# Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution

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## Abstract

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## 1. Introduction

#### Contributions

- The chaotic online differential evolution is included in the two-stage adaptive tuning strategy for the controller gains. This chaotic adaptive tuning strategy can efficiently handle perturbations, uncertainties, noise, and abrupt changes in the references of the closed-loop system.
- The reliability of the proposed CATSCG in future practical applications is confirmed by presenting a nonparametric statistical study that provides more fair and meaningful comparative studies with other tuning approaches.

- In a real application (for example, in the object manipulation task), the system (for instance, the robotic manipulator) to be incorporated into the BLDC motor shaft can be modeled as a dynamic load. So, one of the advantages of the proposed chaotic adaptive tuning strategy in the BLDC motors is that this dynamic load could be considered a dynamic perturbation, such that the proposal could handle such perturbation and others (uncertainties, noise, and reference velocity changes) in a better fashion.
- The proposal assumes an analytical model to represent the system dynamics where its efficiency increases, unlike the model-free tuning approaches where the use of machine learning and reinforcement learning estimates the behavior of the system (or performance function), and so, a trade-off between the model accuracy and computational time must be considered. The model-free tuning approaches tend to increase the computational time when they increase the accuracy, while simple ones affect the precision [53] but reduce the computational time.

## 2. BLDC Motor Dynamics and Speed Controller

## 3. Chaotic Adaptive Tuning Strategy for Controller Gains in BLDC Motors

#### 3.1. Identification Stage in the BLDC Motor

#### 3.2. Predictive Stage in the BLDC Motor

#### 3.3. Chaotic Online Differential Evolution

#### 3.3.1. Differential Evolution

Algorithm 1: Differential evolution (DE) |

- $\alpha $ is the vector ${\chi}_{\alpha}$ to be mutated in (26) and can be the vector ${\chi}_{r}^{G}$ selected randomly from the population (rand), the vector ${\chi}_{b}^{G}$ as the best alternative in the population (best), or as the current vector ${\chi}_{i}^{G}$ plus the scaled difference (using a predefined scaling factor $S\in [0,1]$) between it and one of the previous ones (respectively, current-to-rand and current-to-best [60]).
- $\beta $ is the number of the scaled vector differences used in mutation (26), where $F\in [0,1]$ is an established scaling factor and ${r}_{1},{r}_{2},\dots ,{r}_{2\beta -1},{r}_{2\beta}$ are randomly selected vectors from the current population ${\mathrm{X}}^{G}$, such that $i\ne {r}_{1}\ne {r}_{2}\ne \cdots \ne {r}_{2\beta -1}\ne {r}_{2\beta}$.
- $\gamma $ is the crossover strategy and can be typically binomial (bin) or exponential (exp), as shown in (27) and (28), respectively, where j denotes the j-th design variable in a vector, ${j}_{rand}$ is a number of a randomly chosen design variable, $CR\in [0,1]$ is a predetermined crossover rate, and $rand(0,1)$ generates a random number in $[0,1]$.

- If ${\chi}_{i}^{G}$ and ${\mu}_{i}^{G}$ are feasible, i.e., both meet the constraints of the optimization problem, then the fittest solution is the one with the best objective function value (the lowest value for minimization).
- If ${\chi}_{i}^{G}$ is feasible and ${\mu}_{i}^{G}$ is unfeasible, then ${\chi}_{i}^{G}$ is preferred and vice versa.
- If both ${\chi}_{i}^{G}$ and ${\mu}_{i}^{G}$ are unfeasible, then the one that satisfies a greater number of constraints is preferred.
- If both ${\chi}_{i}^{G}$ and ${\mu}_{i}^{G}$ are unfeasible and meet the same number of constraints, then the fittest solution is selected randomly.

#### 3.3.2. Elitist Online Adaptation

- (a)
- Starting an optimization process from scratch as soon as an environmental change is produced or at fixed update intervals (e.g., every $\Delta \overline{t}$). This approach usually requires more computational resources but is affordable when the time between changes or the fixed intervals are large.
- (b)
- Using the experience gained from past optimization processes to adapt the solutions to the environmental changes instead of restarting the optimization when the time between those changes or a fixed update interval are short enough. In this case, the optimization speeds up but must enhance the diversity of candidate solutions.

#### 3.3.3. Chaotic Initialization

#### 3.4. Integrating CODE with the BLDC Motor Adaptive Tuning Strategy

Algorithm 2: Chaotic adaptive tuning strategy for controller gains (CATSCG) in the closed-loop system of a BLDC motor. |

## 4. Results and Discussion

#### 4.1. Details of the Experiment

#### 4.2. Discussion of the Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Acronyms

DC | direct current |

BLDC | brushless direct current |

PMSM | permanent magnet synchronous motor |

back-EMF | back-electromagnetic force |

PD | proportional derivative |

PI | proportional integral |

PID | proportional integral derivative |

ATCB | adaptive tuning for controller in BLDC motors |

FL | fuzzy logic |

ANFIS | adaptive neuro-fuzzy inference system |

FOPD | fractional order PD |

FOPI | fractional order PI |

FOPID | fractional order PID |

VcPMSMd | vector-controlled PMSM drive |

IC | intelligent control |

RMSE | root mean square error |

ISE | integral square error |

IAE | integral absolute error |

ITSE | integral time-weighted square error |

ITAE | integral time-weighted absolute error |

SSE | steady-state error |

ISU | integral square control signal |

SE | speed error |

MO | maximum overshoot |

RT | rise time |

ST | settling time |

SL | semiconductor lifetime |

TDPC | time domain performance criteria, such as MO, SSE, ST and RT |

AxCOsc | axis-current oscillations |

TR | torque ripple |

CATSCG | chaotic adaptive tuning strategy for controller gains |

CODE | chaotic online differential evolution |

DE | differential evolution |

ODE | online differential evolution |

NSGA-II | non-dominated sorting generic algorithm ii |

GA | genetic algorithm |

OGA | online genetic algorithm |

PSO | particle swarm optimization |

OPSO | online particle swarm optimization |

ABC | artificial bee colony |

FA | firefly algorithm |

BFO | bacterial foraging optimization |

FPA | flower pollination algorithm |

CS | cuckoo search |

GOA | grasshopper optimization algorithm |

SBX | simulated binary crossover |

BA | bat algorithm |

MSA | moth swarm algorithm |

FAMA | fast adaptive memetic algorithm |

ALO | antlion optimization |

RFNN | recurrent fuzzy neural network |

NOC | normal operating conditions |

DOC | disturbed operating conditions |

CT | convergence time of the optimizer |

DRT | disturbance rejection test |

CRPT | changed reference position test |

NCD | non-conclusive data |

## Nomenclature

$\theta \in R$ | angular position |

$w\in R$ | angular velocity |

$r\in R$ | phase resistance |

$l\in R$ | phase inductance |

${b}_{0}\in R$ | viscous friction constant |

${k}_{m}\in R$ | torque constant |

$J\in R$ | rotor inertia |

$\gamma \in a,b,c$ | winding of phase |

${i}_{\gamma}\in R$ | phase current $\gamma $ |

${e}_{\gamma}\in R$ | trapezoidal back-EMF induced in the winding of phase $\gamma $ |

${V}_{\gamma \gamma}\in R$ | phase to phase voltage |

${i}_{\gamma}\in R$ | phase current |

${\tau}_{L}\in R$ | load torque |

${\tau}_{e}\in R$ | total torque |

$R\in R$ | phase to phase resistance |

$L\in R$ | phase to phase inductance |

${k}_{e}\in R$ | back-EMF constant |

$\overline{e}\left(\theta \right):R\to R$ | trapezoidal shape function |

$P\in R$ | number of pole pairs |

$x\in {R}^{5}$ | state vector |

$\overline{x}\in {R}^{5}$ | desired state vector |

u | control system |

$\widehat{u}$ | control system in the predictive stage |

$\Theta \in {R}^{7}$ | motor parameter vector |

$\overline{\Theta}\in {R}^{7}$ | parameter vector of the motor model |

$K\in {R}^{2}$ | PI control gains |

$\overline{\eta}\left(\theta \right):R\to R$ | inverter commutation function |

${t}_{0}$, ${\overline{t}}_{0}\in R$ | initial time |

$t\in R$ | time |

$\overline{t}\in R$ | time sequence where the proposed CATSCG is carried out |

$\Delta t\in R$ | integration time |

$\Delta \overline{t}\in R$ | time interval between two tuning process |

$\Delta w\in R$ | backward/forward time window |

${n}_{\mathbb{N}}\in R$ | number of times that the time is split |

${n}_{\overline{\mathbb{N}}}\in R$ | number of times that the time is split to carry out the proposed CATSCG |

${n}_{w}\in R$ | number of integration steps required in the backward/forward time window |

${J}_{I}$, ${J}_{p}$ | objective function for the identification and predictive stages |

$NP$ | population size |

D | size of design variable vector |

G | current generation |

${G}_{max}$ | maximum generations |

$CR$ | DE’s crossover probability |

F | scale factor for DE |

S | second scale factor for DE |

${p}_{c}$, ${\eta}_{c}$ | probability and density probability in SBX crossover operator for GA |

${p}_{m}$, ${\eta}_{m}$ | probability and density probability in polynomial mutation for GA |

$\overline{\omega}$, ${\overline{\omega}}_{min}$, ${\overline{\omega}}_{max}$ | Inertia weight, and initial and final inertia weight |

${C}_{1}$, ${C}_{2}$ | PSO’s individual and collective factors |

${\mathrm{X}}^{G}$ | population in the generation G of the DE algorithm |

${\chi}_{i}^{G}$, ${\mu}_{i}^{G}$, ${\nu}_{i}^{G}$ | the i-th parent, offspring, and mutant vectors |

in the generation G, respectively | |

${\chi}_{i,j}^{G}$ | the j-th parameter of the i-th parent vector in the generation G |

${\chi}_{min/max,j}$ | the j-th design variable bound (minimum or maximum) |

z | state vector of the Lozi chaotic dynamics |

$\rho $ | statistical significance for non-parametric tests |

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**Figure 3.**Schematic diagram of the proposed chaotic adaptive tuning strategy for controller gains in the BLDC motor.

**Figure 4.**Time horizon of the proposed chaotic adaptive tuning strategy for controller gains in the BLDC motor.

**Figure 5.**Lozi chaotic map with $a=1.7$, $b=0.5$, ${z}_{1}\left(0\right)=0$, and ${z}_{2}\left(0\right)=0$ after 10,000 iterations.

**Figure 6.**Speed output for the best and worst runs of the adaptive controller tuning, considering NOC.

**Figure 7.**Speed output for the best and worst runs of the adaptive controller tuning, considering DOC.

**Figure 8.**Control action for the best and worst runs of the adaptive controller tuning, considering NOC.

**Figure 9.**Control action for the best and worst runs of the adaptive controller tuning, considering DOC.

**Figure 10.**Phase voltages in the motor obtained with the best and worst runs of CATSCG, considering NOC.

**Figure 11.**Phase voltages in the motor obtained with the best and worst runs of CATSCG, considering DOC.

**Table 1.**Investigations related to the BLDC motor controller tuning problem using metaheuristic algorithms.

Ref. | Used Controller | Tuning Parameters | Design Objective * | Employed Algorithms | Optimization Process |
---|---|---|---|---|---|

[10] | PID | PID controller gains | SE | PSO | Offline |

[30] | VcPMSMd | Speed and current controller gains, coefficients | SE in the settling phase, | FAMA, GA | Offline and Online |

in the velocity filter and voltage compensators | MO, RT, AxCOsc | Simplex Method | |||

[31] | FL | Membership function parameters | SE, SL | PSO | Online |

[32] | PI | PI controller gains | MO, ST, SE | DE, Modified DE | Online |

[33] | Online ANFIS, PID, | Learning rate, forgetting factor, steepest descent, | RMSE+IAE+ITAE+ISE | BAT, GA, PSO | Offline |

Fuzzy PID, Adaptive FL | momentum constant, PID, Fuzzy, and FL controller gains | ||||

[34] | Fuzzy PID supervised on-line RFNN | Learning rate, dynamic factor, node number | RMSE, IAE, ITAE, ISE | GA, PSO, ACO, BA, ALO | Offline |

[35] | PI | PI controller gains | IAE, ISU | PSO with five inertia weight adjustment strategies | Offline |

[36] | ANFIS | Learning rate, forgetting factor, | ISE, TDPC | BFO | Offline |

steepest descent momentum constant | BAT, PSO | ||||

[37] | PID–type FL | PID controller gains | IAE | GA | Offline |

[38] | PID | PID controller gains | ISE | FPA, PSO, FA Ziegler–Nichols | Offline |

[39] | PID | PID controller gains | ISE | GOA | Offline |

[40] | Fuzzy PD/PID | PD/PID controller gains, membership function parameters, coefficient of the consequent part of fuzzy PD/PID controller | RMSE, IAE, ITAE, ISE | PSO, CS, BAT | Offline |

[41] | FOPID | PID controller gains, fractional-orders | RMS | BAT, Modified GA, MSA, ABC | Offline |

[42] | FOPI, PI | PI controller gains, fractional-orders | ITAE | Offline | |

[43] | PI | PI controller gains | MO, ST | GA | Offline |

[44] | PID | PID controller gains | ITSE | PSO, BFO | Offline |

[45] | FOPD | PI controller gains, fractional-orders | TR | Jaya | Offline |

[46] | Online ANFIS, PID, offline ANFIS | Learning rate, forgetting factor, steepest descent momentum constant | RMSE, MO | Hybrid GA-PSO | Offline |

[47] | FOPID | PID controller gains, fractional-orders | TR | FA, GA | Offline |

Switching Interval | Sequence Number | Switch Closed | ${\mathit{V}}_{\mathit{aN}}$ | Phase Voltage $\raisebox{1ex}{${\mathit{V}}_{\mathit{s}}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ ${\mathit{V}}_{\mathit{bN}}$ | ${\mathit{V}}_{\mathit{cN}}$ | Voltage Flowing in Sequence |
---|---|---|---|---|---|---|

$0-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ | 1 | ${Q}_{5},{Q}_{3}$ | off | − | + | |

$\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | 2 | ${Q}_{5},{Q}_{1}$ | + | − | off | |

$\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ | 3 | ${Q}_{6},{Q}_{1}$ | + | off | − | |

$\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.-\raisebox{1ex}{$7\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ | 4 | ${Q}_{6},{Q}_{2}$ | off | + | − | |

$\raisebox{1ex}{$7\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.-\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | 5 | ${Q}_{4},{Q}_{2}$ | − | + | off | |

$\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$11\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ | 6 | ${Q}_{4},{Q}_{3}$ | − | off | + | |

$\raisebox{1ex}{$11\pi $}\!\left/ \!\raisebox{-1ex}{$6$}\right.-2\pi $ | 7 | ${Q}_{5},{Q}_{3}$ | off | − | + |

**Table 3.**Nominal parameters and characteristics of the BLDC motor obtained from the maxon flat brushless motor EC 90 with part number 607327.

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

${b}_{0}$ | $3.1288\times {10}^{-4}$$(\mathrm{kg}.{\mathrm{m}}^{2})$ |

J | $5.0600\times {10}^{-4}$$(\mathrm{kg}.{\mathrm{m}}^{2})$ |

L | $1.0700\times {10}^{-3}$$\left(\mathrm{H}\right)$ |

R | $0.8440$$(\Omega )$ |

${k}_{m}$ | $0.2310$$(\mathrm{N}.\mathrm{m}/\mathrm{A})$ |

${k}_{e}$ | $0.2310$$(\mathrm{V}.\mathrm{s}/\mathrm{rad})$ |

${\tau}_{L}$ | $0.0000$$(\mathrm{N}.\mathrm{m})$ |

P | 11 |

Power | $260\left(\mathrm{W}\right)$ |

Nominal voltage | 48 $\left(\mathrm{V}\right)$ |

No load speed | 1960 $\left(\mathrm{rpm}\right)$ |

No load current | 278 $\left(\mathrm{mA}\right)$ |

Nominal speed: | 1670 $\left(\mathrm{rpm}\right)$ |

Nominal torque: | 964 $\left(\mathrm{mNm}\right)$ |

Nominal current | $4.06$$\left(\mathrm{A}\right)$ |

Stall torque | 13100 $\left(\mathrm{mNm}\right)$ |

Stall current | 56.9 $\left(\mathrm{A}\right)$ |

Cond. | Strategy | Mean (ISE) | STD (ISE) | Min (ISE) | Max (ISE) |
---|---|---|---|---|---|

NOC | CATSCG | 25.0690 | 0.5515 | 24.7177 | 26.7540 |

ATCB/ODE | 25.7099 | 0.6303 | 24.8184 | 26.5760 | |

ATCB/OGA | 25.6657 | 0.6827 | 24.7470 | 26.4691 | |

ATCB/OPSO | 26.1632 | 0.4290 | 24.8573 | 26.9761 | |

DOC | CATSCG | 28.2791 | 1.0214 | 27.3197 | 31.3868 |

ATCB/ODE | 28.7739 | 0.9620 | 27.2317 | 30.6307 | |

ATCB/OGA | 28.8977 | 0.8481 | 27.1414 | 30.0556 | |

ATCB/OPSO | 29.5448 | 0.6452 | 27.7060 | 30.9074 |

**Table 5.**Results of the pairwise Wilcoxon signed-rank test over the values of ISE for the adaptive controller tuning.

Cond. | Test | ${\mathit{R}}_{+}$ | ${\mathit{R}}_{-}$ | p-Value |
---|---|---|---|---|

NOC | CATSCG vs. ATCB/ODE | 390 | 75 | $\mathbf{0}.\mathbf{0007}\times {\mathbf{10}}^{\mathbf{0}}$ |

CATSCG vs. ATCB/OGA | 359 | 106 | $\mathbf{0}.\mathbf{0081}\times {\mathbf{10}}^{\mathbf{0}}$ | |

CATSCG vs. ATCB/OPSO | 451 | 14 | $2.0489\times {10}^{-7}$ | |

ATCB/ODE vs. ATCB/OGA | 222 | 243 | $0.8393\times {10}^{0}$ | |

ATCB/ODE vs. ATCB/OPSO | 377 | 88 | $\mathbf{0}.\mathbf{0021}\times {\mathbf{10}}^{\mathbf{0}}$ | |

ATCB/OGA vs. ATCB/OPSO | 375 | 90 | $\mathbf{0}.\mathbf{0025}\times {\mathbf{10}}^{\mathbf{0}}$ | |

DOC | CATSCG vs. ATCB/ODE | 330 | 135 | $\mathbf{0}.\mathbf{0449}\times {\mathbf{10}}^{\mathbf{0}}$ |

CATSCG vs. ATCB/OGA | 342 | 123 | $\mathbf{0}.\mathbf{0234}\times {\mathbf{10}}^{\mathbf{0}}$ | |

CATSCG vs. ATCB/OPSO | 434 | 31 | $\mathbf{4}.\mathbf{4219}\times {\mathbf{10}}^{-\mathbf{6}}$ | |

ATCB/ODE vs. ATCB/OGA | 239 | 226 | $0.9032\times {10}^{0}$ | |

ATCB/ODE vs. ATCB/OPSO | 400 | 65 | $\mathbf{0}.\mathbf{0002}\times {\mathbf{10}}^{\mathbf{0}}$ | |

ATCB/OGA vs. ATCB/OPSO | 423 | 42 | $\mathbf{2}.\mathbf{0798}\times {\mathbf{10}}^{-\mathbf{5}}$ |

**Table 6.**Summary of the pairwise Wilcoxon signed-rank test over the values of ISE for the adaptive controller tuning.

Strategy | Wins under NOC | Wins under DOC | Total Wins |
---|---|---|---|

CATSCG | 3 | 3 | 6 |

ATCB/ODE | 1 | 1 | 2 |

ATCB/OGA | 1 | 1 | 2 |

ATCB/OPSO | 0 | 0 | 0 |

**Table 7.**Results of the multi-comparative Friedman test over the values of ISE for the adaptive controller tuning.

Cond. | Strategy | Rank | Statistic | p-Value |
---|---|---|---|---|

NOC | CATSCG | 1.667 | 23.76 | $2.8034\times {10}^{-5}$ |

ATCB/ODE | 2.667 | |||

ATCB/OGA | 2.4 | |||

ATCB/OPSO | 3.267 | |||

DOC | CATSCG | 1.9 | 21 | $1.0528\times {10}^{-4}$ |

ATCB/ODE | 2.467 | |||

ATCB/OGA | 2.267 | |||

ATCB/OPSO | 3.367 |

**Table 8.**Results of the post hoc Friedman test over the values of ISE for the adaptive controller tuning.

Cond. | Test | Unadjusted | Holm | Shaffer | Bergmann | z |
---|---|---|---|---|---|---|

NOC | CATSCG vs. ATCB/ODE | $\mathbf{2}.\mathbf{6998}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{1}.\mathbf{3499}\times {\mathbf{10}}^{-\mathbf{2}}$ | $\mathbf{8}.\mathbf{0994}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{8}.\mathbf{0994}\times {\mathbf{10}}^{-\mathbf{3}}$ | −3 |

CATSCG vs. ATCB/OGA | $\mathbf{2}.\mathbf{7807}\times {\mathbf{10}}^{-\mathbf{2}}$ | $8.3421\times {10}^{-2}$ | $8.3421\times {10}^{-2}$ | $5.5614\times {10}^{-2}$ | −2.2 | |

CATSCG vs. ATCB/OPSO | $\mathbf{1}.\mathbf{5867}\times {\mathbf{10}}^{-\mathbf{6}}$ | $\mathbf{9}.\mathbf{5199}\times {\mathbf{10}}^{-\mathbf{6}}$ | $\mathbf{9}.\mathbf{5199}\times {\mathbf{10}}^{-\mathbf{6}}$ | $\mathbf{9}.\mathbf{5199}\times {\mathbf{10}}^{-\mathbf{6}}$ | −4.8 | |

ATCB/ODE vs. ATCB/OGA | 4.2371E-01 | $4.2371\times {10}^{-1}$ | $4.2371\times {10}^{-1}$ | $4.2371\times {10}^{-1}$ | 0.8 | |

ATCB/ODE vs. ATCB/OPSO | $7.1861\times {10}^{-2}$ | $1.4372\times {10}^{-1}$ | $1.4372\times {10}^{-1}$ | $7.1861\times {10}^{-2}$ | −1.8 | |

ATCB/OGA vs. ATCB/OPSO | $\mathbf{9}.\mathbf{3224}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{3}.\mathbf{7290}\times {\mathbf{10}}^{-\mathbf{2}}$ | $\mathbf{2}.\mathbf{7967}\times {\mathbf{10}}^{-\mathbf{2}}$ | $\mathbf{2}.\mathbf{7967}\times {\mathbf{10}}^{-\mathbf{2}}$ | −2.6 | |

DOC | CATSCG vs. ATCB/ODE | $8.9131\times {10}^{-2}$ | $2.6739\times {10}^{-1}$ | $2.6739\times {10}^{-1}$ | $2.6739\times {10}^{-1}$ | −1.7 |

CATSCG vs. ATCB/OGA | $2.7133\times {10}^{-1}$ | $5.4266\times {10}^{-1}$ | $5.4266\times {10}^{-1}$ | $2.7133\times {10}^{-1}$ | −1.1 | |

CATSCG vs. ATCB/OPSO | $\mathbf{1}.\mathbf{0825}\times {\mathbf{10}}^{-\mathbf{5}}$ | $\mathbf{6}.\mathbf{4951}\times {\mathbf{10}}^{-\mathbf{5}}$ | $\mathbf{6}.\mathbf{4951}\times {\mathbf{10}}^{-\mathbf{5}}$ | $\mathbf{6}.\mathbf{4951}\times {\mathbf{10}}^{-\mathbf{5}}$ | −4.4 | |

ATCB/ODE vs. ATCB/OGA | $5.4851\times {10}^{-1}$ | $5.4851\times {10}^{-1}$ | $5.4851\times {10}^{-1}$ | $5.4851\times {10}^{-1}$ | 0.6 | |

ATCB/ODE vs. ATCB/OPSO | $\mathbf{6}.\mathbf{9339}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{2}.\mathbf{7736}\times {\mathbf{10}}^{-\mathbf{2}}$ | $\mathbf{2}.\mathbf{0802}\times {\mathbf{10}}^{-\mathbf{2}}$ | $\mathbf{1}.\mathbf{3868}\times {\mathbf{10}}^{-\mathbf{2}}$ | −2.7 | |

ATCB/OGA vs. ATCB/OPSO | $\mathbf{9}.\mathbf{6685}\times {\mathbf{10}}^{-\mathbf{4}}$ | $\mathbf{4}.\mathbf{8342}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{2}.\mathbf{9005}\times {\mathbf{10}}^{-\mathbf{3}}$ | $\mathbf{2}.\mathbf{9005}\times {\mathbf{10}}^{-\mathbf{3}}$ | −3.3 |

**Table 9.**Summary of the post hoc Friedman test over the values of ISE for the adaptive controller tuning.

Strategy | Wins under NOC | Wins under DOC | Total Wins |
---|---|---|---|

CATSCG | 9 | 4 | 13 |

ATCB/ODE | 0 | 4 | 4 |

ATCB/OGA | 4 | 4 | 8 |

ATCB/OPSO | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Rodríguez-Molina, A.; Villarreal-Cervantes, M.G.; Serrano-Pérez, O.; Solís-Romero, J.; Silva-Ortigoza, R. Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution. *Mathematics* **2022**, *10*, 1977.
https://doi.org/10.3390/math10121977

**AMA Style**

Rodríguez-Molina A, Villarreal-Cervantes MG, Serrano-Pérez O, Solís-Romero J, Silva-Ortigoza R. Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution. *Mathematics*. 2022; 10(12):1977.
https://doi.org/10.3390/math10121977

**Chicago/Turabian Style**

Rodríguez-Molina, Alejandro, Miguel Gabriel Villarreal-Cervantes, Omar Serrano-Pérez, José Solís-Romero, and Ramón Silva-Ortigoza. 2022. "Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution" *Mathematics* 10, no. 12: 1977.
https://doi.org/10.3390/math10121977