Intelligent Robust Controllers Applied to an Auxiliary Energy System for Electric Vehicles

Abstract

This paper presents two intelligent robust control strategies applied to manage the dynamics of a DC-DC bidirectional buck–boost converter, which is used in conjunction with a supercapacitor as an auxiliary energy system (AES) for regenerative braking in electric vehicles. The Neural Inverse Optimal Controller (NIOC) and the Neural Sliding Mode Controller (NSMC) utilize identifiers based on Recurrent High-Order Neural Networks (RHONNs) trained with the Extended Kalman Filter (EKF) to track voltage and current references from the converter circuit. Additionally, a driving cycle test tailored specifically for typical urban driving in electric vehicles (EVs) is implemented to validate the efficacy of the proposed controller and energy improvement strategy. The proposed NSMC and NIOC are compared with a PI controller; furthermore, an induction motor and its corresponding three-phase inverter are incorporated into the EV control scheme which is implemented in Matlab/Simulink using the “Simscape Electrical” toolbox. The Mean Squared Error (MSE) is computed to validate the performance of the neural controllers. Additionally, the improvement in the State of Charge (SOC) for an electric vehicle battery through the control of buck–boost converter dynamics is addressed. Finally, several robustness tests against parameter changes in the converter are conducted, along with their corresponding performance indices.

1. Introduction

Since they significantly lower CO₂ emissions when compared to traditional fuel cars, electric vehicles (EVs) have emerged as an excellent alternative for transportation. Nevertheless, a key problem that needs to be resolved is the quick loss of energy from electric car batteries and the amount of time spent driving. Several approaches have been developed to enhance energy storage while taking into account various driving scenarios. Some of the most popular typical urban driving cycles used to evaluate the behavior of EVs were created using vehicles with combustion engines and varied measuring criteria [1]. Consequently, some experts developed a more accurate typical urban driving cycle that gathers information using an EV and improves its reliability with artificial intelligence (AI) methods such as support vector machines with k means [2]. Validating the results in EV energy storage systems requires studying these tactics under real-world circumstances.

Improvements in the energy storage system are crucial for increasing the battery’s state of charge (SOC) and the driving range of EV. Power converters are frequently used in energy systems due to their simple development and various configurations that respond to system requirements, such as buck, boost, and buck–boost [3]. In particular, DC-DC converters are the most utilized topologies for EV energy storage devices. Furthermore, bidirectional DC-DC converters have significant advantages in terms of the efficiency, control, and voltage consumption determined by the operating requirements of the system and the chosen nonisolated or isolated topology architecture [4]. Thus, many control approaches have been introduced with fair results. In [5], a linear control strategy for a DC converter is proposed; this contribution is based on the DC-DC buck–boost circuit controlled with a state feedback and integral control approach. However, when the operating point is changed, then the control parameters must be changed too to obtain a new design.

Nonlinear predictive control is applied in [6] to determine the ideal torque to recover energy while braking, increasing the range of driving of the EV. A Sliding Mode Controller (SMC) is developed in [7], where a EV system with a nonisolated step-up DC-DC converter presents better results compared to a PI controller. Additionally, the SMC offers a better regulation of voltage and rapid settling time. However, many of these works developed their control strategies based on a mathematical model approximation of the storage system.

In [8], many of the control strategies applied to DC-DC converters such as sliding mode, fuzzy logic, PI, and Artificial Neural Networks (ANNs) are presented and reviewed. Nevertheless, strategies employing neural controllers have demonstrated significant advantages over conventional control approaches by not needing previous knowledge of the system and the ability to handle complex and large data. Recurrent High-Order Neural Networks (RHONNs) are a great alternative to ANNs because they allow updating the weights of the network based on the error passing the outputs of each node back to the input during the process, which makes them better than static ANNs in the modeling of nonlinear systems and control applications. The computation of the optimal weights of the RHONN is obtained with the Extended Kalman Filter (EKF).

Intelligent Robust Controllers proposed in this paper use a neural identifier developed with a RHONN trained with the EKF that is employed to control the dynamics of a bidirectional buck–boost converter for an EV architecture; the first one is a Neural Inverse Optimal Controller (NIOC), and the second one is a Neural Sliding Mode Controller (NSMC), both enhancing the battery state of charge (SOC). NIOC and NSMC are described in [9,10] and validated by simulation using Simulink/Simscape from MATLAB R2020a, generating reference signals to test them using a regenerative braking system for a DC motor. In this paper, instead of developing a signal reference empirically, we implement a driving cycle test for typical urban driving exclusively developed for EVs to validate the quality of the proposed controllers and energy improvement strategy using a tracking reference signal more accurate to the real behavior of the EV energy requirements. In addition, an induction motor and the corresponding three-phase inverter are now used instead of a DC motor as reported in [9,10]. The proposed intelligent robust controllers allow the control signal to be updated when the identifier weights are updated. Furthermore, the control strategy is robust because the tracking stays without significant changes despite parametric variations. The proposed controllers are compared with the performance of the PI controllers previously implemented in [11]; the results obtained prove the enhancement of the trajectory tracking of voltage with the neural controllers.

This paper is structured in the following manner: Section 2 outlines the material and methods that have been implemented to generate this work. Section 3 presents the configuration of the EV system, where the basics of the bidirectional buck–boost converter are described. In addition, the EV architecture proposed for this work is also explained. Section 4 presents a mathematical background work that includes the presentation of the discrete-time NIOC and SMC concepts, the EKF for ANN training, and the RHONN identification. Section 5 describes the neural controller design considering the buck–boost converter mathematical model which has been simplified. Moreover, the NIOC and NSMC equations and design for the dynamic control are presented, including the parameters selected for the controllers. The trajectory tracking results are obtained via simulation using real data of velocity and time corresponding to the driving cycle test implemented in Simulink using the “Simscape Electrical” in Section 6. Additionally, the comparison for both intelligent robust controllers and the PI controllers performance is carried out. Additionally, the Mean Squared Error (MSE) is computed for both the voltage and current dynamics. In Section 7, several tests are conducted to determine the robustness of the neural controller against parameter changes in the buck–boost converter system. The MSE for the five tests is calculated, and the results are analyzed for steady-state (SS) conditions as well as for the entire duration of the tests. Finally, in Section 8 and Section 9, the discussions and conclusions of the presented work are described, including future works.

2. Materials and Methods

This work presents two control strategies: NIOC and NSMC. The main goal is to improve the SOC of an EV battery by tracking the dynamics of the converter with the neural controllers developed based on RHONNs. The results are achieved following this methodology and steps:

• We propose the combination of two energy systems to the benchmark of EV architecture described in [11]. The main energy system (MES) consists of a battery bank that powers an induction motor via a three-phase inverter, and the auxiliary energy system (AES), which includes a bidirectional buck–boost converter and a supercapacitor as illustrated in Figure 1, Figure 2 and Figure 3.

  Table 1. AES and MES parameters of the simulation.

  Description	Unit
  Buck–Boost converter Capacitance 1	$4\times {10}^{-9}$F
  Buck–Boost converter Capacitance 2	$1\times {10}^{-15}$F
  Buck–Boost Converter inductor	$13\times {10}^{-1}$H
  Supercapacitor Voltage $V_{sc}$	330 V
  Supercapacitor Capacitance $C_{sc}$	$50\times {10}^{-3}$F
  Battery Bank Voltage $V_{bat}$	550 V
  Initial SOC	$80\%$
  MOSFET PWM frequency	5000 Hz
  Sampling Time	$1\times {10}^{-4}$ s

  and

  Table 2. Parameters of the induction motor.

  Description	Unit
  Nominal Power	3730 VA
  Voltage (line–line)	$460V_{rms}$
  Frequency	60 Hz
  Stator Resistance	$1.115\Omega$
  Stator Inductance	$0.005974$ H
  Rotor Resistance	$1.083\Omega$
  Rotor Inductance	$0.005974$ H
  Power	5 HP
  Rated Speed RPM	1750 RPM

  provide information on the circuit component values.
• The bidirectional buck–boost converter is our main plant to control. However, as the converter is an extremely nonlinear system, a RHONN trained with the EKF is employed to identify the voltage and current dynamics of the converter as shown in Figure 4. In addition, the identifier is used to develop two strategies of control to track the desired current and voltage required to evaluate the controller’s performance in an EV driving cycle [2], for which the induction motor proposed in the EV architecture follows.
• The proposed EV energy system architecture and controller designed were developed and validated by numerical simulation Matlab/Simulink with toolbox “Simscape Electrical” as presented in Section 5. Section 6 contains the corresponding equations and figures.
• The obtained results of the neural identification, the tracking of voltage and current trajectory using the NIOC and NSMC, the improvement in the SOC of the battery, and the MSE obtained for each of the controllers are presented in Section 6, including a comparison of the results with a PI controller as in [11].
• Robustness tests with parameter changes using NIOC and NSMC are included. Furthermore, a comparison based on MSE is presented in Section 7.

3. EV System Configuration

3.1. Bidirectional Buck–Boost DC-DC Converter

The buck–boost converter is a prominent option for EV applications in the wide variety of power converters. Having the ability to increase and decrease the voltage output depending on the Pulse Width Modulation (PWM) signal is the main advantage for helping the system obtain the energy required during the EV driving cycle. Research on the application of bidirectional buck–boost converters in electric vehicles (EVs) has shown that this power device can enhance the EV powertrain by providing the necessary energy for the motor’s speed and extending the driving range across various driving cycles and scenarios. In addition, real-time applications with high-voltage bidirectional converters were developed in [12]. In contrast, connecting multiple battery cells in series to achieve the required voltage for the motor increases the likelihood of failure of battery bank. Therefore, implementing a bidirectional buck–boost converter, which can alternate between two modes of operation based on the direction of current flow, is a suitable solution for EVs with high voltage demands. Figure 1 illustrates the bidirectional buck–boost converter circuit in a nonisolated configuration, which allows the current to flow around both sides of the circuit without restrictions. The elements that compose the buck–boost converter are two capacitors, two MOSFETs, one inductor, and a battery bank connected as the input energy device and a supercapacitor connected to the output of the circuit. The supercapacitor has the capability of rapidly charging and discharging its energy storage, offering great results in regenerative braking systems applications. The voltage in the capacitor $C_2$ is equal to $V_{bat}$, as both elements are connected in parallel. The same situation occurs in $C_1$, where the voltage is equal to $V_{sc}$. The MOSFETs need a control signal to overpass the bias of the gates $(G_1,G_2)$ input to allow the current flows from drain $(D_1,D_2)$ to source $(S_1,S_2)$ respectively. This is accomplished by using the PWM signal that is produced when the control signal for the boost or buck operation mode is acquired. Moreover, the current direction in inductor L will help to transfer the energy storage to the supercapacitor or vice versa.

The DC-DC converter operation modes can be described with the equivalent circuits presented in Figure 2. For the buck operation mode, when the signal $U_2>0$, $MOSFE{T}_2$ is activated, and the current $i_L$ flows to $V_{sc}$. The circuit is now in series and the current $i_L$ is the same in all elements of the circuit. Furthermore, when the signal $U_2=0$, the $MOSFE{T}_2$ is turned off, and the current flowing through the $D_2$ to $S_2$ dissipates its energy stored in $i_L$ to $V_{sc}$. For the boost operation mode, when the signal $U_1>0$, $MOSFE{T}_1$ is activated, and the current $i_L$ flows from $D_1$ to $S_1$ to $C_1$ and the inductor L. However, the generated energy is stored in the inductor L. When $MOSFE{T}_1$ is off and $MOSFE{T}_2$ is on, the energy storage in L and the voltage in capacitor $C_1$ are transferred to $V_{bat}$ across $MOSFE{T}_2$.

Furthermore, as the EV accelerates, the boost function is activated, and when it decelerates, the buck function is activated; thus, the battery bank performance is improved.

3.2. EV Architecture

The EV architecture considered for this work is presented in Figure 3, where the energy system of the EV is divided into two sections. In addition to the AES, there is the MES. The AES combines an energy storage device, such as a supercapacitor, with a power device, such as a bidirectional converter, whereas the MES is constructed from a battery bank. Furthermore, an induction motor with a three-phase inverter is considered. The implementation of an AES allows energy recovery to the battery bank during the deceleration of the induction motor, as the supercapacitor is capable of rapidly charging and discharging its energy to provide the amount required to move the vehicle at the desired speed.

The two energy systems operate in parallel. The MES system supplies energy to the bidirectional buck–boost converter, whose output is connected to the supercapacitor, an integral component of the AES system. The primary function of the AES is to provide power to the EV during acceleration and deceleration, thereby extending the state of charge of the MES. The operating voltage of the supercapacitor corresponds to the energy consumption of the induction motor, thereby linking the energy stored in the AES with the energy generated by the motor.

One of the key advantages of employing this architecture, which incorporates a bidirectional buck–boost converter, is its ability to both increase and decrease output voltage. Additionally, this configuration enables energy recovery during changes in operational mode through the inductor current flow. Furthermore, several real-world implementations or at least real-time prototypes have been demonstrated, integrating the converter with other auxiliary systems to optimize performance [11,13].

However, several challenges may arise in implementing this architecture in real-world applications. These challenges include the speed of the control signal required to overcome the MOSFET bias, as well as the precise selection of key components such as the inductor, capacitor, battery bank, and supercapacitor. These components must be capable of meeting the performance demands of the induction motor, or any motor used in EVs. Additionally, the rapid development of new design boards with greater processing capacity offers potential solutions to these issues. These advancements could enable the implementation of complex ANNs to accurately approximate highly nonlinear systems, such as power converters.

4. Mathematical Preliminaries

4.1. Recurrent High-Order Neural Networks

The ANNs have demonstrated that they can approximate continuous functions and are a good option in the modeling of static systems even with just one hidden layer. ANNs are considered universal approximators, and their capacity to generalize from unknown data makes their implementation attractive in a variety of applications [14]. However, the RHONNs perform better in control applications because more neurons are present. Identifying nonlinear systems involves modifying the parameters of a chosen model associated with an adaptive law. Ref. [15] presents an identifier based on RHONNs with a series-parallel setup to approximate the state variables of a general nonlinear system:

$$\begin{array}{c}\hfill {\chi }_{i,k+1}={\omega }_i^T{\varphi }_i\left(x_k\right)+{\overline{\omega }}_i^T{\phi }_i(x_k,u_k)\end{array}$$

(1)

the estimated state of the ith neuron, ${\chi }_{i,k+1}$, is used to determine the ith component of the state vector $x_k$, where $x_k=[x_{1,k},\cdots ,x_{n,k}]$ are the measured dynamics of the plant, ${\varphi }_i$ and ${\phi }_i$ are linear functions of the state vector used for the approximation of nonlinear systems, $u_k$ is an input vector of the RHONN model, ${\omega }_{i,k}\in {\Re }^{L_i}$ are the RHONN weights, and ${\overline{\omega }}_{i,k}$ are fixed coefficients of the RHONNs. Furthermore, $u_k\in {\Re }^m$ and ${\varphi }_i(x_k,u_k,k)\in {\Re }^{L_i}$ are defined in [14]

$$\begin{array}{c}\hfill {\varphi }_i(x_k,u_k,k)=\left[\begin{array}{c}{\varphi }_{i_1}\\ {\varphi }_{i_k}\\ ⋮\\ {\varphi }_{i_{L_i,k}}\end{array}\right]=\left[\begin{array}{c}\prod _{j\in I_1}{\zeta }_{ij}^{d_{ij\left(1\right)}}\\ \prod _{j\in I_2}{\zeta }_{ij}^{d_{ij\left(2\right)}}\\ ⋮\\ \prod _{j\in I_{L_i}}{\zeta }_{ij}^{d_{ij\left(L_i\right)}}\end{array}\right]\end{array}$$

(2)

with the variables of ${\varphi }_i(x_k,u_k,k)$, $I_1,I_2,\cdots ,$ and $I_{L_i}$ as nonordered subsets; $d_{ij,k}$ as integers $>0$; $L_i$ as the number of the RHONN connection $1,2,3\cdots ,n+m$; the dimension n of the RHONN state; and m as the input dimension. The given values ${\zeta }_i$ are as follows:

$$\begin{array}{c}\hfill {\zeta }_i=\left[\begin{array}{c}{\zeta }_{i_1}\\ ⋮\\ {\zeta }_{i_n}\\ {\zeta }_{i_{n+1}}\\ ⋮\\ {\zeta }_{i_{n+m}}\end{array}\right]=\left[\begin{array}{c}S\left(x_1\right)\\ ⋮\\ S_{x_n}\\ u_{1,k}\\ ⋮\\ u_{m,k}\end{array}\right]\end{array}$$

(3)

where the input vector of the RHONN is defined as $u_k=[u_{1,k},u_{2,k},\cdots ,u_{m,k}^T]$. Moreover, the hyperbolic function $S\left(x_k\right)$ is the activation function selected for the neural network defined as

$$\begin{array}{c}\hfill S\left(x_k\right)={\alpha }_{i}tanh\left({\beta }_i{x}_k\right)\end{array}$$

(4)

where $x_k$ is designated as the state variable, whereas ${\alpha }_i$ and ${\beta }_i$ are constants >0.

4.2. Extended Kalman Filter

In RHONN training, the EKF proves to be a very precise and efficient method for identifying a variety of nonlinear systems while reducing the difference between the measured and estimated states of the plant. Additionally, lots of studies have demonstrated benefits over other training techniques, like back-propagation.The EKF provides prediction error minimization and the identification of optimal weight values [14]. The algorithm model is defined as follows.

$$\begin{array}{c}\hfill w_{i,k+1}=w_{i,k}+{\eta }_i{K}_{i,k}{e}_{i,k}\end{array}$$

(5)

$$\begin{array}{c}\hfill K_{i,k}=P_{i,k}{H}_{i,k}{M}_{i,k}\end{array}$$

(6)

$$\begin{array}{c}\hfill P_{i,k+1}=P_{i,k}-K_{i,k}{H}_{i,k}^T{P}_{i,k}+Q_{i,k}\end{array}$$

(7)

$$\begin{array}{c}\hfill e_{i,k}=x_{i,k}-{\chi }_{i,k}i=1,2,\cdots ,n\end{array}$$

(8)

$$\begin{array}{c}\hfill M_{i,k}={\left[R_{i,k}+H_{i,k}^T{P}_{i,k}{H}_{i,k}\right]}^{-1}\end{array}$$

(9)

where $Q_{i,k}\in {\mathbb{R}}^{L_i\times L_i}$ and $R_{i,k}\in {\mathbb{R}}^{m\times m}$ are set as positive definite constant matrices, $P_{i,k}\in {\mathbb{R}}^{L_i\times L_i}$ is a diagonal matrix with adjustable values, and $e_{i,k}\in \mathbb{R}$ is set as the identification error to be optimized. $K_{i,k}\in {\mathbb{R}}^{L_i\times m}$ is defined as the “Kalman matrix”. $H_{i,k}\in {\mathbb{R}}^{L_i\times m}$ is the product of the adjustable weights of the neural identifiers:

$$\begin{array}{c}\hfill H_{ij,k}={\left[\begin{array}{c}{\displaystyle \frac{\partial {\chi }_{i,x}}{\partial w_{ij,k}}}\end{array}\right]}_{w_{i,k}=w_{i,k+1}}^T\end{array}$$

(10)

where i and j are defined as $i=1,\dots ,n$ and $j=1,\dots ,L_i$, and k is the iteration number. $P_i$, $Q_i$, and $R_i$ are diagonal matrices. It is important to note that $H_{i,k}$, $K_{i,k}$, and $P_{i,k}$ for the EKF are bounded. Therefore, there exist constants $H_i^{*}>0$, $K_i^{*}>0$, and $P_i^{*}>0$, resulting in

$$\begin{array}{c}\hfill \Vert H_{i,k}\Vert \le H_i^{*}\end{array}$$

(11)

$$\begin{array}{c}\hfill \Vert K_{i,k}\Vert \le K_i^{*}\end{array}$$

(12)

$$\begin{array}{c}\hfill \Vert P_{i,k}\Vert \le P_i^{*}\end{array}$$

(13)

The training of the neural networks is implemented online considering a series-parallel configuration as presented in Figure 4.

More details on the demonstration of this stability of RHONN and EKF are provided in detail in [14,16].

4.3. Discrete-Time Inverse Optimal Control

The reference [17] presents a contribution utilizing Discrete-Time Inverse Optimal Control to regulate the output voltage of a boost converter by employing inverse optimal control alongside gain scheduling. The proposed methodology aids in determining the essential parameters for implementing the inverse optimal control under specified conditions. Furthermore, this approach adapts to variations in sampling time, input variables, output voltage, and reference signal sequences. It also effectively mitigates the impact of parameter disturbances within the state equations of the boost converter.

Reference [18] introduces an inverse optimal control approach leveraging the EKF algorithm to address the optimal control problem for discrete-time affine nonlinear systems. The primary objective of inverse optimal control is to bypass the complex process of solving the Hamilton–Jacobi–Bellman equation, which is typically required in conventional nonlinear optimal control problems. The effectiveness and feasibility of the proposed approach are demonstrated through simulations on a nonlinear system model, as well as a real-time laboratory experiment. The controller is implemented on a professional control board to stabilize a DC-DC boost converter while minimizing a relevant cost function. The experimental outcomes underscore the applicability and efficiency of the EKF-based inverse optimal control in real-time control systems, even those with very short time constants.

As stated in [19], consider the following discrete-time nonlinear system:

$$x_{k+1}=f\left(x_k\right)+B\left(x_k\right)u\left(x_k\right)+d\left(x_k\right)$$

(14)

$$y_k=h\left(x_k\right)$$

(15)

Let $x_k$, $u_k$, and $y_k$ be the state variables, input vectors, and controlled output vectors, respectively. The functions $f\left(x_k\right)$, $B\left(x_k\right)$, $h\left(x_k\right)$, and $d\left(x_k\right)$ are constrained and smooth. In addition, $d\left(x_k\right)$ is considered a term of disturbances present in the system. If $y_k$ encompasses the entire state vector and the goal is to guide the controlled dynamics to follow specific trajectories, then the tracking error is given by the following expression:

$$e_{k+1}=x_{k+1}-x_{ref,k+1}$$

(16)

The desired trajectory vector is defined as $x_{ref,k}$. The error at $k+1$ is expressed as

$$e_{k+1}=f(x_k,k)+B\left(x_k\right)u(x_k,k)+d\left(x_k\right)-x_{ref,k+1}$$

(17)

The Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE) is used to minimize the cost function to determine the optimal solution. However, some of these equations are challenging to solve [20]. The trajectory tracking relies on the cost function of system (17)

$$J\left(e_k\right)=\sum _{k=0}^{\infty }(l\left(e_k\right)+u_k^{T}R{u}_k)$$

(18)

Let $J:{\Re }^n\to {\Re }^{+}$ be a performance measure, $l:{\Re }^n\to {\Re }^{+}$ a positive semidefinite function, and $R:{\Re }^n\to {\Re }^{n\times m}$ a positive definite real symmetric matrix. The optimal cost function, denoted by $J^{*}$, is defined as the Lyapunov function $V\left(e_k\right)$. This function is time invariant and must satisfy the discrete-time Bellman equation presented below:

$$V\left(e_k\right)=\underset{u_k}{min}l\left(e_k\right)+u_k^{T}R\left(e_k\right)u_k+V\left(e_{k+1}\right)$$

(19)

Therefore, let us state the Hamiltonian equation as follows:

$$H(e_k,u_k)=l\left(e_k\right)+u_k^{T}R\left(e_k\right)u_k+V\left(e_{k+1}\right)-V\left(e_k\right)$$

(20)

The optimal control function is found by using the equation $H(e_k,u_k)=0$, and $u_k$ [19] is utilized to calculate the gradient of the right-hand side of (20). Consequently,

$$u_k^{*}=-{\displaystyle \frac{1}{2}}R{\left(e_k\right)}^{-1}B{\left(x_k\right)}^T{\displaystyle \frac{\partial V\left(e_{k+1}\right)}{\partial e_{k+1}}}$$

(21)

in which $u_k*$ is the optimal control rule, and $V\left(0\right)=0$ is the condition that limits the function $V\left(e_k\right)$ that needs to be met. With the help of (21) in (19), the HJB equation in discrete time is

$$V\left(e_k\right)={\displaystyle \frac{1}{4}}{\displaystyle \frac{\partial V^T\left(e_{k+1}\right)}{\partial e_{k+1}}}R{\left(e_k\right)}^{-1}B{\left(x_k\right)}^T{\displaystyle \frac{\partial V\left(e_{k+1}\right)}{\partial e_{k+1}}}+l\left(e_k\right)+V\left(e_{k+1}\right)$$

(22)

The solution to the HJB PDE (22) for $V\left(e_k\right)$ is challenging to obtain. The control law is derived using the discrete-time inverse optimal control (IOC) technique in conjunction with a Lyapunov function as referenced in [20]. To formulate the aforementioned problem within the framework of IOC, the forthcoming definition is introduced.

Definition 1. 

A control law like the one defined in (21) for a discrete-time nonlinear system (14) can be identified as globally IOC stabilizing if and only if the following hold

(1)

When $e_k=0$ in (21) ensures that the global asymptotic stability is achieved;

(2)

The objective function (18) with $V\left(e_k\right)>0$ is minimized and guarantees that

$$\overline{V}:=V\left(e_{k+1}\right)-V\left(e_k\right)+u_k^{*}B\left(x_k\right)u_k^{*}\le 0.$$

(23)

Accordingly, the synthesis of IOC is predicated on utilizing $V\left(e_k\right)$ as delineated in the preceding definition. Thereafter, we have the following.

Definition 2 

([19]). $V\left(x_k\right)$ is selected as a radially bounded function $V\left(x_k\right)>0$, conditioning that for a $x_k$, there exist $u_k$ and

$$\Delta V(e_k,u_k)<0$$

(24)

considering $V\left(e_k\right)$ as a control Lyapunov function (CLF) that needs to meet the conditions described in (1) and (2) of Definition 1. The resulting CLF is presented as

$$V\left(e_k\right)=-{\displaystyle \frac{1}{2}}{e}_k^{T}P\left(e_k\right)$$

(25)

where $P\in {\Re }^{n\times n}$ and subsequently, this matrix $P=P^T>0$. The control signal provided in (21) ensures the stability of the equilibrium point $e_k=0$ in (17) by choosing a suitable matrix P. Moreover, the crucial cost function given in (18) is optimized by the control law (21) in combination with (25), which is understood as an inverse optimal control law for (14). Furthermore, the IOC law can be constructed subsequently by utilizing (21) to (25):

$$u_k^{*}=-{\displaystyle \frac{1}{2}}{\left(R+{\displaystyle \frac{1}{2}}B{\left(x_k\right)}^{T}PB\left(x_k\right)\right)}^{-1}B{\left(x_k\right)}^{T}P(f\left(x_k\right)-x_{ref,k+1})$$

(26)

P and R are positive-definite matrices. Ref. [19] provides a full explanation of the NIOC synthesis method. Achieving the best performance inside the discrete-time IOC framework necessitates prior knowledge of all model parameters, which is sometimes impossible in real-time applications. Furthermore, because it is based on a theoretical model, this control strategy is inherently susceptible to parameter changes and external disturbances. To address these constraints, an RHONN identifier is introduced and trained online using an EKF.

4.4. Discrete-Time Sliding Mode Control

One of the notable attributes of SMC discerned over the past few decades is its robustness in the presence of various perturbations. Consider the nonlinear system presented in [21]

$$\dot{x}=f(x,u,t)$$

(27)

In this context, let $f\left(x\right)$ be a bounded constant function $f\left(x\right)$ such that

$$\left|f\left(x\right)\right|<f_o,$$

and let a control law with switching behavior be employed to minimize the error between a reference input r and the current state x, where the error is given by

$$e=r-x.$$

The control law u is then defined as follows:

$$u=\left\{\begin{array}{c}{u}_{0}ifs\left(x\right)\ge 0\\ -u_{0}ifs\left(x\right)<0\end{array}\right.$$

(28)

where $u_0$ and $s\left(x\right)$ represent the upper control bound and sliding surface, respectively. The $u_0$ changes its value in the function of the sliding surface at x. A continuous-time system with scalar SMC is described in Figure 5, where the state of $x\left(t\right)$ initializes from a $x(t=0)$ point, achieving the trajectory of the $s\left(x\right)$ in finite time $t_{sm}$ and remaining on the surface afterwards.

The function is derived for each sampling point $t_j=k\Delta t$, where $k=1,2\cdots$, resulting in a discrete-time representation of the previous control system (27):

$$x_{k+1}=F\left(x_k\right)$$

(29)

with the initial state at $t=\left(t_{sm}\right)$, the path aligns with the sliding manifold when $s\left(x\right(t\left)\right)$, or $k_m\ge t_{sm}/\Delta t$ [21].

$$s\left(x_k\right)=0\forall k\ge k_{sm}$$

(30)

This operation is characterized as discrete-time sliding mode. Accordingly, based on (27), with a fixed control u and any initial condition $x\left(0\right)$, the closed-form solution can be expressed as follows [15]:

$$x\left(t\right)=F\left(x\right(0),u)$$

(31)

To achieve $s\left(x_{k+1}\right)$ with the constant state control signal, it is essential to calculate $u_k$ at each discrete sampling point k. The nonlinear system can be described with a corresponding discrete-time representation as in [15]

$$\begin{array}{cc}\hfill x_{k+1}& =f\left(x_k\right)+B\left(x_k\right)u_k\left(x_k\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill y_{k+1}& =h\left(x_k\right)\hfill \end{array}$$

(33)

where $x_k$ denotes the state of the system expressed in vector form, $u_k$ represents the input to the system, and $y_{k+1}$ is the estimated output vector requiring regulation. Moreover, B is a control matrix of fixed weights, while $f\left(x_k\right)$ and $h\left(x_k\right)$ are field vectors that are bounded and smooth. The tracking error is specified as follows to ensure the tracking of the desired $\widehat{y}\left(k\right)$:

$$\begin{array}{c}\hfill s_k=x_k-x_{ref,k}\end{array}$$

(34)

The sliding manifold can be determined by evaluating at sample time $(k+1)$:

$$\begin{array}{c}\hfill s_{k+1}=f\left(x_k\right)+B\left(x_k\right)u_c\left(x_k\right)-x_{ref,k+1}\end{array}$$

(35)

The NSMC law $u_c$ guarantees that the tracking error (34) for (32) and (33) is bounded ensuring that the system’s output converges to a neighborhood of the respective desired trajectory. The complete description of this statement’s proof is found in [15]

$$\begin{array}{c}\hfill u_c(x_k,k)=u_{eq}(x_k,k)+u_n(x_k,k)\end{array}$$

(36)

Given $s_{k+1}=0$, the equivalent control $u_{eq}(x_k,k)$ is calculated as outlined in [21]:

$$\begin{array}{c}\hfill u_{eq}(x_k,k)=-{B\left(x_k\right)}^{-1}\left[f\left(x_k\right)-x_{ref,k+1}\right]\end{array}$$

(37)

the term $u_n(x_k,k)$ serves as a stabilizing component that must be introduced to ensure asymptotic convergence to the sliding manifold as established in [15]:

$$\begin{array}{c}\hfill u_n(x_k,k)={B\left(x_k\right)}^{-1}\left(S_{sc}{s}_k\right)\end{array}$$

(38)

where $S_{sc}$ is designed as a Schur matrix.

Considering the boundedness of the control signal where $\Vert u_c(x_k,k)\Vert <u_0$, $u_0>0$, the following control law is established as in [21]:

$$\begin{array}{c}\hfill u_n(x_k,k)=\left\{\begin{array}{cc}{u}_c(x_k,k)& if∥u_c(x_k,k)∥<u_0\\ \\ u_0{\displaystyle \frac{u_{eq}(x_k,k)}{∥u_{eq}(x_k,k)∥}}& if∥u_c(x_k,k)∥\ge u_0\end{array}\right.\end{array}$$

(39)

where the Euclidean norm of the term $u_c(x_k,k)$ is denoted as $∥•∥$.

5. Neural Controllers Design

In this section, we describe the design of the two control strategies for the tracking of the desired trajectories in the converter.

5.1. Bidirectional Buck–Boost Converter Model

A DC-to-DC bidirectional converter is achieved by combining boost and buck converters models. Whereas the second is used in discharge-only scenarios, the first is employed in charge-only conditions. Using the Euler discretization approach, the boost converter model is defined using the work in [15,21]:

$$\begin{array}{cc}\hfill x_{1,k+1}& =x_{1,k}-{\displaystyle \frac{ts}{c_1}}{x}_{2,k}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill x_{2,k+1}& =x_{2,k}+{\displaystyle \frac{ts}{L}}{V}_{bat}{u}_1-{\displaystyle \frac{ts}{L}}{x}_{1,k}\hfill \end{array}$$

(41)

The buck operation mode of the converter can be described as follows:

$$\begin{array}{cc}\hfill x_{1,k+1}& =x_{1,k}+{\displaystyle \frac{ts}{c_2}}{x}_{2,k}{u}_2\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill x_{2,k+1}& =x_{2,k}-{\displaystyle \frac{ts}{L}}{x}_{1,k}{u}_2\hfill \end{array}$$

(43)

Introducing $v_1$ and $v_2$ as two logical variables which take the values of 0 or 1 in an alternate form, Equations (40)–(43) can be summarized as

$$\begin{array}{cc}\hfill x_{1,k+1}& =x_{1,k}+(v_2-1){\displaystyle \frac{ts}{c_1}}{x}_{2,k}+{\displaystyle \frac{ts}{c_2}}{x}_{2,k}(1-v_1)u_2\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill x_{2,k+1}& =x_{2,k}+{\displaystyle \frac{ts}{L}}{V}_{bat}(1-v_2)u_1-{\displaystyle \frac{ts}{L}}{x}_{1,k}{v}_1-{\displaystyle \frac{ts}{L}}{x}_{1,k}(1-v_1)u_2\hfill \end{array}$$

(45)

where $V_{bat}$ is the voltage in the battery bank, $u_1$ is the input to the MOSFET 1 and $u_2$ is the input for the MOSFET 2. The converter’s output voltage is $x_{1,k}$, the inductor’s output current is $x_{2,k}$, $t_s$ is the sampling time, L and $C_1$ and $C_2$ are the inductance and capacitances measured in Henry $\left(H\right)$ and farads $\left(F\right)$ units, respectively. When $v_1=1$ and $v_2=0$, the boost mode is activated. On the other hand, when $v_1=0$ and $v_2=1$, the buck mode is activated.

5.2. Neural Identifier Design

Using Equation (1) and the procedures given in Section 4.1, the dynamics of the buck–boost converter are approximated using a RHONN, which ensures the charging and discharging operating modes and controls the current flow. Given the similarity of the buck and boost converter models and the adaptability of the RHONN, the following single identification would be optimal for each scenario:

$$\begin{array}{cc}\hfill {\widehat{\chi }}_{1,k}& ={\omega }_{11,k}S\left(x_{1,k}\right)+{\omega }_{12,k}S\left(x_{2,k}\right)\hfill \\ & +{\omega }_{13,k}S\left(x_{1,k}\right)S\left(x_{2,k}\right)+{\varpi }_1{x}_{2,k}+u\left(x_{1,k}\right)\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\widehat{\chi }}_{2,k}& ={\omega }_{21,k}S\left(x_{2,k}\right)+{\omega }_{22,k}S\left(x_{1,k}\right)\hfill \\ & +{\omega }_{22,k}S\left(x_{1,k}\right)S\left(x_{2,k}\right)+{\varpi }_{2}u\left(x_{2,k}\right)\hfill \end{array}$$

(47)

where the vector ${[x_{1,k},x_{2,k}]}^T$ is employed to obtain ${[{\widehat{\chi }}_{1,k},{\widehat{\chi }}_{2,k}]}^T$ defined as the approximated dynamics. As stated before, the voltage in the output of the converter is denoted by $x_{1,k}$, while the converter’s output current is set to $x_{2,k}$. For each stage of the system, u is selected as the control signal, with ${\varpi }_1$ and ${\varpi }_2$ operating as constant fixed weights.

5.3. Neural Inverse Optimal Controller

After the identification of the buck–boost converter is functioning properly, the NIOC can be obtained. The neural identifier-based controller has the ability to measure some dynamics more accurately, which is difficult for traditional sensors to read. The power converter’s control scheme has to control the dynamics of both the voltage and current for it to transmit and store energy to the battery bank effectively.

Voltage NIOC

The NIOC approach is used to track the voltage of the power converter. The following formula serves to determine the voltage error at $k+1$ using (46):

$$\begin{array}{ccc}\hfill {\widehat{e}}_{x_1,k+1}& =& {\omega }_{11,k}S\left(x_1\right)+{\omega }_{12,k}S\left(x_2\right)\hfill \\ & +& {\omega }_{13,k}S\left(x_1\right)S\left(x_2\right)+{\varpi }_1{x}_2-x_{1ref,k+1}\hfill \end{array}$$

(48)

Subsequently, the bidirectional buck–boost current reference for the supercapacitor is determined by applying the NIOC control law. The control law determined in this step is employed as the current reference signal for tracking:

$$\begin{array}{c}\hfill u_{NIOC}(x_1,k)=-{\displaystyle \frac{1}{2}}{\left(R_1+{\displaystyle \frac{1}{2}}B{\left(x_{1,k}\right)}^T{P}_{1}B\left(x_{1,k}\right)\right)}^{-1}B{\left(x_{1,k}\right)}^T{P}_2{\widehat{e}}_{x_1,k+1}\end{array}$$

(49)

The control method for the regenerative braking system using the current reference generator and NIOC is depicted in Figure 6.

Current NIOC

In the controller design, the suggested controller is employed for current trajectory tracking. For the current dynamics, the tracking error at $k+1$ is calculated as

$$\begin{array}{ccc}\hfill {\widehat{e}}_{x_2,k+1}& =& {\omega }_{21,k}S\left(x_2\right)+{\omega }_{22,k}S\left(x_1\right)\hfill \\ & +& {\omega }_{23,k}S\left(x_1\right)S\left(x_2\right){\varpi }_2{u}_k-x_{2ref,k+1}\hfill \end{array}$$

(50)

Afterwards, employing analogous procedures as in Section 4.3, the equivalent NIOC is determined as follows:

$$\begin{array}{c}\hfill u_{NIOC}(x_2,k)=-{\displaystyle \frac{1}{2}}{\left(R_2+{\displaystyle \frac{1}{2}}B{\left(x_{2,k}\right)}^T{P}_{2}B\left(x_{2,k}\right)\right)}^{-1}B{\left(x_{2,k}\right)}^T{P}_2{\widehat{e}}_{x_2,k+1}\end{array}$$

(51)

The equivalent NIOC control equations can be described in a matrix form:

$$\begin{array}{c}\hfill U=-{\displaystyle \frac{1}{2}}{\left(R+{\displaystyle \frac{1}{2}}{B}^{T}PB\right)}^{-1}{B}^{T}PE\end{array}$$

(52)

where $U=\left[\begin{array}{c}{u}_{NIOC}(x_1,k)\\ u_{NIOC}(x_2,k)\end{array}\right]$, $R=\left[\begin{array}{cc}R1& 0\\ 0& R2\end{array}\right]$, $B=\left[\begin{array}{cc}B\left(x_{1,k}\right)& 0\\ 0& B\left(x_{2,k}\right)\end{array}\right]$, $P=\left[\begin{array}{cc}{P}_1& 0\\ 0& P_2\end{array}\right]$ and $E=\left[\begin{array}{c}{\widehat{e}}_{x_1,k+1}\\ {\widehat{e}}_{x_2,k+1}\end{array}\right]$ where the matrix R and P have to be positive definite. It is worth mentioning that, as a control law is proposed for each dynamical variable, R and P are scalars. Due to this fact, R and P are easily determined empirically [17] as follows:

$$\begin{array}{cc}\hfill P=\left[\begin{array}{cc}50& 0\\ 0& 1000\end{array}\right]& R=\left[\begin{array}{cc}0.05& 0\\ 0& 0.0001\end{array}\right]\end{array}$$

Furthermore, from Equations (46) and (47), it is easy to see that

$$\begin{array}{c}\hfill B=\left[\begin{array}{cc}1& 0\\ 0& {\varpi }_2\end{array}\right]\end{array}$$

with ${\varpi }_2=0.001$.

5.4. Neural Sliding Mode Controller

The NSMC is developed with the obtained neural identifier. This controller is used to control the voltage and current output in the converter output.

Voltage NSMC

The sliding surface at iteration $(k+1)$ of the voltage control ${\widehat{x}}_{1,k}$ is determined as a result of the steps in (34)–(39):

$$\begin{array}{cc}\hfill s_{x_{1,k+1}}& ={\omega }_{11,k}S\left(x_2\right)+{\omega }_{12,k}S\left(x_1\right)+{\omega }_{13,k}S\left(x_{1,k}\right)S\left(x_{2,k}\right)\hfill \\ & +{\varpi }_1{x}_{2,k}+u\left(x_{1,k}\right)-x_{1ref,k+1}\hfill \end{array}$$

(53)

The equivalent control law for ${\widehat{x}}_{1,k}$ is determined as follows:

$$\begin{array}{cc}\hfill u_{eq}(x_1,k)& =-{\displaystyle \frac{1}{{\varpi }_1}}[{\omega }_{11,k}S\left(x_2\right)+{\omega }_{12,k}S\left(x_1\right)+{\omega }_{13,k}S\left(x_{1,k}\right)S\left(x_{2,k}\right)-x_{1ref,k+1}]\hfill \end{array}$$

(54)

$$\begin{array}{c}\hfill u_c\left(x_{1,k}\right)=\left\{\begin{array}{cc}{u}_t(x_1,k)& if∥u_t(x_1,k)∥<u_0\\ \\ u_0{\displaystyle \frac{u_{eq}(x_1,k)}{∥u_{eq}(x_1,k)∥}}& if∥u_t(x_1,k)∥\ge u_0\end{array}\right.\end{array}$$

(55)

The $u_t(x_1,k)$ is set as

$$\begin{array}{c}\hfill u_t(x_1,k)=u_{eq}(x_1,k)+u_n(x_1,k)\end{array}$$

(56)

where a square matrix $S_{sc}\in \Re$ with $∥S_{sc}∥<1$ eigenvalues of restriction is introduced. $u_0$ is an upper bound of control, where $u_0>0$, and finally, with $u_n(x_1,k)=-\left(S_{sc}{s}_{x_{1,k+1}}\right)$ as an stabilizing term.

Current NSMC

The development of the sliding surface and control for ${\widehat{x}}_{2,k}$ is obtained as follows:

$$\begin{array}{cc}\hfill s_{x_{2,k+1}}& ={\omega }_{21,k}S\left(x_2\right)+{\omega }_{22,k}S\left(x_1\right)\hfill \\ \hfill & +{\omega }_{23,k}S\left(x_1\right)S\left(x_2\right){\varpi }_2{u}_k-x_{2ref,k+1}\hfill \end{array}$$

(57)

The current control is obtained as follows:

$$\begin{array}{cc}\hfill u_{eq}(x_2,k)& =-{\displaystyle \frac{1}{{\varpi }_2}}[{\omega }_{21,k}S\left(x_2\right)+{\omega }_{22,k}S\left(x_1\right)+{\omega }_{2,3,k}S\left(x_1\right)S\left(x_2\right)-x_{2ref,k+1}]\hfill \end{array}$$

(58)

and the NSMC is implemented as follows:

$$\begin{array}{c}\hfill u_c(x_2,k)=\left\{\begin{array}{cc}{u}_t(x_2,k)& if∥u_t(x_2,k)∥<u_0\\ \\ u_0{\displaystyle \frac{u_{eq}(x_2,k)}{∥u_{eq}(x_2,k)∥}}& if∥u_t(x_2,k)∥\ge u_0\end{array}\right.\end{array}$$

(59)

The $u_t(x_2,k)$ is set as

$$\begin{array}{c}\hfill u_t(x_2,k)=u_{eq}(x_2,k)+u_n(x_2,k)\end{array}$$

(60)

where $S_{sc}$ is still a real entries $m\times n\therefore m=n$ matrix, and the eigenvalues are constrained with $∥S_{sc}∥<1$. Moreover, the stabilizing term $u_n(x_2,k)$ of the NSMC is defined as $u_n(x_2,k)=-\left(S_{sc}{s}_{x_{2,k+1}}\right)$. Finally, the control limit $u_0$ achieves the upper stabilization condition with $u_0>0$.

The selection of these control approaches is primarily based on the advantages they offer over traditional control schemes, such as the PI or state feedback integral controller [5], which has been widely used in numerous studies with satisfactory results. However, many of these studies rely on simulations employing linear models of power converters. This can be unreliable or even hazardous, as these models only approximate the actual behavior of the system. It is well known that DC-DC converters exhibit highly nonlinear dynamics. The proposed control strategies are developed using stabilization techniques, such as the Lyapunov function, in which a cost function is optimized. Furthermore, this framework enables the integration of evolutionary algorithms to determine optimal parameters that ensure system stability. When combined with ANNs, such as the RHONN employed in this work, the reliability of both system dynamics and control implementation is significantly enhanced.

5.5. Voltage Reference Generator

A reference generator is required to recover energy through regenerative braking and to produce the necessary voltage for either accelerating or decelerating the induction motor. The resulting reference voltage signal corresponds to both the controlled voltage $x_1$ in the buck–boost converter and the supercapacitor. A definition based on the work–energy theorem and its application to voltage generation can be found in [10,11,22].

Definition 3. 

The work performed on a particle as it moves from point A to point B results in an increase in its kinetic energy.

The mathematical formula for the statement is expressed as

$$W_{A\to B}={\int }_A^{B}P{d}_t$$

(61)

In this case, the application of this theorem to an induction motor is considered, and the energy for the time interval $t\in [k\delta ,(k+1\left)\delta \right)]$ can be calculated as follows [11]:

$$E\left(t\right)={\int }_{k\delta }^t{P}_{k}d\xi +E_k$$

(62)

The induction motor power is defined as $P=\tau \omega$ with $\tau$ as the motor torque and $\omega$ as the speed. The supercapacitor operates in two modes: charge and discharge. The following equation describes the energy of the supercapacitor during the charging mode:

$$E_C^{ref-c}\left(t\right)=-k_p{\int }_{k\delta }^{t}sa{t}_1\left(P\right)d\xi +E_{ck}$$

(63)

The following equation describes the energy of the supercapacitor during the discharging mode:

$$E_C^{ref-d}\left(t\right)=k_p{\int }_{k\delta }^{t}sa{t}_2\left(P\right)d\xi +E_{ck}$$

(64)

Let $E_C^{ref-c}\left(t\right)$, $E_C^{ref-d}\left(t\right)$ and $0<k_p<1$ represent the reference energy for the charging mode, discharging mode, and the lost energy, respectively. The function $sa{t}_1$ is defined within the range $(-\infty ,0)$, while $sa{t}_2$ is defined within the range of $(0,\infty )$. The sum of both energy references, as described in (63) and (64), yields the supercapacitor energy reference:

$$E_C=E_C^{ref-c}+E_C^{ref-d}$$

(65)

Moreover, the energy of the supercapacitor can be equally described as

$$Ec={\displaystyle \frac{C_{sc}{V}_{cr}^2}{2}}$$

(66)

where $V_{cr}$ represents the voltage stored in the supercapacitor, $C_{sc}$ denotes its capacitance, and $E_c$ represents its energy storage as a function of the motor power P. Using (63), the formula to obtain the reference voltage is defined as

$$\begin{array}{c}\hfill V_{cr}=\sqrt{{\displaystyle \frac{2E_C}{C_{sc}}}}\end{array}$$

(67)

The reference generator can be easily implemented considering that the power and speed of the induction motor encompass the supercapacitor capability of the energy storage.The primary objective is to determine the required energy during the acceleration and deceleration phases of the EV motor. Additionally, to define the charge reference, the energy contained in the super capacitors must be a function of the energy generated by the induction motor when it operates as a generator.

The generated voltage reference $V_{cr}$ is then used to estimate the corresponding current reference $i_{reference}$ with the first neural controller (NIOC or NSMC) for the capacitor, enabling it to track and compute the duty cycle for the MOSFET gate signals $G_1$ and $G_2$ as described in Section 3. This is achieved using the resulting control signals $u_{NIOC}(x_2,k)$ and $u_c(x_2,k)$ for current control in the buck–boost converter.

The duty cycle for the NIOC is determined by the following formula:

$$\begin{array}{c}\hfill \left(u_1{u}_2\right)=\left\{\begin{array}{cc}(\parallel u_{NIOC}(x_2,k)\parallel 0)& if{u}_{NIOC}(x_2,k)<0\\ \\ (0\parallel u_{NIOC}(x_2,k)\parallel )& if{u}_{NIOC}(x_2,k)>0\\ \\ \left(00\right)& if{u}_{NIOC}(x_2,k)=0\end{array}\right.\end{array}$$

(68)

and the duty cycle for NSMC:

$$\begin{array}{c}\hfill \left(u_1{u}_2\right)=\left\{\begin{array}{cc}(\parallel u_c(x_2,k)\parallel 0)& if{u}_c(x_2,k)<0\\ \\ (0\parallel uc(x_2,k)\parallel )& if{u}_c(x_2,k)>0\\ \\ \left(00\right)& if{u}_c(x_2,k)=0\end{array}\right.\end{array}$$

(69)

The schematic diagram of the EV regenerative braking system, neural identification of the system, intelligent robust controller operation, the designed reference generator block, and the PWM generator used to determine the duty cycle of the converter are presented in Figure 6. The simulation employs this representation as its model for the two cases presented in this work.

6. Results

The results of these simulations are divided into the tracking trajectories of the voltage and current, and the enhancement obtained with the intelligent controllers developed with the RHONN identifier. In addition, each method presents the weight adjustment of the system identification, the control signal, and the improvement in SOC resulting from the control strategies employed. Additionally, the MSE is calculated using the results obtained and compared with those from a PI controller as demonstrated in [11].

6.1. Trajectory Tracking with NIOC

In this section, to track the required voltage and current in the EV driving cycle scenario, the NIOC is selected. The scenario in which the EV operates is the driving cycle for EVs developed in [2], where the main objective is to study the results obtained with the driving motion during 1200 seconds (20 min). The driving cycle is illustrated in Figure 7. Furthermore, the specifications of the buck–boost converter and induction motor are presented in Table 1 and Table 2. These parameters have been adapted to operate with the induction motor and real data instead of the DC motor simulated with synthetic data as presented in [10].

The tracking of the voltage reference generated with (67) and the current reference with (49) are presented in Figure 8 and Figure 9, where is easy to see that the trajectory is followed in both cases and maintains the tracking during the complete 1200 seconds of operation. In addition, the NIOC control signals are presented in Figure 10, where the controllers achieve the desired value to track the trajectory at $t\approx 3$ seconds. Furthermore, Figure 10a illustrates the complete control signal for the voltage $x_1$ during the entire driving cycle, while Figure 10b shows how the controller behaves during the first 4 seconds. On the other hand, Figure 10c,d are similar cases, but for the current $x_2$. Figure 11a shows the weight adjustment of the RHONN identifier during the entire driving cycle, and Figure 11b illustrates the first 20 seconds of the adjustment of the system identification. Finally, in Figure 12, the comparison in the SOC of the battery used in the EV architecture with and without AES is shown, where the NIOC improves the state of charge over the MES as the only energy system of the EV. The difference obtained between the SOC with and without AES is important, as this means that the lifetime of the battery and the driving range for an EV have been extended.

6.2. Trajectory Tracking with NSMC

The NSMC is chosen in this simulation to track the necessary voltage and current in the case of an EV driving cycle. The controlled operation of the buck–boost converter is shown in Figure 13 and Figure 14; the tracked voltage signal is the one that is generated by (67). The NSMC can follow the desired voltage for the supercapacitor in a short amount of time. As presented in the diagram in Figure 6, the $i_{reference}$ is generated through the voltage control, resulting in the tracking signal followed by the NSMC in the current of the converter. Similar to voltage tracking, the current tracking has great performance in obtaining the current necessary. It is necessary to mention that the current in the buck–boost converter behaves in the manner of the $G_1$ and $G_2$ state and the frequency at which the MOSFET operates.

The control signals obtained with the NSMC are presented in Figure 15, where (a) represents the control applied to the system during the whole driving cycle of 1200 seconds, while (b) is a zoom of the control signal, where the fast response of the $u_c\left(x_{1,k}\right)$ at $t\approx 2$ seconds is depicted. Furthermore, (c) illustrates the control signal $u_c\left(x_{2,k}\right)$ during the complete operation of the EVs induction motor, and (d) is the zoom of the control signal, for which, despite the significant adjustment values at the beginning of the operation, the control has a fast response of $t\approx 2$ seconds.

The weight adjustment during the neural identification and the NSMC control is presented in Figure 16a, while (b) depicts the weight adjustment during the first 20 seconds. Furthermore, one of the major enhancements provided by this NSMC is the battery’s state of charge. The comparison of the EV system with AES and without the AES is presented in Figure 17a, where the behavior during all the driving of the EV is illustrated. On the other hand, Figure 17b demonstrates better energy conservation by having the EV configuration with the AES at the final seconds of the EV operation. The difference obtained between the SOC with and without AES is important, as this means that the lifetime of the battery and the driving range for an EV have been extended by employing the regenerative braking systems control proposed in Figure 6.

6.3. Comparison with PI Controller

The same scenario of the EV enforced with the AES using a PI is carried out, where the main results demonstrate that the typical controller is capable of tracking the voltage $V_{cr}$ and current $i_{reference}$ of the buck–boost converter but with decreased performance. Following the same control scheme of the EV architecture in Figure 6 but changing the controller for a PI without the neural identifier offers a smooth signal to track in the current dynamics, but the desired voltage reference is tracked for a long time compared with the NIOC and NSMC, as it is around $t=3$ seconds and $t=2$ seconds respectively, while the PI controller at $t=30$ seconds also has a big error between the reference voltage and the measured one in the buck–boost converter. In Figure 18a the comparison of the NIOC vs. NSMC vs. PI controller is presented, where the neural controllers have better tracking over the conventional. In Figure 18b,c, it is easy to see that the neural controller gives better tracking results than the PI controller, which takes several seconds to track the voltage reference and the overall behavior is more accurate using the NSMC. Additionally, in Figure 19a, the buck–boost converter controlled voltage dynamic is illustrated. Figure 19b shows that the PI controller track the desired voltage reference in $t\approx 30$ seconds. In Figure 20, the current reference is generated with the PI controller, which results in a different signal for tracking than the one with the NIOC and NSMC as presented in Figure 14. Figure 20b illustrates that for the PI track, the desired current signal also is $t\approx 30$ seconds.

The Mean Squared Error (MSE) between the three controllers is presented below, where the NSMC gives a reliable error for the application in batteries and energy management systems for EVs despite the fact that the MSE in the control of the current with NIOC and PI is less than that with the NSMC as mentioned above; the tracking of the voltage is not reliable enough.

The MSE can be calculated using the next equation:

$$\begin{array}{c}\hfill \mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(Y_i-{\widehat{Y}}_i)}^2\end{array}$$

(70)

where the total number of data points for computation is represented by n, the values acquired at the data points are indicated by $Y_i$, and the reference values at the data points are represented by ${\widehat{Y}}_i$.

The MSE obtained is presented in

Table 3. MSE for voltage and current.

Controller	MSE Value for Voltage	MSE Value for Current
NIOC	$40.9070$	$0.0078$
NSMC	$30.9922$	$51.3660$
PI	$195.8461$	$0.0151$

. Discussing these results, it is easy to see that the PI gives a small MSE in the current tracking; nevertheless, a big MSE for the voltage is obtained. On the other hand, the NIOC offers an MSE of $40.9070$ for voltage and $0.0078$ for current tracking; for the manner of this study, the current regulation is important to determine the necessary duty cycle for the application. Moreover, the NSMC gives an MSE that is similar for the tracking of the dynamics in the obtained results illustrated in Figure 13 and Figure 14. In addition, the time of response to track the desired current and voltage in both neural controllers is faster than that of the PI controller.

7. Robustness Test

The NSMC was applied in [10] to an EV architecture featuring a DC motor for regenerative braking, considering various scenarios, though without utilizing real-world data. In contrast, this paper applies both the NSMC and NIOC, where an induction motor and its corresponding three-phase inverter circuit are considered for regenerative braking in an EV using real driving cycle data. While the tracking performance of the proposed controllers has been demonstrated with favorable results, it is crucial to assess their operational capability under varying conditions, such as variations in the components of the bidirectional DC-DC converter. To address this, robustness tests are presented for both the NIOC and NSMC, respectively, with parametric tests conducted to evaluate the robustness of each neural controller. These tests are detailed in the following sections.

7.1. Robustness Test with Parameter Changes Using NIOC

It is important to mention that in some of these tests, modifications in the variables of the buck–boost converter result in the corresponding adjustments of the control signals $u_{NIOC}(x_1,k)$ and $u_c(x_1,k)$, which serve as reference signals for the current controller. As a consequence, the values of these control signals may not be directly comparable, and slight variations in the output current magnitude can occur.

• In this initial demonstration, the same $V_{cr}$ and $i_{current}$ obtained through the steps outlined in Section 5.5 using (67) are employed in the simulation, with a focus on the variations in the tracking signal between $t=40$ seconds and $t=100$ seconds, where significant acceleration and deceleration occur. Furthermore, Figure 21 and Figure 22 illustrate the behavior of the tracking trajectories during the simulation, using the selected values for the components of the buck–boost converter as defined in Table 1. In this case, the tracking signal shows favorable performance, accurately following the desired references without any issues.
• In the second robustness test, the elements of the buck–boost converter remain the same, while the inductor changes to $L=13\times {10}^{-2}H$ as illustrated in Figure 23 and Figure 24. The results demonstrate that the controllers are capable of following the desired tracking signals despite the changes in the inductance, resulting in a different behavior in the current though the converter.
• In the third test, a new configuration change is made to the buck–boost converter. Specifically, the inductor is now selected as $L=65\times {10}^{-1}H$, which is five times larger than the original value, while the other elements of the converter remain unchanged. However, the effective tracking of these signals is achieved despite the parameter changes as illustrated in Figure 25 and Figure 26.
• In the fourth robustness test, a more significant change in the configuration of the buck–boost converter is introduced, where both the inductor L and the capacitor $C_1$ are modified. The capacitor, initially set to $C_1=4\times {10}^{-9}F$ in the original simulation settings, now has the values $L=13\times {10}^{-1}H$ and $C_1=4\times {10}^{-3}F$ for this test. The system’s performance under these new conditions is illustrated in Figure 27 and Figure 28, where the voltage tracking is successfully achieved. The output signal $x_1$ exhibits similar performance to that shown in Figure 21, demonstrating favorable results.
• In this last test, the NIOC demonstrates that the robustness, again, changes in the parameters of the buck–boost converter. In this case, the elements that suffer changes are again the inductor and capacitor, but now the new values considered for this test are $L=13\times {10}^{-2}H$ and $C_1=4\times {10}^{-7}F$. Figure 29 and Figure 30 illustrate the results for both the voltage and current.
• Finally, a more graphical representation of all the tests performed is provided in Figure 31 and Figure 32, where the effects of the parameter changes can be analyzed. The modifications introduced in each test are also highlighted in the zoomed-in sections of the figures, offering a clearer demonstration of the system’s response to these changes.

7.2. Robustness Test with Parameter Changes Using NSMC

In this section, the same changes are made in the buck–boost converter using the NSMC to analyze its robustness and operation under different scenarios.

• The initial analysis utilizes the original parameters of the buck–boost converter, for which previous results have already indicated favorable outcomes. However, it is essential to present these results, as they will serve as a baseline for comparison with the outcomes obtained from variations in the parameters of the buck–boost converter.
• The results employing the changes in the inductor $L=13\times {10}^{-2}H$ with the NSMC are depicted in Figure 33 and Figure 34, where the behavior of the system and the tracking signals under the changes result in positive results for the purpose of this study.
• The results obtained with the updated inductor value of $L=65\times {10}^{-1}H$ are presented in Figure 35 and Figure 36. It is evident from these figures that the voltage reference signal is accurately tracked, with the current tracking signal also following closely.
• The results of the changing inductor and capacitance values to $L=13\times {10}^{-1}H$ and $C_1=4\times {10}^{-3}F$ in Figure 37 and Figure 38 illustrate the accurate operation of the controller with the changes in two parameters of the system.
• The results obtained with changes in the inductor and capacitance, where $L=1\times {10}^{-2}H$ and $C_1=4\times {10}^{-7}F$ are presented in Figure 39 and Figure 40. A small variation in the current of the buck–boost converter is observed. However, the current follows the behavior of the reference current signal generated by the voltage control. This is a result of the capacity of the circuit.
• Finally, the comparison of all results is presented in Figure 41 and Figure 42 for the tracking of the reference signals during the changing of parameters. In the voltage tracking with different parameters, the voltage is accurately tracked. On the other hand, the current tracking may vary, as the reference to track is not always the same, as it depends on the voltage control signal $u_c(x_1,k)$, which is the same case as with the NIOC controller.

7.3. Performance Indices of the System Behavior over Parameters Changes

As a final metric for comparing robustness against parameter variations, two tables are provided. These tables present the MSE values obtained for both the NIOC and NSMC over the time interval $t=0$ seconds to $t=100$ seconds, accounting for the impact of the parameter changes on the convergence time. The effects of parameter variations when employing the NIOC are depicted in Figure 43 and Figure 44, whereas Figure 45 and Figure 46 illustrate the corresponding effects using the NSMC. In all cases, subfigure (a) represents the scenario with a parameter change of $L=13\times {10}^{-1}H$, subfigure (b) corresponds to $L=13\times {10}^{-2}H$, subfigure (c) depicts $L=65\times {10}^{-1}H$, subfigure (d) shows the case of $L=13\times {10}^{-1}H$ with $C_1=4\times {10}^{-3}F$, and subfigure (e) illustrates $L=13\times {10}^{-2}H$ combined with $C_1=4\times {10}^{-7}F$. It is worth mentioning that as described before in Section 5.5 in Figure 43, Figure 44, Figure 45 and Figure 46, the current reference signal $i_{reference}$ is the result of the voltage control, and this makes the system unable to follow the desired track during the first seconds of the operation. However, when the voltage reference is tracked, the current signal reference is generated and tracked as well.

In addition,

Table 4. MSE for voltage and current with change in the parameters of the converter with NIOC.

Parameter Changes	MSE Value for Voltage	MSE Value for Current
$L=13\times {10}^{-1}H$	$763.8465$	$178.8690$
$L=13\times {10}^{-2}H$	$97.5746$	$23.0422$
$L=65\times {10}^{-1}H$	$704.1140$	$3.7356\times {10}^3$
$L=13\times {10}^{-1}H{C}_1=4\times {10}^{-3}F$	$883.4956$	$205.3395$
$L=13\times {10}^{-2}H{C}_1=4\times {10}^{-7}F$	$97.5737$	$23.0421$

presents the MSE values obtained using the NIOC, while

Table 5. MSE for voltage and current with change in the parameters of the converter with NSMC.

Parameter Changes	MSE Value for Voltage	MSE Value for Current
$L=13\times {10}^{-1}H$	$371.8693$	$616.3767$
$L=13\times {10}^{-2}H$	$49.6044$	$54.1294$
$L=65\times {10}^{-1}H$	$1.8125\times {10}^3$	$3.0724\times {10}^3$
$L=13\times {10}^{-1}H{C}_1=4\times {10}^{-3}F$	$415.0718$	$682.8270$
$L=13\times {10}^{-2}H{C}_1=4\times {10}^{-7}F$	$49.6046$	$54.1283$

provides the MSE results achieved with the NSMC over a simulation period of 100 seconds. These results are essential for comparing the performance of the neural controllers. Specifically, the NSMC demonstrates superior performance in controlling the voltage dynamics, whereas the NIOC yields better MSE results for current tracking.

Furthermore, the SS analysis of the MSE during $t=40$ seconds to $t=100$ seconds is presented in

Table 6. MSE for voltage and current in steady state with change in the parameters of the converter with NIOC.

Parameter Changes	MSE Value for Voltage in SS	MSE Value for Current in SS
$L=13\times {10}^{-1}H$	$5.6113\times {10}^{-4}$	$0.0017$
$L=13\times {10}^{-2}H$	$0.0012$	$0.0535$
$L=65\times {10}^{-1}H$	$5.5427\times {10}^{-4}$	$0.0011$
$L=13\times {10}^{-1}H{C}_1=4\times {10}^{-3}F$	$5.6050\times {10}^{-4}$	$0.0017$
$L=13\times {10}^{-2}H{C}_1=4\times {10}^{-7}F$	$0.0012$	$0.0536$

and

Table 7. MSE for voltage and current in steady state with change in the parameters of the converter with NSMC.

Parameter Changes	MSE Value for Voltage in SS	MSE Value for Current in SS
$L=13\times {10}^{-1}H$	$0.0013$	$8.4347\times {10}^{-4}$
$L=13\times {10}^{-2}H$	$0.0035$	$0.0840$
$L=65\times {10}^{-1}H$	$0.0013$	$3.3949\times {10}^{-5}$
$L=13\times {10}^{-1}H{C}_1=4\times {10}^{-3}F$	$0.0014$	$0.0011$
$L=13\times {10}^{-2}H{C}_1=4\times {10}^{-7}F$	$0.0034$	$0.0839$

, where the NIOC has better results in the voltage tracking compared to the NSMC, while the NSMC has better numbers in the current tracking. However, both results reflect the performance of the neural controller in tracking the dynamics of the buck–boost converter.

The results, along with the comparison of the MSE between the system in SS and the system over all the time of simulation ($t=100$ seconds), provide valuable insights for a more comprehensive evaluation of the performance of the controllers proposed in this paper. In both scenarios, it is evident that the MSE remains significantly lower during steady-state operation compared to the period when controller adjustments are still in progress. Despite the relatively short convergence time, these adjustments contribute to an increase in the error metrics during the transition phase.

Furthermore, the improvement in tracking performance with the neural controllers, as well as the robustness against parameter variations in the system, serve as strong indicators of the advantages of the proposed NIOC and NSMC presented in this paper. These metrics further demonstrate the effectiveness of the controllers in maintaining accurate performance under varying conditions.

8. Discussion

This work aims to improve the SOC of the battery bank and build an NIOC and NSMC for a buck–boost converter that has a bidirectional configuration for an electric vehicle application. In addition, a driving cycle test specifically designed for electric vehicles (EVs) is used to validate the effectiveness of the suggested controllers and the energy enhancement methodology. This approach is crucial given that previous research has focused primarily on energy management strategies tailored to typical urban driving cycles, which were originally developed for vehicles with combustion engines. Furthermore, the main objectives of the control scheme have been achieved, such as tracking the voltage $V_{cr}$ generated with the induction motor outputs, the $i_{reference}$ tracked in the manner described in Figure 6, the setup strategy for regenerative braking, and the improvement in battery charge during the driving cycle.

Additionally, the NIOC and NSMC exhibit faster response times in tracking the desired trajectories in both dynamics, making them more suitable for this application compared to the PI controller, despite the PI achieving a small MSE in some cases.

Moreover, robustness tests are conducted to evaluate the performance of the neural controllers under parameter variations in the buck–boost converter, with trajectory tracking results demonstrating successful performance. Furthermore, the MSE calculations for the system over the entire 100 seconds period, as well as during SS, are presented and analyzed. In some cases, the NIOC outperforms the NSMC, while in others, the NSMC yields better results than the NIOC. Despite these variations, both neural controllers maintain consistently strong performance.

These results increase our interest in the management of energy storage systems for electric vehicles, such as the supercapacitor that greatly helps in regenerative braking systems and energy acquisition using neural controllers.

The authors are encouraged to extend the obtained results through real-time applications. As a future work, various topologies and designs will be considered for the DC converter circuit. In the first stage, a DC converter utilizing a MOSFET will be implemented, where both a PI controller and a state-feedback integral controller will be applied. These will be compared against NSMC and NIOC controllers, employing a RHONN identifier trained with EKF. In the second stage, the DC converter and EV architecture used in this study, along with NSMC and NIOC controllers, will be applied to a DC motor in real-time, using RHONN identifiers with EKF training, and compared with alternative intelligent control strategies. This final application will be replicated using an induction motor and its corresponding three-phase inverter.

In the medium term, after our intelligent robust controllers are probed in a prototype, these controllers will be applied in an electric motorcycle, and an embedded system will be employed to achieve this objective using an electronic board, where intelligent controller algorithms will be programmed and stored in a software platform of a digital electronic device.

9. Conclusions

The neural controllers proposed in this work have demonstrated significant advantages over conventional control strategies for bidirectional buck–boost converters, as well as robustness against parameter variations. A control strategy that incorporates neural identifiers and other artificial intelligence (AI) techniques could prove more reliable in industrial applications, as these identifiers can estimate information that is difficult to measure with traditional sensors. Moreover, the substantial impact and rapid growth of AI will assist engineers and experts in developing new and improved approaches to enhance the performance of energy and driving systems in EVs. The obtained results motivated the authors to apply the intelligent robust controllers in real-time in an experimental prototype.