Issue 
Mechanics & Industry
Volume 25, 2024



Article Number  8  
Number of page(s)  12  
DOI  https://doi.org/10.1051/meca/2024006  
Published online  06 March 2024 
Regular Article
Precise and coordinated gearshift control for AMT gearshift system equipped with two linear actuators
^{1}
School of Aeronautical Engineering, Taizhou University, Taizhou, China
^{2}
School of Transportation and Vehicle Engineering, Shandong University of Technology, 266 West Xincun Road, Zibo, 255000, PR China
^{3}
School of Mechanical Engineering and Automation, Zhejiang SciTech University, Hangzhou, China
^{*} email: 441421855@qq.com
Received:
25
August
2022
Accepted:
29
January
2024
A type of automated manual transmission gearshift system that uses two linear actuators to perform gearshift events is presented. The shifting mechanism can be made simpler by using linear actuator to drive the shift fork directly. The mathematical model of the actuator is built and analyzed. Coordinated control of the engagement and disengagement processes is created based on a detailed analysis of the gearshift process in order to shorten the shift time. The force characteristics test reveals the nonlinear output characteristics. The precise displacement control need for coordinated control and stable control requirement for the shifting operation are satisfied by a linear extended state observer based controller. To verify the novel gearshift system and the control method, a test bench and control system are constructed. The results of the simulations indicate that the displacement control's accuracy is enhanced. As the speed difference reaches 500 r/min, the testing results demonstrate a noticeably reduced of shift force fluctuation and jerk generated during the gearshift process, from 20 m/s^{3} to 3.1 m/s^{3}. With the improved gearshift system and control method, the gearshift coordinated control can also save 25 ms of shift time.
Key words: Automated manual transmission / linear actuator / linear extended state observer / coordinated control / gearshift
© S. Lin et al., Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Multiple DC motors are typically used by the traditional automated manual transmission (AMT) gearshift mechanism to carry out gear change events [1]. Before power is sent to the shift rail and shift fork, further intermediate devices are needed to convert the rotational action of the DC motor into linear motion. Furthermore, the DC motors' output torque can be boosted through the employment of reduction gears. It is obvious that the structure of this kind of gearshift mechanism can be optimized. Furthermore, the hysteresis of the intermediate mechanisms limit the gearshift system's dynamic capability. Consequently, this study designs a type of direct drive gearshift system that makes use of two linear actuators. Thanks to its quick dynamic performance, simple structure, and linear movement, the linear actuator finds extensive use as an actuator in industrial applications, including machine tools, robotic drivers, electromagnetic valves, vehicle suspension, and linear compressors [2–5].
AMTs have several advantages over other automated transmission types, including high efficiency, low cost, simple design, and control criteria. The shift quality and power interruption issue are its drawbacks. When employed in an internal combustion engine vehicle (ICEV), coordinated management of the engine, AMT, and clutch is the most efficient approach to increase performance. Appropriate control techniques are usually designed to minimize shift time and jerk. The clutch is typically cancelled when an AMT is used in an electric car. This makes it easier to produce quick and smooth gearshift performance because of the drive motor's quick adjustment capability, which includes torque and speed. Because of this, numerous research use AMT to increase the EVs' drive range [6–9]. Meanwhile, it is also possible to enhance the EVs' dynamic performance.
To meet the increasing requirements, AMT gearshift performance needs to be improved regardless of whether the application object is an ICEV or an EV. With the exception of the gearshift actuators, AMT and manual transmissions have a similar construction. In AMT, a variety of actuator typesincluding hydraulic and electric actuatorsare commonly used to implement gearshift. However, because of its high density, quick dynamic performance, and linear movement form, linear actuator is more suitable for the AMT gearshift system. Additionally, the inventions of AMT structure are a significant subject. In study [10], hybridized automated manual transmission (HAMT) is proposed using the torque gap filler (TGF) concept. In order to supply power during the shifting process, an additional power flow line is added. Literature [11] presents a similar architecture along with an ideal controller to improve performance. Literature [12] introduces a type of seamless twospeed AMT that makes use of a singlestage epicyclical gear system. The TCUcontrolled brake band and friction disk clutch allow the AMT to achieve smooth gearshift events. A twostage planetary gear set with common sun, common ring gears, and brake mechanisms is used in study [13] to manage the power flow. A superior shift control technique is used to ensure a seamless gearshift. Moreover, in literature [14], a new concept of harpoonshift synchronizer equipped clutchless AMT is introduced, and the frictionrelated drawbacks of traditional synchronizers are overcomed. Twospeed IAMT is designed in study [15], the synchronizer and shift fork are replaced by controllable overrunning clutch system, and the gearshift is realized through the separation and engagement of inverse dry clutch. A kind of 2DOF actuator is developed to execute gearshift in authors' previous work [16].
Researchers are also interested in how effectively the synchronizers work. Research [17] shows that synchronizer cone angles can be optimized to lower the maximum focal surface temperature. Moreover, multicone synchronizers with carbon friction linings outperform singlecone synchronizers in terms of performance [18]. In study [19], a novel friction model is developed to comprehend the frictional behavior throughout the synchronization process quantitatively. Additionally, a novel springbased synchronizer is presented in study [20] that uses a torque spring to deliver torque to synchronize the speed difference between the shaft and the target gear.
Despite the advances in structure, optimization of control method is another way to improve the gearshift performance. In study [21], a coordinated control method based on model predictive control (MPC) is designed to improve gearshift quality, and the fast compensation of driving torque is realized by combining the prediction model and quadratic programming method. A gear shifting concept based on dynamic programming which considers the shifting loss caused by shifting actuators is proposed in study [22], and the system efficiency is improved distinctly. In the literature [23], a position and force switching control scheme is developed for gear engagement. At the start and finish of the gear engagement process, the combination sleeve is controlled using a sliding mode control method to rapidly reach the desired position, and the synchronizing step is handled by a force controller. Simulation and experiment results verify the effectiveness of the control method. The dynamic management of a clutchless AMT in EVs is the subject of literature [24]. Clutchless gear changing is achieved by combining synchronizer mechanism actuation with closed loop motor speed and torque regulation. Study [25] proposes a threeparameter gearshift scheduling strategy which uses vehicle speed, acceleration and road grade to evaluate EV performance under varying driving conditions on flat and hilly roads. The results show that 16.5% of energy can be saved. Furthermore, literature [26] develops modelbased gearshift strategies that are focused on the torque phase and the inertia phase, respectively. The torque phase's control target is the optimization of vehicle jerk and friction work, while the inertia phase's focus is solely on reducing friction work. Their further work proposes three planning torque trajectories to control the electric machine torque so as to achieve optimal gear shifting including the vehicle jerk and shift duration [27]. In addition, research [28] uses a modelbased reinforcement learning algorithm to calibrate the gearshift controller parameters, which is advantageous for investigating different gearshift control approaches. Nevertheless, there aren't many studies on how to optimize the control of the gearshift actuator, particularly when using linear actuators.
The efficiency of the gearshift system can be further increased by applying ELAs. This study first presents the unique shifting mechanism for AMT, including its operation and transmission connections. Additionally, a test bench is created to confirm the benefits of the gearshift system based on ELAs. Second, the gearshift system's characteristics are examined, and two primary control methods are offered with the goal of optimizing the gearshift quality. Owing to the ELAs' quick dynamic performance, a coordinated control method is proposed in order to shorten the gearshift time. This strategy is based on a detailed analysis of the gear engagement and disengagement process. The second strategy is a control method based on linear extended state observers (LESO), which can achieve accurate displacement control and stable force control. Ultimately, through simulation and experimentation, the benefits of the new gearshift system and the potency of the control schemes are confirmed.
2 The AMT gearshift system based on linear actuators
The gearshift driver in a traditional electric gearshift system is usually DC motor, and in order to convert rotational momentum into linear motion, intermediate devices are used. To improve simplicity and dynamic performance, the structure may be improved. The AMT shifting system can benefit from linear motion produced by an electromagnetic linear actuator (ELA). Therefore, in this work, gearshift events, such as the gear engagement and disengagement processes, are handled by ELA. The structure of the new gearshift system is shown in Figure 1a, and the installation on the transmission is presented in Figure 1b. It adopts two ELAs, and each ELA is in charge of two speeds. The ELA's output shafts attach to shift rails directly, transferring linear motion to shift forks and synchronizers without the need for additional intermediary components. Because the motion transfer path is straightforward, it makes sense to enhance the shifting process' dynamic performance. Besides, the moving mass of the gearshift system will also be reduced. Two displacement sensors are chosen in order to guarantee accurate displacement control during gear changes. Furthermore, the two decoupled ELAs of the new gearshift system can operate concurrently, allowing the next gear to engage and the present gear to disengage at the same time, enabling coordinated control. The gearshift time is an important index of the gearshift performance, and the precise overlap of the disengagement process and engagement process can reduce part of the shift time.
Studying the static and dynamic properties of the ELA is important because it helps to achieve improved shift quality through accurate displacement control. As seen in Figure 2, the ELA's mathematical model is built. The I, R, L and U represent the current, the resistance, the inductance and the voltage of the coil respectively. E_{emf} represents the back electromotive force (EMF), F_{mag} is the electromagnetic force, k_{m} is the force coefficient, m is the moving mass, x is the displacement, F_{f} is the friction force.
There are three subsystems in the coupled mathematical model: the mechanical, magnetic, and electric subsystems. Additionally provided are the subsystems' interaction relationships. The electromagnetic force is only related to current I if electromagnetic force coefficient k_{m} is a constant. However, the forcedisplacementcurrent characteristics of the ELA experiment findings, shown in Figure 3, show that the coefficient k_{m} varies, which would cause the ELA to be unsteady and have imprecise displacement control during the shifting process. The variation range of the k_{m} is between 40.1 and 43.4 N/A when the input current I is 10 A; the k_{m} is between 39.8 and 42.3 N/A when I is 20 A; the k_{m} is between 38.9 and 41.7 N/A when I is 30 A. It appears that the force characteristics are nonlinear. The nonlinearity would impact both the stable force control during the synchronization phase and the displacement control during the nonsynchronization phase. Consequently, in order to solve the issue, a suitable control strategy must be developed. Additionally, the values of the electromechanical time constant, which is 0.98 ms, and the electrical time constant, which is 0.7 ms, are obtained through experimentation. The design of the control method benefits from the small time constants, which reflect the quick dynamic features.
Fig. 1 Directdrive electromagnetic gearshift system. 
Fig. 2 Coupled mathematical model of the ELA. 
Fig. 3 Forcedisplacementcurrent characteristics. 
3 Coordinated control of the ELAs
AMT's primary drawback is the torque interruption issue that arises during gearshift process. It not only affects the car's dynamic performance but also lessens passenger comfort. Conventional AMT utilizes a minimum of two DC motors for executing gearshift events, while intermediary devices, such as motion converters and reduction gears, are outfitted to achieve linear gearshift movement. The driving force is increased by reduction gears, and motion converters are employed to transform rotational movement into linear movement, enabling the forward and backward movement of the shift fork and synchronizers. Coordinated control of DC motors is challenging due to their complex structure. In this design, it is possible to reduce the torque interruption time by applying coordinated control of two ELAs during the gearshift process due to the fast dynamic performance of the ELA.
Before designing the coordinated control approach, it is necessary to explain the detailed process of the gearshift events. The five normal states of the gear engagement process are depicted in Figure 4. The sleeve remains in the middle of the two gears in the initial state, and there is a space between it and the synchronizer ring. Second, the shift force drives the sleeve forward until it makes contact with the synchronizer ring. The synchronizing torque will then have the effect of eliminating the speed differences. As demonstrated in state 4, at stage 3, the synchronizer ring will turn slightly to allow the sleeve to advance once again to the target gear. At last, the sleeve makes contact with the targeted gear.
The entire process consists of displacement and undisplaced stages, each with its own set of control requirements. Additionally, the controller is complicated by excessive gear engagement process states. Thus, Figure 5 concludes the controloriented simplified gear engagement process. Furthermore, the engagement and disengagement processes run contrary to one another. In a short period of time, the sleeve changes from state 3 to state 1, and the control requirements can be met with the appropriate disengagement force.
In Figure 6a, a typical gear change procedure is simulated. An ELA handles each step of the process; the engagement process starts after the disengagement process. In this design, the representative disengagement time ranges from 60 to 80 ms. However, the torque transfer pauses when the sleeve moves backwards until it disengages from the current gear, as shown in state 2 in Figure 7. At this point, the engagement process can begin, allowing it to overlap with the disengagement process for the duration indicated in Figure 6b. It is evident that the overall shifting time is shortened, and the shortened time is reliant upon the overlap time. Different contact situations between the target gear and the sleeve are indicated by the dashed line in Figure 6. The sleeve engages with the target gear directly since its teeth are not in contact with the target gear's teeth. Precise displacement control of the ELAs is necessary for the coordinated management of the two processes. Since the distance is 4 mm, the engagement process can begin as soon as 4 mm remain in the disengagement distance. Figure 8 shows the coordinated control process flow chart, where S represents the distance.
Fig. 4 Five states of the gear engagement process. 
Fig. 5 Controloriented simplified gearshift process. 
Fig. 6 Coordinated control of the disengagement and engagement process. 
Fig. 7 Disengagement process. 
Fig. 8 Coordinated displacement control flow chart. 
4 Controller design for the ELAs
There are a variety of factors that could contribute to the force characteristics' nonlinearity, including manufacturing defects and friction. Furthermore, the output performance will also be affected by the coils' and permanent magnets' varying relative positions. However, it is hard to quantify each of these influencing factors, which makes it challenging to eliminate the influences. Nevertheless, by employing appropriate control techniques, these effects can be minimized. Therefore, the robustness and the precision of the displacement control should be taken into account simultaneously in the controller design. PID technology is widely used in motor motion control. Its robustness needs to be strengthened in complex application environments, nevertheless. Since it is derived from the PID technique, the active disturbance rejection control (ADRC) method combines the benefits of the PID method with nonlinear state error feedback, tracking differential, and extended state observer [29]. However, the ADRC design uses an extended state observer (ESO), allowing for the estimation and elimination of internal and external errors, nonlinearities, and variances through appropriate compensation. As a result, the integratorcascaded system replaces the original complex controlled object, simplifying the control system architecture. The drawback of the ADRC method lies in the adoption of nonlinear function so that the mathematical form is difficult to understand. Besides, the regulation of at least eight coupled parameters makes it inappropriate for industrial applications. Gao optimizes the ADRC method and linear functions are adopted to replace the three part of the ADRC [30], and the tracking differential is replaced by a PD controller. Therefore, linear ADRC method is developed for industrial applications, and the control parameters need tuning is reduced to three.
For linear control system described in equation (1), it is easy to build the state observer as shown in equation (2), in which L is the gain matrix, Z is the measurement matrix and it measures the value of X, A is state matrix, B is input matrix, C is output matrix, U is input variables matrix.
$$\{\begin{array}{c}X=AX+BU\\ Y=CX\end{array}$$(1)
$$\stackrel{.}{Z}}=AZL(CZY)+BU=(ALC)Z+LY+BU$$(2)
The estimation error E can be described as
$$\dot{E}=\dot{Z}\dot{X}=\left(ALC\right)Z+LY+BUAXBU=\left(ZX\right)\left(ALC\right)=\left(ALC\right)E$$(3)
Apparently, if (ALC) is a stable matrix, the estimation error E will approach to zero, and the measurement matrix Z can estimate the original state variable X accurately.
The mathematical model of the gearshift system is rewritten as
$$\{\begin{array}{c}I=\frac{R}{L}I\frac{{k}_{m}}{m}{v}_{s}+\frac{U}{L}\\ {v}_{s}=\frac{{k}_{m}}{m}I\frac{c}{m}{v}_{}\\ S={v}_{s}\end{array}$$(4)
where v_{s} is the velocity of the sleeve, c is the viscous friction damping coefficient, S is the displacement of the sleeve.
Displacement accuracy is a significant aspect that affects shift performance in addition to shift time. The shift process's accuracy in terms of displacement will also be impacted by the nonlinear features of the shift force. Hence, suppose S is x_{1}, v_{s} is x_{2}, u is input variable, and the equation of state is
$$\{\begin{array}{c}{x}_{1}={x}_{2}\hfill \\ {x}_{2}=f\left({x}_{1},{x}_{2},w,t\right)+bu\hfill \\ y={x}_{1}\hfill \end{array}$$(5)
f(x_{1}, x_{2}, w, t) is the nonlinear function of the controlled system that includes uncertainties, nonlinearities and disturbances, and w is external disturbance. Since these influence factors are hard to measure, and also cannot be modeled, it can be considered as a new state variable x_{3}. Therefore, equation (5) is rewritten as
$$\{\begin{array}{c}{x}_{1}={x}_{2}\hfill \\ {x}_{2}={x}_{3}+{b}_{0}u\hfill \\ {x}_{3}=g\left({x}_{1},{x}_{2},w,t\right)\hfill \\ y={x}_{1}\hfill \end{array}$$(6)
in which$g\left({x}_{1},{x}_{2},w,t\right)=f\left({x}_{1},{x}_{2},w,t\right),{b}_{0}\approx b$ From the above equation, it can be deduced that
$$\mathbf{A}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right),\mathbf{B}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill {b}_{0}\hfill \\ \hfill 0\hfill \end{array}\right),\mathbf{C}=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\mathbf{L}=\left(\begin{array}{c}\hfill {\beta}_{1}\hfill \\ \hfill {\beta}_{2}\hfill \\ \hfill {\beta}_{3}\hfill \end{array}\right),\mathbf{L}\mathbf{c}=\left(\begin{array}{c}\hfill {\beta}_{1}\hfill \\ \hfill {\beta}_{2}\hfill \\ \hfill {\beta}_{3}\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),\text{and}\mathbf{A}\mathbf{Z}\mathbf{L}\mathbf{E}=\left(\begin{array}{l}{z}_{2}{\beta}_{1}e\\ \begin{array}{cc}\hfill {z}_{3}\hfill & \hfill {\beta}_{2}e\hfill \end{array}\\ {\beta}_{3}e\end{array}\right).\text{AS}$$
a result, the linear extended state observer (LESO) equations are obtained as
$$\left[\begin{array}{c}{\stackrel{}{z}}_{1}\\ {\dot{z}}_{1}\\ {\dot{z}}_{1}\end{array}\right]=\left[\begin{array}{c}\begin{array}{ccc}{\beta}_{1}& 1& 0\end{array}\\ \begin{array}{ccc}{\beta}_{2}& 0& 1\end{array}\\ \begin{array}{ccc}{\beta}_{3}& 0& 0\end{array}\end{array}\right]+\left[\begin{array}{c}0\\ {b}_{0}\\ 0\end{array}\right]u+\left[\begin{array}{c}{\beta}_{1}\\ {\beta}_{2}\\ {\beta}_{3}\end{array}\right]y$$(7)
z_{1}, z_{2} and z_{3} are estimated values of the x_{1}, x_{2} and the total disturbances respectively. β_{1}, β_{2} and β_{3} are observer gains. The matrix (ALC) should be a stable matrix so that the LESO is effective, and hence, the characteristic values should be negative.
$$\left\lambda I(ALC)\right={\lambda}^{3}+{\beta}_{1}{\lambda}^{2}+{\beta}_{2}\lambda +{\beta}_{3}={(\lambda +{\omega}_{0})}^{3}$$(8)
ω_{0} is designed to simplify the adjustment of variables β_{1}, β_{2} and β_{3}, and ω_{0} is the bandwidth of the observer. The relationship can be deduced from equation (8) as${\beta}_{1}=3{\omega}_{0},{\beta}_{2}=3{\omega}_{0}^{2},{\beta}_{3}={\omega}_{0}^{3}$.
Usually employed to balance quick dynamic response and overshoot, the tracking differentiator module will lengthen the adjustment time. However, the dynamic response remains unaffected while the overshoot can be reduced using the linear state error feedback module. Thus, to build the active disturbance rejection controller, the linear extended state observer and linear state error feedback module are integrated.
For the controlled system as shown in equation (5), the input variable u can be designed as
$$u=\frac{\left({z}_{3}+{u}_{0}\right)}{{b}_{0}}$$(9)
and the controlled system is changed into
$$\{\begin{array}{c}{\dot{x}}_{1}={\dot{x}}_{2}\\ {\dot{x}}_{2}={u}_{0}\\ y={x}_{1}\end{array}$$(10)
A PD controller is suitable for such a double integrator system,
$${u}_{0}={k}_{p}({x}^{*}{z}_{1}){k}_{d}{z}_{2}$$(11)
where x^{*}is the target value, k_{p} is proportional coefficient, k_{d} is differential coefficient. The transfer function is
$$G(s)=\frac{{k}_{p}}{{s}^{2}+{k}_{d}s+{k}_{p}}$$(12)
Another advantage of the improved LESO controller is that it is insensitive to the slight variation of control parameters which inherits from the ADRC method. Therefore, in order to simplify the tuning of control parameters, the PD variables can be selected as ${k}_{p}={\omega}_{c}^{2}$, k_{d} = 2ω_{c}, and ω_{c} is the bandwidth of the controller [31,32]. As a result, the input variable u is rewritten as
$$u=\frac{\left({\omega}_{c}^{2}({x}^{*}{z}_{1})2{\omega}_{c}{z}_{2}{z}_{3}\right)}{{b}_{0}}$$(13)
Put β_{1} = 3ω_{0}, ${\beta}_{2}=3{\omega}_{0}^{2}$, ${\beta}_{3}={\omega}_{0}^{3}$ into equation (7), and the transfer functions of each variable is obtained,
$$\{\begin{array}{l}{z}_{1}(s)=\frac{3{\omega}_{0}{s}^{2}+2{\omega}_{0}^{2}+{\omega}_{0}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}y(s)+\frac{{b}_{0}s}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\\ {z}_{2}(s)=\frac{3{\omega}_{0}{s}^{2}+{\omega}_{0}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}y(s)+\frac{{b}_{0}(s+3{\omega}_{0})}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\\ {z}_{3}(s)=\frac{{\omega}_{0}^{3}s}{{\left(s+{\omega}_{0}\right)}^{3}}y(s)+\frac{{b}_{0}{\omega}_{0}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\end{array}$$(14)
The errors of each order of the LESO is
$$\{\begin{array}{c}{e}_{1}={z}_{1}y\\ {e}_{2}={z}_{2}\dot{y}\\ {e}_{3}={z}_{3}f\end{array}$$(15)
The nonlinear function $f={\dot{x}}_{2}{b}_{0}u=\dot{y}{b}_{0}u$ therefore, the observe errors are
$$\{\begin{array}{l}{e}_{1}(s)=\frac{{b}_{0}s}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\frac{{s}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}y(s)\\ {e}_{2}(s)=\frac{{b}_{0}s}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\frac{(3{\omega}_{0}+s){s}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}y(s)\\ {e}_{3}(s)={b}_{0}\frac{{\left(s+{\omega}_{0}\right)}^{3}{\omega}_{0}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}u(s)\frac{{\left(s+{\omega}_{0}\right)}^{3}{\omega}_{0}^{3}}{{\left(s+{\omega}_{0}\right)}^{3}}{s}^{2}y(s)\end{array}$$(16)
Suppose the input variable u and output variable y are suffers from positive disturbance, and the steadystate error
$$\{\begin{array}{l}{e}_{1}={\displaystyle \underset{s\to 0}{\text{lim}}}s{e}_{1}(s)={\displaystyle \underset{s\to 0}{\text{lim}}}\frac{{b}_{0}su(s){s}^{3}y(s)}{{\left(s+{\omega}_{0}\right)}^{3}}s=0\\ {e}_{2}={\displaystyle \underset{s\to 0}{\text{lim}}}s{e}_{2}(s)={\displaystyle \underset{s\to 0}{\text{lim}}}\frac{{b}_{0}su(s)(3{\omega}_{0}+s){s}^{3}y(s)}{{\left(s+{\omega}_{0}\right)}^{3}}s=0\\ {e}_{3}={\displaystyle \underset{s\to 0}{\text{lim}}}s{e}_{3}(s)={\displaystyle \underset{s\to 0}{\text{lim}}}\frac{{b}_{0}({(s+{\omega}_{0})}^{3}{\omega}_{0}^{3})u(s)({(s+{\omega}_{0})}^{3}{\omega}_{0}^{3}){s}^{2}y(s)}{{\left(s+{\omega}_{0}\right)}^{3}}s=0\end{array}$$(17)
The LESO appears to be convergent and capable of accurately estimating state variables and system disturbances. The dynamic performance is also analyzed. Suppose the b_{0} = 0, and step signal y = 1 is applied. The z_{1} in equation 14 can be transformed into
$${Z}_{1}=\left(\frac{1}{s}\frac{1}{s+{\omega}_{0}}+\frac{2{\omega}_{0}}{{\left(s+{\omega}_{0}\right)}^{2}}\frac{{\omega}_{0}^{2}}{{\left(s+{\omega}_{0}\right)}^{3}}\right)$$(18)
And after the inverse Laplace transform,
$${Z}_{1}(t)=1\left(\frac{1}{2}{\omega}_{0}^{2}{t}^{2}2{\omega}_{0}t+1\right){e}^{{\omega}_{0}t}$$(19)
Suppose ${\dot{Z}}_{1}\left(t\right)=0,$ the extreme points are solved as
$$\{\begin{array}{l}{t}_{1}=\frac{3\sqrt{3}}{{\omega}_{0}}\\ {t}_{2}=\frac{3+\sqrt{3}}{{\omega}_{0}}\end{array}$$(20)
According to the equation (20), parameter ω_{0} has an effect on the tracking speed of the LESO, and large ω_{0} is beneficial to improve the dynamic performance. However, overlarge ω_{0} will amplify the observation noise. The control parameters of the improved LESO are ω_{0}, ω_{c} and b_{0}. For the most of industrial control objectives, the relationship between ω_{0} and ω_{c} can be determined, and the variation range is ω_{0} = (3∼5) ω_{c} [31]. The improved LESO controller is presented in Figure 9.
Simulations are carried out to verify the performance of the designed controller. Figure 10 shows the step response performance, and a PID controller is created as a comparison group. The LESO controller has a response time of 16.3 ms, which is almost 2 ms faster than the PID controller. The step response of the PID controller after parameter change to nearly equal the time of 16.3 ms with the LESO controller is represented by the curve with the label PID_overshoot. However, as displacement overshoot will result in a severe impact on synchronizer components, the shifting process' displacement control is undesirable. Furthermore, the controllers' robustness has been confirmed too. The shifting system's moving mass, represented by parameter m, grows by 20%. Figure 10 also displays the controllers' step responses, which are denoted by the labels PID_robust and LESO_robust. The LESO and PID both respond slowly, and there is a minor overshoot for both controllers. However, there are differences in the overshoot's amplitude. For PID controllers, the overshoot value is 12%, while for LESO controllers, it is 1.6%. Furthermore, the PID controller's steady state error is greater than 1.5%, but it is insignificant when the LESO control is used.
In addition, Figure 11 displays the two controllers' tracking performance. As shown in Figure 11b, the maximum tracking error values for the PID and LESO controllers, respectively, are 0.041 mm and 0.026 mm, indicating that they can both track the sinusoid curve with good accuracy. Next, the sinusoid curve's amplitude and frequency are adjusted, and Figure 12 displays the tracking performance. The LESO controller's highest tracking error value is 0.03 mm, which is nearly identical to the tracking performance displayed in Figure 11b. However, when using a PID controller, the maximum tracking error value is greater than 0.2 mm. It is clear that compared to PID controller, LESO controller is less sensitive to changes of the controlled system's parameters. Furthermore, it is evident that the shifting system's dynamic response is quick, indicating that shortening the gearshift time is advantageous.
Fig. 9 Improved linear extended state observer (LESO) controller. 
Fig. 10 Simulation results of the step response of the two controllers. 
Fig. 11 Tracking performance of the two controllers. 
Fig. 12 Tracking performance of the two controller after parameter variation. 
5 Experiments and discussions
As seen in Figure 13, the test bench is made up of a drive motor, a flywheel, two ELAs, two torque sensors, a gearbox, and other auxiliary components. On the test bench, a fourspeed transmission is installed, with each ELA controlling two speeds. In order to replicate various operating situations, the test bench has adjustable equivalent input and output inertia. The core controller is an ARM cortexM0 chip, and ahead of the core controller sampling the sensor signals, each one is filtered. The drive motor provides the input torque and speed, while the flywheel is used to simulate the vehicle's inertia. On the test bench, comparative shifting events are performed to confirm the effectiveness of the LESO control method.
First of all, gearshift events are executed without the use of a control system. Because the voltage applied to the ELA remains constant during the shifting operation, the ELA's maximum output force is also regulated. There are two different types of speed differences that have been chosen: 200 rpm and 500 rpm. Figure 14 presents the outcomes of the experiment. When the speed difference is 200 r/min, the displacement and shift force are represented by the labels D200 and F200, respectively, and 500 r/min by D500 and F500. The shift times are, respectively, 203.8 and 98.2 ms. While the shift time is acceptable, the shift force curve fluctuates quite a lot, which transfers to the output shaft and creates a significant jerk while the synchronization process is processing. When the speed difference is 500 r/min, the maximum jerk exceeds 20 m/s^{3}, and when the speed difference is 200 r/min, it is 15.3 m/s^{3}. Additionally, at the end of state 1 shown in Figure 5, the shift force overshoots by almost 6%, causing the sleeve to make contact with the synchronizer ring. This shortens the lifespan of these parts since an appropriate method isn't used to restrict the displacement.
The gearshift events performance under PID control is depicted in Figure 15, where the input and output shafts are rotating at different speeds of 500 r/min. PID displacement control is used to implement states 1 and 3, and as the displacement is nearly constant during this process of synchronization, the ELA is supplied with a constant voltage. When compared to the performance shown in Figure 14, there is a noticeable decrease in fluctuation, and the maximum jerk drops to 6.3 m/s^{3} during the shift force's ascent stage. The shift time has decreased slightly to 196.5 ms in the meantime. The overshoot is still noticeable, though. To enhance the shifting quality, additional optimization is needed to reduce the fluctuation brought on by vibration, nonlinearity, and other disturbances during the synchronization process.
The shifting performance with the LESO control method is displayed in Figures 16 and 17. Compared to the two experiment results previously mentioned, the shift force is more stable. With the consistent shift force helping during the synchronization process, the shift time decreases to 189.5 ms, and at 200 r/min, the duration is 108.7 ms. The greatest jerk for a speed differential of 200 r/min and 500 r/min is 2.6 m/s^{3} and 3.1 m/s^{3}, respectively. The slight standstill of the displacement around 6 mm indicates that the contact of the spline teeth of the sleeve and the target gear happens, as the state 4 in Figure 4. From the comparative experiment results, it is clear that with the LESO control method, the nonlinear output characteristics of the ELA are weakened to an acceptable level.
Based on the suggested control strategy, two ELAs are tested with coordinated control tests, and the outcomes are displayed in Figure 18. The disengagement time is nearly 55 ms under the control of PID method when the displacement reaches point C. When the disengagement displacement reaches 4 mm, as shown by B in the Figure, the engagement process begins. The time t_{2} from B to C is saved since the typical engagement procedure often begins at the C point. After numerous trials, the fluctuation range of t_{2} is 25 ms to 31 ms. Furthermore, since the sleeve for engagement needs an additional 20 ms to reach the target gear's synchronizer ring before the torque transmit, it is still safe to perform the advanced coordinated control. This is because the engagement process begins at the A point, where the sleeve's spline teeth are already out of contact with those of the current gear. In Figure 17, the advanced engagement curve is depicted by a dashed line, and the additional time saved (t_{1}) is over 12 ms. However, the advanced coordinated control requires more precise and robustness displacement control, it needs further investigation to insure the thorough success of the coordinated control.
Fig. 13 Gearshift test bench. 
Fig. 14 Gearshift performance without control method. 
Fig. 15 Gearshift performance with PID control method. 
Fig. 16 Gearshift performance with LESO control method (500 r/min). 
Fig. 17 Gearshift performance with LESO control method (200 r/min). 
Fig. 18 Coordinated gearshift control performance. 
6 Conclusions
This paper presents a novel gearshift system that uses two electromagnetic linear actuators (ELAs) to perform gearshift events. The ELAs are installed with the transmission, and they have a direct connection to the shift rail, eliminating the need for any intermediate mechanisms. Hence, the structure of the new gearshift system is simple and reliable. Each ELA is in charge of two speeds so that coordinated control of disengagement process and engagement process is feasible. Coordinated control strategy is designed after specific analysis of the gear engagement process. The whole gearshift process is divided into three phases, which is eliminating the gap phase, synchronization phase and engage phase. To cut down on shift time, the elimination phase begins when there is 4 mm remaining in the disengagement displacement. This indicates that the sleeve's spline teeth are no longer in contact with the current gear, avoiding any torque transfer. The beginning point of the engagement process and accurate displacement control are two elements that are crucial to the coordinated gearshift control. Consequently, an appropriate displacement control method is investigated.
In the meantime, the gearshift system's mathematical model is explained, and the ELA's forcedisplacementcurrent characteristics demonstrate that the output force has a nonlinear appearance, which is undesirable to the enhancement of gearshift performance. Because of its simple and reliable features, linear extended state observer (LESO) based control method is proposed to achieve precise displacement control. The advantages of the LESObased control method over the PID system are shown by the simulation results. The outcomes of the gearshift experiment further confirm the efficiency of the control strategy. When the speed difference is 500 r/min, the gearshift jerk drops from 20 m/s^{3} to 3.1 m/s^{3}, while the shift time remains somewhat constant. Coordinated control experiments also show that it can reduce the gearshift time by at least 25 ms. Additionally, the benefits of the innovative ELAsbased gearshift system are confirmed. Further investigation will focus on the matching control when the two ELAs based 4speed AMT is integrated with a drive motor for electric vehicles.
Funding
This work was funded by the National Natural Science Foundation of China (grant number 51905364 and 51975341) and the Natural Science Foundation of Zhejiang Province, China (grant number LTY20E050002).
Conflict of interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and publication of this article.
Data availability
The data presented in this study are available on request from the corresponding author.
Author contribution statement
The author contributions are: Conceptualization S.L. and B.L., methodology S.L., software, simulation, analysis S.L. and M.T., validation S.L., M.T., and L.W., writing S.L.
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Cite this article as: S. Lin, B. Li, M. Tang, L. Wang, Precise and coordinated gearshift control for AMT gearshift system equipped with two linear actuators, Mechanics & Industry 25, 8 (2024)
All Figures
Fig. 1 Directdrive electromagnetic gearshift system. 

In the text 
Fig. 2 Coupled mathematical model of the ELA. 

In the text 
Fig. 3 Forcedisplacementcurrent characteristics. 

In the text 
Fig. 4 Five states of the gear engagement process. 

In the text 
Fig. 5 Controloriented simplified gearshift process. 

In the text 
Fig. 6 Coordinated control of the disengagement and engagement process. 

In the text 
Fig. 7 Disengagement process. 

In the text 
Fig. 8 Coordinated displacement control flow chart. 

In the text 
Fig. 9 Improved linear extended state observer (LESO) controller. 

In the text 
Fig. 10 Simulation results of the step response of the two controllers. 

In the text 
Fig. 11 Tracking performance of the two controllers. 

In the text 
Fig. 12 Tracking performance of the two controller after parameter variation. 

In the text 
Fig. 13 Gearshift test bench. 

In the text 
Fig. 14 Gearshift performance without control method. 

In the text 
Fig. 15 Gearshift performance with PID control method. 

In the text 
Fig. 16 Gearshift performance with LESO control method (500 r/min). 

In the text 
Fig. 17 Gearshift performance with LESO control method (200 r/min). 

In the text 
Fig. 18 Coordinated gearshift control performance. 

In the text 
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