic control based spiral dynamic algorithm (Scopus-Indexed) | 2 | 1 | 25 |

[34] | 2021 | Optimization of Fuzzy Logic Based Virtual Pilot for Wargaming (Scopus-Indexed) | 2 | 1 | 25 |

[35] | 2021 | LQR and Fuzzy Logic Control for the Three-Area Power System (Scopus-Indexed) | 5 | 5 | 25 |

[36] | 2021 | Smart Homes as Enablers for Depression Pre-Diagnosis Using PHQ-9 on HMI through Fuzzy Logic Decision System (Scopus-Indexed) | 2 | 1 | 20 |

[37] | 2021 | Pineapple maturity classifier using image processing and fuzzy logic (Scopus-Indexed) | 3 | 1 | 18 |

[38] | 2021 | Algorithm for Preventing the Spread of COVID-19 in Airports and Air Routes by Applying Fuzzy Logic and a Markov Chain (Scopus-Indexed) | 4 | 1 | 14 |

[39] | 2021 | Artificial Intelligence Search Strategies for Autonomous Underwater Vehicles Applied for Submarine Groundwater Discharge Site Investigation (Scopus-Indexed) | 3 | 2 | 13 |

[40] | 2021 | Intelligent Fault Detection and Identification Approach for Analog Electronic Circuits Based on Fuzzy Logic Classifier (Scopus-Indexed) | 3 | 1 | 12 |

[32] | 2021 | GPS Data Correction Based on Fuzzy Logic for Tracking Land Vehicles: Fuzzy system 2 (Scopus-Indexed) | 2 | 1 | 9 |

[41] | 2021 | Optimum Design of a Composite Optical Receiver by Taguchi and Fuzzy Logic Methods (Scopus-Indexed) | 3 | 1 | 9 |

[42] | 2022 | SOC Balancing and Coordinated Control Based on Adaptive Droop Coefficient Algorithm for Energy Storage Units in DC Microgrid (Scopus-Indexed) | 3 | 3 | 9 |

[43] | 2021 | Fuzzy Logic in Selection of Maritime Search and Rescue Units (Scopus-Indexed) | 2 | 1 | 6 |

**Table 2.**Fuzzy relations of the developed static FLC [105].

Error (E_{k}) | ||||||||
---|---|---|---|---|---|---|---|---|

NB | NM | NS | ZE | PS | PM | PB | ||

Rate of Change of Error(dE_{k}) | NB | PB | PB | PB | PS | PS | PS | ZE |

NM | PB | PM | PM | PS | PS | ZE | NS | |

NS | PB | PM | PS | ZE | ZE | NS | NS | |

ZE | PM | PS | ZE | ZE | ZE | NS | NS | |

PS | PM | PS | ZE | ZE | NS | NS | NM | |

PM | PS | PS | NS | NS | NS | NS | NM | |

PB | ZE | NS | NS | NM | NB | NB | NB |

C_{i} (Optimized Peak Value or the Center Value) | j | σ_{i} (Optimized Standard Deviation) | j |
---|---|---|---|

$gai{n}_{{c}_{1}}=-\frac{1}{{\alpha}_{1}}{G}_{{\left(Best\right)}_{j}}\times 5000$ | 1 | $gai{n}_{{\sigma}_{1}}=\frac{1}{{\beta}_{1}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 2 |

$gai{n}_{{c}_{2}}=-\frac{1}{{\alpha}_{2}}{G}_{{\left(Best\right)}_{j}}\times 3333$ | 3 | $gai{n}_{{\sigma}_{2}}=\frac{1}{{\beta}_{2}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 4 |

$gai{n}_{{c}_{3}}=-\frac{1}{{\alpha}_{3}}{G}_{{\left(Best\right)}_{j}}\times 1667$ | 5 | $gai{n}_{{\sigma}_{3}}=\frac{1}{{\beta}_{3}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 6 |

$gai{n}_{{c}_{4}}=\pm \frac{1}{{\alpha}_{4}}{G}_{{\left(Best\right)}_{j}}$ | 7 | $gai{n}_{{\sigma}_{4}}=\frac{1}{{\beta}_{4}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 8 |

$gai{n}_{{c}_{5}}=\frac{1}{{\alpha}_{5}}{G}_{{\left(Best\right)}_{j}}\times 1667$ | 9 | $gai{n}_{{\sigma}_{5}}=\frac{1}{{\beta}_{5}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 10 |

$gai{n}_{{c}_{6}}=\frac{1}{{\alpha}_{6}}{G}_{{\left(Best\right)}_{j}}\times 3333$ | 11 | $gai{n}_{{\sigma}_{6}}=\frac{G}{{\beta}_{6}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 12 |

$gai{n}_{{c}_{7}}=\frac{1}{{\alpha}_{7}}{G}_{{\left(Best\right)}_{j}}\times 5000$ | 13 | $gai{n}_{{\sigma}_{7}}=\frac{G}{{\beta}_{7}}{G}_{{\left(Best\right)}_{j}}\times 707$ | 14 |

Optimized Gain Factors for the Input Variable E_{k} (Error) | ||
---|---|---|

c_{i}(Optimized Peak Value) | σ_{i} (Optimized Standard Deviation or the Width of the Curve) | Membership Function (MF) |

${c}_{{E}_{1}}=-5000\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{1}}}\right)\times gai{n}_{{c}_{1}}$ | ${\sigma}_{{E}_{1}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{1}}}\right)\times gai{n}_{{\sigma}_{1}}$ | NB |

${c}_{{E}_{2}}=-3333\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{2}}}\right)\times gai{n}_{{c}_{2}}$ | ${\sigma}_{{E}_{2}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{2}}}\right)\times gai{n}_{{\sigma}_{2}}$ | NM |

${c}_{{E}_{3}}=-1667\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{3}}}\right)\times gai{n}_{{c}_{3}}$ | ${\sigma}_{{E}_{3}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{3}}}\right)\times gai{n}_{{\sigma}_{3}}$ | NS |

${c}_{{E}_{4}}=\left(\frac{{\lambda}_{E}}{{p}_{{E}_{4}}}\right)\times gai{n}_{{c}_{4}}$ | ${\sigma}_{{E}_{4}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{4}}}\right)\times gai{n}_{{\sigma}_{4}}$ | Z |

${c}_{{E}_{5}}=1667\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{5}}}\right)\times gai{n}_{{c}_{5}}$ | ${\sigma}_{{E}_{5}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{5}}}\right)\times gai{n}_{{\sigma}_{5}}$ | PS |

${c}_{{E}_{6}}=3333\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{6}}}\right)\times gai{n}_{{c}_{6}}$ | ${\sigma}_{{E}_{6}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{6}}}\right)\times gai{n}_{{\sigma}_{6}}$ | PM |

${c}_{{E}_{7}}=5000\pm \left(\frac{{\lambda}_{E}}{{p}_{{E}_{7}}}\right)\times gai{n}_{{c}_{7}}$ | ${\sigma}_{{E}_{7}}=707\pm \left(\frac{{\lambda}_{E}}{{q}_{{E}_{7}}}\right)\times gai{n}_{{\sigma}_{7}}$ | PB |

Optimized Gain Factors for the Input Variable dE_{k} (Error) | ||
---|---|---|

c_{i}(Optimized Peak Value) | σ_{i}(Optimized Standard Deviation or the Width of the Curve) | Membership Function (MF) |

${c}_{d{E}_{1}}=-5000\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{1}}}\right)\times gai{n}_{{c}_{1}}$ | ${\sigma}_{d{E}_{1}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{1}}}\right)\times gai{n}_{{\sigma}_{1}}$ | NB |

${c}_{d{E}_{2}}=-3333\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{2}}}\right)\times gai{n}_{{c}_{2}}$ | ${\sigma}_{d{E}_{2}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{2}}}\right)\times gai{n}_{{\sigma}_{2}}$ | NM |

${c}_{d{E}_{3}}=-1667\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{3}}}\right)\times gai{n}_{{c}_{3}}$ | ${\sigma}_{d{E}_{3}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{3}}}\right)\times gai{n}_{{\sigma}_{3}}$ | NS |

${c}_{d{E}_{4}}=\left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{4}}}\right)\times gai{n}_{{c}_{4}}$ | ${\sigma}_{d{E}_{4}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{4}}}\right)\times gai{n}_{{\sigma}_{4}}$ | Z |

${c}_{d{E}_{5}}=1667\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{5}}}\right)\times gai{n}_{{c}_{5}}$ | ${\sigma}_{d{E}_{5}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{5}}}\right)\times gai{n}_{{\sigma}_{5}}$ | PS |

${c}_{d{E}_{6}}=3333\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{6}}}\right)\times gai{n}_{{c}_{6}}$ | ${\sigma}_{d{E}_{6}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{6}}}\right)\times gai{n}_{{\sigma}_{6}}$ | PM |

${c}_{d{E}_{7}}=5000\pm \left(\frac{{\lambda}_{dE}}{{p}_{d{E}_{7}}}\right)\times gai{n}_{{c}_{7}}$ | ${\sigma}_{d{E}_{7}}=707\pm \left(\frac{{\lambda}_{dE}}{{q}_{d{E}_{7}}}\right)\times gai{n}_{{\sigma}_{7}}$ | PB |

Optimized Gain Factors for the Input Variable U_{k} (Error) | ||
---|---|---|

c_{i}(Optimized Peak Value) | σ_{i}(Optimized Standard Deviation or the Width of the Curve) | Membership Function (MF) |

${c}_{{U}_{1}}=-5000\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{1}}}\right)\times gai{n}_{{c}_{1}}$ | ${\sigma}_{{U}_{1}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{1}}}\right)\times gai{n}_{{\sigma}_{1}}$ | NB |

${c}_{{U}_{2}}=-3333\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{2}}}\right)\times gai{n}_{{c}_{2}}$ | ${\sigma}_{{U}_{2}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{2}}}\right)\times gai{n}_{{\sigma}_{2}}$ | NM |

${c}_{{U}_{3}}=-1667\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{3}}}\right)\times gai{n}_{{c}_{3}}$ | ${\sigma}_{{U}_{3}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{3}}}\right)\times gai{n}_{{\sigma}_{3}}$ | NS |

${c}_{{U}_{4}}=\left(\frac{{\lambda}_{U}}{{p}_{{U}_{4}}}\right)\times gai{n}_{{c}_{4}}$ | ${\sigma}_{{U}_{4}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{4}}}\right)\times gai{n}_{{\sigma}_{4}}$ | Z |

${c}_{{U}_{5}}=1667\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{5}}}\right)\times gai{n}_{{c}_{5}}$ | ${\sigma}_{{U}_{5}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{5}}}\right)\times gai{n}_{{\sigma}_{5}}$ | PS |

${c}_{{U}_{6}}=3333\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{6}}}\right)\times gai{n}_{{c}_{6}}$ | ${\sigma}_{{U}_{6}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{6}}}\right)\times gai{n}_{{\sigma}_{6}}$ | PM |

${c}_{{U}_{7}}=5000\pm \left(\frac{{\lambda}_{U}}{{p}_{{U}_{7}}}\right)\times gai{n}_{{c}_{7}}$ | ${\sigma}_{{U}_{7}}=707\pm \left(\frac{{\lambda}_{U}}{{q}_{{U}_{7}}}\right)\times gai{n}_{{\sigma}_{7}}$ | PB |

Linguistic Variables | |||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZE | PS | PM | PB | |

Rounded decimal value in Axis 18, 19 and 20. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

Fuzzy variables | : | ${E}_{k}$ | $d{E}_{k}$ | ${U}_{k}$ |

Decimal representation | : | 3 | 5 | 6 |

Corresponding fuzzy rule | : | $\mathrm{If}\text{}\mathrm{the}\text{}{E}_{k}$$\text{}\mathrm{is}\text{}\mathbf{NS}\text{}\mathrm{and}\text{}\mathrm{the}\text{}d{E}_{k}$$\text{}\mathrm{is}\text{}\mathbf{PS}\text{}\mathrm{then}\text{}{U}_{k}$ is PM |

TSK-FL Controller Outputs | ||||||
---|---|---|---|---|---|---|

Linguistic Variables | nPop | ${\mathit{\omega}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{\omega}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{\mathit{f}}$ |

PVS | 5 | 0.800 | 0.100 | 0.975 | 0.975 | None |

PS | 10 | 0.825 | 0.300 | 0.985 | 0.985 | 0.009 |

PM | 15 | 0.875 | 0.400 | 0.995 | 0.995 | None |

PB | 20 | 0.900 | 0.500 | 1.000 | 1.000 | None |

No. | Inputs | Outputs | ||||||
---|---|---|---|---|---|---|---|---|

E_{k} | dE_{k} | nPop | ω_{max} | ω_{min} | a_{1} | a_{2} | a_{f} | |

01 | PVS | PVS | PS | PM | PM | PM | PM | None |

02 | PS | PVS | PS | PM | PM | PM | PM | None |

03 | PM | PVS | PM | PB | PM | PM | PM | None |

04 | PB | PVS | PB | PB | PB | PB | PB | PS |

05 | PVS | PS | PS | PM | PM | PS | PS | None |

06 | PS | PS | PM | PM | PM | PM | PM | None |

07 | PM | PS | PM | PB | PB | PM | PM | None |

08 | PB | PS | PB | PB | PB | PB | PB | PS |

09 | PVS | PM | PM | PM | PM | PM | PM | None |

10 | PS | PM | PM | PB | PM | PM | PM | None |

11 | PM | PM | PM | PB | PB | PM | PM | None |

12 | PB | PM | PB | PB | PB | PB | PB | PS |

13 | PVS | PB | PM | PM | PM | PM | PM | None |

14 | PS | PB | PM | PB | PB | PM | PM | None |

15 | PM | PB | PB | PB | PB | PM | PM | None |

16 | PB | PB | PB | PB | PB | PB | PB | PS |

**Table 11.**Steady-state error percentage (Ess%) of each independent wheel [104].

Wheel of the Rover | Steady-State Error (E_{ss}) (r.p.m.) | Steady-State Error % (E_{ss}%) (r.p.m.) |
---|---|---|

Front-Left (FL) | 58.16 | 5.24 |

Front-Right (FR) | 68.81 | 6.27 |

Back-Left (BL) | −28.65 | −2.95 |

Back-Right (BR) | −51.82 | −5.36 |

Reference | Similar Research Works Recently Published | Compared Parameter(s) | Advantages of the Proposed Metaheuristic FLC System | Established Mechanism to Verify the Control Strategy | ||||
---|---|---|---|---|---|---|---|---|

Research Title and the Published Year | Control Strategy | The Controlled Physical Phenomenon | Controlled Parameter(s) of the Research Work | Controlled Parameter(s) of the Proposed Metaheuristic FLC | The Mechanism Used in the Research Work | The Mechanism Used in the Proposed Metaheuristic FLC | ||

[111] | Research on Torque Distribution of Four-Wheel Independent Drive Off-Road Vehicle Based on PRLS Road Slope Estimation. 2021. | PRLS Road Slope Estimation. | Wheel slip and orientation of the vehicle. | Wheel torque distribution. The maximum translational velocity was tested at around 25 km/h. The average wheel slip of all four wheels was 0.8 The maximum wheel torque was achieved at 2.4 kN·m | The angular speed of each wheel. The angular torque of each wheel. The desired orientation of the rover under high-speed conditions (sudden acceleration and deceleration). The rover had lateral stability, longitudinal stability and radial stability under high-speed conditions. The top recorded translational speed of the rover was approximately 90 km/h. The maximum translational (longitudinal) acceleration on wet grass slippery surface ($0.01\le \mu \le 0.4$) was 3.4 ${\mathrm{ms}}^{-2}$. The recorded wheel slip of the rover was less than 0.35. | The proposed metaheuristic FLC is independent of mathematical governing equation(s). | Hardware-in-the-loop real-time simulation and real vehicle tests. | The proposed dynamic metaheuristic FLC was tested via a four-wheel independent-drive electric rover model. Figure A9 and Figure A10. |

[112] | Adaptive Fuzzy Type-II Controller for Wheeled Mobile Robot with Disturbances and Wheel slips. 2021. | Adaptive Fuzzy Type-II Control mechanism. | Wheel slip and trajectory follower. | Wheel torque distribution. The maximum recorded translational velocity was around 12.4 m/s. | As the authors stated: “the control scheme is the complication in the mathematic proof”. The proposed metaheuristic FLC system is independent of the system-governing equation(s). | Simulation setup. The authors have done a simulation with two types of referencetrajectory: elliptical and Trifolium shapes. | ||

[113] | Control for four-wheel independently driven electric vehicles to improve steering performance using ${H}_{\infty}$. and Moore–Penrose theory. 2019. | H_∞ and Moore–Penrose theory. In this case, the authors developed a “logarithmic functional relationship between wheel cornering stiffness”. | Wheel slip and orientation (yaw moment) of the vehicle. | Regulated the wheel cornering stiffness. Controlled the yaw moment of the 4WID EV. | As the authors stated, the “decrease of adhesive force caused by the wear of the tyre could change the vehicle’s dynamic property, and the design of a more robust controller adjusting to a varying vehicle system would bring some new challenges”. This issue is not a problem for the proposed metaheuristic FLC because it is independent of the system-governing equation(s)/(mathematical model) | A simulation test setup has been established for the following three cornering stiffness (${S}_{c\alpha}$) categories.Category 1: If ${S}_{c\alpha}>1$ Category 2: If ${S}_{c\alpha}<1$ Category 3: If ${S}_{c\alpha}=1$ | ||

[114] | A New Torque Distribution Control for Four-Wheel Independent-Drive Electric Vehicles. 2021. | Torque distribution control. | Vehicle stability and handling performance, especially under extreme driving conditions. | Wheel torque distribution. Torque control was considered to achieve the desired yaw moment of the 4WIDEV. As the authors stated, they made “quicker and fuller use of lateral force to generate yaw moment and gained better vehicle stability”. | In this similar research work, an “ideal motion state estimator” was developed. However, when the mathematical model needed to become more realistic, all system information needed to be captured. The proposed FLC was tested in real-time through a hardware application (4WDI ER) and compared to similar research work. | HIL simulation has been utilized by the authors to verify the effectiveness of the proposed optimaltorque distribution approach (two approaches have been considered). Approach 1: Sine with Dwell: The initial speed was set to 80 km/h. The friction coefficient was 0.8. Approach 2: Double Lane Change Closed-loop simulations have been conducted at a constant speed of 60 km/h. The friction coefficient was 0.8. | ||

[115] | A new application for fast prediction and protection of electrical drive wheel speed using machine learning methodology. 2022. | Artificial neural network (ANN) coupled with particle swarm optimization (ANN-PSO). | Steering angle and steering ahead are achieved via an electronic differential control. | Angular velocity and wheel slip. The longitudinal forces, lateral forces and radial forces. The maximum recorded translational velocity was around 80 km/h. The maximum wheel torque was $\approx $ 138 N·m (total wheel torque) | In this similar research work to stabilize a vehicle under uncertain conditions, the wheel speed or torque have to be regulated. Therefore, to achieve the desired electric current or voltage of the permanent magnet synchronize motor (PMSM), Lyapunov’s stability analysis theory is taken into consideration. The proposed metaheuristic FLC is independent of non-linear mathematical (Lyapunov’s, etc.) models. | The electric rear-wheel-drive PMSM speed regulation is simulated using the DTFCcommand. | ||

[116] | A Novel Longitudinal Speed Estimator for Four-Wheel Slip in Snowy Conditions. 2021. | Longitudinal vehicle speed estimator based on fuzzy logic control. | Wheel angular velocity/torque and wheel slip. | Angular velocity and wheel slip. The translational velocity of the vehicle. Longitudinal acceleration of the vehicle. | The authors stated in similar research work that “the estimated result is not accurate in high-slip conditions”. However, when considering the proposed FLC mechanism, the observed test results show that the controller performed at an admirable level. The maximum translational speed of the rover is approximately 90 km/h, while synchronizing the all-wheel speed to achieve a fixed orientation. The average kinetic friction coefficient is around 0.1. Therefore, the proposed FLC has the ability to perform well under high wheel slip conditions. | Experimental and simulation tests have been carried out.Three driving condition cases have been taken into consideration. 1st Case: No wheel Slip. 2nd Case: At least one-wheel slips. 3rd Case: All four wheels slip. | ||

[117] | Torque Vectoring Control of RWID Electric Vehicle for Reducing Driving-Wheel Slippage Energy Dissipation in Cornering. 2021. | Vector control mechanism. | Wheel angular velocity/torque and wheel slip. | Longitudinal linear stiffness of each driving wheel. The initial differential torque. Tire slippage energy dissipation. Acceleration slip regulation (ASR). | Simulations of typical maneuvering have been considered. | |||

[118] | Research on Anti-Skid Control Strategy for Four-Wheel Independent Drive Electric Vehicle, 2021. | Fuzzy PID Control strategy (Artificial Intelligence and classical control-based control strategy). | Anti-skid control. Wheel slip rate in real time. | Angular velocity and wheel slip. The maximum electric vehicle driving translational velocity was around 10 km/h. The driving torque of each independent driving wheel was 500 N·m. | The authors stated that “he entire road surface identification process is in line with the assumptions”. The proposed FLC has the ability to compensate for unexpected disturbances. | Based on Carsim and MATLAB/Simulink, the vehicle dynamics model, tire model and driving anti-skid control simulation model(s) have been established. |

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**MDPI and ACS Style**

Jayetileke, H.R.; de Mel, W.R.; Mukhopadhyay, S.C.
Real-Time Metaheuristic Algorithm for Dynamic Fuzzification, De-Fuzzification and Fuzzy Reasoning Processes. *Appl. Sci.* **2022**, *12*, 8242.
https://doi.org/10.3390/app12168242

**AMA Style**

Jayetileke HR, de Mel WR, Mukhopadhyay SC.
Real-Time Metaheuristic Algorithm for Dynamic Fuzzification, De-Fuzzification and Fuzzy Reasoning Processes. *Applied Sciences*. 2022; 12(16):8242.
https://doi.org/10.3390/app12168242

**Chicago/Turabian Style**

Jayetileke, Hasitha R., W. R. de Mel, and Subhas Chandra Mukhopadhyay.
2022. "Real-Time Metaheuristic Algorithm for Dynamic Fuzzification, De-Fuzzification and Fuzzy Reasoning Processes" *Applied Sciences* 12, no. 16: 8242.
https://doi.org/10.3390/app12168242