2.1. The Configuration of EPS System

An autonomous vehicle (AV) system utilizes a combination of sensors, including cameras, lidar, radar, and other perception systems. These sensors gather real-time data about the surrounding environment, road conditions, and potential obstacles. This sensor data is processed by the vehicle control unit segment and used to generate a comprehensive understanding of the vehicle’s position, motion, and desired trajectory. Based on the navigation unit sensors and a pre-determined navigation plan, advanced control algorithms are employed to calculate the optimal steering commands. In the EPS system unit after received a command, the steering control unit (SCU) calculates the required torque for the electric motor, which directly applies torque to the steering rack. The structure of the EPS system and mechanism can be seen in Fig. 1.

Fig. 1. The EPS system and mechanism in an AV.

2.2. ANN Method Concept

Characterizing and identifying systems is a fundamental task in systems theory, where the former involves establishing the mathematical representation of a system. This means the main challenge in SI entails determining an appropriate model structure of the experimental data to be a representation of the real system. Creating a model that can capture the dynamic behavior of the system can be done through an understanding of the essential elements of the system. And since we do not know the essential element or parameter of the physical system, we can’t write the model directly using the first principle and physics. In this situation, a black-box approach can be used. The black-box SI approach is basically constructed as a suitable identification model as shown in Fig. 2, which is subjected to the same input $u(t)$ as the plant, produce an output $\hat{y}(t)$ that approximates $y(t)$ according to desire sense, and $e(t)$ is the error between desire sense and the approximates output.

Fig. 2. General SI scheme.

The formulation of the SI model can be categorized into linear and nonlinear. In nonlinear SI, the relationship between the input $u(t)$ and the output $y(t)$ of the system can be expressed using nonlinear equations. The general form of a nonlinear SI can be written as

where $y(t)$ is the system output at time $t$, $u(t)$ denotes the input to the system at time $t$, $\theta$ represents the vector of unknown parameters that need to be estimated, $f$ is a known set of functions that relate the input and parameters to the output, and $e(t)$ represents the model error at time $t$.

This study encountered EPS with its nonlinearities behavior and proposed a nonlinear method in SI of its dynamic. The objective is to achieve a realization or approximation of the underlying dynamics $f$ by using ANN. To accomplish this, let’s introduce the input vector $v(t)$ for the network which is a confined number of the past inputs and the output as

In practical applications, these decisions are often influenced by the availability of prior information pertaining or training set to the plant under identification that can be written as

Based on the data, we deduce the correlation or connection as

where $f$ is a function deduced from Eq. (2). In the presence of unknown physical parameters within the system, we can examine the relationship between $y$ and $v$.

The equation representation in Eq. (1) is simplified of a general non-linear system of the NARMAX model as

where $y(t -1), y(t -2), ..., y(t -n)$ denotes the past output values, $u(t -1), u(t -2), ..., u(t -m)$ represents the past input values, and $e(t -1), e(t -2), ..., e(t - p)$ denotes the past error or residual values. The output $y(t)$ on Eq. (1) is a function of the current input $u(t)$, the unknown parameters $\theta$, and an error term $e(t)$. This equation suggests a direct relationship between the current input and output, potentially with some error or noise. On the other hand, Eq. (5) represents a more complex nonlinear model that the output $y(t)$ depends not only on the current input $u(t)$ and the unknown parameters $\theta$ but also on the past outputs $y(t -1)$, $y(t -2), ..., y(t -n)$, and the past inputs and errors. This equation incorporates a memory or feedback mechanism, allowing for the consideration of the system’s history and potentially capturing dynamic or temporal dependencies.

In the context of SI, an ANN is used to approximate the underlying dynamics or relationship between input and output variables of a system based on observed data. It serves as a flexible and adaptive model that can capture nonlinearities and complex interactions in the system’s behavior. Learning processes in ANN refer to the mechanisms by which ANN adapts and improves their performance based on the input data and desired outputs. In Fig. 3, we can observe an illustration depicting the process of learning and error correction in an ANN.

Fig. 3. Illustrating the learning process and error correction: (a) An ANN with one output, (b) neuron output signal flow graph, and (c) flow on forward and backward propagation.

Assuming the network depicted in Fig. 3(a) comprises a single hidden layer, with inputs represented by the vector $x(z)$, and an output layer containing a neuron labeled $k$. This neuron in the output layer is driven by a signal vector $h(z)$ generated by the previous layer, and $y_k(z)$ corresponds to the predicted output associated with this configuration. In this context, the variable $z$ signifies a discrete time step, representing the iterative stages involved in adjusting the interconnection weights of the neurons within the network.

Now, let’s examine a simpler scenario involving a neuron-$k$ located in the output layer, serving as the only computational neuron, with a designated signal vector $h(z)$ as shown in detail in Fig. 3(b). After the signals have undergone forward propagation, the output of the ANN is compared to a target output represented by $g_k(z)$. This comparison leads to the derivation of an error signal, labeled as $e_k(z)$. By definition, this error signal quantifies the difference between the ANN’s output and the desired target output. Consequently, the error output signal at the output neuron-$k$ during the $z$th iteration or time step is formally expressed as

To ensure that errors in both positive and negative directions are treated equally, the error needs to be square, we thus have

The learning process entails iteratively modifying the weights of the network’s connections. The term iteration is defined as a complete cycle of calculations (forward and backward passes) as illustrated in Fig. 3(c). This objective is to achieve a minimize of error criterion by minimizing a cost function. Normalizing the sum of squared error with respect to the set size $N$, we get

According to the BP algorithm, applied a weight’s correction $\Delta W_{kN}(z)$ to the weight $W_{kN}(z)$ which is proportional to the partial derivative $E_k(z)$ with respect to $W_{kN}(z)$, we get

The $W_{kN}(z)$ is the weight of the neurons in between the output layer and hidden layer (see Fig. 3(b)) and $\Delta W_{kN}(z)$ is the tuning of the weight $W_{kN}$ at time step $z$. The correction $\Delta W_{kN}(z)$ applied to $W_{kN}(z)$ is defined by

where $\eta$ is learning rate. Having computed the weight adjustment $\Delta W_{kN}(z)$, the updated value of the weight $W_{kN}(z)$ is defined by

where $W_{kn}(z+1)$ described a new weight.

4.1. EPS Data Acquisition Preparation

In order to provide datasets for EPS SI using ANN, the commutation code that has been selected and its sequence algorithm that had been uploaded to a microcontroller is performed. The microcontroller is connected to an upper computer through serial communication to record the motor response based on some input signal given. For the code generation, we use MATLAB/Simulink integrated with NXP model-based design toolbox (MBDT). To control the motor speed is done by a PWM technique. The 12VDC of a maximum input voltage is given to rotate the motor due to the motor selection that meets the vehicle voltage environment. The actual motor angle position as feedback is measured by a magnetic position sensor AS5247U through the index pulse output-A/B index (ABI) interface. The EPS steering input command is given by the upper computer and at the same time the data acquisition is recorded every 0.032 seconds. This experimental data collection scheme can be seen in Fig. 7.

Fig. 7. Experimental data collection scheme.

Fig. 8 illustrates the experimental setup, including all hardware and components involved in data recording. The upper computer handles data recording via UART communication using FreeMASTER software. A 50A power supply provides power to the microcontroller, power inverter, and BLDCM. Additionally, Fig. 9 shows the EPS MATLAB/Simulink design block code, which reflects the implementation of the data acquisition scheme for recording.

Fig. 8. Experimental hardware and components setup for data recording.

Fig. 9. EPS MATLAB/Simulink design block code for EPS data recording.

4.2. EPS Data Acquisition Results and Data Preprocessing

The datasets for training, validation and test are recorded from the real EPS as described previously, the form of step variations, sine wave, square wave, and triangular wave were recorded for the EPS SI in this research. These datasets configurations are shown in Table 1.

In the further step, the datasets that had been taken above are then normalized with min-max scaling in the $\pm 1$ range of scale. It is determined that the data in range from 0 to $+1$ is the clockwise direction data and the data range below 0 to $-1$ is the counterclockwise direction data. Data normalization is performed to help improve the convergence and stability of optimization algorithms. The choice of normalization method in this research mathematically can be written as

where $a$ and $b$ are the desired range which is $a$ is $-1$ and $b$ is 1, $x$ is the scalar that needs to be scaled, $\min x$ or $\max x$ is the minimum and maximum of the vectors data.

4.3. Training Procedures and Model Evaluation

In this research, MATLAB Editor was used to create the code for training, validating, and testing the ANN model. MATLAB Editor is a powerful scientific computing tool with advanced debugging capabilities. To explore various ANN structure possibilities and understand the complex performance trade-offs, several tests were conducted. The details of the test configurations are provided in Table 2.

Table 1 shows that the first dataset was used for training, the second dataset was used for validation, and the remaining three datasets were used for testing. We conducted two sets of learning iterations, each with a different number of times that the optimization algorithm updated the model’s parameters. We also selected two different values of the learning rate, which determines the step size or the rate at which the algorithm updates the model’s parameters in the direction of the steepest descent of the loss function.

Introduced the term "model order", which refers to the number of recurrent units or time steps used in this proposed network architecture. For example, in a simple ANN with a model order of 1, the network only considers the current input and the information from the previous time step to predict the output at the next time step. In contrast, if the model order is set to 5, the ANN will take into account the current input and the information from the past five steps to make predictions. Ultimately, three sets of experiments with different numbers of neurons in the hidden layer to investigate the effect on generalization capabilities were conducted. These experiments were designed to assess the performance and generalization capabilities of the proposed ANN structure.

The ANN was trained using the data collection of step variations. During the training phase, the model learned from the training set to make predictions and minimize prediction errors using BP techniques. this process helps to obtain global optimal network parameters. After training the ANN to produce a model based on the training dataset, it is essential to evaluate its performance on a separate validation dataset. The validation dataset is distinct from the training data and helps assess how well the model generalizes to new data. Next, the model is tested on some unseen data, known as the test datasets. This final evaluation provides a realistic estimation of how well the model will perform in real-world scenarios. The different types of datasets are described in detail in Table 1 and the final matrices of the weighs after the training process are summarized as follows:

Next, after receiving the predicted model results, we can calculate the percentage of the model’s best fits, also known as the goodness of fit, between the EPS model’s predictions and the EPS actual measured data. This can be done using the MSE concept. In the context of SI, the model’s predicted output values are compared to the actual measured output values, and the MSE is computed to quantify the differences between the two.

Fig. 10. Performance evaluation algorithm for a predictive model.

Lower values of MSE indicate a better fit, meaning the model’s predictions closely match the measured data. The percentage of best fits can be calculated based on the ratio of the minimized error MSE for the model to the total variance of the measured output data. The percentage indicates how well the model captures the variation in the data. A higher percentage indicates a better fit and a closer match between the model and the actual data. This performance evaluation algorithm for the predictive model is shown in Fig. 10.

5.1. Analysis of the EPS TFE System

One common linear method used for SI is the TFE. This method is used to model and estimate the dynamics of a linear time-invariant (LTI) system, where the relationship between inputs and outputs remains constant over time and is unaffected by changes in the system’s operating conditions. TFE provides a mathematical representation of the system’s input $U(s)$-output $Y(s)$ relationship in the frequency domain.

where $\omega_n$ is the natural frequency and $\zeta$ is the damping ratio.

The same provided angle position observations data as described in Table 1 in this research are given to validate, and test the model produced by the TFE. To measure the model’s best fits, we also use the MSE method. The percentage of best fits are calculated based on the ratio of the minimized error MSE for the model to the total variance of the measured output data. This TFE resulted 87.6%, 92.1%, and 87.3% of best fits input-output model of given input signals of sine wave, square wave, and triangular wave, respectively. Fig. 11 shows the performance results between EPS actual observe outputs and the transfer function model’s estimation.

Fig. 11. The time response results for original EPS and simplified 2nd order TFE model (with structure: number of poles is 2, number of zero is 0, search method through gradient descent with maximum iterations of 100 and tolerance of 0.1): (a) Sine wave response, (b) square wave response, and (c) triangular response.

5.2. Analysis of the EPS ANN Results

Selecting an ANN structure requires careful consideration of factors such as task complexity, data characteristics, overfitting, computational resources, training time and generalization capabilities. All configurations described in Table 2 have a fast-training time process, but the setting structure number 11 is a good construction because it strikes a balance between model complexity and the problem’s requirements. It has a fast-training time and is able to achieve good generalization performance. The test model’s result for setting structure number 11 at learning rate of 0.1 is shown in Fig. 12.

Fig. 12. The time response results for original EPS and identified ANN model (network topology 5-5-1, learning iteration=100, and $\eta = 0.1$): (a) Sine wave response, (b) square wave response, and (c) triangular response.

Based on the test results, the best-fit model in this SI study was achieved using an artificial neural network (ANN) with design configuration number 11, as shown in Fig. 12. This configuration, detailed in Table 2, features a network topology of 5-5-1 and a learning rate of 0.1. At both learning rates of 0.1 and 0.01, the model achieved an optimal fit, with predictions matching the actual measured data by over 99.6% across all testing data.

The data in Table 2 also highlight the importance of selecting an appropriate model order, as it significantly impacts the ANN’s ability to capture temporal dependencies and make accurate predictions. A higher model order allows the ANN to consider longer-term dependencies in the data, but it also increases computational complexity and the risk of overfitting, where the model memorizes noise rather than learning general patterns. Configuration number 11 uses a model order of 2, which strikes a good balance between capturing dependencies and avoiding overfitting.

The selection of the number of iterations is also crucial. It controls the computational process and balances the trade-offs between computation time and solution accuracy. For example, comparing data from configurations 1 and 10, we see that at 100 iterations, the optimization algorithm had sufficient time to update the model’s parameters and enhance its performance. Finally, choosing an appropriate learning rate is essential. A higher learning rate allows the optimization algorithm to make larger updates to the model’s parameters, potentially leading to faster convergence during training. However, if the learning rate is too high, the optimization process may overshoot the optimal parameters and fail to converge. Conversely, a very low learning rate may result in slow convergence and extended training times.