Fast Finite-Time Composite Controller for Vehicle Steer-by-Wire Systems with Communication Delays

By inergency On Mar 26, 2024

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1. Introduction

In recent decades, the automotive industry has experienced a profound transformation, propelled by significant technological advancements [1,2]. The emergence of x-by-wire technology (i.e., x-by-wire including steer/drive/brake-by-wire) has become a focal point in current research, reflecting the industry’s dynamic evolution [3]. One such advancement is the introduction of steer-by-wire (SBW) technology that replaces the traditional mechanical steering mechanism with electronic signals. The concept of steering by wire represents a departure from the traditional design of vehicles, where a mechanical link directly transmits the input of the driver to the vehicle’s wheels. Instead, SBW systems rely on electronic sensors, actuators, and control units to interpret and execute steering commands. This technology promises several advantages, such as increased design flexibility by reducing weight, enhanced driving experience, and the potential for advanced driver assistance systems and functionalities associated with automotive driving [4]. The SBW systems can be integrated with intelligent collision avoidance algorithms that anticipate potential hazards and automatically adjust the steering to avoid collisions, thereby reducing the likelihood of road traffic accidents. As the automotive industry charts its course toward a future characterized by connected and autonomous vehicles, SBW emerges as a pivotal component in the pursuit of these transformative goals.

The domain of steer-by-wire systems has witnessed an extensive exploration of diverse control strategies aimed at enhancing the dynamic performance and safety of SBW vehicles [5,6,7,8,9]. In [10], an adaptive sliding mode control (SMC) method was presented for the vehicle SBW system to address the trajectory tracking problem and enhance robustness against diverse road conditions. Similarly Wang et al. [11] presented an adaptive terminal SMC (TSMC) scheme that operates in the existence of parameter uncertainties and changes in driving conditions. Sun et al. [12] and Sun et al. [13] both proposed adaptive SMC (ASMC) concepts for vehicle SBW systems. Sun et al. [13] emphasized a nested adaptive super-twisting SMC (NASTSMC), while Sun et al. [12] focused on the ASMC. The shared objective of both designs is to enhance tracking accuracy and robustness. Shi et al. [14] investigated a strategy based on a fractional-order SMC (FOSMC) with an extended state observer (ESO), specifically designed to handle challenges related to parameter perturbation and external interference within the dynamical model. Liang et al. [15] implemented an adaptive scheme for compensating friction torque within the SBW system of a vehicle. Shukla et al. [16] specifically addressed the issue of state-dependent uncertainties in the SBW systems, introducing an adaptive control framework designed to effectively manage these uncertainties and external disturbances without requiring prior knowledge of their structures or bounds.

Some studies have explored the implementation of advanced control strategies for the SBW systems. For example, Ye and Wang [17] investigated robust adaptive integral TSMC (AITSMC) strategies, incorporating an extreme learning machine (ELM) estimator for handling lumped uncertainties. Similarly, Zhang et al. [18] constructed an active front-steering control scheme that combines adaptive recursive integral TSMC in the top controller and fast nonsingular TSMC (FNTSMC) with the ELM estimator in the bottom controller to enhance steering control performance and ensure a faster convergence rate. In a recent study, Zhao et al. [19] applied an observer-based discrete-time cascaded control strategy designed to address the challenge of lateral stabilization in SBW vehicles amidst uncertainties and disturbances. Wang et al. [20] developed a neural output feedback control with predefined performance and composite learning, ensuring transient and steady-state performance within specified boundaries. Li et al. [21] explored trajectory tracking control for four-wheel independently actuated electric vehicles equipped with the SBW systems. This control strategy employs a robust H_∞ dynamic output feedback approach, integrating the dynamics of SBW devices into a polyhedral linear parameter-varying trajectory tracking error model. These control approaches enhanced tracking accuracy to a certain degree; nevertheless, they overlooked the impact of time delays, leading to a reduction in the robustness of SBW systems.

The technology behind (SBW) systems offers greater flexibility and control but is not without its challenges, particularly in dealing with time delays within the system [22]. Time delays can occur at various stages, including signal processing, communication, and actuation [23]. These delays, especially in communication, significantly impact the performance of the SBW systems, leading to sluggish response times and compromising the vehicle’s ability to navigate swiftly and precisely. Such compromises not only affect the effectiveness of SBW systems but also raise safety concerns, particularly in critical situations where delays pose inherent risks [22]. Therefore, it is imperative to comprehensively address the challenges posed by time delays in SBW systems to ensure their smooth integration into the automotive industry. While time delays are recognized as significant contributors to instability and poor performance in various systems [24], their specific impact on SBW vehicles has received limited attention in the research. Recent efforts, such as the work by Yang et al. [25], have focused on introducing adaptive fast super twisting sliding mode control (SMC) based on time-delay estimation to mitigate the challenges posed by inaccurate modeling and variable perturbations. To tackle challenges arising from significant random delays in the steering systems, Zhang et al. [26] proposed a layered time-delay robust control strategy. This strategy integrates a lower controller to minimize the tracking error and ensure stability, along with an upper controller employing the TSMC approach to enhance vehicle yaw stability. Nevertheless, the strategy does not explicitly account for model uncertainties and parameter variations, which are crucial factors for ensuring robust stability and performance. However, there is a significant shortage of studies that address trajectory tracking control algorithms for the SBW systems while taking into account the time delays in the transmission channel. Furthermore, the methods mentioned above can only ensure the slow asymptotic convergence of the SBW system states.

Therefore, proposing a fast finite-time composite controller (FFTCC) for the uncertain SBWs that are subject to communication delays has motivated us to contribute to this field. This controller strategy is designed to enable the front wheel angle of the SBW system to rapidly and precisely follow the desired command input from the hand wheel within a finite time, independent of time delays and other uncertainties. The main contributions of the current paper can be summarized as follows:

A dynamical model for the SBW system is systematically formulated to incorporate the inherent time delays in the transmission channel connecting the hand wheel and the steering actuation module. Moreover, the model accommodates parametric system uncertainties and external disturbances.
A new control strategy, denoted by the FFTCC, is devised to address the challenge of rapid finite-time convergence of tracking errors in the time-delayed SBW systems. This proposed fast finite-time convergent observer-based control is specifically designed to accommodate the time delays inherent in the transmission channel, ensuring robust performance in different communication scenarios.
A new fast-scaling finite-time ESO is constructed to estimate unmeasured velocity variables and the unknown overall disturbances in rapid finite-time instances. By integrating the unmeasured variable and the lumped perturbations into the proposed FFTCC, the proposed composite control scheme is explicitly realized. The overall closed-loop stability is proved as global finite time by the Lyapunov theory.
The effectiveness of our designed controller is rigorously evaluated under three distinct scenarios, providing a comprehensive assessment of its performance. To validate its efficacy, simulation results are compared against two benchmark control methods—scaling ADRC (SADRC) and well-known ADRC. This comparative analysis serves to underscore the advantages and advancements offered by the introduced fast finite-time convergent observer-based control.

In this paper, the structure of the remaining sections is as follows: In Section 2, the mathematical modeling of the SBW system is delved into. Section 3 navigates through the design of a fast finite-time controller via error feedback and introduces the construction of a fast finite-time convergent composite controller. Section 4 extends our discussion to simulation results by offering a detailed presentation and analysis of simulation outcomes. In the final section, the findings are summarized, and the conclusion is presented.

4. Simulation Results

To validate the designed controllers, simulation tests are performed within the MATLAB/SIMULINK environment. The SBW model parameters in (1) are specified in Table 1. In addition, the flow chart presented in Figure 3 is followed in this section to apply the proposed method in the SBW system. Furthermore, to demonstrate the benefit of the control method introduced in this paper, the simulation results of both the ADRC and SADRC methods are compared with the proposed one and presented in this section. For the proposed controller and the comparative ADRC and SADRC methods [43], the sampling time in all three methods is identical and set to

T_{s} = 4 \times 10^{- 3}

sec. Moreover, the controller gains of the proposed method, the ADRC method, and the SADRC method are selected and presented in Table 2. As listed in Table 2, to accomplish the aim of the finite-time convergence of estimation and tracking errors of the SBW, as stated in the Proof of Theorem 2 and Remark 5, we set the observer and control parameters to

α = 0.14

and

L = 1.2

Then, the simulation results and corresponding analyses are presented in three different scenarios. Please note that to compare the robustness with various controllers in the SBW system, the controller gains remain constant across all three cases. Furthermore, in these three cases, both the Coulomb friction torque

τ_{c}

, as presented in Equation (1), and the self-aligning torque

τ_{s e l}

, as shown in Equation (3), are taken into account. To represent the changing road conditions during the simulation,

ρ_{τ}

is given to determine

τ_{s e l}

as follows

$\begin{matrix} ρ_{τ} = & \{\begin{matrix} 155, & 0 < t \leq 20 \sec, Snowy road \\ 585, & 20 < t \leq 40 \sec, Wet asphalt road \\ 960, & 40 < t \leq 60 \sec, Dry asphalt road . \end{matrix} \end{matrix}$

(85)

4.1. Case 1: Nominal Steering for a Sinusoidal Reference Following the Input and Output Time Delay

In this case, the tracking performance of the SBW system is examined under the nominal dynamics with time delays. For the purpose of conducting this case study, both the parametric uncertainties defined in (2) and the external disturbance

d (t)

specified in (5) have been set to zero. In addition, the input and output time lags illustrated in Figure 1 have been allotted values of

τ_{i} = 1 \times 10^{- 3}

sec and

τ_{o} = 2 \times 10^{- 3}

sec, correspondingly. Moreover, the aggregate of

τ_{i}

and

τ_{o}

is ensured to be smaller than the sampling time

T_{s}

. Furthermore, as illustrated in Figure 5a, the reference angle of the front wheels has a sinusoidal pattern. In Figure 4a–d, the estimation errors for

x_{1}

x_{2}

x_{3}

, and

u (t - τ)

are illustrated, showcasing the exceptional estimation capability demonstrated by the proposed approach when it is contrasted with the methods of ADRC and SADRC. It can be observed that the estimation error of the proposed approach is narrower than that of the two alternative comparison techniques, whereas the rate of convergence for estimation in the proposed approach is higher. Furthermore, it is noteworthy that small peaking phenomenon arises within the estimated errors, primarily stemming from the high observer gains. Nevertheless, the observer gains cannot be set too low since it is also necessary to ensure a rapid tracking rate, minimal tracking error, and robust system performance. In Figure 5a–c, the graphs present the curves of the steering angle

θ_{s}

, the tracking error of

θ_{s}

, and the control input

u (t)

under the proposed control approach as well as the competing methods. In Figure 5a,b, it is apparent that the rate of convergence achieved through the proposed control technique surpasses that of ADRC and SADRC, while the tracking error in the proposed approach is diminished, as the effect of time lags is efficiently reduced in the proposed control strategy. Furthermore, Figure 5c demonstrates that the proposed control method exhibits a faster response in the control input compared to the other two comparison methods, whereas the maximum values of the control input among all three methods remain similar.

4.2. Case 2: Uncertain Steering for a Sinusoidal Reference Following the Input and Output Time Delay

Compared with Case 1, the parameter uncertainties stated in (2) are newly taken into account in Case 2. The transmission time delays are set to

τ_{i} = 2 \times 10^{- 3}

s and

τ_{o} = 2 \times 10^{- 3}

s, respectively. Hence, the combination of

τ_{i}

and

τ_{o}

is equal to the sampling time

T_{s}

. In Figure 6a–d, the corresponding estimation errors of

x_{1}

x_{2}

x_{3}

, and

u (t - τ)

are presented. We can observe that the proposed control strategy continues to demonstrate superior estimation performance compared to those of the two comparison methods, exhibiting a swifter estimation convergence rate and smaller estimation errors. Furthermore, it is evident from Figure 7a–c that the tracking error of

θ_{s}

within the proposed control framework maintains a lower magnitude than that observed in the ADRC and SADRC methods, while the convergence speed of the tracking error and the responsiveness of the control input in the proposed control technique are enhanced.

4.3. Case 3: Uncertain Steering for a Sinusoidal Reference Following the Time-Varying Delays and External Disturbances

Compared with Case 2, not only the parameter uncertainties given in Equation (2) are taken into account but also the time-varying delays

τ_{i} = (0.001 + 0.001 \sin t)

s and

τ_{o} = (0.001 + 0.001 \sin t)

s and the external disturbance

d (t) = \sin t

Nm. The simulation results of the steering angle

θ_{s}

, the tracking error of

θ_{s}

, and the control input for Case 3 are presented in Figure 8a–c. A comparison of Figure 8a–c reveals that the proposed control method continues to outperform the other two comparison methods. This indicates that the proposed control method effectively compensates for parameter uncertainties and external disturbance amidst time-varying input and output delays. In practical applications, if the front wheels track their reference slowly or with large tracking errors, it can lead to delayed or inaccurate path changes in the vehicle, particularly in dangerous traffic conditions. This situation poses a significant safety risk. As shown in Figure 5b, Figure 7b, and Figure 8b, and their corresponding Case 1, Case 2, and Case 3 analyses, the proposed control scheme offers improved driving safety in the SBW system compared with two existing control algorithms. This is achieved through swifter convergence rates and diminished tracking errors of the front wheels.

5. Conclusions

In an effort to enhance the performance of the steering-by-wire (SBW) system, this paper proposes a novel robust fast finite-time composite control algorithm. This algorithm is designed to precisely track the front wheel angles in response to the desired command from the hand wheel within a fast finite time, overcoming challenges posed by time delays and uncertainties. The proposed controller was examined across three distinct scenarios: nominal steering involving input and output time delays; uncertain steering entailing both uncertainties and transmission time delay; and the final scenario encompassing uncertainties, transmission time delay, and external disturbances. To verify the outperformance of our proposed method, simulations are conducted, and the results are compared with two benchmark control methods, namely, ADRC and SADRC. The simulation results indicate the superiority of the proposed controller over the two comparative methods. Lastly, the findings emphasize the effectiveness of the proposed control method in entirely addressing parameter uncertainties and external disturbances, even in challenging conditions characterized by time-varying input and output delays. Moreover, the discerned accelerated convergence rate of the front wheels’ tracking errors, coupled with the reduction in the tracking errors, provides compelling evidence that the proposed method enhances driving safety in the SBW system, even when facing various uncertainties. While we acknowledge the importance of experimental validation, we view this work as a foundational step. Future research will undoubtedly include experimental studies to bridge the gap between simulation and reality, ensuring the robustness and effectiveness of the proposed controller in actual automotive scenarios. Furthermore, to reduce noise measurement caused by the sensitivity of sensors, the noise-free FFTCC scheme will be studied.

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