A Finite-Time Tracking Control Scheme Using an Adaptive Sliding-Mode Observer of an Automotive Electric Power Steering Angle Subjected to Lumped Disturbance

Abstract

Steering angle control in self-driving cars is usually organized in layers combining trajectory planning, path tracking, and low-level actuator control. The steering controller converts the planned path into a desired steering angle and then ensures accurate tracking by the electric power steering (EPS). However, automotive electric power steering (AEPS) systems face many problems caused by model uncertainties, disturbances, and unknown system dynamics. In this paper, a robust finite-time control strategy based on an adaptive backstepping scheme is proposed to handle these problems. First, radial basis function neural networks (NNs) are designed to approximate the unknown system dynamics. Then, an adaptive sliding-mode disturbance observer (ASMDO) is introduced to address the impacts of the lumped disturbance. Enhanced control performance for the AEPS system is implemented using a combination of the above technologies. Numerical simulations and a hardware-in-the-loop (HIL) experimental verification are performed to demonstrate the significant improvement in performance achieved using the proposed strategy.

1. Introduction

Automotive electric power steering (AEPS) systems are designed to assist drivers by reducing the steering effort required and by enhancing vehicle safety [1,2,3]. AEPS systems can be classified into two types: hydraulic, and electric steering control, with the latter being more widely used. Fast and accurate tracking of the desired command signal in AEPS systems can significantly improve drivability, fuel efficiency, and overall control comfort. Consequently, many researchers have focused on developing robust control strategies to achieve these objectives [4,5]. However, AEPS systems are often challenged by factors such as model uncertainties, transmission friction, external disturbances, and unmodeled dynamics, all of which can degrade control performance [6,7]. Addressing these issues is essential to ensure satisfactory tracking and reliable system behaviors.

Most previous works have omitted unknown system dynamics from AEPS dynamic behaviors, to simplify the control design [8]. Nevertheless, intelligent approximation approaches such as fuzzy inference systems and artificial neural networks (ANNs) have been explored to address the limitations posed by unknown nonlinear dynamics [9,10], [11,12,13]. In [14], an ANN-based identifier was employed to address the unknown dynamics of an AEPS system, with the Levenberg–Marquardt algorithm being used to enhance tracking accuracy and maintain optimal EPS performance; however, disturbance effects were not considered. In [15], a neuroadaptive approach incorporating a simplified type-2 radial basis function network was proposed to approximate unknown functions in an AEPS system, but experimental validation was not provided. In [16], a fuzzy logic system combined with a genetic algorithm was utilized to optimize controller parameters, thereby improving system stability and adaptability under complex conditions; however, finite-time convergence was not achieved. In [17], the issue of unknown nonlinear frictional resistance in AEPS, which led to poor dynamic performance, was addressed using a PID-backpropagation neural network with proportional self-tuning. However, the development of a neural network-based observer for AEPS systems has not yet been investigated, particularly for handling the simultaneous challenges of unknown system dynamics and external disturbances.

In the design of AEPS control schemes, incorporating observers is essential for estimating lumped disturbance and uncertainty. A disturbance observer is typically employed to address the effects of disturbances and uncertainties, thereby enhancing overall control performance [18,19]. In [20], a combination of a least-squares state variable filter and an instrumental-variable state filter algorithm was used to eliminate model mismatches and achieve satisfactory tracking performance in an AEPS system. In [21], Lubna et al. proposed a disturbance rejection scheme to enhance the robustness of AEPS. Moreover, the integration of an observer into a variable-gain sliding-mode steering angle controller was designed to address issues such as dead zone, chattering, and static friction in the steering system. In [22], an extended state observer (ESO) was employed to estimate both state variables and unknown disturbances in an AEPS system. Subsequently, the ESO was combined with an adaptive robust control scheme to improve control performance. However, the aforementioned studies were limited to simulation-based evaluations and lacked experimental validation on a real AEPS system.

Many control approaches have been introduced for AEPS systems; these include PID [23], sliding-mode control (SMC) [24], adaptive backstepping control [25,26], model predictive control [27], and intelligent control [28], among others. Due to the advantages of the backstepping (BS) technique in control design, various BS-based schemes have been effectively applied to AEPS and other nonlinear systems to achieve stable performance under lumped disturbances [29,30,31,32]. In [33], a novel combination of the backstepping technique and a fuzzy logic system was developed to mitigate the effects of errors and phase difference phenomena in an AEPS system, with control performance being evaluated and validated under several simulation conditions. In [34], a backstepping-based quasi-SMC was proposed for electric vehicle systems. In [35], an adaptive backstepping control was reported to solve the impacts of parameters and disturbances for vehicle active steering. In [36], a disturbance-rejection-based adaptive fuzzy sliding-mode control strategy was presented, and it was demonstrated that achieving satisfactory steering-angle tracking performance is critical for EPS systems, particularly in safety-critical applications. Recent studies [37] have introduced adaptive output-constrained control methods for EPS systems under variable steering load conditions to meet prescribed performance requirements. However, when using these approaches, the tracking error is only guaranteed to be bounded; there is no guarantee of convergence in finite time.

Motivated by the above-mentioned discussion, the main goal of this study is to design a finite-time tracking control scheme using an adaptive sliding-mode observer of an automotive electric power steering angle subjected to lumped disturbance. The innovation of this paper can be stated in the following points:

(1) Compared to previous works [14,15,16], a robust finite-time control strategy based on an adaptive backstepping control (ABSC) scheme is proposed to handle model uncertainties, disturbances, and unknown system dynamics in an AEPS system.

(2) Radial basis function neural networks (RBFNNs) are designed to approximate the unknown system dynamics. An adaptive sliding-mode disturbance observer (ASMDO) to compensate for the effects of lumped disturbances is investigated. The system stability achieved using the proposed strategy is demonstrated using the Lyapunov theorem.

(3) Simulation studies and hardware-in-the-loop (HIL) experimental validation are conducted to demonstrate the significant performance improvements achieved by the proposed methodology.

The structure of the article is organized as follows: Section 2 presents the modelling of the AEPS system and the control objectives. Section 3 details the design of the finite-time backstepping controller and the adaptive sliding-mode disturbance observer (ASMDO). Simulation studies and experimental studies using a hardware-in-the-loop (HIL) system are detailed in Section 4. Finally, conclusions drawn are presented in Section 5.

2. Description and Mathematical Modelling

The AEPS under study contains a direct-current motor integrated with a position (angle) sensor, a steering column, a steering wheel, and a power supply. In addition, a transmission mechanism is deployed to reduce speed and boost torque. The topology of the AEPS system is displayed in Figure 1. This system’s mathematical model results from the above-mentioned components, as detailed below [25]:

$$\left\{\begin{array}{c}\ddot{\phi }=-{\displaystyle \frac{b_{\mathrm{e}}}{I_{\mathrm{e}}}}\dot{\phi }+{\displaystyle \frac{K_T}{I_{\mathrm{e}}}}i-{\displaystyle \frac{1}{I_{\mathrm{e}}}}{T}_L\\ \dot{i}=-{\displaystyle \frac{K_e}{L}}\dot{\phi }-{\displaystyle \frac{R}{L}}i+{\displaystyle \frac{V_{\mathrm{in}}}{L}}u\end{array}\right.$$

(1)

where φ, $\dot{\phi }$, and $\ddot{\phi }$ represent the position, speed, and acceleration, respectively, of the DC motor; u defines the duty cycle of pulse-width modulation (PWM); V_in is the DC voltage supply; L, R, and i define armature inductance, resistance, and armature current, respectively; K_T and K_e define the torque factor and the back electromotive force factor, respectively; and I_e, T_L, and b_e represent, respectively, the equivalent inertia, the EPS load torque, and the equivalent damping of the EPS system.

The system state variables are selected as follows: $\epsilon ={\left[\begin{array}{ccc}\phi & \dot{\phi }& {\displaystyle \frac{K_T}{I_e}}i\end{array}\right]}^T={\left[\begin{array}{ccc}{\epsilon }_1& {\epsilon }_2& {\epsilon }_3\end{array}\right]}^T$. The EPS system can be formulated as follows:

$$\left\{\begin{array}{l}{\dot{\epsilon }}_1={\epsilon }_2\hfill \\ {\dot{\epsilon }}_2={\epsilon }_3+m_1{\epsilon }_2+d_1\hfill \\ {\dot{\epsilon }}_3=qu+m_2{\epsilon }_2+m_3{\epsilon }_3+d_2\hfill \end{array}\right.$$

(2)

where q = K_T·V_in/(L·I_e), m₁ = −b_e/I_e, m₂ = −K_T·K_e/(L·I_e), and m₃ = −R/L denote the known factors; d₁ = −T_L/I_e + Δm₁ε₂ and d₂ = Δm₂ε₂ + Δm₃ε₃ are the lumped disturbances; and Δm_i (i = 1, 2, 3) are the parametric uncertainties.

Remark 1.

The objective of control methodology is to propose an adaptive neural network controller that the automotive electric power steering angle precisely follows the desired angle signal as closely as possible.

3. Controller Algorithm

In this section, a combination of a backstepping approach, an adaptive control law, and a disturbance estimation is given to build a control law and achieve high tracking performance. A schematic diagram of the proposed algorithm is presented in Figure 2. The step-by-step control implementation is described as follows.

3.1. Adaptive Sliding-Mode Disturbance Observer (ASMDO)

Lemma 1 

([12,13,38]). For an arbitrary nonlinear function g, there is a RBF neural network that can approximate g, such as the following: $f_i\left(\epsilon \right)=W_i{}^{*T}\theta +{\varsigma }_i,i=2,3$. Herein, θ(ε) = [θ₁(ε), θ₂(ε), …,  θ_m(ε)]^T is a Gaussian function that obeys ‖θ‖ ≤ 1; σ(ε) defines an NN approximation error; |σ(ε)| < σ_m(ε) and σ_m(ε) > 0; and $W_i^{*T}$ denotes an ideal NN weight vector. The expression can be written as follows:

$${\sigma }_i=\arg \min \left\{\sup \left|f_i\left(\epsilon \right)-W_i^{*T}{\theta }_i\left(\epsilon \right)\right|\right\}$$

(3)

where ${\theta }_i\left({\epsilon }_{in}\right)={\exp }^{-{\displaystyle \frac{{‖{\epsilon }_{in}-{\omega }_j‖}^2}{2b_j^2}}}$; ε_in denotes the input vector of NN; $j=1÷L$; L is the number of basic functions; and ω_j and b_j are the center and width, respectively, of the Gaussian function.

We define $f_2\left(\epsilon \right)=m_1{\epsilon }_2\mathrm{and}{f}_3\left(\epsilon \right)=m_2{\epsilon }_2+m_3{\epsilon }_3$ as the unknown nonlinear functions. Using RBF neural networks to approximate the continuous function f_i, we obtain the following:

$${\hat{f}}_i\left(\epsilon \right)={\hat{W}}_i^T{\theta }_i\left(\epsilon \right)$$

(4)

where ${\hat{W}}_i={\hat{W}}_i-{\hat{W}}_i^{*}, {\hat{W}}_i,\mathrm{and} {\tilde{f}}_i={\hat{f}}_i-f_i={\tilde{W}}_i{}^T\theta _i-{\sigma }_i,i=2,3$ define the weight estimation error, the estimated value, and the approximation error, respectively, of the nonlinear function f_i.

Lemma 2

 ([39]). For the positive arbitrary c₀, Young’s inequality can be described as follows:

$$x_1{x}_2\le {\displaystyle \frac{c_0^p}{p}}{\left|x_1\right|}^p+{\displaystyle \frac{1}{q{c}_0^q}}{\left|x_2\right|}^q$$

(5)

where $x_1,x_2\in R,p,q>1\mathrm{and}\left(q-1\right)\left(p-1\right)=1$.

An adaptive sliding-mode disturbance observer (ASMDO) is introduced to estimate and compensate for lumped disturbances in the control design. Compared with conventional observers, including disturbance observers [21] and extended state observers [22], the ASMDO provides advantages such as reduced reliance on auxiliary variables and integrated neural network modeling. Consequently, by properly tuning the observer gains, the ASMDO is capable of estimating the lumped uncertainties with high accuracy. The ASMDO can be established as follows:

$$\left\{\begin{array}{l}{\dot{\xi }}_2=-{\mu }_2{s}_2-D_1\mathrm{sgn}\left(s_2\right)-{\eta }_2{s}_2^{{\alpha }_2}+{\hat{W}}_2^T{\theta }_2+{\epsilon }_3\hfill \\ {\dot{\xi }}_3=-{\mu }_3{s}_3-D_2\mathrm{sgn}\left(s_3\right)-{\eta }_3{s}_3^{{\alpha }_3}+{\hat{W}}_3^T{\theta }_3+qu\hfill \end{array}\right.$$

(6)

where $s_i={\xi }_i-{\epsilon }_i,i=2,3$ denote the auxiliary variables; ${\mu }_i,D_{i-1},\mathrm{and}{\eta }_i,i=2,3$ define the positive constants; and ${\alpha }_i=2l_i+1/2l_i+3,l_i\in {\rm N}$.

The matched and mismatched disturbances can be estimated by

$${\hat{d}}_{i-1}=-{\mu }_2{s}_i-D_{i-1}\mathrm{sgn}\left(s_i\right)-{\eta }_i{s}_i^{{\alpha }_i},\hspace{1em}i\in \{2,3\}$$

(7)

where ${\hat{d}}_i$ denotes the estimated disturbance value.

The adaptive law is defined as follows:

$${\dot{\hat{W}}}_i={\Gamma }_i\left({\theta }_i{s}_i-{\chi }_i{\hat{W}}_i\right),i\in \{2,3\}$$

(8)

where ${\Gamma }_i\mathrm{and}{\chi }_i$ denote the constant gains.

The Lyapunov function (LFU) can be chosen as follows:

$$V_{ASOBi}={\displaystyle \frac{1}{2}}{s}_i^2+{\displaystyle \frac{1}{2{\Gamma }_i}}{\tilde{W}}_i^T{\tilde{W}}_i,i=2,3$$

(9)

The time derivative of V_ASMDOi can be calculated by

$$\begin{array}{l}{\dot{V}}_{ASOBi}=s_i{\dot{s}}_i+{\Gamma }_i^{-1}{\tilde{W}}_i^T{\dot{\tilde{W}}}_i=s_i\left({\dot{\xi }}_i-{\dot{\epsilon }}_i\right)+{\Gamma }_i^{-1}{\tilde{W}}_i^T{\dot{\hat{W}}}_i\\ =s_i\left(-{\mu }_i{s}_i-D_{i-1}\mathrm{sgn}\left(s_i\right)-{\eta }_i{s}_i^{{\alpha }_i}+{\hat{W}}_i^T{\theta }_i-W_i{}^{*T}\theta _i-{\varsigma }_i-d_{i-1}\right)+{\Gamma }_i^{-1}{\tilde{W}}_i^T{\dot{\hat{W}}}_i\\ =-{\mu }_i{s}_i^2-D_{i-1}\left|s_i\right|-{\eta }_i{s}_i^{{\alpha }_i+1}-s_i{d}_{i-1}-s_i{\varsigma }_i+{\tilde{W}}_i^T\left({\theta }_i{s}_i+{\Gamma }_i^{-1}{\dot{\hat{W}}}_i\right)\\ \le -{\mu }_i{s}_i^2-\left(D_{i-1}-\left|d_{i-1}\right|\right)\left|s_i\right|-{\eta }_i{s}_i^{{\alpha }_i+1}-s_i{\varsigma }_i-{\chi }_i{\tilde{W}}_i^T{\hat{W}}_i\end{array}$$

(10)

Applying Young’s inequality, the following is yielded:

$$\begin{array}{l}-s_i{\varsigma }_i\le {\displaystyle \frac{1}{2}}{s}_i^2+{\displaystyle \frac{1}{2}}{\varsigma }_{im}^2\\ -{\displaystyle \sum _{i=2}^3{\chi }_i{\tilde{W}}_i^T{\hat{W}}_i}\le {\displaystyle \sum _{i=2}^3\left(-\frac{{\chi }_i}{2}{‖{\tilde{W}}_i‖}^2+\frac{{\chi }_i}{2}{‖W_i^{*}‖}^2\right)}\end{array}$$

(11)

Substituting Equation (11) into Equation (10), one obtains

$$\begin{array}{l}{\dot{V}}_{ASOB}={\displaystyle \sum _{i=2}^3{\dot{V}}_{ASOBi}}\le -{\displaystyle \sum _{i=2}^3\left({\mu }_i-\frac{1}{2}\right)s_i^2}-{\displaystyle \sum _{i=2}^3\frac{{\chi }_i}{2}{‖{\tilde{W}}_i‖}^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=2}^3{\varsigma }_{im}^2}+{\displaystyle \sum _{i=2}^3\frac{{\chi }_i}{2}{‖W_i^{*}‖}^2}\\ {\dot{V}}_{ASOB}\le -\Pi {\dot{V}}_{ASOB}+\Psi \end{array}$$

(12)

where $\Pi =min\left({\mu }_i-{\displaystyle \frac{1}{2}},{\displaystyle \frac{{\chi }_i}{2}}\right)\mathrm{and}\Psi ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=2}^3{\varsigma }_{im}^2}+{\displaystyle \sum _{i=2}^3\frac{{\chi }_i}{2}{‖W_i^{*}‖}^2}$.

Theorem 1.

Considering the presented system (2), under the ASMDO designed using Equations (6) and (7) and the adaptive law set out in Equation (8), the disturbance approximation errors of d_i−1 and ${\tilde{W}}_i,i=2,3$ are bounded and go to the small region, i.e., $‖{\tilde{W}}_i‖\le w_{im},\left|{\tilde{d}}_{i-1}\right|\le d_{im},i=2,3;w_{im}>0,d_{im}>0$.

Proof of Theorem 1.

It is revealed from Equation (12) that $\underset{t\to \infty }{\lim }{V}_{ASOB}={\displaystyle \raisebox{1ex}{$\Psi $}\!\left/ \!\raisebox{-1ex}{$\Pi $}\right.}$. This implies that the Lyapunov function V_ASMDO in (9) is convergent. Therefore, Theorem 1 is demonstrated. □

3.2. Synthesis of Adaptive Finite-Time Backstepping Control (ABSC) Design

Step 1: The tracking error and adaptive law are proposed as follows:

$$\begin{array}{l}{z}_1={\epsilon }_1-{\epsilon }_{1d}\\ z_i={\epsilon }_i-{\beta }_{i-1},i=2,3\end{array}$$

(13)

where ε_1d defines the target signal and β_i is the virtual law, to be defined later.

Differentiating (13) with respect to time yields the following:

$${\dot{z}}_1={\epsilon }_2-{\dot{\epsilon }}_{1d}$$

(14)

The intermediate control law is now introduced, as follows:

$${\beta }_1=\left(-{\rho }_{11}{z}_1-{\rho }_{12}{z}_1^{{\lambda }_{12}}-{\rho }_{13}{z}_1^{{\lambda }_{13}}\right)+{\dot{\epsilon }}_{1d}$$

(15)

where $0<{\lambda }_{12}<1\mathrm{and}{\lambda }_{13}>1$, ${\rho }_{11},{\rho }_{12},\mathrm{and}{\rho }_{13}$ define the positive scalars.

The Lyapunov function can be designed as follows:

$$V_1={\displaystyle \frac{1}{2}}{z}_1^2$$

(16)

Taking the derivative of (16) yields the following:

$$\begin{array}{l}{\dot{V}}_1=z_1{\dot{z}}_1=z_1\left(z_2+{\beta }_1-{\dot{\epsilon }}_{1d}\right)\\ =z_1{z}_2-z_1\left({\rho }_{11}{z}_1+{\rho }_{12}{z}_1^{{\lambda }_{12}}+{\rho }_{13}{z}_1^{{\lambda }_{13}}\right)\end{array}$$

(17)

Step 2: Differentiating the error between the virtual signal η₁ and ξ₂ in (13) with respect to time, we get

$${\dot{z}}_2={\epsilon }_3+f_2+d_1-{\dot{\beta }}_1=z_3+{\beta }_2+f_2+d_1-{\dot{\beta }}_1$$

(18)

We now propose the virtual control law, as follows:

$${\beta }_2=\left(-{\rho }_{21}{z}_2-{\rho }_{22}{z}_2^{{\lambda }_{22}}-{\rho }_{23}{z}_2^{{\lambda }_{23}}\right)-{\hat{f}}_2-{\hat{d}}_1-z_1+{\dot{\beta }}_1$$

(19)

where $0<{\lambda }_{22}<1\mathrm{and}{\lambda }_{23}>1$; and ${\rho }_{21},{\rho }_{22},\mathrm{and}{\rho }_{23}$ define the positive scalars.

The LFC in Step 2 can be formulated as follows:

$$V_2=V_1+{\displaystyle \frac{1}{2}}{z}_2^2$$

(20)

Taking the derivative of (20) yields the following:

$$\begin{array}{l}{\dot{V}}_2={\dot{V}}_1+z_2{\dot{z}}_2={\dot{V}}_1+z_2\left(z_3+{\beta }_2+f_2+d_1-{\dot{\beta }}_1\right)\\ =z_2{z}_3-{\displaystyle \sum _{i=1}^2{z}_i\left({\rho }_{i1}{z}_i+{\rho }_{i2}{z}_i^{{\lambda }_{i2}}+{\rho }_{i3}{z}_2^{{\lambda }_{i3}}\right)}-z_2{\tilde{f}}_2-z_2{\tilde{d}}_1\end{array}$$

(21)

Step 3: Differentiating the error between the virtual signal η₂ and ξ₃ in (13) with respect to time, we get

$${\dot{z}}_3={\dot{\epsilon }}_3-{\dot{\beta }}_2=qu+f_3+d_2-{\dot{\beta }}_2$$

(22)

We propose the virtual control law as follows:

$$u={\displaystyle \frac{1}{q}}\left(-{\rho }_{31}{z}_3-{\rho }_{32}{z}_3^{{\lambda }_{32}}-{\rho }_{33}{z}_3^{{\lambda }_{33}}-{\hat{f}}_3-{\hat{d}}_2-z_2+{\dot{\beta }}_2\right)$$

(23)

where $0<{\lambda }_{32}<1\mathrm{and}{\lambda }_{33}>1$, ${\rho }_{31},{\rho }_{32},\mathrm{and}{\rho }_{33}$ define the positive scalars.

The Lyapunov candidate function in step 3 can be formulated as follows:

$$V_3=V_2+{\displaystyle \frac{1}{2}}{z}_3^2$$

(24)

Taking the derivative of (24), the following is deduced:

$$\begin{array}{l}{\dot{V}}_3={\dot{V}}_2+z_3{\dot{z}}_3\\ ={\dot{V}}_2+z_3\left(qu+f_3+d_2-{\dot{\beta }}_2\right)\\ =-{\displaystyle \sum _{i=1}^3{z}_i\left({\rho }_{i1}{z}_i+{\rho }_{i2}{z}_i^{{\lambda }_{i2}}+{\rho }_{i3}{z}_i^{{\lambda }_{i3}}\right)}-{\displaystyle \sum _{i=2}^3{z}_i{\tilde{f}}_i}-{\displaystyle \sum _{i=2}^3{z}_i{\tilde{d}}_{i-1}}\end{array}$$

(25)

Selecting V = V₃ as the Lyapunov function, we obtain

$$\dot{V}\le -{\displaystyle \sum _{i=1}^3{\rho }_{i1}{z}_i^2}-{\displaystyle \sum _{i=1}^3{\rho }_{i2}{z}_i^{{\lambda }_{i2}+1}}-{\displaystyle \sum _{i=2}^3{z}_i{\tilde{f}}_i}-{\displaystyle \sum _{i=2}^3{z}_i{\tilde{d}}_{i-1}}$$

(26)

From Theorem 1, utilizing Young’s inequality, we obtain

$$\begin{array}{l}-{\displaystyle \sum _{i=1}^3{z}_i{\tilde{f}}_i}\le {\displaystyle \sum _{i=1}^3\left(\frac{1}{2}{z}_i^2+\frac{1}{2}{f}_{im}^2\right)}\\ -{\displaystyle \sum _{i=2}^3{z}_i{\tilde{d}}_{i-1}}\le {\displaystyle \sum _{i=2}^3\left(\frac{1}{2}{z}_i^2+\frac{1}{2}{d}_{im}^2\right)}\end{array}$$

(27)

where $f_{im}={\mathrm{w}}_{im}+{\sigma }_{im}$.

Substituting (27) into (26), the following is yielded:

$$\begin{array}{l}\dot{V}\le -{\displaystyle \sum _{i=1}^3\left({\rho }_{i1}-1\right)z_i^2}-{\displaystyle \sum _{i=1}^3{\rho }_{i2}{\left(z_i^2\right)}^{{\lambda }_t}}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^3{f}_{im}^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{d}_{im}^2}\\ \dot{V}\le -{\Pi }_{1}V-{\Pi }_2{V}^{{\displaystyle \frac{{\lambda }_{i2}+1}{2}}}+{\Psi }_1\end{array}$$

(28)

where ${\Pi }_1={\lambda }_{min}\left({\rho }_{i1}-1\right),{\Pi }_2={\lambda }_{min}\left({\rho }_{i2}\right),{\lambda }_t={\displaystyle \frac{{\lambda }_{i2}+1}{2}},$ and ${\Psi }_1={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^3{f}_{im}^2}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^2{d}_{im}^2}$.

Theorem 2.

Considering the EPS system (2), under the presented ASMDO (6), as well as the adaption law (8), the intermediate control law (15) and (19), and the actual controller (23), the error state variables $z_i$ in (13) and system states of the EPS system satisfy that $z_i\to 0\mathrm{as}t\to \infty ,i=1,2,3$ in finite time.

Remark 2.

It is noted that finite-time convergence is particularly critical for electric power steering systems, due to their safety-critical role and real-time operational requirements. Unlike asymptotic convergence, which only guarantees error reduction over an infinite time horizon, finite-time convergence ensures that errors in steering-angle and torque tracking are eliminated within a known and bounded time. This property is essential for rapid driver–vehicle interaction, effective disturbance rejection, and safe operation under dynamic driving conditions, especially in autonomous and steer-by-wire applications.

Proof of Theorem 2.

It is revealed from Equation (28) that the residual set of the solution of the AEPS system is presented by

$$\begin{array}{l}\underset{t\to T_r}{\lim }V(x)\le \min \left\{{\displaystyle \frac{{\Psi }_1}{\left(1-{\theta }_0\right){\Pi }_1}},{\left({\displaystyle \frac{{\Psi }_1}{\left(1-{\theta }_0\right){\Pi }_2}}\right)}^{{\displaystyle \frac{1}{{\lambda }_t}}}\right\}\\ \mathrm{where}{\theta }_0\mathrm{satisfies}0<{\theta }_0<1.\end{array}$$

(29)

where ${\theta }_0\mathrm{satisfies}0<{\theta }_0<1.$

We can now state that V is bounded, and convergence time is obtained as in [40], as follows:

$$T_r\le \max \left\{t_0+{\displaystyle \frac{1}{{\theta }_0{\Pi }_1(1-{\lambda }_t)}}\ln {\displaystyle \frac{{\theta }_0{\Pi }_1{V}^{1-{\lambda }_t}\left(t_0\right)+{\Pi }_2}{{\Pi }_2}}\right.\left.t_0+{\displaystyle \frac{1}{{\Pi }_1(1-{\lambda }_t)}}\ln {\displaystyle \frac{{\Pi }_1{V}^{1-{\lambda }_t}\left(t_0\right)+{\theta }_0{\Pi }_2}{{\theta }_0{\Pi }_2}}\right\}$$

(30)

Therefore, Theorem 2 is verified. □

4. Results and Discussion

4.1. Simulation Results

In a MATLAB/SIMULINK 2024a environment, simulation studies of two cases are performed to demonstrate the advantage of the suggested methodology. The simulation time is T_f = 15 s. The first reference tracking trajectory is given as ε_1d = 60(1 − cos(0.75πt)(1 − e^−t), where t ∈ [0, T_f]. The second reference, i.e., a step signal, is given as ε_1d = 20 if (t ≤ 5 s) and ε_1d = 50 if (5 < t ≤ T_f). The AEPS system parameters in simulations are presented as follows: V_in = 12 V, L_a = 0.006 H, K_e = K_T = 0.05 V/(rad/s), R = 0.35 Ω, I_e = 0.0002 kg·m², and b_e = 0.0001 Nm/(rad/s).

Three other control strategies are used for the purpose of performance comparison, including a PI controller (C1), as follows:

$$u=k_p\left({\epsilon }_{1d}-{\epsilon }_1\right)+k_i{\displaystyle \underset{0}{\overset{t}{\int }}\left({\epsilon }_{1d}-{\epsilon }_1\right)dt}$$

(31)

where k_p = 12, k_i = 6, and a sliding-mode controller with super twisting algorithm (C2) is expressed as follows:

$$u=-k_1{\left|{\epsilon }_{1d}-{\epsilon }_1\right|}^{1/2}sign\left({\epsilon }_{1d}-{\epsilon }_1\right)-k_2{\displaystyle \underset{0}{\overset{t}{\int }}sign\left({\epsilon }_{1d}-{\epsilon }_1\right)dt}$$

(32)

where k₁ = 5.2, k₂ = 4.5, and an adaptive backstepping controller without disturbance observer (C3) is expressed as follows:

$$\begin{array}{l}{\dot{\hat{W}}}_i={\Gamma }_i\left({\theta }_i{s}_i-{\chi }_i{\hat{W}}_i\right),i\in \{2,3\}\\ {\hat{d}}_{i-1}=-{\mu }_2{s}_i-D_{i-1}\mathrm{sgn}\left(s_i\right)-{\eta }_i{s}_i^{{\alpha }_i},\hspace{1em}i\in \{2,3\}\\ {\beta }_1=\left(-{\rho }_{11}{z}_1-{\rho }_{12}{z}_1^{{\lambda }_{12}}-{\rho }_{13}{z}_1^{{\lambda }_{13}}\right)+{\dot{\epsilon }}_{1d}\\ {\beta }_2=\left(-{\rho }_{21}{z}_2-{\rho }_{22}{z}_2^{{\lambda }_{22}}-{\rho }_{23}{z}_2^{{\lambda }_{23}}\right)-{\hat{f}}_2-{\hat{d}}_1-z_1+{\dot{\beta }}_1\\ u={\displaystyle \frac{1}{q}}\left(-{\rho }_{31}{z}_3-{\rho }_{32}{z}_3^{{\lambda }_{32}}-{\rho }_{33}{z}_3^{{\lambda }_{33}}-{\hat{f}}_3-{\hat{d}}_2-z_2+{\dot{\beta }}_2\right)\end{array}$$

(33)

where the control gain of C3 is chosen to be the same as that of the proposed strategy. The gain parameters of the proposed methodology are designed as follows:

$$\begin{array}{l}{\Gamma }_i=5,{\chi }_i=0.3,{\mu }_i=0.6,D_1=10,D_2=50,{\alpha }_i=9/11,{\rho }_{11}=25,{\rho }_{12}=10,{\rho }_{13}=0.6,{\lambda }_{12}={\lambda }_{22}={\lambda }_{32}=7/6,\\ {\lambda }_{13}={\lambda }_{23}={\lambda }_{33}=6/7,{\rho }_{21}=85,{\rho }_{22}=10,{\lambda }_{12}=7/6,{\rho }_{23}=0.6,{\rho }_{31}=65,{\rho }_{32}=10,{\lambda }_{32}=7/6,{\rho }_{33}=0.6.\end{array}$$

Remark 3.

Based on the developed control law in (33), guidelines for parameter selection are provided as follows: First, the finite-time backstepping control parameters ρ_1i, ρ_2i, ρ_3i, (i = 1, 2, 3) are selected. Next, the observer gains ${\mu }_i,D_{i-1},\mathrm{and}{\eta }_i,i=2,3$ are determined. Then, the adaptation rates of the neural network weights ${\Gamma }_i\mathrm{and}{\chi }_i$ (i = 2, 3) are tuned. Finally, the influence of these parameters on the behavior of the AEPS system can be assessed through simulation studies and experimental validation.

Case 1.

The simulation results for this case study are provided in Figure 3 and Figure 4. From the first subgraph of Figure 3, the tracking control performance under the influence of disturbances and uncertainties may be evaluated. It is evident that the proposed algorithm provides more precise control than other control methods. The second subgraph of Figure 3 presents the tracking errors of the various controllers. The C1 controller performs significantly worse than any other controller, exhibiting the largest tracking error. Although C2 and C3 demonstrate better tracking performance than C1, their errors remain larger than those obtained using the proposed method. The third subgraph of Figure 3 shows the control action signals of the steering angles. To further compare the performance of the four controllers, quantitative metrics—including root mean square error ($e_{rms}$) and maximum error ($e_{max}$)—are provided in 

Table 1. Quantitative metrics of four controllers in simulation and experiment.

		C1	C2	C3	Proposed
Case 1	eRMS	1.7342	0.5785	0.3201	0.0956
	emax	3.1146	0.9853	0.4659	0.1415
Case 2	eRMS	3.0129	2.4942	0.8656	0.1011
	emax	9.2841	8.1470	6.3301	3.4153
Experiment	eRMS	2.5322	2.8165	2.1269	1.0354
	emax	7.1206	5.6041	4.4750	2.5618

, in addition to the graphical results. It can be observed that the proposed strategy achieves the lowest values of $e_{rms}$ and $e_{max}$.

The effective estimation capability of the ASMDO is illustrated in Figure 4. The ASMDO demonstrates superior stability and accuracy, contributing to the enhanced control performance of the proposed strategy.

Case 2.

The simulation results for this case study are presented in Figure 5 and Figure 6. In the first subgraph of Figure 5, the results indicate that all four schemes can effectively track their reference signals. However, the control responses of C1, C2, and C3 all exhibit significant overshoot, particularly around 0 to 5 s where the desired signal changes rapidly. For ease of comparison, the tracking errors of the four controllers are displayed in the second subgraph of Figure 5. It is evident that the proposed method exhibits the lowest tracking error and exhibits less overshoot compared to the other control methods. Finally, the control signals are presented in the third subgraph of Figure 5.

The disturbance estimation curves are presented in Figure 6. These show that both matched and mismatched disturbances are successfully estimated and fed into the proposed control algorithm. As a result, the tracking performance of the suggested method is significantly enhanced. Therefore, the simulation results reconfirm that the proposed scheme is both effective and feasible for the AEPS system.

4.2. Experimental Results

Hardware-in-the-loop testing of the AEPS system is utilized to verify the efficiency of the suggested strategy, as displayed in Figure 7. Figure 7a gives an overview of the experiment platform. Figure 7b shows the connection of the electrical and mechanical parts. Herein, the Arduino Mega 2560 microcontroller serves as the central controller of the system, receiving input from encoder sensors mounted on the motors to monitor their speed and position. The microcontroller interfaces with a MATLAB/SIMULINK-integrated computer, receiving encoder feedback and transmitting control commands to the MDS40B motor driver, which adjusts each motor’s speed and direction using PWM signals. The system employs a 12V MAXON DC motor connected to the steering wheel via a gear mechanism. By adjusting the current supplied to the motor, the control circuit governs the steering wheel’s movement, ensuring precise and efficient operation of the model. Details regarding the equipment specifications and sensor precision are provided in

Table 2. Hardware specifications of the real AEPS system.

Components	Parameters	Specification
DC motor	Type	DCX-26L
	Rated power	22 [W]
	Dimensions (D × L)	Ø26 × 57 [mm]
	Nominal torque	46.1 [Nm]
	Speed constant	445 r/min/V
	Commutation	Precious metal brushes
Encoder	Type	ENX 16 EASY
	Supply voltage	+4.5 to +5 [V]
	Number of channels	3 (ChA, ChB, ChI)
	Counts per turn (N)	1024
	Dimensions (D × L)	Ø15.8 × 8.5 [mm]
	Number of pins	10
Arduino	Model	AVR ATmega 2560 (8 bit)
	Input supply	7 to 12 [V]
	Number of DI/DO pins	54
	Number of AI pins	16
	Number of PWM pins	12
Battery	Model	TROY TTX5L
	Voltage	12 [V]
	Capacity	3.5 [Ah]
	Dimensions (L × W × H)	114 × 70 × 86 [mm]
DC motor driver	Type	SmartDrive40 MDS40B
	Maximum current	40 [A]
	Input voltage	10 to 45 [V]
	Dimensions (L × W)	124 × 107 [mm]
	Input modes	Analog, PWM

.

In the experiment, the sampling time is set to 0.01 s. For data acquisition, the output position of the steering wheel is measured by the ENX 16 EASY encoder and sent to an Arduino, which communicates with a PC running MATLAB/Simulink through the Real-Time Windows Target Toolbox. The actual position is collected via the ‘To workspace’ block in MATLAB/Simulink, then compared with the desired signal to calculate the tracking error. This error is then fed into the proposed control algorithm developed in MATLAB/Simulink, which generates the control signal for the SmartDrive40 MDS40B motor driver. All signals—including the actual position, desired signal, tracking error, control input, and others—are collected using the ‘To Workspace’ block in MATLAB/Simulink and then plotted using commands in the MATLAB command window.

The reference trajectory signal, i.e., a multistep signal, is given as ε_1d = 50 if (t ≤ 5 s), ε_1d = 55 if (5 ≤ t ≤ 10 s), and ε_1d = 60 if (5 < t ≤ T_f).

The experimental results are depicted in Figure 8, Figure 9 and Figure 10, and are consistent with the simulation results. As shown in Figure 8, the tracking performance of all four controllers is satisfactory. However, the proposed controller, indicated by the purple line, achieves the highest accuracy. Figure 9 displays the tracking errors of different strategies. It can be observed that the proposed scheme yields a tracking error lower than that obtained using any existing control method.

Figure 10 presents the disturbance estimation results. These demonstrate that the ASMDO effectively estimates the lumped disturbance. Consequently, the impact of the disturbance is mitigated, leading to improved control performance using the proposed methodology. Overall, the results confirm that the proposed control scheme outperforms existing approaches.

5. Conclusions

For this work, we proposed an adaptive finite-time backstepping control-based disturbance observer for an AEPS system with unknown dynamics and disturbance. We deployed an RBFNN to approximate the unknown dynamics function of the AEPS system. Meanwhile, an adaptive sliding-mode disturbance observer (ASMDO) was constructed to estimate the lumped disturbance. By integrating the RBFNN and ASMDO into the finite-time backstepping control scheme, performance control was significantly improved. The system stability of the closed-loop AEPS system was confirmed by the Lyapunov principle. Finally, from simulation and HIL experiment results, we concluded that the proposed control effectively tracked the desired trajectory, with excellent performance being exhibited. In future work, the optimal tuning of control gains for the proposed control strategy will be investigated to further enhance control performance. In addition, sensor fault handling and the implementation of finite-time steering angle tracking on a real autonomous vehicle will be pursued.