Longitudinal Tire Force Estimation Method for 4WIDEV Based on Data-Driven Modified Recursive Subspace Identification Algorithm

Abstract

For the longitudinal tire force estimation problem of four-wheel independent drive electric vehicles (4WIDEVs), traditional model-based observers have limitations such as high modeling complexity and strong parameter sensitivity, while pure data-driven methods are susceptible to noise interference and have insufficient generalization ability. Therefore, this study proposes a joint estimation framework that integrates data-driven and modified recursive subspace identification algorithms. Firstly, based on the electromechanical coupling mechanism, an electric drive wheel dynamics model (EDWM) is constructed, and multidimensional driving data is collected through a chassis dynamometer experimental platform. Secondly, an improved proportional integral observer (PIO) is designed to decouple the longitudinal force from the system input into a state variable, and a subspace identification recursive algorithm based on correction term with forgetting factor (CFF-SIR) is introduced to suppress the residual influence of historical data and enhance the ability to track time-varying parameters. The simulation and experimental results show that under complex working conditions without noise and interference, with noise influence (5% white noise), and with interference (5% irregular signal), the mean and mean square error of longitudinal force estimation under the CFF-SIR algorithm are significantly reduced compared to the correction-based subspace identification recursive (C-SIR) algorithm, and the comprehensive estimation accuracy is improved by 8.37%. It can provide a high-precision and highly adaptive longitudinal force estimation solution for vehicle dynamics control and intelligent driving systems.

1. Introduction

With the development of the automotive industry, performance optimization and safety control of vehicles have become increasingly important. Accurate vehicle parameters and status information are crucial for fields such as vehicle dynamics control, fault diagnosis, and intelligent driving. The mechanism model-based vehicle state estimation method is currently one of the most commonly used methods for soft sensing of vehicle states [1,2]. However, the balance between modeling complexity and state estimation accuracy is an important issue in the design process of model-based observers [3,4,5]. On the one hand, although complex high-dimensional vehicle dynamics models can more accurately describe the vehicle state, there are problems such as difficulty in accurately measuring parameters, difficulty in model initialization and calibration, and heavy computational burden. In addition, some vehicle parameters have weak observability, which makes it difficult to achieve the expected calibration effect and limits their practical application [6,7]. On the other hand, some studies simplify the model structure or use low-order models, such as vehicle dynamics models based on two or three degrees of freedom, for state estimation, which reduces computational complexity but sacrifices accuracy [8,9].

Traditional physics model-based methods face problems such as large model errors and difficulty in accurately measuring parameters when facing complex and changing practical working conditions [10,11,12,13]. However, data-driven methods provide a new approach to solve these problems. In data-driven methods, neural networks have attracted much attention due to their powerful nonlinear mapping and learning capabilities. For example, multi-layer perceptrons can compress time-series field measurement data to achieve long-term accurate prediction of vehicle states, while long short-term memory networks [14] are suitable for processing long-term dependencies in time-series data and can be used for estimating state variables such as vehicle yaw rate and center of mass sideslip angle. Support vector machine [15] is based on statistical learning theory, which analyzes and predicts data by finding the optimal classification hyperplane or regression function. It has certain applications in classification and regression problems in vehicle parameter identification and state estimation. Kalman filtering and its improved algorithms are also commonly used methods. The extended Kalman filter [16,17,18] estimates the state by linearizing nonlinear systems, but its accuracy is limited in strongly nonlinear systems; the unscented Kalman filter uses an unscented transform to approximate the probability density distribution, resulting in higher estimation accuracy for nonlinear systems. In addition, deep learning [19,20] is gradually emerging in the fields of vehicle parameter identification and state estimation, which can automatically extract complex features from data and improve the accuracy and robustness of estimation.

However, data-driven state estimation methods also face many challenges. Data is the foundation of data-driven methods, but in practical applications, data often has problems such as noise and missing values, which affect data quality [21,22,23]. To this end, researchers use data preprocessing techniques such as data cleaning, denoising, interpolation, etc., to improve data quality. At the same time, data augmentation techniques are used to simulate different working conditions or driving scenarios to generate more training data and expand the dataset [24,25]. Mon, YJ et al. [14] used long short-term memory neural networks to store high-fidelity data of sequence signals, and improved the control accuracy of self-balancing vehicles through neural network weight optimization. Chor, WT et al. [24] proposed a vehicle quality estimation scheme based on the multi-forgetting factor recursive least squares algorithm. To address the issue of increased data inaccuracy at low sampling rates, a robust recursive algorithm and a reconstructed longitudinal dynamics model were combined to reduce estimation bias. In addition, the generalization ability of the model is crucial, and data-driven models need to adapt to different vehicle models, operating conditions, and environmental conditions. To improve generalization ability, researchers use techniques such as multi-source data fusion and transfer learning to enable the model to maintain good performance under different data distributions. Furthermore, vehicle parameter identification and state estimation typically require real-time performance to meet the requirements of vehicle control and safety systems [26,27,28]. Therefore, researchers continuously optimize algorithm structures and computational efficiency, using lightweight neural network architectures and improved Kalman filtering algorithms, etc., to reduce computational complexity and improve model running speed.

The fusion of data-driven and model-based identification algorithms has demonstrated many significant advantages [29,30,31,32,33]. The fusion of the two cleverly compensates for the errors of using them separately, thereby improving the accuracy of vehicle state estimation [34,35,36,37]. Zhang, Y et al. [19] proposed an estimator for intelligent tire force soft measurement and built tire force estimation models under different sensor target configurations based on radial basis function neural networks, significantly improving the parameter sensitivity and generalization performance of the model. In terms of enhancing generalization ability, fusion algorithms reduce reliance on a large amount of comprehensive and high-quality training data and utilize prior knowledge of the model to supplement and derive data, enabling the algorithm to maintain good estimation performance even in conditions where data is scarce or insufficiently covered [38,39,40]. This enables the algorithm to better adapt to different vehicles and complex real-world conditions, greatly improving its generalization ability. In terms of improving computational efficiency, model-based methods provide prior information and constraints for parameter searches in data-driven algorithms, narrowing down the search scope and reducing computational complexity [41,42]. Meanwhile, by integrating with model-based methods, physical models can be used to describe the dynamics of some vehicles, simplifying the structure and input–output relationship of data-driven models and reducing the complexity and computational burden of the models. In terms of enhancing physical interpretability, the model-based part integrates the physical laws and dynamic characteristics of the vehicle into the algorithm, making it have clear physical meaning and interpretability. The data-driven part is also constrained and guided by physical laws after fusion, avoiding unreasonable estimation results that violate physical reality, thereby enhancing the credibility and interpretability of the algorithm [43,44].

In response to the above challenges, this study focuses on the longitudinal tire force estimation problem of 4WIDEV and proposes a longitudinal force estimation framework that combines data-driven and modified recursive subspace identification algorithms, aiming to balance the interpretability of the mechanism model with the dynamic adaptability of the data-driven algorithm. The research contribution of this paper can be summarized as follows. (1) An electric drive wheel model is constructed, driven data is collected based on the chassis dynamometer experimental platform, and research on a tire longitudinal force estimation method based on a modified recursive subspace identification algorithm is conducted. (2) An improved proportional integral observer structure is proposed, which dynamically modifies the data compression matrix through matrix decomposition techniques to enhance the algorithm’s ability to dynamically track time-varying parameters, and a subspace identification recursive algorithm based on forgetting factor correction is designed. (3) By constructing an EDWM and an improved proportional integral observer (PIO) and combining it with a dynamic correction recursive subspace identification algorithm (CFF-SIR), high-precision dynamic decoupling and robust estimation of tire longitudinal force are achieved. As a whole, this study provides a solution for longitudinal force estimation of four-wheel independent drive electric vehicles that balances accuracy, robustness, and engineering applicability, laying a theoretical foundation for vehicle dynamics control and intelligent driving technology.

The structure of the rest of the full text is as follows: Section 2 elaborates on the mechanism modeling and data acquisition methods of an electric drive wheel; Section 3 proposes an improved observer design and a modified recursive subspace identification algorithm; Section 4 verifies the effectiveness of the algorithm through simulation and experimentation; and Section 5 summarizes the research results and looks forward to future directions.

2. Mechanism Model Construction and Identification Data Collection

2.1. Mathematical Mechanism Model of Tire Longitudinal Force Based on Electromechanical Coupling Driving Relationship

Given the unique structural features and driving performance advantages of 4WIDEVs, this study considers each electric drive wheel, consisting of a hub motor and tires, as a relatively independent driving module. Based on this independent driving module, the construction of a local model was carried out, and the resulting model was named the EDWM. The electric drive wheel model is presented in Figure 1. As for the entire electric drive wheel model, its rotational dynamics equation can be expressed in the following form:

$$J_1{\dot{\omega }}_j=T_{Lj}-F_{xj}r\left(j=1,2,3,4\right),$$

(1)

where F_xj is the longitudinal force of the tire, ω_j is the wheel speed, J₁ is the moment of inertia of the wheel, r is the effective radius of the electrically driven wheel, and T_Lj is the motor torque. For a single electric drive model of a wheel hub motor, the torque output equation at the drive shaft can be expressed as

$$J_2{\dot{\omega }}_j+b{\omega }_j=K_{\mathrm{t}}{i}_j-T_{Lj},$$

(2)

where J₂ is the rotational inertia of the rotor of the hub motor, b is the equivalent damping constant, K_t is the proportional coefficient of the load torque of the hub motor, and i_j is the bus current of the hub motor. In electric drive wheels, the wheel hub motor is selected as a permanent magnet DC brushless motor, and its equivalent voltage equation can be expressed as

$$U_j=R{i}_j+L{\dot{i}}_j+K_{\mathrm{a}}{\omega }_j,$$

(3)

where U_j is the bus voltage of the hub motor, R is the equivalent resistance of the hub motor, L is the equivalent inductance of the hub motor, and K_a is the back electromotive force coefficient. The simultaneous Equations (1)–(3) yield the electric drive wheel model as follows:

$${\dot{i}}_j=-{\displaystyle \frac{R}{L}}{i}_j-{\displaystyle \frac{K_{\mathrm{a}}}{L}}{\omega }_j+{\displaystyle \frac{1}{L}}{U}_j,$$

(4)

$${\dot{\omega }}_j={\displaystyle \frac{K_{\mathrm{t}}}{J}}{i}_j-{\displaystyle \frac{b}{J}}{\omega }_j-{\displaystyle \frac{r}{J}}{F}_{xj},$$

(5)

where J = J₁ + J₂ is the equivalent moment of inertia of the electric drive wheel.

2.2. Acquisition of Driven Data for Longitudinal Tire Force Identification Model

In the process of constructing the longitudinal force recognition model for electric drive wheels, this study relies on the chassis dynamometer experimental platform to collect multi-dimensional driving data of electric drive wheels. The experimental system architecture and principle are presented in Figure 2. The vehicle selected for the experiment has an electric chassis architecture with four-wheel independent drive, and wheel hub motors are installed at each wheel position to achieve direct drive function. By using a chassis dynamometer experiment, it is possible to capture the relevant parameter information of the electric drive wheel module during vehicle driving.

The distributed drive vehicle chassis control system is developed based on the D2P MotoHawk rapid prototyping platform, and its integrated development environment and production grade ECU hardware architecture are conducive to independent controller development and strategy design. At the same time, the system can be deeply integrated with MATLAB/Simulink joint simulation models, directly calling on industrial grade motor controllers and chassis hardware resources to achieve efficient development of the drive system without the need for secondary development of underlying hardware.

During the experiment, a vehicle drive motor control algorithm model was constructed using the MotoHawk development environment to output specific voltage excitation signals to the actuator, simulating the dynamic response of the drive motor from start-up to stable operation under different constant power requirements. The composite sensing network equipped in the drive system includes real-time monitoring of controller power bus parameters by the current/voltage sensing module of the drive system; a speed detection unit based on the principle of gear pulse counting is used for measuring the angular velocity of motors; and the torque analysis system built into the chassis dynamometer obtains mechanical torque values through mechanical parameter conversion. All dynamic parameters are synchronously collected through multi-channel CAN bus protocol, and dual recording and cross-validation of controller command signals and sensor data are performed using the VehicleSPY onboard network analyzer. The test data collected during the chassis dynamometer test is shown in Figure 3.

2.3. Construction of Longitudinal Tire Force Estimation Mechanism Model

By observing Equations (4) and (5), it can be seen that in the mathematical mechanism model involving the relationship between motor drive and tire longitudinal force, longitudinal force is the input of the system model from the perspective of control theory. In order to estimate the longitudinal force of the tire, it is necessary to convert the longitudinal force from an input quantity to a state quantity. To construct a tire longitudinal force estimation model, Equations (4) and (5) can be expressed as the following system equations:

$$\left\{\begin{array}{l}\dot{x}=Ax+Bu+Dd\\ y=Cx\end{array}\right.,$$

(6)

where x, u, d, and y are the system state variables, known inputs, unknown inputs, and measured values, respectively, and it can be written as $x={\left[\begin{array}{cc}i& \omega \end{array}\right]}^T$, $u=U$, $d=F_x$, $y={\left[\begin{array}{cc}i& \omega \end{array}\right]}^T$. In addition, the parameter matrix can be denoted as $A=\left[\begin{array}{cc}-{\displaystyle \frac{R}{L}}& -{\displaystyle \frac{K_a}{L}}\\ {\displaystyle \frac{K_t}{J}}& -{\displaystyle \frac{b}{J}}\end{array}\right]$, $B=\left[\begin{array}{c}{\displaystyle \frac{1}{L}}\\ 0\end{array}\right]$, $D=\left[\begin{array}{c}0\\ -{\displaystyle \frac{r}{J}}\end{array}\right]$, $C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$.

By analyzing the dynamic Equation (6) of the electrically driven wheel, it can be concluded that there are unmeasured longitudinal force input parameters of the tire in the system model. To effectively obtain the dynamic characteristics of the unmeasurable variable, it is necessary to establish a decoupling mechanism between the state parameters and external inputs to derive the dynamic relationship of tire force. This scheme is based on the electric drive wheel architecture to construct a Luenberger state observer and then uses the state extension method to embed the longitudinal force parameters into the system state space. A tire force dynamic identification model is constructed by introducing a proportional integral observer with an integral compensation mechanism, ultimately achieving input variable decoupling and joint estimation of mechanical parameters. The extended state observation system after algorithm reconstruction can be expressed as

$$\left\{\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+Bu+K_g\left(y-\widehat{y}\right)+D\widehat{d}\\ \dot{\widehat{d}}=K_i\left(y-\widehat{y}\right)\\ \widehat{y}=C\widehat{x}\end{array}\right.,$$

(7)

where $\widehat{x}$ represents system state estimation, $\widehat{d}$ represents unknown input estimation, K_g represents observer gain matrix, and K_i represents the integral gain matrix of longitudinal tire force.

By analyzing the structure of the integral observer constructed by Equation (7), it can be concluded that the existing observer only has an integral gain matrix configured for compensating unknown inputs. This architecture exhibits good estimation accuracy when dealing with slowly changing system state variables, but its tracking performance deteriorates significantly when the system state exhibits significant dynamic characteristics. To enhance the response speed and tracking accuracy of the observer to dynamic processes, this study introduces a proportional gain matrix into the original architecture and constructs an improved proportional integral observer structure. Its mathematical expression is

$$\left\{\begin{array}{l}\dot{\widehat{x}}=A\widehat{x}+Bu+K_g\left(y-\widehat{y}\right)+D\widehat{d}\\ \dot{\widehat{d}}=K_i\left(y-\widehat{y}\right)+K_p\left(\dot{y}-\dot{\widehat{y}}\right)\\ \widehat{y}=C\widehat{x}\end{array}\right.,$$

(8)

where $K_p$ is the proportional gain matrix of longitudinal tire force. The mechanism model for estimating longitudinal force in tires can be constructed in the following form by combining Equation (8)

$$\left\{\begin{array}{l}\dot{X}=\left[\begin{array}{c}\dot{\widehat{x}}\\ {\dot{\widehat{F}}}_x\end{array}\right]=\left[\begin{array}{cc}A& D\\ 0& 0\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {\widehat{F}}_x\end{array}\right]+\left[\begin{array}{c}Bu+K_g\left(y-\widehat{y}\right)\\ K_i\left(y-\widehat{y}\right)+K_p\left(\dot{y}-\dot{\widehat{y}}\right)\end{array}\right]\\ Y=\overline{C}X=\left[\begin{array}{cc}C& 1\end{array}\right]\left[\begin{array}{c}\widehat{x}\\ {\widehat{F}}_x\end{array}\right]\end{array}\right.,$$

(9)

3. Identification of Longitudinal Tire Force Estimation Model Based on Modified Recursive Subspace Identification Algorithm

3.1. Problem Description of Longitudinal Tire Force Estimation Model

The mechanism model used for tire longitudinal force estimation in Equation (9) can be represented as the following discrete subspace model:

$$\left\{\begin{array}{l}{x}_{t+1}=A{x}_t+B{u}_t+w_t\hfill \\ y_t=C{x}_t+D{u}_t+v_t\hfill \end{array}\right.,$$

(10)

where $u_t\in R^m$ is the input, $y_t\in R^l$ is the output, and $x_t\in R^n$ is the state vector of the system, respectively. In addition, $w_t\in R^n$ is the process noise, and $v_t\in R^l$ is the measurement noise, respectively. To ensure the identifiability and convergence of the system, it is assumed that the dimension of the identified system remains unchanged during the time period of interest and that (A, B) is controllable, (A, C) is observable, and the system is the minimum implementation; external input and noise are uncorrelated and meet sufficient excitation conditions. The focus of the subspace identification algorithm is to use the following Hankel matrix relationship to project future data pairs onto past data pairs in order to estimate the generalized observation matrix; that is,

$$Y_{0,i,N}={\Gamma }_i{X}_{0,N}+H_i{U}_{0,i,N}+N_{0,i,N},$$

(11)

where $U_{0,i,N}$ is the input matrix, $Y_{0,i,N}$ is the output matrix, and $N_{0,i,N}$ is the measurement noise Hankel matrix, which can be described as $Y_{0,i,N}=\left[\begin{array}{ccc}{y}_0& \dots & y_{N-i+1}\\ ⋮& \dots & ⋮\\ y_{i-1}& \dots & y_N\end{array}\right]$.

When i is determined, $Y_{0,i,N}$ can be simplified and labeled as $Y_N$, $X_N=\left[x_0\cdots x_{N-i}\right]$ is the state matrix, and the generalized observability matrix $X_N=\left[x_0\cdots x_{N-i}\right]$ and Markov coefficient matrix H are $\Gamma =\left[\begin{array}{c}C\\ CA\\ ⋮\\ C{A}^{i-1}\end{array}\right]$ and $H=\left[\begin{array}{cccc}D& 0& \dots & 0\\ CB& D& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ C{A}^{i-2}B& C{A}^{i-3}B& \dots & D\end{array}\right]$, respectively. The system matrix parameters A, B, C, and D can be calculated from the estimated value $\widehat{\Gamma }$ of the generalized observability matrix. The following two projection matrices are defined as

$$\begin{array}{l}{\Pi }_{U_N}=U_N^{\mathrm{T}}{(U_N{U}_N^{\mathrm{T}})}^{-1}{U}_N\\ {\Pi }_{U_N}^{\perp }=I-{\Pi }_{U_N}\end{array}.$$

(12)

The above projection matrix has the following properties:

$$\begin{array}{l}{\Pi }_{U_N}^{\mathrm{T}}={\Pi }_{U_N}\cdot {({\Pi }_{U_N}^{\perp })}^{\mathrm{T}}={\Pi }_{U_N}^{\perp }\\ U_N{\Pi }_{U_N}^{\perp }=0\\ {\Pi }_{U_N}^{\perp }\times {\Pi }_{U_N}^{\perp }={\Pi }_{U_N}^{\perp }\end{array}.$$

(13)

Equation (11) can be right-hand multiplied by the matrix ${\Pi }_{U_N}^{\perp }$ on both sides to obtain the following equation:

$$Y_N{\Pi }_{U_N}^{\perp }=\Gamma X_N{\Pi }_{U_N}^{\perp }.$$

(14)

By using the singular value decomposition method, the estimated value $\widehat{\Gamma }$ of the generalized observable matrix can be obtained. Aiming at the optimization requirements of the existing recursive N4SID algorithm, an improvement plan is proposed while maintaining computational efficiency. In terms of reducing computational complexity, existing research often uses matrix operation frameworks to construct recursive N4SID algorithms. However, as the data scale expands, the time-varying characteristics of system parameters can lead to dynamic tracking performance degradation of traditional identification algorithms, mainly due to the continuous accumulation effect of historical data. Although the forgetting factor mechanism can moderately improve the ability to identify time-varying parameters through exponential decay weighting, it still cannot completely eliminate the residual influence of old data, in essence. Compared to other strategies, although the sliding window strategy can achieve data truncation, it faces the dual limitations of high sensitivity to window parameters and a heavy real-time computational burden. Based on this, this article proposes a new recursive improvement strategy: dynamically correcting the data compression matrix through matrix decomposition technology and accurately separating the residual components of historical data during the recursive iteration process. Through mathematical deduction, it has been proven that this correction term can fundamentally cut off the correlation path between old data and current estimation, thereby significantly enhancing the algorithm’s dynamic tracking ability for time-varying parameters.

3.2. Design of Subspace Identification Recursive Algorithm Based on Correction Quantity

Assuming that $P_{N-1}^{-1}:=U_{N-1}{U}_{N-1}^{\mathrm{T}}$, ${\mathrm{X}}_{N-1}:=Y_{N-1}{P}_{U_{N-1}^{^}}{Y}_{N-1}^T$ and ${\mathrm{Y}}_N-1U_{N-1}^T$ from the previous step are known, when obtaining a new set of input-output data $\left\{u_N,y_N\right\}$, the data compression matrix ${\Xi }_N$ can be updated as follows:

$$\left\{\begin{array}{l}{\alpha }_N={(1+u_N^{\mathrm{T}}{P}_{N-1}{u}_N)}^{-1}\\ e_N=y_N-Y{U}_{N-1}^{\mathrm{T}}{P}_{N-1}{u}_N\\ P_N=P_{N-1}-{\alpha }_N{P}_{N-1}{u}_N{u}_N^{\mathrm{T}}{P}_{N-1}\\ Y{U}_N^{\mathrm{T}}=Y{U}_{N-1}^{\mathrm{T}}+y_N{u}_N^{\mathrm{T}}\\ E_N=E_{N-1}+{\alpha }_N{e}_N{e}_N^{\mathrm{T}}\end{array}\right..$$

(15)

After obtaining the data compression matrix ${\Xi }_N$, the SVD method can be used to decompose and obtain the estimation result $\widehat{\Gamma }$, which can then be used to estimate the parameter matrices A, B, C, and D of the system. If A and B are the matrices of $p\times (m+n)$ and $q\times (m+n)$ dimensions and satisfy the condition $A=\left[A_1⋮A_2\right]$ and $B=\left[B_1⋮B_2\right]$, then:

$$A{B}^{\mathrm{T}}=A_1{B}_1^{\mathrm{T}}+A_2{B}_2^{\mathrm{T}}.$$

(16)

where $A_1, A_2, B_1, B_2$ are $p\times m, p\times n, q\times m, q\times n$ dimensional matrices. According to the assumption of $A=[A_1⋮A_2]$ and the law of matrix addition, it can be concluded that

$$\begin{array}{l}A=[A_1⋮0_{p\times n}]+[0_{p\times m}⋮A_2]\\ A{B}^{\mathrm{T}}=([A_1:0_{p\times n}]+[0_{p\times m}:A_2])B^{\mathrm{T}}\end{array},$$

(17)

Then, according to the law of matrix multiplication, we can obtain

$$A{B}^{\mathrm{T}}=[A_1:0_{p\times n}]{[B_1:B_2]}^{\mathrm{T}}+[0_{p\times m}:A_2]{[B_1:B_2]}^{\mathrm{T}}=A_1{B}_1^{\mathrm{T}}+A_2{B}_2^{\mathrm{T}}.$$

(18)

According to Equation (15), the recursive calculation of the data compression matrix ${\Xi }_N$ is the key to the recursive N4SID subspace identification algorithm. If the influence $\Delta {\Xi }_N$ of the old data to be removed on the data compression matrix ${\Xi }_N$ in Formula (15) can be separated, it can effectively block the role of the old data and improve the tracking speed of the identification algorithm for time-varying parameters. Based on this idea, this study uses Equations (15) and (16) to implement the calculation steps of the correction amount $\Delta {\Xi }_N$ for removing old data.

Assuming that $Y{U}_{N-1}^T$, $Y{Y}_{N-1}^T$, $P_{N-1}$, and ${\Xi }_{N-1}$ from the previous step are known, the correction amount $\Delta {\Xi }_N$ of the data compression matrix ${\Xi }_N$ has the following relationship:

$$\Delta {\Xi }_N=Y{Y}_1^T-Y{U}^T(P-P_2)U{Y}^T-Y_1{U}_1^T{P}_{2}U{Y}^T-(Y{U}^T-Y_1{U}_1^T)P_2{U}_1{Y}_1^T$$

(19)

where $Y_1$ and $U_1$ are the matrix blocks that need to be removed from the input and output Hankel matrices Y and U in the recursion, respectively, and they can be represented as $P_2={(P^{-1}-U_1{U}_1^T)}^{-1}=P+P{U}_1{U}_1^{T}P{(I-U_1{U}_1^{T}P)}^{-1}$.

To simplify the equation description, the data compression matrix ${\Xi }_N$ in Equation (15) is abbreviated as $\Xi$ afterwards. According to the definition of the data compression matrix ${\Xi }_N$ in Equation (15), the following equation can be obtained:

$${\Xi }_N=Y{\prod }_U^{\perp }{Y}^T=Y{Y}^T-Y{U}^T{(U{U}^T)}^{-1}U{Y}^T,$$

(20)

According to Equation (16), the input Hankel matrix U has the following relationship:

$$U{U}^T=[U_1^T⋮U_2^T]{[U_1^T⋮U_2^T]}^T=U_1{U}_1^T+U_2{U}_2^T,$$

(21)

where U₂ is the matrix composed of the data pairs that the Hankel matrix U needs to retain. Similarly, it can be concluded that

$$\begin{array}{l}Y{U}^T=[Y_1⋮Y_2]{[U_1⋮U_2]}^T=Y_1{U}_1^T+Y_2{U}_2^T\\ Y{Y}^T=[Y_1⋮Y_2]{[Y_1⋮Y_2]}^T=Y_1{Y}_1^T+Y_2{Y}_2^T\end{array}.$$

(22)

Defining that ${P_2}^{-1}=U_2{U_2}^T$, and then according to the matrix inverse lemma, the following equation can be calculated:

$$P_2={(P^{-1}-U_1{U_1}^T)}^{-1}=P+P{U}_1{U_1}^T\times P{(I-U_1{U_1}^{T}P)}^{-1},$$

(23)

The matrix $U_1{U_1}^T$ in Equation (23) can be obtained by inverting the previously recursive matrix P. From Equations (20) to (23), it can be concluded that

$$\begin{array}{l}{\Xi }_N=(Y_1{Y}_1^T+Y_2{Y}_2^T)-(Y_1{U}_1^T+Y_2{U}_2^T)P\times (U_1{Y}_1^T+U_2{Y}_2^T)\\ =(Y_2{Y}_2^T-Y_2{U}_2^T{P}_2{U}_2{Y}_2^T)+(Y_1{Y}_1^T-Y{U}^T(P-P_2)U{Y}^T-\\ Y{U}^T{P}_{2}U{Y}^T-(Y{U}^T-Y{U}^T)P_{2}U{Y}^T)={\Xi }_2+\Delta \Xi \end{array}.$$

(24)

Thus, it can be proven that Equation (19) is obtained. From the above proof process, it can be seen that the recursive step of implementing the data compression matrix correction $\Delta \Xi$ requires first calculating five matrices, namely: $Y_1{U}_1^{\mathrm{T}}$, $U{Y}^{\mathrm{T}}$, $U_1{Y}_1^{\mathrm{T}}$, $Y_1{Y}_1^{\mathrm{T}}$ and $P_2$. If the threshold size is determined, the matrix $Y_1{U}_1^{\mathrm{T}}$ can be directly obtained from the previously recursive matrix $Y{U}^{\mathrm{T}}$, while $U{Y}^{\mathrm{T}}$ and $U_1{Y}_1^{\mathrm{T}}$ are transposes of $Y{U}^{\mathrm{T}}$ and $Y_{\mathrm{l}}{U}_1^{\mathrm{T}}$. Therefore, calculating the correction $\Delta \Xi$ only requires adding the calculation steps of matrix $Y{Y}^{\mathrm{T}}$ and matrix $P_2$.

3.3. Design of Subspace Identification Recursive Algorithm Based on Correction Quantity with Forgetting Factor

The size of the forgetting factor is a key factor determining the performance of subspace identification algorithms under the forgetting factor mechanism. Generally, identification algorithms choose smaller forgetting factors, which have fast tracking speed but are sensitive to noise; choosing a larger forgetting factor makes the identification algorithm less susceptible to noise interference but slower in tracking speed.

On the basis of the recursive algorithm in Equation (19), a new recursive N4SID subspace identification algorithm with a fixed forgetting factor mechanism is designed using a correction amount. This algorithm not only selects a larger forgetting factor and reduces sensitivity to noise, but also improves the tracking speed of the algorithm for time-varying parameters by using the correction amount and can adjust the threshold size of the correction amount to change the tracking speed of the algorithm for time-varying parameters.

Assuming the correction threshold is p and the $Y{U}_{N-1}^T$, $Y{Y}_{N-1}^T$, $P_{N-1}$, and ${\Xi }_{N-1}$ of the previous step are known, the algorithm implementation steps are as follows:

$$\begin{array}{l}{\varphi }_{yN}=T_l{y}_N, {\varphi }_{uN}=T_m{u}_N\\ Y{U}_N^T=\beta Y{U}_{N-1}^T+{\varphi }_{yN}{\varphi }_{uN}^T\\ Y{Y}_N^T=\beta Y{Y}_{N-1}^T+{\varphi }_{yN}{\varphi }_{yN}^T\\ {\alpha }_N={(\beta +{\varphi }_{uN}^T{P}_{N-1}{\varphi }_{yN})}^{-1}\\ P_N=(P_{N-1}-{\alpha }_N{P}_{N-1}{\varphi }_{yN}{\varphi }_{yN}^T{P}_{N-1})/\beta \\ e_N={\varphi }_{yN}-Y{U}_{N-1}^T{P}_{N-1}{\varphi }_{uN}\end{array},$$

(25)

When $N<p$, the following equation can be obtained by recursively solving the subspace matrix:

$${\Xi }_N={\Xi }_{N-1}+{\alpha }_N{e}_N{e}_N^T,$$

(26)

When $N\ge p$, the matrices $P_N^2$ and $\Delta {\Xi }_N$ are calculated, and the data compression matrix ${\Xi }_N$ is corrected, resulting in the following equation:

$$\begin{array}{l}{P}_N^2=P_N+{\beta }^p{P}_N{P}_{N-1}^{-1}{P}_N{(I-{\beta }^p{P}_{N-1}^{-1}{P}_N)}^{-1}\\ \Delta {\Xi }_N={\beta }^{p}Y{Y}_N^T-Y{U}_N^T{P}_N{(Y{U}_N^T)}^T+Y{U}_N^T{P}_N^2{(Y{U}_N^T)}^T-\\ {\beta }^{p}Y{U}_{N-p}^T{P}_N^2{(Y{U}_{N-p}^T)}^T-{\beta }^p(Y{U}_N^T-{\beta }^{p}Y{U}_{N-p}^T)P_N^2{(Y{U}_{N-p}^T)}^T\\ {\Xi }_N={\Xi }_{N-1}+{\alpha }_N{e}_N{e}_N^T-\Delta {\Xi }_N\end{array},$$

(27)

where $T_l=\mathrm{diag}(\sqrt{{\beta }^{i-1}}{I}_1,\dots ,I_l)$, $T_m=\mathrm{diag}(\sqrt{{\beta }^{i-1}}{I}_m,\dots ,I_m)$. The column vector length of the initial Hankel matrix is s-i. When calculating the identification algorithm with a column number interval of $[p\hspace{1em}\hspace{1em}p+s-i)$ for the Hankel matrix, due to the existence of the initial values of $Y{Y}_0^{\mathrm{T}}$, $Y{U}_0^{\mathrm{T}}$, and $P_0$, the correction within this interval can only be indirectly calculated. Therefore, in the recursive process, p + s − i is used as the switch, and the previous method is used for recursion, while the subsequent recursive process increases the correction amount $\Delta {\Xi }_N$.

The following conclusions can be drawn from the recursive steps of the new algorithm: (1) When the length of the column vector N of the recursive data is less than the threshold p, the 4SID identification algorithm with a fixed forgetting factor mechanism is used for recursion; when $N\ge p$, increase the computation step of correction calculation $\Delta {\Xi }_N$ and correct the original data compression matrix ${\Xi }_N$. (2) The magnitude of the correction effect $\Delta {\Xi }_N$ is determined by both the forgetting factor $\beta$ and the threshold p. When choosing a larger forgetting factor to reduce sensitivity to noise and β is fixed, the larger p, the longer the length of the memory data, and the smaller the effect of the correction amount $\Delta {\Xi }_N$. On the contrary, the shorter the length of the memory data, the greater the effect of the correction amount $\Delta {\Xi }_N$. After obtaining the data compression matrix ${\Xi }_N$, the SVD decomposition method is used to obtain the estimated value ${\widehat{\Gamma }}_N$. The estimated values of ${\widehat{A}}_N$ and ${\widehat{C}}_N$ can be obtained according to the following equation

$$\begin{array}{l}{\widehat{C}}_N={\widehat{\Gamma }}_N(1:l,1:n)/\sqrt{{\beta }^{i-1}}\\ {\widehat{A}}_N=\sqrt{\beta }{({\widehat{\Gamma }}_N(1:l(i-1),:))}^{†}{\widehat{\Gamma }}_N(l+1:li,:)\end{array}.$$

(28)

After obtaining ${\widehat{A}}_N$ and ${\widehat{C}}_N$, the Toeplitz matrix ${\widehat{H}}_N$ can be estimated using ${\widehat{\Gamma }}_N$, and then the matrices ${\widehat{B}}_N$ and ${\widehat{D}}_N$ can be estimated using the least squares method.

A performance evaluation method is proposed in this work based on system invariants to address the theoretical characteristics of parameter identification in state space models. At the level of system modeling, the state space equation has the characteristic of parameter matrix reconstruction under similar transformations, which poses a theoretical challenge to the performance evaluation of identification algorithms due to the non-uniqueness of equivalent system descriptions. It is worth noting that although there are multiple equivalent expressions for parameter matrices, they can all converge to a unified standard form after being transformed by specifications, providing a theoretical basis for the construction of evaluation systems. Starting from the essential characteristics of the system, the eigenvalues, as the core invariants reflecting the dynamic characteristics of the system, can effectively characterize the performance of the identification algorithm based on the convergence characteristics between the estimated trajectory and the true value. The existing evaluation system mainly includes two typical methods: one is to compare the convergence process of the true eigenvalue trajectory of the system matrix A with its estimated value $\widehat{A}$. The second approach is to conduct matching analysis based on modal parameters derived from eigenvalues. This study adopts the former to construct an evaluation framework, quantifying the dynamic response ability of the identification algorithm to time-varying parameters through the time-domain tracking accuracy of the eigenvalue trajectory. The iterative steps of the subspace identification recursive algorithm based on the forgetting factor correction are shown in Figure 4.

4. Result Verification

In order to demonstrate the effectiveness of the CFF-SIR algorithm proposed in this study in estimating longitudinal tire forces in vehicles, a comparative verification was first conducted in the joint simulation environment of CarSim and Simulink. At the same time, in order to reflect the application effect and applicability of the proposed estimation method under various working conditions, simulation verification was carried out under three working conditions: no noise and no interference, with noise influence, and with interference influence. Firstly, simulation tests were conducted under J-turn conditions, with the vehicle speed set at 15m/s and the steering wheel angle shown in Figure 5.

The estimated longitudinal tire force under the J-turn condition is shown in Figure 6. From the figure, it can be observed that, without considering the noise and interference encountered during the model identification process, the curves of the estimated longitudinal tire force values obtained by the CFF-SIR and C-SIR algorithms and the actual longitudinal tire force values can generally remain consistent, indicating that both models have high accuracy in identification. In addition, compared to the C-SIR algorithm, the estimation accuracy under the CFF-SIR algorithm is significantly higher, while the C-SIR algorithm has relatively obvious bias and lag in the estimation result curve, resulting in relatively poorer estimation performance.

In order to further evaluate the performance of the proposed estimation model identification method based on the CFF-SIR algorithm in practical application scenarios, a noisy identification data model validation experiment was conducted. At this point, an additional 5% zero mean white noise was added to the identification data, and the joint simulation verification data also added the influence of 5% noise. As shown in the figure, when the identification model is affected by noise interference, the longitudinal tire force estimation values exhibit more obvious oscillation phenomena. Although the estimation result curves under the CFF-SIR and C-SIR algorithms remain roughly unchanged in overall trend, the longitudinal tire force estimation based on C-SIR is more significantly affected by noise, resulting in larger amplitude oscillations and glitches in the state curve. In comparison, the CFF-SIR algorithm still demonstrates higher estimation accuracy in this situation.

On this basis, further research was conducted on the situation where driving data contains interference. Similarly, 5% of irregular interference signals were added to the identification data and joint simulation validation data of the estimation model. Compared with the undisturbed situation, the estimation accuracy under the C-SIR algorithm has decreased overall, and the longitudinal tire force curve has shown a more significant deviation. However, the estimated values under the CFF-SIR algorithm were not significantly affected by interference, and the estimation accuracy was also relatively close to the estimation results under no interference, demonstrating strong anti-interference ability.

Based on the above comparison results, the CFF-SIR algorithm exhibits better estimation accuracy and stability in the absence of noise and interference, with noise influence, and with interference influence, and has significant advantages compared to the C-SIR algorithm. This indicates that the CFF-SIR algorithm can more accurately identify the essential parameters of the longitudinal tire force estimation model in practical applications, thereby obtaining more accurate longitudinal force calculation results.

To further evaluate the accuracy characteristics of the CFF-SIR algorithm in the identification results of the vehicle longitudinal tire force estimation model, this study uses statistical comparison methods to quantitatively analyze the performance indicators of the proposed algorithm. As shown in

Table 1. Mean error and mean square error of estimation results in J-turn maneuver.

Working Condition	Method	Mean Error	Mean Square Error
Without noise or interference	CFF-SIR	0.9337	0.4377
	C-SIR	1.8566	0.9604
With noise	CFF-SIR	0.9886	0.6987
	C-SIR	2.2607	1.8662
With interference	CFF-SIR	0.9218	0.6331
	C-SIR	2.4039	1.4306

, by establishing a dual-index evaluation system of mean error and mean square error, under the J-turn simulation condition, the average deviation of longitudinal tire force estimation obtained by the C-SIR algorithm is 1.8566, and the mean square error of the fluctuation amplitude indicator is 0.9604. In contrast, the improved CFF-SIR algorithm exhibits significant advantages, with the average deviation of its estimates reduced to 0.9337 and the mean square error reduced to 0.4377. In addition, when considering noise, the average estimation error and mean square error under the C-SIR algorithm are 2.2607 and 1.8662, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 0.9886 and 0.6987, respectively. When considering interference, the average estimation error and mean square error under the C-SIR algorithm are 2.4039 and 1.4306, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 0.9218 and 0.6331, respectively. The results indicate that the improved algorithm achieves performance improvements in both model estimation accuracy and result stability, reflecting the application value of the improved CFF-SIR algorithm in identifying estimation models under complex operating conditions.

In order to further reflect the longitudinal tire force estimation effect of the proposed model identification method under more complex vehicle operating conditions, simulation tests were conducted under variable speed sinusoidal steering conditions. The steering angle and vehicle speed of the vehicle under variable speed sinusoidal steering conditions are shown in Figure 7. The longitudinal tire force estimation results obtained under different settings are shown in Figure 8.

As shown in Figure 8, under the variable speed sine-steering condition, the longitudinal tire force of the vehicle changes more dramatically, and there is a scenario of sudden changes in longitudinal tire force during the switching between constant speed and acceleration conditions of the vehicle. In this case, when noise and interference effects are not considered, the longitudinal tire force estimation results under the CFF-SIR algorithm are closer to the measured values. When there is noise influence, there is a certain degree of fluctuation in the longitudinal tire force estimation values under both algorithms, but compared to the C-SIR algorithm, the results under the C-SIR algorithm have more obvious deviations and fluctuations, while the CFF-SIR algorithm is relatively less affected by noise. Similarly, when there is interference, there is a significant estimation amplitude deviation in the longitudinal force under the C-SIR algorithm, while the estimation value under the CFF-SIR algorithm is relatively small.

Next, quantitative statistical analysis was used to calculate the average estimation error and mean square error of the two algorithms under variable speed sinusoidal steering conditions. The results are shown in

Table 2. Mean error and mean square error of estimation results in sine-steering maneuver.

Working Condition	Method	Mean Error	Mean Square Error
Without noise or interference	CFF-SIR	1.1684	0.5311
	C-SIR	2.3588	1.2069
With noise	CFF-SIR	1.2639	0.8682
	C-SIR	2.8692	1.7646
With interference	CFF-SIR	1.3417	0.5934
	C-SIR	2.9902	1.5006

. According to the comparison data of the dual index evaluation system presented in Table 2, in the simulation scenario of variable speed sinusoidal steering, the estimation results under the C-SIR algorithm show an increasing error in the dynamic estimation of longitudinal tire force. Without considering noise and interference, the average error of the longitudinal tire force estimation value remains at the level of 2.3588, while the mean square error quantification index, which characterizes the fluctuation of the error, reaches 1.2069. In the same situation, the average estimation error and mean square error under the CFF-SIR algorithm are 1.1684 and 0.5311, respectively, which are significantly lower than the estimation results under the C-SIR algorithm overall. In addition, when considering noise, the average estimation error and mean square error under the C-SIR algorithm are 2.8692 and 1.7646, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 1.2639 and 0.8682, respectively. When considering interference, the average estimation error and mean square error under the C-SIR algorithm are 2.9902 and 1.5006, respectively, while the average estimation error and mean square error under the CFF-SIR algorithm are 1.3417 and 0.5934, respectively. The comparative data confirms that the algorithm improvement not only significantly improves the accuracy of the estimation results but also effectively controls the degree of deviation of the estimation results throughout the day, highlighting the applicability of the CFF-SIR algorithm in complex scenarios.

On the basis of verification on the joint simulation platform, further use of chassis dynamometer test data would verify the longitudinal tire force estimation effect. During the model validation process, using an independent dataset for validation is crucial to ensuring the reliability of the model. By isolating training data from validation data, the simulation model can objectively evaluate its generalization ability by simulating unknown scenarios that it may face in the future. When the model performs well on the training data, it indicates that the model may have only excessively captured the noise or specific patterns in the training set, rather than truly representing the patterns behind the data. And independent datasets can force models to confront the complexity of the real world.

Therefore, an independent dataset from chassis dynamometer testing was used to validate the longitudinal force estimation model. The D2P MotoHawk rapid prototyping system is a development environment that integrates standard ECU hardware and Matlab/Simulink(R2014a) software, providing a complete solution for motor drive control. The platform directly adopts mature ECU hardware architecture, saving users the workload of developing hardware on their own. During the experiment, using the vehicle dynamics model constructed by MotoHawk, researchers were able to simulate the complete driving conditions of the vehicle from stationary acceleration to constant speed under different throttle opening conditions. The experimental system is equipped with multiple measuring devices: current/voltage sensors installed at the hub motor monitor the electrical parameters of the controller input in real time; the speed measurement is achieved using the gear pulse counting method; the tire force data is collected and processed by a professional chassis dynamometer system. All measurement data and control instructions are transmitted and stored through the CAN bus, with the Vehicle SPY tool specifically responsible for recording bus data. The verification strategy for the longitudinal force estimation results of 4WIDEV is shown in Figure 9. The accuracy of the estimation is verified by comparing the results output by the estimation model with the data collected during testing.

The longitudinal tire force estimation results under the chassis dynamometer test are shown in Figure 10. According to the overall trend and local amplification comparison chart of the estimated result curve, it can be seen that the CFF-SIR algorithm has higher estimation accuracy in three situations: no noise and interference, noise influence, and interference influence. Then, quantitative statistics were also used to compare and analyze the performance of estimation errors, and the results are shown in

Table 3. Mean error and mean square error of estimation results in chassis dynamometer test.

Working Condition	Method	Mean Error	Mean Square Error
Without noise or interference	CFF-SIR	2.3745	1.5968
	C-SIR	3.9118	2.3467
With noise	CFF-SIR	2.6787	1.8487
	C-SIR	4.6788	3.9512
With interference	CFF-SIR	2.6854	1.6738
	C-SIR	4.7903	2.9106

.

In the three cases of no noise and interference, noise influence, and interference influence, the average estimation errors under the C-SIR algorithm are 3.9118, 4.6788, and 4.7903, respectively, while the average estimation errors under the CFF-SIR algorithm are 2.3745, 2.6787, and 2.6854, respectively. In addition, the mean square error under the CFF-SIR algorithm is also less correlated. By comparison, it can be seen that the proposed improved algorithm can effectively suppress the estimation error and error fluctuation of the identification model, and the overall estimation effect has been significantly improved. By combining simulation data and chassis dynamometer test data and calculating their mean, it can be concluded that compared to the C-SIR algorithm, the CFF-SIR algorithm has improved the comprehensive estimation accuracy by 8.37%.

5. Conclusions

This study proposes a joint estimation method that combines data-driven and modified recursive subspace identification algorithms for the longitudinal force estimation problem of tires in four-wheel independent drive electric vehicles. By constructing an electric drive wheel mechanism model and combining an improved proportional integral observer with a dynamically modified recursive subspace identification algorithm, the problem of insufficient estimation accuracy and robustness of traditional methods under noise, interference, and dynamic conditions has been effectively solved. Simulation and experimental verification show that the CFF-SIR algorithm significantly reduces the mean and mean square error of longitudinal force estimation in complex scenarios such as J-turn and variable speed sinusoidal steering compared to the traditional C-SIR algorithm, and maintains high stability in noisy and disturbed environments; the actual test data of the chassis dynamometer further confirms that its comprehensive accuracy has improved by 8.37%. The innovation of this method lies in the collaborative optimization of mechanism model constraints and data-driven compensation, which achieves a high-precision dynamic decoupling estimation of longitudinal forces and provides reliable technical support for active safety control and intelligent driving systems of vehicles.

Future work can explore strategies for adaptively adjusting forgetting factors to further enhance the algorithm’s ability to track time-varying characteristics and noise interference. The size of the forgetting factor can be automatically adjusted based on the statistical characteristics of real-time data or the dynamic change rate of the system, thereby enhancing the response speed to rapidly changing system characteristics while maintaining robustness to noise. This research direction helps to improve the adaptability and accuracy of longitudinal force estimation algorithms in complex and changing practical driving environments. Meanwhile, more real-time deployment experiments can be conducted in future research to verify the performance and reliability of the proposed algorithm in actual vehicle systems. Field testing can be considered on different types of vehicles, covering various driving and environmental conditions such as different road surface types, weather conditions, and driving modes. Through these experiments, more real vehicle data can be collected to further optimize algorithm parameters and verify their feasibility and effectiveness in practical applications. In addition, real-time deployment experiments can also help identify and solve hardware compatibility, computing latency, and resource limitations that may arise during the actual operation of algorithms, laying a solid foundation for the final commercial application of algorithms.