Design of a Robust Controller Based on Barrier Function for Vehicle Steer-by-Wire Systems

Abstract

In this research paper, a recent robust control scheme was proposed and designed for a VSbW (vehicle steer-by-wire) system. Using an integral sliding mode control (ISMC) design based on barrier function (ISMCbf) could improve the robustness of ISMCs. This control scheme, due to the characteristics of the barrier function, can improve the robustness of the proposed controller better than that based on the conventional SMC or integral SMC (ISMC). The ISMCbf scheme exhibits all the benefits of the conventional ISMC with the addition of two main advantages: it does not require prior knowledge of perturbation bounds or their derivatives, and it can effectively eliminate the chattering phenomenon associated with the classical ISMC due to the smooth characteristics of the barrier function. On the other hand, in terms of the design implementation, the ISMCbf is simpler than the ISMC. In this study, the mathematical dynamical model of the VSbW plant was first presented. Then, the control design of the ISMCbf scheme was developed. The numerical results showed that the proposed scheme is superior to the conventional ISMC. The superiority of the proposed ISMCbf controller versus the classical ISM has been evaluated under three different uncertain conditions, and three scenarios can be deduced: a slalom path, quick steering, and shock disturbance rejection. Furthermore, a comparative analysis with other controllers from the literature has further been established to show the effectiveness of the proposed ISMCbf.

1. Introduction

Advancements in technology have significantly impacted various research fields and applications. One area that has experienced notable development with the evolution of technology is drive-by-wire (DBW) technology. DBW technology holds great importance in the automotive industry, as it aims to replace traditional mechanical bonds with controlled electromechanical actuators. This technology, known as “by-wire” technology, offers enhanced freedom in designing force transmission characteristics, thereby improving vehicle maneuverability and stability [1,2]. Currently, DBW technology is widely employed in vehicle production, specifically for brakes and throttle control [2,3].

While implementing DBW technology for brakes and throttle control has become commonplace, steering by wires presents additional challenges for drivers. Nonetheless, it offers numerous advantages over conventional steering systems in terms of active guidance, maneuverability enhancement, and increased stability through improved driver steering input [4]. The use of active steering interventions has attracted researchers and the automotive industries to improve driving behaviors in the modern technology of car manufacturing. In recent years, improvements have been reached using controlled vehicle dynamics for steer-by-wire systems [5].

Accordingly, various control schemes for VSbW (vehicle steer-by-wire) systems have been developed and conducted. In [6], A. Baviskar et al. presented a non-linear observer-based adaptive control design for enhancing the tracking performance of the VSbW model subject to uncertainties in the system parameters. The adaptive control structure consists of elements. The first sub-controller is responsible for the compensation of uncertainties, while the other sub-controller is designed to discard the use of torque measurements. In [7], R. Kazemi and A. Janbakhsh proposed an ASMC (adaptive sliding mode controller) to improve the dynamic performance of the controlled VSbW system. This study considerably enhanced the robustness characteristics and tracking performance of the VSbW system, and the chattering behavior was also significantly reduced. In [8], Q. Xuyun et al. presented a robust control structure based on a LADRC (linear active disturbance rejection control) structure to enhance the performance of the controlled VSbW system. The proposed LADRC has shown good robustness features and improved accuracy under variations of the VSbW system. In [9], P. Zhai et al. presented a PID (proportional–integral–derivative) controller to enhance the tracking performance of the PID-based controlled VSbW system. In [10], an ILC (iterative learning control) approach was suggested by Z. Sun et al. for controlling the VSbW system. In a comparison study with the conventional PI (proportional–integral) controller, the results showed that the ILC control design has a better accuracy and less tracking errors compared to its counterpart. In [11], a SMLC (sliding mode learning control) approach for controlling the VSbW system was proposed by M. Tuan Do et al. Under the proposed control approach, excellent efficiency and steering performance have been verified based on the numerical and experimental results. In the study proposed by H. Wang et al. [12], an exact robust differentiator was designed to estimate the noisy position signal and its derivative. The designed differentiator was integrated with a SMC (sliding mode controller) to enhance the control performance of the VSbW system. As compared to other control strategies in the literature, better accuracy and tracking performance have been presented by the proposed method. In [13], another control approach based on the “Imperialist Competitive Algorithm” was presented by M. Ali et al. for controlling the VSbW system based on the PID controller. The ICA (imperialist competitive algorithm) has been utilized to optimize or tune the terms (gains) of the PID controller for enhancing the performance of the PID-based controlled system. Better tracking performance, better accuracy, and better speed of transient behavior have been obtained using the proposed controller as compared to the other conventional no-optimized PID controller. In [14], an ASMC (adaptive sliding mode control) was proposed by Z. Sun et al. for controlling the VSbW system. The proposed control approach has shown better robustness characteristics compared to other robust controllers, such as the linear H∞ controller and the conventional sliding mode controller (SMC). In [15], a SMADRC (sliding mode-based active disturbance rejection control) was designed by Z. Sun et al. to control the VSbW system. These authors conducted a comparison study between the proposed SMADRC and other control schemes, represented by the PDADRC (PD-based active disturbance rejection controller) strategy and classical SMC. Better efficiency could be obtained using the SMADRC compared to other control structures. In [16], the ATSMC (adaptive terminal SMC) algorithm was developed to control the VSbW model by H. Wang et al. The proposed controller gives better error convergence compared to other suggested control schemes in the literature. In [17,18], PD–PID control structures were developed by Tumari et al. for controlling the VSbW model. This study utilized the Ziegler–Nichols method to set the parameters of PID controller. Their study showcased the capability of the controller to improve system performance. The same researchers suggested the combined use of PID and fuzzy logic control (FLC) algorithms to address steering wheel returnability, directional control, and wheel synchronization in VSbW models [19,20].

L. Pugi et al. presented a general flexible simulation methodology that can be used for the modeling and simulation of various brake systems and power trains. This methodology is based on modular models, which can be customized and reassembled according to applied case-in-use [21]. J. Liang et al. proposed an integrated control framework that combines AFS (active front wheel steering system) and TV (torque vectoring) to guarantee the stability and performance of vehicle lateral motion. A distributed model predictive control has been adopted under system uncertainties. In order to mitigate the system disturbance, a robust H∞ approach was added to suppress the disturbance of the system [22].

The existing literature offers three distinct categories of control strategies for VSbW systems: classical control techniques (e.g., PI, PD, and PID), intelligent control strategies, and robust control strategies. The sliding mode control (SMC) is a notable robust control methodology that ensures system invariance to perturbations during sliding motion [23,24]. The SMC consists of two phases: the reaching and sliding phases [25]. While the reaching phase is affected by model uncertainties and external disturbances [26], the sliding phase reduces system dynamics. Several sliding mode controller strategies have been proposed to address these challenges, including the integral sliding mode controller (ISMC) strategy, which eliminates the reaching phase by enforcing the system state to be on the sliding manifold from the beginning through an equivalent controller [27,28]. Notably, the ISMC does not alter the system order [29]. However, both the SMC and ISMC suffer from chattering, necessitating approximations to mitigate the issue [30]. To overcome these limitations, the integral sliding mode controller based on barrier function (ISMCbf) has been proposed. The ISMCbf inherits the advantages of the ISMC but does not require information on the upper bound of perturbations [31]. Therefore, the key contributions of this study are outlined as follows:

• The development of the ISMC to ensure the minimization of tracking errors in the controlled VSbW system.
• Design of the ISMCbf, which allows the knowledge of lower limit of model uncertainty rather than its upper limit.
• A comparison was made between the proposed control approach and the ISMC strategy based on three main hard scenarios, represented by the slalom path, quick steering, and shock disturbance rejection.
• The control design was devoted to reduce the absolute error to its lower bound.

The structure of this work is organized as follows: Section 2 introduces the mathematical model of the VSbW system, Section 3 gives the details of the control design, Section 4 presents and discusses the obtained results, and Section 5 highlights the points concluded due to numerical simulation and suggests the future direction of research.

2. Dynamic Model

The technology of the steer-by-wire system had been firstly introduced as a prototype model by Chevrolet Corvette in 1997 to discard the direct connection between the car’s wheel and steering wheel. This technology uses controlled electric motors for changing the orientation of wheels and provides the driver with feedback information [32]. Figure 1 shows the conventional vehicle steering system.

As indicated in Figure 2, the conversion to this type of driving utilizes the existing stock components of vehicles except for the intermediate steering shaft. A brushless DC servomotor replaces the intermediate steering shaft to undertake the role of steering actuator. This modification enhances the steering capabilities of the vehicle based on this technology [1].

According to the simplified model of the VSbW, the mathematical model can be described using the following equation [14,15,33]:

$$J\ddot{y}+\rho sign\left(\dot{y}\right)+c \dot{y}+\tau =bu$$

(1)

where $y$ denotes the steering angle of the front wheels, $J$ denotes the moment of inertia, $c$ represents the viscous friction of the steering system, $\rho sign\left(\dot{y}\right)$ represents the Coulomb friction, where $\rho$ is a constant that is associated with Coulomb friction, $\tau$ refers to the self-aligning torque exerted on the front wheels, and $u$ represents the control signal.

The term “b” denotes a scale factor that consists of four elements. The first element is responsible for converting the input voltage of the steering motor into its output torque. The second element represents the gear ratio of the gear head. The third component is the gear ratio of the pinion–rack system. Finally, the fourth scale factor accounts for the transmission of the rack’s linear motion to the steering angle of the front wheels. In this study, the variation in the value of “b” is assumed to be very minimal [14].

Assumption 1.

The value of b slightly varies in this work; hence, it is assumed to be constant.

Assumption 2.

The upper bounds of the model uncertainties, which are indicated by the parameters $J, c,$and$\rho$, are defined and described as follows:

$$\begin{gathered} \left|\Delta J\right|=\left|J-J_0\right| \le J^{\Delta } \\ \left|\Delta c\right|=\left|c-c\right| \le c^{\Delta } \\ \left|\Delta \rho \right|=\left|\rho -{\rho }_0\right| \le {\rho }^{\Delta } \end{gathered}$$

(2)

where${\rho }^{\Delta }$,$c^{\Delta }$, and$J^{\Delta }$denote the upper bounds in model uncertainty.

Assumption 3.

There is no forward velocity at the front wheels. In other words, there is no actual self-aligning torque exerted on them by the ground. To model the self-aligning torque to minimize slip angles, a hyperbolic tangent function$, tanh \left(y\right)$, was applied for this purpose. This can be described using the following expression [6]:

$$\tau =\xi ·tanh \left(y\right)$$

(3)

where $\xi$ is a coefficient that depends on road conditions, and the function $tanh (.)$ refers to the hyperbolic tangent function. Due to Assumption 2, the system’s differential equation in Equation (1) can be rewritten in two terms: the nominal part and the model uncertainty part.

$$\begin{gathered} (\Delta J+J_0)\ddot{y}+(\Delta c+c_0)\dot{y}+(\Delta \rho +{\rho }_0)sign\left(\dot{y}\right)+\tau =bu \\ \ddot{y}=-\frac{c_0}{J_0} \dot{y}-\frac{{\rho }_0}{J_0} sign\left(\dot{y}\right)-\frac{\tau }{J_0}+K u+\Delta \end{gathered}$$

(4)

where

$$\Delta =\frac{{\Delta }_1}{J_0}$$

(5)

$${\Delta }_1=-\Delta J\ddot{y}-\Delta c \dot{y}-\Delta \rho sign\left(\dot{y}\right)$$

(6)

$$K=\frac{b}{J_0}$$

(7)

Equation (4) can be rewritten in state variable form by introducing the state variables ($x_1=y$ and $x_2=\dot{y}$), and the system model can be expressed as follows:

$$\begin{gathered} {\dot{x}}_1=x_2 \\ {\dot{x}}_2=f_n+\Delta +Ku \end{gathered}$$

(8)

The function $f_n$ refers to the nominal term of the system model, which includes the certain parts and it can be expressed using the following equation:

$$f_n=-\frac{c_0}{J_0} x_2-\frac{{\rho }_0}{J_0} sign\left(x_2\right)-\frac{\tau }{J_0}$$

(9)

Additionally, the control design interestingly represents the model VSbW system in terms of error variables. This error variable defines the difference between the measured and reference signals:

$$e_1=x_1-x_d$$

(10)

$$e_2=x_2-{\dot{x}}_d$$

(11)

$$\begin{gathered} {\dot{e}}_1=e_2 \\ {\dot{e}}_2=f_n+\Delta +Ku-{\ddot{x}}_d \end{gathered}$$

(12)

where $x_d$ denotes the reference signal, and $e_1$ and $e_2$ represent the error signal between the reference and measured signals, respectively.

In the next section, two types of sliding mode controllers (ISMC and ISMCbf) will be designed according to the model of Equation (12), and then they will be compared with other control strategies presented in previous works [14,16].

3. Control Design

In this part, two types of sliding mode control, ISMC and ISMCbf, will be presented. The SMC is one of the main control strategies that effectively control linear and non-linear systems operating under disturbances, model uncertainties, or both. There are two phases that can be found in the SMC strategy: the reaching phase and the sliding phase. The perturbations will be compensated and rejected during the sliding phase [34,35,36]. However, during the reaching phase, disturbances and model uncertainties affect the system. Moreover, during the sliding phase, the system order is reduced by one [37]. The ISMC is a new type of sliding mode control strategy that can eliminate the reaching phase. This means that the ISMC enforces the system to be on the sliding surface from the first instance, and the order of the model does not change during the sliding mode [38,39].

Two problems have been reported with the ISMC in many applications. The first problem is the chattering phenomenon caused by the discontinuous part of controller law (u_d), which requires approximation or smooth functions to eliminate such chattering behaviors, while the prior knowledge of the perturbation’s bounds is the second problem which has to be solved [39]. In order to avoid these problems, a new version of the ISMC, represented by the ISMCbf, has been introduced. All the advantages of the ISMC can be found in the new control methodology (ISMCbf). In addition, the ISMCbf has continuous smooth function and, therefore, does not require any type of approximation to avoid high-frequency chattering. The no need of prior knowledge for perturbation bounds is the most important benefit of the ISMCbf. Moreover, the ISMCbf is simple in design, and it only requires one control parameter [40].

3.1. Classical Integral Sliding Mode Control Design for VSbW

The ISMC is a new version of the SMC, which has a robust control characteristics, and it can eliminate the reaching phase by enforcing the system states to be on the sliding manifold right from the beginning [23,25]. Unlike other control approaches, the ISMC does not have a reaching phase, and the order of motion remains unchanged during the sliding mode [27,28]. The main steps involved in designing the ISMC are to design the sliding surface and the control law [30,31]. For the system described by Equation (12), the sliding variable is designed as follows:

$$s=e_2+Z$$

(13)

where $s$ denotes the sliding manifold, sliding variable, or sliding surface, and $Z$ is the integral term, where $Z \in R^1$.

Taking the derivative of Equation (13), one can obtain:

$$\dot{s}={\dot{e}}_2+\dot{Z}$$

(14)

Substituting ${\dot{e}}_2$ into Equation (14) to obtain:

$$\dot{s}=f_n+\Delta +Ku-{\ddot{x}}_d+\dot{Z}$$

(15)

The considered control law design is given by:

$$u=\frac{1}{K} (u_n+u_d)$$

(16)

In this context, $u_n$ represents the nominal controller that addresses the nominal aspect of the plant, while $u_d$ stands for the discontinuous control component that tackles the uncertainty in the dynamic model.

Substituting Equation (16) into Equation (15), the following equation can be obtained:

$$\dot{s}=f_n+\Delta +u_n+u_d-{\ddot{x}}_d+\dot{Z}$$

(17)

The integral part can be assigned to derivative $\dot{Z}$ as follows:

$$\dot{Z}=-u_n-f_n+{\ddot{x}}_d$$

(18)

Accordingly, Equation (17) becomes:

$$\dot{s}=\Delta +u_d$$

(19)

The nominal $\left(u_n\right)$ and discontinuous ${(u}_d)$ parts of the controller ($u$) are designed as follows:

$$u_n=-fn+{\ddot{x}}_d-c_1{e}_1-c_2{e}_2$$

(20)

$$u_d=-Msgn\left(s\right)$$

(21)

where $c_1$ and $c_2$ denote positive constant values selected according to the desired characteristic, and $M$ represents the sliding gain. Moreover, the discontinuous controller is responsible for eliminating the uncertainties.

Lemma 1.

Consider the tracking system defined in Equation (1) with uncertain parameters (6) under the ISMC Equation (16), where$u_n$and$u_d$are defined in Equation (20) and Equation (21), respectively. Then, the tracking error in Equation (12) converges to zero for the reference-commanded VSbW system.

Proof. 

Choose the following positive definite Lyapunov function:

$$v=\frac{1}{2} s^2$$

(22)

Taking the derivative of Equation (22) results in the following:

$$\dot{v}=s\dot{s}$$

(23)

In order to ensure the attractiveness of the trajectory to the sliding manifold, the condition below must be satisfied [30,31]:

$$s \dot{s}<0$$

(24)

After substituting Equation (21) into Equation (19), and using the result of reaching the condition outlined Equation (24), one can obtain:

$$s \dot{s}=s(\Delta -Msgn\left(s\right))$$

(25)

By using the upper bounds I of Equation (2), Equation (25) becomes:

$$s \dot{s}<-\left|s\right|(M-\left|\Delta \right|)$$

(26)

In order to satisfy the condition of Equation (25), the value of M has to be chosen according to the following inequality:

$$M\ge \left|\Delta \right|+\eta$$

(27)

where the constant $\eta$ has a very small, positive real value. □

Lemma 2.

The model is represented from the first instance by a new certain system with the design of the equivalent controller.

Proof. 

When the state trajectories reach the sliding manifold, $s=0$ and $\dot{s}=0$. The discontinuous part of the controller (${[ u_d]}_{eq}$) can be deduced according to Equation (19):

$$\begin{gathered} \dot{s}=\Delta +u_d \\ 0=\Delta +u_d \\ {[ u_d]}_{eq}=-\Delta \end{gathered}$$

(28)

The error dynamics during the sliding phase become:

$$\begin{gathered} {\dot{e}}_2=f_n+\Delta +K(\frac{1}{K}(u_n+{\left[ u_d\right]}_{eq})-{\ddot{x}}_d \\ {\dot{e}}_2=f_n+\Delta +-fn+{\ddot{x}}_d-c_1{e}_1-c_2{e}_2-\Delta -{\ddot{x}}_d \\ {\dot{e}}_1=e_2 \\ {\dot{e}}_2=-c_1{e}_1-c_2{e}_2 \end{gathered}$$

(29)

The main feature of using the ISMC is clarified from Equation (29), where the model is initially represented in terms of a certain system. □

3.2. Integral Sliding Mode Controller Design Based on Barrier Function (ISMCbf) for VSbW

The ISMCbf was applied in this work as it does not require the pre-knowledge of the upper bound of model uncertainties and it provides smooth function, which can inherently reduce the chattering behavior. The key with this control design is to replace the discontinuous control component $\left(u_d\right)$ with a continuous barrier function. The sliding mode control design with barrier function is more straightforward compared to the ISMC approach as it eliminates the need for upper limits for unknown perturbations, which is a requirement in ISMC design [28,38]. This control approach also ensures a continuous control signal, thus avoiding the high-frequency chattering typically induced by the discontinuous control part in the traditional ISMC. As illustrated in Figure 3, there are two variants of barrier functions: positive definite barrier functions (PBFs) and positive semi-definite barrier functions (PSBFs) [40,41,42,43,44]. The following mathematical fundamentals explains the concepts of barrier functions:

Definition 

([28,38,40]). For some fixed positive values, $\epsilon >0$, the barrier function can be expressed in terms of the continuous function $f:x\in [-\epsilon ,\epsilon ]\to g\left(x\right)\in [b,\infty )$ strictly increasing on $[0,\epsilon ].$

• ${lim}_{\left|x\right|}\to \epsilon g\left(x\right)=+\mathrm{\infty }$.
• The function $g\left(x\right)$ has a unique minimum value at zero, such that $g\left(0\right)=b\ge 0$.

There are two different classes of BFs (barrier functions), which can be defined as:

• Positive-definite BFs (PBFs):

$$g_p\left(x\right)=\frac{\epsilon F}{\epsilon -\left|x\right|} , g_p\left(0\right)=F>0$$

(30)

• Positive semi-definite BFs (PSBFs):

$$g_{ps}\left(x\right)=\frac{\left|x\right|}{\epsilon -\left|x\right|} , g_{ps}\left(0\right)=0$$

(31)

Figure 3. The positive-definite barrier function and the positive semi-definite barrier function.

Figure 3. The positive-definite barrier function and the positive semi-definite barrier function.

In this work, the positive semi-definite BF was used in the design of the ISMCbf. For the system of Equation (12), the proposed design of the ISMCbf is not different from the previous ISMC; it only differs in the design of the discontinuous term ($u_d)$, as follows:

$$u_d=-\frac{s}{ϵ-\left|s\right|}$$

(32)

$$u=\frac{1}{K} (u_n+u_d)$$

(33)

As mentioned earlier, the responsibility for eliminating uncertainties returns to the discontinuous control part.

Lemma 3.

Consider the tracking system defined in Equation (1) with uncertain parameters defined by Equation (6). Under the ISMCbf with the control law of Equation (33), where $u_n$and$u_d$are defined in Equation (20) and Equation (32), respectively, the tracking error in Equation (12) converges to zero for the commanded-reference VSbW system.

Proof. 

Choose the following Lyapunov function:

$$v=\frac{1}{2} s^2$$

(34)

By taking its derivative, the result is:

$$\dot{v}=s\dot{s}$$

(35)

The sliding manifold is attractive to solution trajectories when the condition in Equation (24) has to be satisfied [28,38]. After substituting Equation (32) into Equation (33) and substituting the result in Equation (24), the following can be obtained:

$$\begin{gathered} s\dot{s}=s(\Delta -\frac{s}{ϵ-\left|s\right|})<0 \\ s \dot{s}\le -\frac{{\left|s\right|}^2}{\mid ϵ-\left|s\right|\mid }+s \Delta \\ s \dot{s}\le -\left|s\right|(\frac{\left|s\right|}{\left|ϵ-s\right|}-\left|\Delta \right|) \end{gathered}$$

(36)

Therefore, the $s\dot{s}<0$ for $\left|s\right|$ is sufficiently near $ϵ$, where $\frac{\left|s\right|}{\left|ϵ-\left|s\right|\right|}>\left|\Delta \right|$. □

Since the dynamic model has been represented in terms of uncertainty, the discontinuous control part in the ISMCbf can be used to compensate for the uncertainties on the sliding surface, where s = 0 and $\dot{s}$ = 0, as follows:

$$\begin{gathered} \dot{s}=\Delta +u_d \\ 0=\Delta +u_d \\ {[ u_d]}_{eq}=-\Delta \end{gathered}$$

(37)

Eventually, the system dynamics during the sliding phase, based on the discontinuous control part, become:

$$\begin{gathered} {\dot{e}}_2=f_n+\Delta +K(\frac{1}{K}(u_n+{\left[ u_d\right]}_{eq})-{\ddot{x}}_d \\ {\dot{e}}_2=f_n+\Delta +-fn+{\ddot{x}}_d-c_1{e}_1-c_2{e}_2-\Delta -{\ddot{x}}_d \\ {\dot{e}}_1=e_2 \\ {\dot{e}}_2=-c_1{e}_1-c_2{e}_2 \end{gathered}$$

(38)

From the above equation, it is clear that the system dynamics using the ISMCbf are similar to those based on the ISMC. The two controllers lead to the generation of the same results; however, the ISMCbf does not need prior knowledge of bounds for model uncertainty, and it also has a smooth function [28,38].

4. Simulation Results and Discussion

This section showcases and discusses the effectiveness of the ISMC and the ISMCbf via computer simulations using the MATLAB 2016b software programming tool. The controllers have been evaluated according to three cases (scenarios) that mimic the real situation of road conditions. Subsequently, a comparison study has been made between the performance of the ISMCbf controller and some robust control strategies that have been reported in research studies [14,16]. In order to mitigate the chattering effect in the control signal caused by the discontinuous term ${(u}_d)$, an approximation was utilized to replace the signum function in Equation (21):

$$sgn\left(s\right)\approx \frac{s}{\left|s\right|+\gamma }$$

(39)

where γ represents a very small, positive constant. The model’s parameters, along with the control design parameters for both control strategies (ISMC and ISMCbf), are listed in

Table 1. Design parameters of the VSbW model and controllers.

Parameter	Value
$J_0, c_0, {\mathrm{and} \rho }_0$ [16]	$86 {\mathrm{k}\mathrm{g}\mathrm{m}}^2$$, 220 \frac{\mathrm{N}\mathrm{m}\mathrm{s}}{\mathrm{r}\mathrm{a}\mathrm{d}}$, and 4.2 $\mathrm{N}\mathrm{m}$
$b$ [16]	275
$J^{\Delta }, c^{\Delta }$$, \mathrm{and} {\rho }^{\Delta }$ [16]	9, 22, and 0.4
$\epsilon$	0.002
$\eta$	0.001
$\gamma$	0.003

.

4.1. Slalom Path

In this situation, the controller enforces the VSbW system to follow a sinusoidal waveform while working to minimize the tracking error as much as possible. Additionally, the road coefficient ξ, as shown in Equation (3), has been adjusted to depict three distinct road scenarios: a snowy road, a wet asphalt road, and a dry asphalt road. These conditions were sequentially implemented in this system over three separate time periods as follows:

$$\zeta =\left\{\begin{array}{c} 155 0<t\le 20s Snowy road \\ 585 20s<t\le 40s Wet asphalt road\\ 960 20s<t\le 40s Dry asphalt road \end{array}\right.$$

The behavior of coefficient can be shown in Figure 4.

The control performance of the model controlled by the ISMC and the ISMCbf are shown in Figure 5, which presents the steering angle, tracking error, control effort, and sliding surface.

From the above results, it is clear that the two controllers could successfully perform good tracking performance, giving the same levels of control effort and tracking error.

4.2. Quick Steering

In this scenario, the performances of both controllers (ISMC and ISMCbf) have been assessed by conducting the VSbW system to move along a curved-road path. The simulation involved two desired trajectories to be followed using the VSbW system: straight-line motion and maneuvering along a circular path. This desired trajectory was generated by steering the vehicle’s wheel for $15 \mathrm{s}$ of vehicle movement. Accordingly, the value of coefficient ξ for this specific scenario was set to ξ = 950 for both controllers. The effectiveness of these controllers is demonstrated in Figure 6, which showcases the responses of various parameters, such as the steering angle, tracking error, control effort, and sliding manifold. These metrics provide valuable insights into the performance of the controllers under examination.

4.3. Shock Disturbance Rejection

In this scenario, a sudden shock disturbance was introduced to simulate an obstacle, like a bump or a brick, in front of the path of the VSbW system. The VSbW model was commanded to follow a straight path for this specific case. When facing a sudden change, the response of the controlled vehicle aligns to the desired position.

The objective of the controllers based on the ISMC and ISMCbf approaches is to compensate for the effect of the exerted load during the load change period. The controllers could ensure a high convergence rate of tracking errors during this period of load change. To numerically simulate the real situation of vehicle motion, a shock disturbance was applied for a duration of 10 s, while the reference input was set to zero. To eliminate the influence of road conditions, the coefficient ξ, representing the self-aligning torque, was set to ξ = 150, representing an invariant snowy road condition.

Figure 7 provides the performance of both the ISMC and ISMCbf controllers. This figure presents the responses of various parameters, including the steering angle, tracking error, control effort, and sliding manifold.

The above results show that the efficacy of the ISMC and ISMCbf control strategies has been improved in terms of tracking performance and control effort under three different situations. Based on the behavior of the sliding manifold for the ISMCbf, it is clear that the solution trajectory becomes invariant to model uncertainties. One can conclude that the control design based on the ISMCbf is simple and does not need any type of approximation to avoid the high-frequency phenomenon as compared to the classical ISMC.

To validate the superiority of the proposed ISMCbf method over previously published control techniques, a comparison study was conducted with an ASMC [14], a PDADRC, and a SMC [16]. The performances of the proposed and suggested controllers have been reported in

Table 2. The performances of the controllers.

Scenario I: Slalom Trajectory
Control Method	Max Tracking Error (Rad)	Max Control Signal (v)
ISMC	0.012	1.1
SMC	0.055	1.2
PDADRC	0.029	1.5
ASMC	0.028	1.4
ISMCbf	0.012	1.1
Scenario II: Quick steering
ISMC	0.0075	1.45
SMC	0.041	1.5
PDADRC	0.032	1.7
ASMC	0.05	1
ISMCbf	0.0075	1.45
Scenario III: Shock disturbance rejection
ISMC	0.0022	1.45
SMC	0.082	1.5
PDADRC	0.022	1.3
ASMC	-	-
ISMCbf	0.0022	1.45

. The three different road conditions have been taken into account in this comparison study. The metrics for this evaluation are the maximum error value and maximum control efforts. The results in Table 2 indicate that the ISMCbf approach gives better tracking performance and lower control efforts compared to the other control strategies.

5. Conclusions

In this study, a robust continuous control strategy based on the ISMCbf was developed for controlling a vehicle steer-by-wire system. Through numerical simulations, it was observed that the ISMCbf exhibits superior robustness characteristics compared to other control strategies when encounters to uncertainty in system parameters. The ISMCbf effectively mitigates the impact of perturbation terms, enabling the controlled system to behave as if it were operating under nominal conditions right from the beginning. Additionally, the ISMCbf demonstrates smooth functionality, eliminating the need for approximations to avoid the chattering phenomenon often associated with other control techniques. Moreover, the ISMCbf offers simplicity in design by not requiring any prior knowledge of model uncertainty parameters or external disturbances, and it only relies on a single control parameter. Both the ISMCbf and the ISMC demonstrate their ability to minimize the influence of perturbation terms and achieve nominal system behavior.

Further advancements can be explored by suggesting other control techniques, such as adaptive control schemes, observe-based control, backstepping-based control, and active disturbance rejection control for the sake of comparison [45,46,47,48,49,50,51]. Also, an improvement of the proposed controller can be presented by suggesting modern optimization techniques to optimize the design parameters of the controller [52,53,54,55]. Another extension of this study could be to implement the proposed control in a real-time environment, either using a FPGA (field programming gate array) hardware tool, Raspberry-Pi single board computer, or LabVIEW-based data acquisition system [56,57,58,59,60].