Finite-Time Adaptive Control for Electro-Hydraulic Braking Gear Transmission Mechanism with Unilateral Dead Zone Nonlinearity

Abstract

Autonomous vehicles require more precise and reliable braking control, and electro-hydraulic braking (EHB) systems are better adapted to the development of autonomous driving. However, EHB systems inevitably suffer from unilateral dead zone nonlinearity, which adversely affects the position tracking control. Therefore, a finite-time adaptive control strategy was designed for unilateral dead zone nonlinearity. Initially, the unilateral dead zone nonlinearity was reformulated into a matched disturbance term and an unmatched disturbance term to reduce the adverse effects of disturbances, thereby enhancing system controllability. Then, the “complexity explosion” in the design of the control strategy was avoided by command filtering, and the design process of the controller was simplified. Furthermore, the finite-time control theory was employed to boost the system’s convergence speed, thereby enhancing control performance. In order to ensure the stability of the system under the dead zone disturbance, the unknown disturbance terms were estimated. The stability of the control strategy was validated through the finite-time stability theorem and the Lyapunov function. Eventually, simulations and hardware-in-the-loop (HIL) experiments validated the feasibility and availability of the finite-time adaptive control strategy.

1. Introduction

In the context of the rapid advancement of automobile technology [1,2,3,4,5], intelligent driving vehicles have higher requirements for the reliability and accuracy of the braking system. With traditional braking systems exhibiting drawbacks such as braking force delays and extended braking distances, they no longer suffice to meet the evolving requirements of intelligent driving vehicles [6]. As a result, EHB systems have progressively supplanted traditional braking systems due to their swift braking response and precise braking force modulation [7]. Consequently, to adapt more effectively to the progression of intelligent driving vehicles, numerous scholars have conducted extensive research on EHB systems [8,9,10] and gradually recognized the influence of backlash nonlinearity on the system’s performance.

In practical mechanical transmission systems, the presence of backlash nonlinearity tends to cause unnecessary disturbances, significantly impacting the system’s dynamic response, control performance, and overall reliability [11,12]. Given that the gear transmission mechanism is an important part of the EHB system, the detrimental effects of backlash nonlinearity on the control performance of EHB systems are particularly pronounced. This nonlinearity leads to unknown disturbances in the gear transmission mechanism, hindering the realization of precise position tracking control [13,14]. Consequently, designing a control strategy capable of addressing the backlash nonlinearity emerges as a pressing issue in need of resolution.

Considering the effects of backlash nonlinearity on the system, researchers have commonly employed dead zone models to characterize backlash nonlinearity [15,16,17]. However, traditional dead zone models exhibit non-differentiable characteristics within their dead zone ranges, where input signals fail to induce changes in output signals. To address this limitation, a differentiable dead zone model was proposed [18]. On this basis, the concept of “soft degree” was introduced to mitigate oscillation and instability issues [19]. Subsequent research extended the above achievement to mechanical systems featuring asymmetric dead zone nonlinearity, offering better solutions for general backlash nonlinearity systems [20,21]. Yiyun Zhao et al. utilized a differentiable function to approximate the traditional dead zone model in order to address system jitter problems [22]. Despite these advances, unknown disturbances in the system are still not negligible.

Mingjie Cai et al. transformed the dead zone model into a linear system with unknown gain and bounded disturbance and proposed an adaptive neural finite-time control method [23]. An adaptive robust control strategy was proposed to solve the unknown parameters, dead zones, and disturbances [24]. Xiuping Wang et al. proposed an adaptive RBF neural network command filter backstepping control for the unmodeled disturbances in PPMLM [25]. A robust adaptive sliding mode control strategy was designed [26]. This strategy provides strong robustness to uncertain parameters and unknown disturbances. Shihong Ding et al. constructed a new third-order sliding mode control strategy for the effect of unknown disturbances on tracking control [27]. From the above analysis, it can be seen that adaptive control is an important tool for solving the dead zone nonlinearity. However, the design of adaptive control strategies is prone to the problem of “complexity explosion” in high-order nonlinearity systems.

For the problem of “complexity explosion”, researchers have made significant advancements by incorporating command filtering technology [28,29,30,31]. Command filtering technology was introduced into the adaptive control algorithms, which simplified the design of the controller [32,33]. Focusing on uncertain nonlinearity systems with unknown dynamics models, Yongliang Yang et al. presented a robust adaptive control method to solve the issue of “complexity explosion” by utilizing command filtering technology [34]. Additionally, a command filtering adaptive fuzzy control scheme was proposed to address parameter uncertainty within the system and to simplify the controller’s design [35]. Command filtering technology has proven to be a valuable solution for overcoming the “complexity explosion” in adaptive control strategy design. Furthermore, there is a growing recognition of the impact of convergence speed on the control performance of systems.

Considering the influence of convergence speed on system control performance, researchers have introduced finite-time control theory within adaptive control [36,37]. An adaptive finite-time control strategy based on a neural network was proposed, effectively resolving the limitations of traditional control methods in achieving rapid convergence under input saturation conditions [38]. Moreover, an output feedback control was presented based on adaptive finite-time control, which accomplished tracking tasks of the system within finite time [39]. A novel adaptive control strategy was proposed, which used the finite-time control theory to achieve rapid convergence to the desired trajectory [40]. For the unmodeled dynamic nonlinearity systems, a control method integrating adaptive control, fuzzy control, and finite-time control theory was designed, providing a novel solution to control issues in such systems [41]. All of the above control methods have achieved good control results.

It can be seen from the above analysis that the combination of command filtering technology and adaptive control can avoid the issues of “complexity explosion” and simplify the design of the controllers. Meanwhile, by applying the finite-time control theory, the convergence speed of the controller can be improved, thereby enhancing the control performance of the system. Therefore, this paper presents a finite-time adaptive control strategy based on command filtering. The strategy utilizes command filtering to introduce the compensation signals to reduce the system’s tracking error. Then, a power exponent term is introduced through finite-time control theory to make the system converge quickly in finite time. Finally, the control signal is solved by adaptive control. The contributions are as follows:

(1) A finite-time adaptive control strategy is proposed for the unilateral dead zone of the EHB gear transmission mechanism. The “complexity explosion” in the design of the control strategy is avoided by command filtering, and the design difficulty of the control strategy is simplified. Considering the influence of convergence speed on the control performance, this paper incorporates finite-time control theory into the control strategy. This method guarantees that the system converges to the desired trajectory within a finite time.

(2) The unilateral dead zone nonlinearity is transformed into a matched disturbance term and an unmatched disturbance term through a linear transformation. This transformation enhances the controllability of the system. The unknown disturbance terms in the system are estimated to ensure the stability of the system under the dead zone disturbance.

2. System Modeling and Problem Description

The EHB system, an electronic control braking system, comprises essential components such as a power source, a sensor, a brake pedal, a gear transmission mechanism, and a brake master cylinder, as depicted in Figure 1. In contrast to traditional braking systems, the EHB system no longer relies solely on the mechanical connection to realize the transmission of braking force. Instead, it utilizes electronic components to replace some of the mechanical components in the traditional braking system. The hydraulic unit is electronically controlled to manage and allocate braking force effectively. Central to the EHB system is the gear transmission servo system, encompassing the power source and gear transmission mechanism. This system converts the motor’s high-speed rotational motion into low-speed axial translational motion, utilizing gear ratios to amplify torque. However, there are inevitably dead zones between the gears that prevent perfect meshing.

Figure 2 shows the schematic diagram of the unilateral dead zone in Figure 1. Assuming that a is the initial moment of the steady state, at this time, the driving and driven gears are in full contact at the left meshing point; if, at the moment of b, the driving gear begins to decelerate or change direction, the speed difference between the driving and driven gears is generated, and the driving and driven gears are separated and cross the tooth gap; at this time, the driven gear is not under the control of the driving gear, and it begins to move freely; when the driving gear crosses the tooth gap, the positional relationship of the driving and driven gears is shown as c, and the two of them are in full contact at the right meshing point. Therefore, the presence of the dead zone nonlinearity within the gear transmission mechanism poses challenges for precise position tracking control, thereby impacting the braking performance of the EHB system.

The gear transmission mechanism of the EHB system serves as a classic example of a sandwich system, which can be treated as two interconnected subsystems. Considering that the moment of inertia of the two subsystems is highly coincident with the physical reality, the unilateral dead zone nonlinearity under investigation in this paper is mathematically represented by (1).

$$\left\{\begin{array}{l}{J}_l{\ddot{\theta }}_l+C_l{\dot{\theta }}_l=N_{0}D\left[\theta \right]\\ J_m{\ddot{\theta }}_m+C_m{\dot{\theta }}_m=u-D\left[\theta \right],\end{array}\right.$$

(1)

where $J_l$, ${\theta }_l$, and $C_l$ are the moment of inertia and position and viscous friction coefficient of the load end; $J_m$, ${\theta }_m$, and $C_m$ are the moment of inertia and position and viscous friction coefficient of the drive end; $N_0$ is the transmission ratio; $u$ is the input torque; and $\theta ={\theta }_m-N_0{\theta }_l$ is the relative displacement.

In (1), $D\left[\theta \right]$ symbolizes the transmission torque between the load end and the drive end, as illustrated in Figure 3. The mathematical model of $D\left[\theta \right]$ is expressed as

$$D\left[\theta \right]=\left\{\begin{array}{l}h\left(\theta -\alpha \right), \theta \ge \alpha \\ 0, 0<\theta <\alpha , \end{array}\right.$$

(2)

where $h>0$ is the rigidity coefficient and $\alpha >0$ is the backlash width.

Remark 1.

The study in this paper refers to the unilateral dead zone nonlinearity between the time the driver depresses the brake pedal and the end of braking, excluding the process of pedal reset. When the EHB system starts braking, the gear transmission mechanism rotates from the starting point to one side to transfer the torque. When the EHB system ends braking, the gear drive mechanism returns to the starting point in the original path. Therefore, the dead zone in the EHB gear drive mechanism is one-sided.

Remark 2.

In the gear transmission mechanism, all parameters, including $J_l$, ${\theta }_l$, $C_l$, $J_m$, ${\theta }_m$, $C_m$, $h$, $\alpha$, and so forth, are considered unknown. Building upon this premise, the primary control objective of this study is to achieve motion trajectory tracking at the load end, ensuring alignment with the desired trajectory while guaranteeing the overall stability of the system.

To address the unilateral dead zone nonlinearity in the gear transmission mechanism, we undertake a straightforward mathematical transformation of (2) as follows:

$$D\left[\theta \right]=h\theta +d,$$

(3)

where

$$d=\left\{\begin{array}{l}-h\alpha , \theta \ge \alpha \\ -h\theta , 0<\theta <\alpha .\end{array}\right.$$

(4)

We can obtain

$$\left|d\right|\le h\alpha .$$

(5)

Let $x_1={\theta }_l$, $x_2={\dot{\theta }}_l$, $x_3={\theta }_m$, and $x_4={\dot{\theta }}_m$. By substituting (2) and (3) into (1), we derive the following relationship:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=a_1{x}_1+a_2{x}_2+a_3{x}_3+\Delta d_1\\ {\dot{x}}_3=x_4\\ {\dot{x}}_4=b_1{x}_1+b_2{x}_3+b_3{x}_4+b_{4}u+\Delta d_2,\end{array}\right.$$

(6)

where $a_1=-N_0^{2}h/J_l$, $a_2=-C_l/J_l$, $a_3=N_{0}h/J_l$, $b_1=N_{0}h/J_m$, $b_2=-h/J_m$, $b_3=-C_m/J_m$, and $b_4=1/J_m$.

The following relationship can be deduced from (6):

$$\left|\Delta d_1\right|\le {\displaystyle \frac{N_{0}h\alpha }{J_l}}=D_1,$$

(7)

$$\left|\Delta d_2\right|\le {\displaystyle \frac{h\alpha }{J_m}}=D_2,$$

(8)

where $D_1$ and $D_2$ are unknown bounded disturbances.

Remark 3.

The operating conditions of the gear transmission mechanism. Throughout its operation, factors such as friction, gear collisions, and temperature fluctuations produce variations in parameters $J_l$, ${\theta }_l$, $C_l$, $J_m$, ${\theta }_m$, $C_m$, and h, consequently influencing the width α of the unilateral dead zone nonlinearity in the gear transmission mechanism. Hence, we consider parameters$a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$, and$b_4$ in (6) as unknown variables.

3. Controller Design and Stability Analysis

Based on command filtering technology, a finite-time adaptive control strategy is proposed in this chapter to solve the unilateral dead zone nonlinearity in the gear transmission mechanism. The combination of adaptive control with command filtering technology and finite-time control theory effectively avoids the issue of “complexity explosion” in traditional adaptive control and enhances the convergence speed of the system. This method enhances the overall control performance of the EHB system. The control block diagram of the gear transmission mechanism is depicted in Figure 4.

3.1. Design of the Controller

Before designing the controller, we conducted the following coordinate transformations:

$$z_i=x_i-{\overline{\alpha }}_i, i=1,2,3,4$$

(9)

where ${\overline{\alpha }}_1=y_r$ is the desired position signal and ${\overline{\alpha }}_i$ is the output signal and satisfies the following relationship:

$${\epsilon }_i{\dot{\overline{\alpha }}}_i+{\overline{\alpha }}_i={\alpha }_i,{\overline{\alpha }}_i\left(0\right)={\alpha }_i\left(0\right), i=1,2,3,4$$

(10)

where ${\epsilon }_i>0$ and ${\alpha }_i$ are virtual control signals.

Next, the design process of the proposed control strategy is introduced.

Step 1: It can be derived from (9) that

$$z_1=x_1-{\overline{\alpha }}_1=x_1-y_r.$$

(11)

The derivation of (11) is obtained:

$${\dot{z}}_1=x_2-{\dot{\overline{\alpha }}}_1=z_2+{\overline{\alpha }}_2-{\alpha }_2+{\alpha }_2-y_r.$$

(12)

Define the compensation signal ${\xi }_1$ as follows:

$${\dot{\xi }}_1=-k_1{\xi }_1+{\xi }_2+\left({\overline{\alpha }}_2-{\alpha }_2\right)-l_{1}sign\left({\xi }_1\right),$$

(13)

where $k_1>0$ and $l_1>0$.

Define the tracking error signal after command filtering compensation as follows:

$${\chi }_1=z_1-{\xi }_1,$$

(14)

$${\chi }_2=z_2-{\xi }_2.$$

(15)

According to (12), (13), and (14),

$${\dot{\chi }}_1={\chi }_2+{\alpha }_2+k_1{\xi }_1-{\dot{y}}_r+l_{1}sign\left({\xi }_1\right).$$

(16)

Construct the Lyapunov function as

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

(17)

The derivation of (17) is obtained:

$${\dot{V}}_1={\chi }_1{\chi }_2+{\chi }_1{\alpha }_2+{\chi }_1{k}_1{\xi }_1-{\chi }_1{\dot{y}}_r+{\chi }_1{l}_{1}sign\left({\xi }_1\right).$$

(18)

To ensure the stability of the first subsystem, the virtual control signal ${\alpha }_2$ is defined as

$${\alpha }_2=-k_1{z}_1+{\dot{y}}_r-s_1{\chi }_1^r.$$

(19)

Substitute (19) into (18) to obtain

$${\dot{V}}_1=-k_1{\chi }_1^2+{\chi }_1{\chi }_2-s_1{\chi }_1^{r+1}+{\chi }_1{l}_{1}sign\left({\xi }_1\right).$$

(20)

Step 2: It can be derived from (9) that

$$z_2=x_2-{\overline{\alpha }}_2.$$

(21)

The derivation of (21) is obtained:

$$\begin{array}{lll}{\dot{z}}_2& =& {\dot{x}}_2-{\dot{\overline{\alpha }}}_2\\ & =& a_1{x}_1+a_2{x}_2+a_3{x}_3+\Delta d_1-{\dot{\overline{\alpha }}}_2\\ & =& a_1{x}_1+a_2{x}_2+a_3\left(z_3+{\overline{\alpha }}_3-{\alpha }_3+{\alpha }_3\right)+\Delta d_1-{\dot{\overline{\alpha }}}_2.\end{array}$$

(22)

Define the compensation signal ${\xi }_2$ as follows:

$${\dot{\xi }}_2=-k_2{\xi }_2+a_3\left({\overline{\alpha }}_3-{\alpha }_3+{\xi }_3\right)-{\xi }_1-l_{2}sign\left({\xi }_2\right),$$

(23)

where $k_2>0$ and $l_2>0$.

Define the tracking error signal after command filtering compensation as follows:

$${\chi }_2=z_2-{\xi }_2,$$

(24)

$${\chi }_3=z_3-{\xi }_3.$$

(25)

According to (22), (23), and (24),

$${\dot{\chi }}_2=a_1{x}_1+a_2{x}_2+a_3{\chi }_3+a_3{\alpha }_3+\Delta d_1-{\dot{\overline{\alpha }}}_2+k_2{\xi }_2+{\xi }_1+l_{2}sign\left({\xi }_2\right).$$

(26)

Construct the Lyapunov function as

$$V_2=V_1+{\displaystyle \frac{1}{2}}{\chi }_2^2+{\displaystyle \frac{1}{2\gamma }}{\tilde{D}}_1^2.$$

(27)

where $\gamma >0$, ${\tilde{D}}_1=D_1-{\widehat{D}}_1$, ${\widehat{D}}_1$ is the estimate of $D_1$, and $\left|\Delta d_1\right|<D_1$.

The derivation of (27) is obtained as follows:

$$\begin{array}{ll}{\dot{V}}_2& ={\dot{V}}_1+{\chi }_2{a}_1{x}_1+{\chi }_2{a}_2{x}_2+{\chi }_2{a}_3{\chi }_3+{\chi }_2{a}_3{\alpha }_3+{\chi }_2\Delta d_1\\ & -{\chi }_2{\dot{\overline{\alpha }}}_2+{\chi }_2{k}_2{\xi }_2+{\chi }_2{\xi }_1+{\chi }_2{l}_{2}sign\left({\xi }_2\right)-{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1{\dot{\widehat{D}}}_1.\end{array}$$

(28)

To ensure the stability of the first subsystem, the virtual control signal ${\alpha }_3$ is defined as

$${\alpha }_3={\displaystyle \frac{1}{a_3}}\left(-a_1{x}_1-a_2{x}_2-\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right){\widehat{D}}_1+{\dot{\overline{\alpha }}}_2-k_2{z}_2-z_1-s_2{\chi }_2^r\right).$$

(29)

where $c_2>0$.

Substitute (29) into (28) to obtain

$$\begin{array}{ll}{\dot{V}}_2& =-k_1{\chi }_1^2-k_2{\chi }_2^2+a_3{\chi }_2{\chi }_3+{\chi }_2\Delta d_1-{\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)\left(D_1-{\tilde{D}}_1\right)-{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1{\dot{\widehat{D}}}_1\\ & -s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}+{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right).\\ & \le -k_1{\chi }_1^2-k_2{\chi }_2^2+a_3{\chi }_2{\chi }_3+\left[\left|{\chi }_2\right|-{\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)\right]D_1+{\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right){\tilde{D}}_1\\ & -{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1{\dot{\widehat{D}}}_1-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}+{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right)\\ & =-k_1{\chi }_1^2-k_2{\chi }_2^2+a_3{\chi }_2{\chi }_3-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}+{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right)\\ & +0{.2785c}_2{D}_1+{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1\left(-{\dot{\widehat{D}}}_1+\gamma {\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)\right).\end{array}$$

(30)

For any $c>0$ and $c\in R$, the following relation holds:

$$0\le \left|\epsilon \right|-\epsilon \tanh \left({\displaystyle \frac{\epsilon }{c}}\right)\le lc.$$

(31)

We can obtain $k=e^{-(k+1)}$ after a simple calculation.

Step 3: It can be derived from (9) that

$$z_3=x_3-{\overline{\alpha }}_3.$$

(32)

The derivation of (32) is obtained:

$${\dot{z}}_3=x_4-{\dot{\overline{\alpha }}}_3=z_4+{\overline{\alpha }}_4-{\alpha }_4+{\alpha }_4-{\dot{\overline{\alpha }}}_3.$$

(33)

Define the compensation signal ${\xi }_3$ as follows:

$${\dot{\xi }}_3=-k_3{\xi }_3+{\overline{\alpha }}_4-{\alpha }_4+{\xi }_4-a_3{\xi }_2-l_{3}sign\left({\xi }_3\right),$$

(34)

where $k_3>0$ and $l_3>0$.

Define the tracking error signal after command filtering compensation as follows:

$${\chi }_3=z_3-{\xi }_3,$$

(35)

$${\chi }_4=z_4-{\xi }_4.$$

(36)

According to (33), (34), and (35),

$${\dot{\chi }}_3={\chi }_4+{\alpha }_4-{\dot{\overline{\alpha }}}_3+k_3{\xi }_3+a_3{\xi }_2+l_{3}sign\left({\xi }_3\right).$$

(37)

Construct the Lyapunov function as

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

(38)

The derivation of (38) is obtained:

$$\begin{array}{ll}{\dot{V}}_3& ={\dot{V}}_2+{\chi }_3{\chi }_4+{\chi }_3{\alpha }_4-{\chi }_3{\dot{\overline{\alpha }}}_3\\ & +{\chi }_3{k}_3{\xi }_3-{\chi }_3{\xi }_4+{\chi }_3{a}_3{\xi }_2+{\chi }_3{l}_{3}sign\left({\xi }_3\right).\end{array}$$

(39)

To ensure the stability of the third subsystem, the virtual control signal ${\alpha }_4$ is defined as

$${\alpha }_4=-k_3{z}_3+{\dot{\overline{\alpha }}}_3-a_3{z}_2-s_3{\chi }_3^r.$$

(40)

Substitute (40) into (39) to obtain

$$\begin{array}{ll}{\dot{V}}_3& =-k_1{\chi }_1^2-k_2{\chi }_2^2-k_3{\chi }_3^2+{\chi }_3{\chi }_4-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}\\ & -s_3{\chi }_3^{r+1}+{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right)+{\chi }_3{l}_{3}sign\left({\xi }_3\right)\\ & +0.2785c_2{D}_1+{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1\left(-{\dot{\widehat{D}}}_1+\gamma {\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)\right).\end{array}$$

(41)

Step 4: It can be derived from (9) that

$$z_4=x_4-{\overline{\alpha }}_4.$$

(42)

The derivation of (42) is obtained:

$${\dot{z}}_4={\dot{x}}_4-{\dot{\overline{\alpha }}}_4=b_1{x}_1+b_2{x}_3+b_3{x}_4+b_{4}u+\Delta d_2-{\dot{\overline{\alpha }}}_4.$$

(43)

Define the compensation signal ${\xi }_4$ as follows:

$${\dot{\xi }}_4=-k_4{\xi }_4-{\xi }_3-l_{4}sign\left({\xi }_4\right),$$

(44)

where $k_4>0$ and $l_4>0$.

Define the tracking error signal after command filtering compensation as follows:

$${\chi }_4=z_4-{\xi }_4.$$

(45)

According to (43), (44), and (45),

$$\begin{array}{ll}{\dot{\chi }}_4& =b_1{x}_1+b_2{x}_3+b_3{x}_4+b_{4}u+\Delta d_2\\ & -{\dot{\overline{\alpha }}}_4+k_4{\xi }_4+{\xi }_3+l_{4}sign\left({\xi }_4\right).\end{array}$$

(46)

Construct the Lyapunov function as

$$V_4=V_3+{\displaystyle \frac{1}{2}}{\chi }_4^2+{\displaystyle \frac{1}{2\gamma }}{\tilde{D}}_2^2.$$

(47)

where $\gamma >0$, ${\tilde{D}}_2=D_2-{\widehat{D}}_2$, ${\widehat{D}}_2$ is the estimate of $D_2$, and $\left|\Delta d_2\right|<D_2$.

The derivation of (47) is obtained:

$$\begin{array}{ll}{\dot{V}}_4& ={\dot{V}}_3+{\chi }_4{b}_1{x}_1+{\chi }_4{b}_2{x}_3+{\chi }_4{b}_3{x}_4+{\chi }_4{b}_{4}u+{\chi }_4\Delta d_2\\ & -{\chi }_4{\dot{\overline{\alpha }}}_4+{\chi }_4{k}_4{\xi }_4+{\chi }_4{\xi }_3+{\chi }_4{l}_{4}sign\left({\xi }_4\right)-{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_2{\dot{\widehat{D}}}_2.\end{array}$$

(48)

From (48), the actual control signal $u$ is defined as

$$u={\displaystyle \frac{1}{b_4}}\left(-b_1{x}_1-b_2{x}_3-b_3{x}_4-\tanh \left({\displaystyle \frac{{\chi }_4}{c_4}}\right){\widehat{D}}_2+{\dot{\overline{\alpha }}}_4-k_4{z}_4-z_3-s_4{\chi }_4^r\right).$$

(49)

where $c_4>0$.

According to (49), we can design parameter updating laws as

$${\dot{\widehat{D}}}_1=\gamma {\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)-{\lambda }_1{\widehat{D}}_1,$$

(50)

$${\dot{\widehat{D}}}_2=\gamma {\chi }_4\tanh \left({\displaystyle \frac{{\chi }_4}{c_4}}\right)-{\lambda }_2{\widehat{D}}_2.$$

(51)

where ${\lambda }_1>0$ and ${\lambda }_2>0$.

The control signal block diagram of the control strategy proposed in this paper is shown in Figure 5.

3.2. Stability Analysis

Theorem 1.

According to the actual control signal (49); virtual control signals (19), (29), and (40); and parameter updating laws (50) and (51), we can obtain the following conclusions:

(1) All signals of the gear transmission system remain bounded;

(2) The tracking error of the system  ${\chi }_i=z_i-{\xi }_i$ converges to a compact set.

Lemma 1

([42]). If $V>0$ and there exist design constants, $\sigma >0$, $\beta >0$, $0<p<1$, and $0<b<\infty$ satisfy the following condition:

$$\dot{V}\le -\sigma V-\beta V^p+b,$$

(52)

Then, system  $\dot{x}=f\left(x,u\right)$ converges and achieves global stability within a finite time.

Proof. 

Substitute (49) into (48) to obtain

$$\begin{array}{ll}{\dot{V}}_4& =-k_1{\chi }_1^2-k_2{\chi }_2^2-k_3{\chi }_3^2-k_4{\chi }_4^2-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}-s_3{\chi }_3^{r+1}-s_4{\chi }_4^{r+1}\\ & +{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right)+{\chi }_3{l}_{3}sign\left({\xi }_3\right)+{\chi }_4{l}_{4}sign\left({\xi }_4\right)\\ & +0.2785c_2{D}_1+0.2785c_4{D}_2+{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_1\left(-{\dot{\widehat{D}}}_1+\gamma {\chi }_2\tanh \left({\displaystyle \frac{{\chi }_2}{c_2}}\right)\right)\\ & +{\displaystyle \frac{1}{\gamma }}{\tilde{D}}_2\left(-{\dot{\widehat{D}}}_2+\gamma {\chi }_4\tanh \left({\displaystyle \frac{{\chi }_4}{c_4}}\right)\right).\end{array}$$

(53)

Substitute (50) and (51) into (53) to obtain

$$\begin{array}{ll}{\dot{V}}_4& =-k_1{\chi }_1^2-k_2{\chi }_2^2-k_3{\chi }_3^2-k_4{\chi }_4^2-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}-s_3{\chi }_3^{r+1}-s_4{\chi }_4^{r+1}\\ & +{\chi }_1{l}_{1}sign\left({\xi }_1\right)+{\chi }_2{l}_{2}sign\left({\xi }_2\right)+{\chi }_3{l}_{3}sign\left({\xi }_3\right)+{\chi }_4{l}_{4}sign\left({\xi }_4\right)\\ & +0.2785c_2{D}_1+0.2785c_4{D}_2+{\displaystyle \frac{1}{\gamma }}{\lambda }_1{\tilde{D}}_1{\widehat{D}}_1+{\displaystyle \frac{1}{\gamma }}{\lambda }_2{\tilde{D}}_2{\widehat{D}}_2.\end{array}$$

(54)

According to Young’s inequality,

$$v_i{l}_{i}sign\left({\xi }_i\right)\le {\displaystyle \frac{l_i}{2}}{v}_i^2+{\displaystyle \frac{l_i}{2}}{\left[sign\left({\xi }_i\right)\right]}^2\le {\displaystyle \frac{l_i}{2}}{v}_i^2+{\displaystyle \frac{l_i}{2}}.$$

(55)

Further solving (54), the following is obtained:

$$\begin{array}{ll}{\dot{V}}_4& \le -k_1{\chi }_1^2-k_2{\chi }_2^2-k_3{\chi }_3^2-k_4{\chi }_4^2-s_1{\chi }_1^{r+1}-s_2{\chi }_2^{r+1}-s_3{\chi }_3^{r+1}-s_4{\chi }_4^{r+1}\\ & +{\displaystyle \frac{l_1}{2}}{\chi }_1^2+{\displaystyle \frac{l_1}{2}}+{\displaystyle \frac{l_2}{2}}{\chi }_2^2+{\displaystyle \frac{l_2}{2}}+{\displaystyle \frac{l_3}{2}}{\chi }_3^2+{\displaystyle \frac{l_3}{2}}+{\displaystyle \frac{l_4}{2}}{\chi }_4^2+{\displaystyle \frac{l_4}{2}}+0.2785c_2{D}_1\\ & +0.2785c_4{D}_2-{\displaystyle \frac{1}{2\gamma }}{\lambda }_1{\tilde{D}}_1^2+{\displaystyle \frac{1}{2\gamma }}{\lambda }_1{D}_1^2-{\displaystyle \frac{1}{2\gamma }}{\lambda }_2{\tilde{D}}_2^2+{\displaystyle \frac{1}{2\gamma }}{\lambda }_2{D}_2^2\\ & \le -\sigma V_4-\beta V_4^{{\displaystyle \frac{r+1}{2}}}+b,\end{array}$$

(56)

where

$$\sigma =\min \left\{2k_1-l_1,2k_2-l_2,2k_3-l_3,2k_4-l_4,{\lambda }_1,{\lambda }_2\right\},$$

(57)

$$\beta =\min \left\{s_1\cdot 2^{{\displaystyle \frac{r+1}{2}}},s_2\cdot 2^{{\displaystyle \frac{r+1}{2}}},s_3\cdot 2^{{\displaystyle \frac{r+1}{2}}},s_4\cdot 2^{{\displaystyle \frac{r+1}{2}}},\right\},$$

(58)

$$b={\displaystyle \frac{l_1}{2}}+{\displaystyle \frac{l_2}{2}}+{\displaystyle \frac{l_3}{2}}+{\displaystyle \frac{l_4}{2}}+0.2785c_2{D}_1+0.2785c_4{D}_2+{\displaystyle \frac{1}{2\gamma }}{\lambda }_1{D}_1^2+{\displaystyle \frac{1}{2\gamma }}{\lambda }_2{D}_2^2,$$

(59)

and the following relationship is used in (56):

$$\tilde{P}\widehat{P}=\tilde{P}\left(P-\tilde{P}\right)=-{\tilde{P}}^2+\tilde{P}P\le -{\tilde{P}}^2+{\displaystyle \frac{1}{2}}{\tilde{P}}^2+{\displaystyle \frac{1}{2}}{P}^2\le -{\displaystyle \frac{1}{2}}{\tilde{P}}^2+{\displaystyle \frac{1}{2}}{P}^2.$$

(60)

□

Based on Lemma 1, we can obtain the gradual convergence of the motion trajectory of the controlled system within finite time as $V_4$ approaches

$$V^p\le b/\left(1-\eta \right)\beta .$$

(61)

Hence, the motion trajectory of the controlled system is bounded in finite time as follows:

$$‖{\chi }_i‖\le \sqrt{2{\left({\displaystyle \frac{b}{\left(1-\lambda \right)\beta }}\right)}^{{\displaystyle \frac{1}{p}}}}.$$

(62)

The maximum settling time is as follows:

$$T_1\le {\displaystyle \frac{1}{\sigma \left(1-p\right)}}\ln {\displaystyle \frac{\sigma V^{1-p}\left(x_0\right)+\eta \beta }{\eta \beta }}.$$

(63)

It can be seen from ${\chi }_i=z_i-{\xi }_i$ that if ${\xi }_i$ is convergent in finite time, then $z_i$ is also convergent in finite time.

Choosing the Lyapunov function

$$V_5={\displaystyle \sum _{i=1}^4\frac{1}{2}{\xi }_i^2},$$

(64)

we obtain

$$\begin{array}{lll}{\dot{V}}_5& =& {\xi }_1{\dot{\xi }}_1+{\xi }_2{\dot{\xi }}_2+{\xi }_3{\dot{\xi }}_3+{\xi }_4{\dot{\xi }}_4\\ & =& -k_1{{\xi }_1}^2+{\xi }_1{\xi }_2+{\xi }_1\left({\overline{\alpha }}_2-{\alpha }_2\right)-{\xi }_1{l}_{1}sign\left({\xi }_1\right)\\ & & -k_2{{\xi }_2}^2+a_3{\xi }_2\left({\overline{\alpha }}_3-{\alpha }_3\right)+a_3{\xi }_2{\xi }_3-{\xi }_1{\xi }_2-{\xi }_2{l}_{2}sign\left({\xi }_2\right)\\ & & -k_3{{\xi }_3}^2+{\xi }_3\left({\overline{\alpha }}_4-{\alpha }_4\right)+{\xi }_3{\xi }_4-a_3{\xi }_2{\xi }_3-{\xi }_3{l}_{3}sign\left({\xi }_3\right)\\ & & -k_4{{\xi }_4}^2-{\xi }_3{\xi }_4-{\xi }_4{l}_{4}sign\left({\xi }_4\right)\\ & =& -{\displaystyle \sum _{i=1}^4{k}_i}{{\xi }_i}^2-{\displaystyle \sum _{i=1}^3{l}_i}\left|{\xi }_i\right|+{\xi }_1\left({\overline{\alpha }}_2-{\alpha }_2\right)+a_3{\xi }_2\left({\overline{\alpha }}_3-{\alpha }_3\right)+{\xi }_3\left({\overline{\alpha }}_4-{\alpha }_4\right).\end{array}$$

(65)

According to the Lemma in [43], it can achieve $‖{\overline{\alpha }}_{i+1}-{\alpha }_{i+1}‖<{\eta }_i$ (${\eta }_i$ is a known constant) in finite time $T_2$. Therefore, by selecting appropriate parameters that satisfy $l_i>{\eta }_i$, the following inequality holds:

$$\begin{array}{ll}{\dot{V}}_5& \le -{\displaystyle \sum _{i-1}^4{k}_i}{{\xi }_i}^2-\left(l_1-{\eta }_1\right)\left|{\xi }_1\right|-\left(l_2-a_3{\eta }_2\right)\left|{\xi }_2\right|-\left(l_3-{\eta }_3\right)\left|{\xi }_3\right|-l_4{\xi }_4\\ & \le -K_m{V}_5-d{V}_5^{{\displaystyle \frac{1}{2}}},\end{array}$$

(66)

where $K_m=2\min \left\{k_i\right\}$ and $d=\sqrt{2}\min \left\{l_1-{\eta }_1,l_2-a_3{\eta }_2,l_3-{\eta }_3\right\}$. From Lemma 1, we can conclude that ${\xi }_i$ converges to the origin in finite time $T_3$. Therefore, $z_i$ is finitely stable in $T=T_1+T_2+T_3$.

Therefore, from the above analysis, it can be deduced that ${\chi }_1$, ${\chi }_2$, ${\chi }_3$, ${\chi }_4$, ${\tilde{D}}_1$, and ${\tilde{D}}_2$ are bounded. Considering the boundedness of ${\chi }_1$ and (14), we can conclude that $z_1$ and ${\xi }_1$ are bounded. Similarly, based on the boundedness of ${\chi }_2$ and (24), we can infer that $z_2$ and ${\xi }_2$ are bounded. Additionally, by observing the boundedness of ${\chi }_3$ and (35), we can deduce that $z_3$ and ${\xi }_3$ are bounded. Furthermore, from the boundedness of ${\chi }_4$ and (45), it follows that $z_4$ and ${\xi }_4$ are bounded. Moreover, with the boundedness of $x_1$, $x_2$, $x_3$, ${\tilde{D}}_2$, $z_3$, $z_4$, ${\chi }_4$, and (49), we can obtain that $u$ is bounded. Consequently, the controlled system can be inferred to be bounded.

4. Simulation Results

This paper proposes a finite-time adaptive control strategy. Although this control strategy introduces a power exponent term $-s_i{\chi }_i^r$ based on the finite-time control theory, the design steps and complexity are essentially the same as those of the control strategy without finite-time control theory.

To verify the feasibility and availability of the finite-time adaptive control strategy, simulation experiments on the control strategy are conducted within the 2018b MATLAB/Simulink environment in this section. In simulation experiments, we established the simulation model of the gear transmission mechanism and configured the simulation parameters as shown in

Table 1. Parameters of simulation experiments.

Symbol	Parameters	Value	Units
$J_l$	Moment of inertia of the load end	0.5	kg·m2
$J_m$	Moment of inertia of the driving end	0.01	kg·m2
$C_l$	Viscous friction coefficient of the load end	0.12	Nm/rad
$C_m$	Viscous friction coefficient of the driving end	0.1	Nm/rad
$N_0$	Transmission ratio	5	Nm/rad
$h$	Rigidity coefficient	0.2	null

.

In experiment 1, the desired trajectory adopts a slope function with the slope and rise time set to 1 and 1 s. Once the maximum value of 1 rad is attained, the function value remains constant, and the simulation continues until 5 s. In experiment 2, the desired trajectory adopts a sinusoidal absolute value function with the amplitude and frequency set to 2 rad and 2.5 rad/s. The simulation runs until 5 s. In experiment 3, the desired trajectory adopts a step function. The simulation runs until 15 s.

Remark 4.

To validate the feasibility and availability of the finite-time adaptive control strategy proposed in this paper, we conduct a comparative analysis with the control strategy without finite-time control theory, denoted by “*”.

• Experiment 1: The desired trajectory in experiment 1 is shown in (61).

  $$x_d=t.$$

  (67)

The simulation results are depicted in Figure 6. Observing Figure 6a, it is evident that the tracking performance of both strategies is comparable and meets the design requirements. However, as can be seen from the local magnification diagram in Figure 6a, the tracking performance of the control strategy in this paper is better and can converge in finite time to 1. Figure 6b illustrates the tracking error of $x_1$ and $x_1^{\ast }$. The tracking error of $x_1$ is approximately −0.002~0.002 rad, which is roughly 2/3 smaller than $x_1^{\ast }$. Moreover, the former can rapidly converge to 0 in a finite time.

Figure 6c illustrates the tracking performance of $x_2$ and $x_2^{\ast }$. Due to the turning points of the slope function at 0 s and 1 s, the tracking performance of $x_2$ and $x_2^{\ast }$ is adversely affected. However, as can be seen from the local magnification diagram in Figure 6c, it is evident that $x_2$ can converge to the steady state within a finite time, showing better tracking performance. In contrast, $x_2^{\ast }$ shows significant fluctuations at the turning points, leading to less effective tracking and longer convergence times. The tracking error curves of $x_2$ and $x_2^{\ast }$ in Figure 6d are synchronized with the tracking performance curves of $x_2$ and $x_2^{\ast }$ in Figure 6c. Notably, the former displays a smaller tracking error and faster convergence speed.

In Figure 6e, the comparison of compensation signals ${\xi }_1$ and ${\xi }_1^{\ast }$ is presented. Through analysis, it can be found that the trends in the compensation signals ${\xi }_1$ and ${\xi }_1^{\ast }$ closely mirror those of Figure 6b,d. However, ${\xi }_1$ shows fewer fluctuations at the turning points, a smaller fluctuation range, and the ability to converge to the desired value in a finite time. Figure 6f shows a comparison of the command filtering output signals ${\overline{\alpha }}_2$ and ${\overline{\alpha }}_2^{\ast }$. It can be seen that the output signal ${\overline{\alpha }}_2$ of the control strategy with finite-time control theory is more stable.

• Experiment 2: The desired trajectory in experiment 2 is shown in (62).

  $$x_d=\left|2\sin \left(2.5t\right)\right|.$$

  (68)

The simulation results are depicted in Figure 7. Figure 7a indicates that both tracking performances are nearly identical, meeting the design requirements. However, the local magnification diagram in Figure 7a reveals that the motion trajectory of the control strategy with finite-time control theory aligns more closely with the desired trajectory $x_d$. Figure 7b presents a comparison of the tracking error of $x_1$ and $x_1^{\ast }$. It is evident from Figure 7b that when the rotation angle of the load end deviates from 0, the error of both $x_1$ and $x_1^{\ast }$ remains close to 0. However, when the rotation angle of the load end is 0, the error range of $x_1$ and $x_1^{\ast }$ is about −0.018~0 rad and −0.063~0.025 rad, respectively. In comparison to the control strategy without finite-time control theory, this control strategy demonstrates a noteworthy improvement, with the error fluctuation range reduced by approximately 4/5.

The tracking performance curves of $x_2$ and $x_2^{\ast }$ are illustrated in Figure 7c. Compared with $x_2^{\ast }$, $x_2$ has a better tracking performance and shorter convergence time when the rotation angle of the load end is near 0. The tracking error curves of $x_2$ and $x_2^{\ast }$ are depicted in Figure 7d. These curves change as $x_2$ and $x_2^{\ast }$ in Figure 7c change. From the above analysis, the control strategy proposed exhibits smaller tracking errors and faster convergence, demonstrating its availability in enhancing tracking performance.

Figure 7e shows a comparison of the compensation signals ${\xi }_1$ and ${\xi }_1^{\ast }$. It is evident that the compensation signal ${\xi }_1^{\ast }$ has an obvious fluctuation and the fluctuation range is large. In comparison to ${\xi }_1^{\ast }$, ${\xi }_1$ is more in line with the design requirements. Figure 7f shows a comparison of the command filtering output signals ${\overline{\alpha }}_2$ and ${\overline{\alpha }}_2^{\ast }$. ${\overline{\alpha }}_2$ and ${\overline{\alpha }}_2^{\ast }$ are basically consistent with the trend shown in Figure 7c.

• Experiment 3: The desired trajectory in experiment 3 is a step function. The value of the function in 0~0.5 s is 0, the value of the function in 0.5~1 s is 0.2, and the value of the function in 1~1.5 s is 0.1. The period is set to 1.5 s and ends after 5 s.

The simulation results are depicted in Figure 8. Figure 8a indicates the tracking performance of both. The local magnification diagram in Figure 8a reveals that the motion trajectory of the control strategy with finite-time control theory is better with a faster response. Figure 8b presents a comparison of the tracking error of $x_1$ and $x_1^{\ast }$. It is evident from Figure 8b that the error of $x_1$ is more consistent with the design than the error of $x_1^{\ast }$.

The tracking performance curves of $x_2$ and $x_2^{\ast }$ are illustrated in Figure 8c. As we can see from the comparison of $x_2$ and $x_2^{\ast }$, $x_2$ has faster convergence and desirable tracking effects when the rotation angle of the load end is near 0. Figure 8d illustrates the tracking error curve of Figure 8c. The trends of $x_2$ and $x_2^{\ast }$ in Figure 8d resemble those in Figure 8b, with $x_2$ exhibiting superior convergence. In conclusion, the control strategy proposed in this paper effectively mitigates the impact of unilateral dead zone nonlinearity in the EHB gear mechanism.

In Figure 8e, the comparison of compensation signals ${\xi }_1$ and ${\xi }_1^{\ast }$ is presented. By comparing ${\xi }_1$ with ${\xi }_1^{\ast }$, we find that compensation signal ${\xi }_1$ is the most appropriate for the design of the controller. Figure 8f presents a comparison of the command filtering output signals ${\overline{\alpha }}_2$ and ${\overline{\alpha }}_2^{\ast }$. The two are almost identical except for some differences in fluctuations.

From the analysis of the three simulation experiments above, we observe that the state variables $x_1$, $x_1^{\ast }$, $x_2$, and $x_2^{\ast }$ and the tracking errors $z_1$, $z_1^{\ast }$, $z_2$, and $z_2^{\ast }$ all converge to their expected values within a finite time. However, the control strategy with finite-time control theory demonstrates faster convergence, a reduced fluctuation range, and superior tracking performance. Hence, the control strategy with finite-time control theory proposed in this paper for addressing the unilateral dead zone nonlinearity in the gear transmission mechanism effectively enhances convergence speed and elevates tracking performance.

5. HIL Experiments and Results

In the previous chapters, we design a control strategy to address the unilateral dead zone nonlinearity in the gear transmission mechanism and conduct a stability analysis of the proposed approach. Then, the control strategy is validated through MATLAB/Simulink simulations. Considering the differences between the actual engineering and the simulation results, we perform pertinent HIL experiments on the EHB system test platform to further validate the feasibility and availability of the control strategy.

The EHB system test platform used in this chapter is depicted in Figure 9, comprising the upper computer, the lower computer NI-PXI, the monitor, and the EHB system. Given the direct connection of the drive end of the gear transmission mechanism to the PMSM, angle information of the PMSM is obtained via an angle sensor to derive the angle information of the drive end. Finally, utilizing the transmission ratio relation of the gear transmission mechanism, the angle information of the load end is computed. Based on this transformational process, we conducted two HIL experiments to verify the feasibility and availability of the control strategy.

In the first HIL experiment, we adopt the slope function as the expected trajectory. The tracking performance of $x_1$ and $x_1^{\ast }$, the tracking error of $x_1$ and $x_1^{\ast }$, and the master-cylinder push rod displacement are shown in Figure 10a–c, respectively. As depicted in Figure 10a, compared with the tracking trajectory without finite-time control, the control strategy proposed in this paper is closer to the desired trajectory. When 1 s is reached, the control strategy in this paper can converge to 1 rad in a short time, which is not available for the control strategy without finite-time control. Therefore, the control strategy proposed successfully tracks the desired trajectory within a finite time, exhibiting superior tracking performance. It can be seen from Figure 10b that the errors of $x_1$ and $x_1^{\ast }$ within the range of 0~1 s are 0~0.8 rad and 0~0.16 rad, respectively, converging to the desired values after 1 s, with the error stabilizing near 0. However, the error of $x_1$ converges to 0 faster, the error is smaller, and the control performance is better. Figure 10c illustrates the displacement of the master-cylinder push rod after $x_1$ reaches the desired value of 1 rad, with a peak value of about 0.75 mm.

In the second HIL experiment, we adopt the sinusoidal absolute value function as the expected trajectory. The tracking performance of $x_1$ and $x_1^{\ast }$, tracking error of $x_1$ and $x_1^{\ast }$, and the master-cylinder push rod displacement are shown in Figure 11a, Figure 11b, and Figure 11c, respectively. As can be seen from the local magnification diagram shown in Figure 11a, the convergence speed and tracking accuracy of the control strategy proposed in this paper are better than those of the control strategy without finite-time control theory at the corner. From Figure 11b, the error in $x_1$ fluctuates between −0.06 and ~0.102 rad, while the error in $x_1^{\ast }$ fluctuates between −0.2 and ~0.225 rad. It is evident that the error fluctuation range of the former is about 3/5 smaller. The master-cylinder push rod displacement is illustrated in Figure 11c. The change trend of the master-cylinder push rod displacement is similar to that in Figure 11a, and the peak value is about 1.5 mm.

The above two HIL experiments demonstrate that the control strategy presented in this paper can rapidly converge system state variables and errors under different working conditions. In conclusion, the finite-time adaptive control strategy based on command filtering designed in this paper proves to be both feasible and effective for addressing the unilateral dead zone nonlinearity position tracking of the gear transmission mechanism. This strategy adequately compensates for the impact of dead zone nonlinearity, enhancing the tracking accuracy of the system.

6. Conclusions

In this paper, the unilateral dead zone nonlinearity problem of the gear transmission mechanism is described, and the unilateral dead zone model is established. Secondly, utilizing command filtering to avoid the issue of “complexity explosion” in the finite-time adaptive control strategy design simplifies the controller design process. Then, the finite-time stability analysis of the control strategy is carried out, which proves that the system is convergent and globally stable in finite time. Finally, we have carried out relevant simulation experiments, and the results show that the control strategy in this paper effectively compensates for the influence of unilateral dead zone nonlinearity on the EHB gear transmission servo system, which proves its feasibility and availability.

In the future, we will continue to use more advanced sensors to collect information for better feedback control. To illustrate the practicality of the control strategy designed in this paper, we will conduct accurate vehicle tests under various working conditions.