# Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for Electric Vehicles

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## Abstract

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## 1. Introduction

_{2}emissions in a certain period of analysis compared to conventional fossil fuel vehicles. The commitment with the environmental impact has developed new strategies to recycle battery packs such as lithium ion batteries [2], for which demand has increased in recent years. The lead acid battery recycling process has demonstrated great performance, as well as the necessary chemical procedure.

## 2. Materials and Methods

- The main objective of this paper is to improve the energy storage and power consumption of the regenerative brake system. To achieve this, the current and voltage dynamics can be controlled using a bidirectional converter. The regenerative braking system and the neural controller have been simulated and verified using the Simscape Electrical from Matlab (Matlab R2020a, Simulink, 1994–2024 ©The MathWorks, Inc., Natick, MA, USA).
- The simulation implements the mathematical representation of the buck–boost converter from [21].
- The RHONN philosophy is employed to design the identification process, which is then trained with the EKF and UKF algorithms.
- The NSMC is developed to track the trajectories in the AES chosen for the case study. In the first scenario, a signal is selected to validate the identification and regulation of the buck–boost converter dynamics. The signal is variable, and designed to operate within the range of the AES (supercapacitor) and MES (battery bank) parameters. In the second scenario, the complete control framework implemented in the simulation is verified. The DC motor parameters are considered for building a reference generator that tracks the EV system requirements.
- The results of the NSMC with EKF and UKF are presented, and the enhancement of the battery banks with the implemented AES is demonstrated.
- The NSMC with EKF and UKF is validated to demonstrated the robustness of the system under different conditions. Additionally, the comparison of the NSMC’s robustness with a PI controller based on the control strategy [14] is presented.
- The MSEs and EESs obtained with UKF and EKF are compared with a PI control strategy [14] to identify the best performance in regenerative braking simulation and robustness of the control strategies.

## 3. DC-DC Bidirectional Buck–Boost Converter

**Buck mode:**When this mode is activated, there is a reduction in the output voltage related to the input voltage. ${T}_{1}$ is deactivated and ${T}_{2}$ is activated, resulting in the transfer of the energy coming from the capacitor voltage ${V}_{c}$ to the supercapacitor voltage ${V}_{sc}$. Once ${T}_{2}$ is deactivated, the energy stored in the capacitor flows C to the supercapacitor employing the current capacitor ${i}_{c}$. Furthermore, the inductor L is charged with a fragment of the same current capacitor ${i}_{c}$ energy. Likewise, when ${T}_{2}$ is deactivated, the L current is discharged into ${V}_{c}$ via ${D}_{1}$, guiding this current energy in the flow of C [14].

**Boost operation:**During activation of this mode, the output voltage increases concerning the input voltage. ${T}_{1}$ is ON and ${T}_{2}$ is OFF, and the energy coming from supercapacitor $Vsc$ is transferred to battery bank $Vc$. As soon as ${T}_{1}$ is activated, the capacitor’s energy is taken and stored in L. On the other hand, when ${T}_{1}$ is activated, the energy in L is driving into the capacitor C across ${D}_{2}$ into the battery bank.

## 4. Mathematical Preliminaries

#### 4.1. Discrete-Time Sliding Mode Control

#### 4.2. Discrete-Time Recurrent High-Order Neural Networks

#### 4.3. Extended Kalman Filter

#### 4.4. Unscented Kalman Filter

## 5. System Modeling and Sliding Mode Neural Control

#### 5.1. Buck–Boost Converter Model

#### 5.2. Neural Controller Design

#### 5.3. Discrete Time Sliding Mode Control

**Voltage NSMC**

**Current NSMC**

#### 5.4. Reference Generator Development

## 6. Simulation Results

#### 6.1. RHONN Validation

#### 6.2. NSMC Trajectory Tracking Results Using UKF

#### 6.3. NSMC Trajectory Tracking Results Using EKF

#### 6.4. Regenerative Braking System Control Using a Reference Generator with UKF

#### 6.5. Regenerative Braking System Control Using a Reference Generator with EKF

#### 6.6. Robustness Test

#### 6.6.1. Gaussian Noise

**Robustness Test with NSMC and UKF**

**Robustness Test with NSMC and EKF**

**Robustness Test with PI**

#### 6.6.2. Changes in the Parameters of the Buck–Boost Converter

**NSMC with UKF for the tracking of dynamics with changes in buck-boost converter**

**NSMC with EKF for the tracking of dynamics with changes in buck–boost converter**

## 7. Comparative Analysis between Proposed NSMC Variants and PI Controller

**n**is the number of data points to evaluate the computation, ${Y}_{i}$ represents the values obtained evaluated at the data points, and ${\widehat{Y}}_{i}$ represents the reference values at the data points.

#### 7.1. MSE and EES of a Time-Varying Signal Controlled with NSMC

#### 7.2. MSE and EES of the Complete Regenerative Braking System Controlled with NSMC

#### 7.3. MSE and EES of Robustness Test

#### 7.4. MSE and EES of Robustness Test with Changes in the Buck–Boost Converter Parameters

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NSMC | Neural Sliding Mode Controller |

SMC | Sliding Mode Controller |

NIOC | Neural Inverse Optimal Controller |

IOC | Inverse Optimal Control |

PWM | Pulse Width Modulation |

UKF | Unscented Kalman Filter |

EKF | Extended Kalman Filter |

EV | Electric Vehicle |

RHONN | Recurrent High-Order Neural Network |

MES | Main Energy System |

AES | Auxiliary Energy System |

DC | Direct Current |

PI | Proportional–Integral |

SOC | State of Charge |

MSE | Mean Squared Error |

EES | Energy of the Error Signal |

Variable | Description |

$i=1,2,3,\dots ,n$ | index for name states |

$k=1,2,3,\dots ,n$ | index for name RHONN states |

${x}_{i,k}$ | neural network adjustable synaptic weights |

${\omega}_{i,k}$ | fixed weights |

${x}_{k}={\left[{x}_{1,k},{x}_{2,k},\dots ,{x}_{n,k}\right]}^{T}$ | state vector |

${\mathcal{X}}_{i,k+1}$ | state of the ith neuron that identifies the ith component of the state vector ${x}_{k}$ |

${\varphi}_{i}$, ${\phi}_{i}$ | linear functions of the state vector |

${u}_{k}$ | vector input of the RHONN model |

${d}_{ij,k}$ | non-negative integers |

${L}_{i}$ | number of the connection |

${I}_{{L}_{i}}$ | collection of non-ordered subsets $1,2,\dots ,n+m$ |

n | the state dimension |

m | the input dimension |

${\zeta}_{i}$ | defined as [23] |

Extended Kalman Filter | |

${e}_{i}\in \mathbb{R}$ | the identification error to be minimized |

${\eta}_{i}$ | a parameter of the training algorithm design |

${K}_{i,k}\in {\mathbb{R}}^{{L}_{i}\times Li}$ | the Kalman gain matrix |

${Q}_{i,k}\in {\mathbb{R}}^{{L}_{i}\times Li}$ | the measurement noise covariance matrix |

${R}_{i,k}\in {\mathbb{R}}^{m\times m}$ | the state noise covariance matrix |

${P}_{i}\in {\mathbb{R}}^{{L}_{i}\times {L}_{i}}$ | the prediction error matrix |

${H}_{i}\in {\mathbb{R}}^{{L}_{i}\times m}$ | the measurement matrix, which is defined as the resulting state derivative with respect to the adjustable weights of the neural identifiers |

Unscented Kalman Filter | |

${e}_{i,k}^{j}$ | the identification error |

${P}_{i,k}^{j}$ | the prediction error covariance matrix |

${P}_{i,k}^{jyy}$ | the covariance of predicted output matrix |

${P}_{i,k}^{jxy}$ | the output matrix |

${w}_{i,k}^{j}$ | the j-th weight of the i-th subsystem |

${\eta}_{i}^{jc}$ | a parameter of design |

${\mathcal{X}}_{i,k}^{j}$ | the j-th state of the plant |

${x}_{i,k}^{j}$ | the j-th state of the neural network |

L | the number of states |

${K}_{i,k}^{j}$ | the Kalman gain matrix |

${Q}_{i,k}^{j}$ | the measurement noise covariance matrix |

${R}_{i,k}^{j}$ | the state noise covariance matrix |

${x}_{i,k}^{j-}$ | the predicted state mean |

${y}_{i,k}^{j-}$ | the predicted output mean |

${X}_{i,k|k-1}^{j}$ | sigma-points propagated through prediction |

${\mathcal{Y}}_{i,k|k-1}^{j}$ | sigma-points propagated through observation |

Buck–Boost Converter Model | |

${x}_{1,k}$ | the buck–boost converter output voltage |

${x}_{2,k}$ | the buck–boost converter output current |

${\mathrm{U}}_{btt}$ | the voltage in the battery |

L | the inductance in henry (H) |

R | the load resistance in ohms ($\mathsf{\Omega}$) |

C | the capacitor in farads (F) |

${t}_{s}$ | the sample time |

$Gn$ | Gaussian Noise |

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**Figure 36.**Tracking trajectory with changes in the buck–boost converter for voltage ${x}_{1}$ with UKF.

**Figure 37.**Tracking trajectory with changes in the buck–boost converter for current ${x}_{2}$ with UKF.

**Figure 38.**Tracking trajectory with changes in the buck–boost converter for voltage ${x}_{1}$ with EKF.

**Figure 39.**Tracking trajectory with changes in the buck–boost converter for current ${x}_{2}$ with EKF.

**Figure 45.**Tracking trajectory with the three controllers for voltage ${X}_{1}$ with regenerative braking system.

**Figure 46.**Tracking trajectory with the three controllers for current ${X}_{2}$ with regenerative braking system.

Extended Kalman Filter (EKF) | Unscented Kalman Filter (UKF) |
---|---|

Design parameters | |

Analytical Jacobian matrices ${F}_{n},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{H}_{n}$ | Scaling parameters $\begin{array}{c}\hfill \alpha ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\kappa ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\lambda ,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\eta}_{i}^{m},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\eta}_{i}^{c}\end{array}$ |

Initialization | |

${Q}_{n},\phantom{\rule{1.em}{0ex}}{R}_{n},\phantom{\rule{1.em}{0ex}}{\widehat{x}}_{0},\phantom{\rule{1.em}{0ex}}{P}_{0}$ | |

Prediction | |

$\begin{array}{ccc}\hfill {\widehat{x}}_{{k}^{-}}& =& f\left({\widehat{x}}_{k},{u}_{k}\right)\hfill \\ \\ \hfill {P}_{{k}^{-}}& =& {F}_{k}{P}_{k}{F}_{k}^{\top}+{Q}_{k}\hfill \end{array}$ | $\begin{array}{ccc}\hfill {X}_{i,{k}^{-}}& =& f\left({X}_{i,k},{u}_{k}\right)\hfill \\ \hfill {\widehat{x}}_{{k}^{-}}& =& {\displaystyle \sum _{\mathfrak{i}=0}^{2L}}{\eta}_{\mathfrak{i}}^{m}{X}_{i,k-1}\hfill \\ \hfill {P}_{{k}^{-}}& =& {Q}_{k}+{\displaystyle \sum _{\mathfrak{i}=0}^{2L}}{\eta}_{i}^{m}\left({X}_{i,{k}^{-}}-{\widehat{x}}_{{k}^{-}}\right)\hfill \\ & & {\left({X}_{i,{k}^{-}}-{\widehat{x}}_{{k}^{-}}\right)}^{\top}\hfill \end{array}$ |

Observation | |

$\begin{array}{ccc}\hfill {\widehat{y}}_{{k}^{-}}& =& h\left({\widehat{x}}_{{k}^{-}}\right)\hfill \\ \\ \hfill {P}_{n}^{yy}& =& {H}_{n}{P}_{{k}^{-}}{H}_{n}^{\top}+{R}_{n}\hfill \\ \\ \hfill {P}_{n}^{xy}& =& {P}_{{k}^{-}}{H}_{n}^{\top}\hfill \end{array}$ | $\begin{array}{ccc}\hfill {\mathcal{Y}}_{i,{k}^{-}}& =& h\left({X}_{i,k},{u}_{k}\right)\hfill \\ \hfill {\widehat{y}}_{k-1}& =& {\displaystyle \sum _{\mathfrak{i}=0}^{2L}}{\eta}_{\mathfrak{i}}^{m}{\mathcal{Y}}_{i,{k}^{-}}\hfill \\ \hfill {P}_{n}^{xy}& =& {\displaystyle \sum _{\mathfrak{i}=0}^{2L}}{\eta}_{\mathfrak{i}}^{m}\left({X}_{i,{k}^{-}}-{\widehat{x}}_{{k}^{-}}\right)\hfill \\ & & {\left({\mathcal{Y}}_{i,{k}^{-}}-{\widehat{y}}_{{k}^{-}}\right)}^{\top}\hfill \\ \hfill {P}_{n}^{yy}& =& {R}_{n}+{\displaystyle \sum _{\mathfrak{i}=0}^{2L}}{\eta}_{\mathbf{i}}^{m}\left({\mathcal{Y}}_{i,{k}^{-}}-{\widehat{y}}_{{k}^{-}}\right)\hfill \\ & & {\left({\mathcal{Y}}_{i,{k}^{-}}-{\widehat{y}}_{{k}^{-}}\right)}^{\top}\hfill \end{array}$ |

Update | |

$\begin{array}{ccc}\hfill {K}_{n}& =& {P}_{n}^{xy}{\left({P}_{n}^{yy}\right)}^{-1}\hfill \\ \hfill {\widehat{x}}_{n}& =& {\widehat{x}}_{{k}^{-}}+{K}_{n}\left({y}_{n}-{\widehat{y}}_{{k}^{-}}\right)\hfill \\ \hfill {P}_{n}& =& {P}_{{k}^{-}}-{K}_{n}{P}_{n}^{yy}{K}_{n}^{\top}\hfill \end{array}$ |

Description | Unit |
---|---|

Buck–boost converter resistance R | $50\phantom{\rule{0.277778em}{0ex}}\mathsf{\Omega}$ |

Buck–boost converter inductance L | 13 mH |

Buck–boost converter capacitance ${C}_{1}$ | 2 mF |

Buck–boost converter capacitance ${C}_{2}$ | 1 $\mathsf{\mu}$F |

Supercapacitor voltage ${V}_{sc}$ | 350 V |

Battery bank voltage ${V}_{c}$ | 500 V |

Initial SOC | 80% |

Sampling time $\left({t}_{s}\right)$ | 0.1 $\mathsf{\mu}$ seg |

Chirp Signals Parameters | |||
---|---|---|---|

Controller | Initial Frequency | Target Time (s) | Frequency at Target Time (Hz) |

Chirp signal for voltage | 0.5 Hz | 25 | 0.6 Hz |

Chirp signal for current | 0.5 Hz | 10 | 0.5 Hz |

Description | Unit |
---|---|

Armature resistance ${R}_{a}$ | $2.581\phantom{\rule{0.277778em}{0ex}}\mathsf{\Omega}$ |

Armature inductance ${L}_{a}$ | $0.028$ H |

Field resistance ${R}_{f}$ | $281.3\phantom{\rule{0.277778em}{0ex}}\mathsf{\Omega}$ |

Field inductance ${L}_{f}$ | 156 H |

Field-armature mutual inductance ${L}_{af}$ | $0.9483$ H |

Power ${P}_{w}$ | 5 H_{p} |

DC voltage | 240 v |

Rated speed rpm | 1750 rpm |

Field voltage | 300 v |

Description | Unit |
---|---|

Buck–boost converter resistance R | $50\phantom{\rule{0.277778em}{0ex}}\mathsf{\Omega}$ |

Buck–boost converter inductance L | 13 mH |

Buck–boost converter capacitance ${C}_{1}$ | 14 mF |

Buck–boost converter capacitance ${C}_{2}$ | 6 $\mathsf{\mu}$ F |

Supercapacitor voltage ${V}_{sc}$ | 350 V |

Battery bank voltage ${V}_{c}$ | 500 V |

Initial SOC | 80% |

Sampling time $\left({t}_{s}\right)$ | $0.1$ $\mathsf{\mu}$ seg |

MSE and EES of Tracking Trajectories in ${\mathit{x}}_{1}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | 72.9529 | $3.647\times {10}^{8}$ |

NSMC with EKF | 1.0585 | $5.2923\times {10}^{6}$ |

PI | 1.7403 | $8.7016\times {10}^{6}$ |

MSE and EES of Tracking Trajectories in ${\mathit{x}}_{2}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | $1.0427\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | $5.2134\times {10}^{10}$ |

NSMC with EKF | 0.1362 | $6.8084\times {10}^{5}$ |

PI | 270.1132 | $1.3506\times {10}^{9}$ |

MSE and EES of Tracking Trajectories in ${\mathit{x}}_{1}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | 45.2714 | $2.2636\times {10}^{8}$ |

NSMC with EKF | 1.0321 | $5.1604\times {10}^{6}$ |

PI | 132.4538 | $6.6227\times {10}^{8}$ |

MSE and EES of Tracking Trajectories in ${\mathit{x}}_{2}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | 1.9679 | $9.8395\times {10}^{6}$ |

NSMC with EKF | 8.9223 | $4.4611\times {10}^{7}$ |

PI | $1.8035\times {10}^{5}$ | $9.0176\times {10}^{11}$ |

MSE and EES of Tracking Trajectories with Gaussian noise in ${\mathit{x}}_{1}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | 73.4682 | $3.6734\times {10}^{8}$ |

NSMC with EKF | 45.7588 | $2.2879\times {10}^{8}$ |

PI | $132.9545$ | $6.65\times {10}^{8}$ |

MSE and EES of Tracking Trajectories with Gaussian noise in ${\mathit{x}}_{2}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | $1.0435\times {10}^{4}$ | $5.2174\times {10}^{10}$ |

NSMC with EKF | $2.780\times {10}^{4}$ | $1.390\times {10}^{10}$ |

PI | $1.80\times {10}^{5}$ | $9.02\times {10}^{11}$ |

MSE and EES with Changes in Buck–Boost Converter Parameters in ${\mathit{x}}_{1}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | $73.7266$ | $3.6863\times {10}^{8}$ |

NSMC with EKF | $46.0728$ | $2.3036\times {10}^{8}$ |

PI | $134.9163$ | $6.7458\times {10}^{8}$ |

MSE and EES with Changes in Buck–Boost Converter Parameters in ${\mathit{x}}_{2}$ | ||
---|---|---|

Controller | MSE Value | EES Value |

NSMC with UKF | $1.0488\times {10}^{4}$ | $5.2439\times {10}^{10}$ |

NSMC with EKF | $2.8186\times {10}^{3}$ | $1.4093\times {10}^{10}$ |

PI | $1.8475\times {10}^{5}$ | $9.2374\times {10}^{11}$ |

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## Share and Cite

**MDPI and ACS Style**

Ruz-Hernandez, J.A.; Garcia-Hernandez, R.; Ruz Canul, M.A.; Guerra, J.F.; Rullan-Lara, J.-L.; Vior-Franco, J.R.
Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for Electric Vehicles. *World Electr. Veh. J.* **2024**, *15*, 48.
https://doi.org/10.3390/wevj15020048

**AMA Style**

Ruz-Hernandez JA, Garcia-Hernandez R, Ruz Canul MA, Guerra JF, Rullan-Lara J-L, Vior-Franco JR.
Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for Electric Vehicles. *World Electric Vehicle Journal*. 2024; 15(2):48.
https://doi.org/10.3390/wevj15020048

**Chicago/Turabian Style**

Ruz-Hernandez, Jose A., Ramon Garcia-Hernandez, Mario Antonio Ruz Canul, Juan F. Guerra, Jose-Luis Rullan-Lara, and Jaime R. Vior-Franco.
2024. "Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for Electric Vehicles" *World Electric Vehicle Journal* 15, no. 2: 48.
https://doi.org/10.3390/wevj15020048