Position Control of Gear Shift based on Sliding Mode Active Disturbance Rejection Control for Small-Sized Tractors

Abstract

A modified sliding mode active disturbance rejection control (MSMADRC) position system is designed for small-sized tractors to solve the longer shift time and reduced shifting quality because of the inaccurately motor control used for the automatic mechanical transmission (AMT) gear shift actuator. Firstly, the control model of the motor with total disturbance is established. Then an extended observer is presented to monitor the unmodeled dynamics and various disturbances of the system in real time, at the same time the extended state and the system feedback variables are constructed as the system variables of the sliding mode control (SMC) algorithm. Secondly, a sliding mode surface instead of the nonlinear control law in the active disturbance rejection control (ADRC) algorithm is designed, which realizes the fast and accurate tracking of the position. What's more, the stability of the control system is proved by Lyapunov theory. Lastly, the simulation results demonstrate that the position control precision by MSMADRC is higher 37% than by SMC and higher 75% than by ADRC. Furthermore, the response speed of MSMADRC is the fastest, it only takes about 0.7s.

Introduction

Agricultural tractors can be used as a power source for various field operations and is regarded as one of the vital implement in farm mechanization and as a versatile machine both for agriculture and farmer. ¹ With the development of modern agriculture, tractors are rapidly developing towards automation and intellectualization. ² Automatic gear shifting is therefore one of the important development trends for agricultural tractors. ³ As we know, agricultural tractors have to cope with working conditions which are more complex and demanding than those experienced by other ground vehicles. Usually, tractors move on uneven soils at low speed but with large traction forces, so they are equipped with power shift transmission to ensure flexibility at each speed in different working conditions. If manual gear shifting is adopted, repeated gear shifting will not only cause operator's discomfort because of its excessive limbs intervention but also reduce the life of clutch. What's more, tractors will stop unexpectedly because of inappropriate gear shift failing to transmit continuous power to the driving wheels. ⁴ Therefore, to improve work efficiency and reduce driving effort, power shift transmissions enabling the automatic shifting of gears are gradually proposed and used.^5,6 Automatic gear shifting really can reduce driver work intensity for repetitive shifting operations. It can free the driver from the frequent gear shift and make the driver focus more on the operation of the implements. As a result, Automatic gear shifting not only improves the quality and efficiency of field work but also reduces fuel and power loss caused by improper manual gear shifting operations. ⁷

As for the design of a control system for automatic gear shifting, there are some achievements for ground vehicles, but only little dealing with gear shift for agricultural tractors are available. The main reason is that there are different demands for the gear shift because of the variability of the load and broad range of working conditions. Li et al ² proposed a novel power shift transmission based on hydro-viscous drive technology and control strategy for automatic starting of the tractor matching the power-shift transmission. To achieve high transmission efficiency, they established the powertrain dynamical model, analyzed the shifting processes and proposed coordinated control strategies for upshift and downshift processes with changing multiple clutches. ⁸ In their recent research paper, ⁹ four-parameter gear shifting schedules were developed for tractor with different types of agricultural implements and recursive least squares algorithm was used for real time identification of the changing characteristic of draught force, by which realized the automatic gear shift and energy saving. For a high-power agricultural tractor, Mara Tanelli et al. ¹⁰ designed a transmission control system based on single clutch and double clutch gear shifts respectively. The control system can provide good shifting performance in all operating conditions. Satyam Raikwar et al. ¹¹ proposed a power shuttle transmission system for an agricultural tractor by using MATLAB as a simulation platform. They established the model of the individual component of power shuttle transmission system and integrated them as a whole vehicle model.

Usually, hydraulic power regulated by proportional solenoid valve is used to control the gears and realize the power shift transmission for giant tractors. For small-sized tractors, electric shift with advantages of high speed and reduced the shift time is alternative mode to realize gear shift, it only uses the motor as the power source. Clutch separation, combination and shift action can be achieved through the control of the motor. But the control accuracy of this kind of shift is low and needs to be properly controlled, that is also the focus of the research on electronic shift control. In Zhang et al. ¹² study, to meet the speed and stability requirements of the motor control system, they employed a fuzzy PID control algorithm to the motor control and greatly improved the shift quality. SMC has been recently used in a number of studies for controlling the position of motor. Shenye et al. ¹³ designed a sliding mode controller to track the position of AMT shift actuator. Li et al. ¹⁴ proposed a position and force switching control scheme for gear shift based on SMC method. The combination sleeve can reach the desired position quickly with a small resistance at the beginning and the end of gear shift process. Temporelli et al. ¹⁵ proposed a dual SMC to control clutch pressure with the electromechanical clutch actuator. In, ¹⁶ an adaptive sliding-mode-control-based model predictive torque control for induction motor was proposed and the optimized exponential reaching law can adjust the switching gain adaptively. In ¹⁷ sensorless field oriented control based on switching linear feedback structure of sliding mode control plugged with model reference adaptive system was presented for induction motor. Homaeinezhad et al. ¹⁸ proposed a position braking tracking control based on discrete time SMC, by which the DC motor can be controlled in position tracking mode and torque tracking mode respectively. To control the position of a DC motor, SMC based on uncertainty and disturbance estimator is presented. ¹⁹ What's more, ADRC has the advantages of active disturbance rejection and handling parameter uncertainties, which make it a strong candidate for industry control applications. ²⁰ Hao et al. ²¹ proposed a linear/nonlinear active disturbance rejection control switching control strategy and a parameter tuning strategy. To cope with the high-order disturbance and enhance the dynamic tracking performance of the servo systems, Liu et al. ²² put forward an integrated method, in which finite-time convergent nonlinear function and novel extended state observer are designed. Yang et al. ²³ proposed an ADRC strategy based on improved particle swarm optimization-genetic algorithm. And first-order and the second-order ADRC are designed for the different order of the bearingless induction motor system respectively. Sun et al. ²⁴ used ADRC to adjust the motor torque to control the regenerative current accurately in the current closed-loop control mode. Madonski et al. ²⁵ designed an ADRC with novel characterized of concise and practically appealing form of the control action for DC motor system. The control system demonstrated performance of improved tracking precision and disturbance rejection. For the established fifth order system model of the valve-controlled hydraulic motor, in literature ²⁶ an ADRC strategy was designed and the grey wolf algorithm was used to set the parameters of the three parts in ADRC. It can be seen that strategies based on SMC and ADRC have been shown great performance and promises. But the structure of ADRC is complex and needs to tune a bunch of parameters, which makes it difficult to use in practice. And it is critical for SMC to cope with the chattering caused by the switching motion of the sliding surfaces.

Automatic gear shifting is important to dynamics and economic performance for agricultural tractors. To improve the accuracy and speed of the shifting gear motor used for gearshifting as well as to cope with the influence of the uncertain parameters such as the variation of the internal parameters of the motor and the external gear shifting resistance, a position control of gear shift system for small-sized tractors adopting electric shift is presented. Motivated by the above achievements, borrowing ideas from SMC and ADRC theory, the advantage of SMC that can be integrated with other control algorithms to compensate for the weakness of each other is developed. A MSMADRC based algorithm is proposed to design the shift motor control system. The method is simple and easy to be programmed into a digital real - time embedded system. For clear illustration, the paper is organized as follows: Section ‘Analysis of gear shift’ introduces the gear shift system. The control structure and controller design method are detailed in Section ‘Control law design’. In Section ‘Stability analysis’, system stability is analyzed. Simulations are carried out to demonstrate the validity of the proposed scheme in Section ‘Simulation results’. Finally, some conclusions are drawn in Section ‘Conclusion’.

Analysis of gear shift

Principle of gear shift actuator

AMT for the small-sized SL350 wheeled tractor is designed by adding a set of electronic mechanical equipment on the basis of the traditional manual transmission. And motor is selected as power source for the gear shifting. The structure of gear shifting system is shown in Figure 1, which consists of worm, gear, shift motor and transmission control unit (TCU).

Figure 1.

The structure of gear shift system.

The shift motor will rotate after receiving the command from the TCU. And the torque output from the motor is transmitted to the sector gear after reduced the speed by the worm and gear. The movement of the sector gear is transmitted through the transmission rod to the rotation of the shift finger. Then the shift finger drives the reverse shift fork to complete the shift operation.Considering the working conditions full of vibration and direct displacement signal requirement, angular displacement sensor is preferable choice rather than the encoder, so angular displacement sensor installed at the end of worm gear shaft is used to measure the angular displacement of worm. The electric gear shifting system inherits the advantages of traditional manual transmission and combines simple structure with high transmission efficiency and low manufacturing cost.

Shift motor modeling

As for the gear shifting system, motor is one of the key element affecting the shift time. In this paper a low-voltage and cheap DC motor is selected as shift motor, it demonstrates higher reliability, easier initialization processes and more flexibility in position compared to AC motors. ²⁷ And the angular displacement of the motor is controlled by the input torque. The dynamic equation of the motor can be written as

$$J\ddot{\theta }(t)=u(t)-B\dot{\theta }(t)-D(t)$$

(1)

Where J is the inertia load, $\mathrm{kg}\cdot {\mathrm{m}}^2$ , $\theta$ is the angular displacement of the motor, $\mathrm{rad}$ , that is the output of the system, $u(t)$ is the control input torque, $\mathrm{N}\cdot \mathrm{m}$ , B is the viscous friction coefficient, $\mathrm{N}\cdot \mathrm{m}/(\mathrm{rad}/\mathrm{s})$ , $D(t)$ is the total disturbance which is unmodeled, such as external disturbances and unmodeled dynamics. Therefore, shift time may be longer due to the influence of the uncertain parameters which will reduce the shifting quality greatly. Therefore, demand for control precision and shift speed is put forward to the electric gear shift.

Control law design

Structure of proposed control system

As mentioned above, the output angular displacement of the shift motor is crucial to realize the precise gear shift, the control accuracy and robustness to the load changes of the motor control system are more important. So the objective of position control of the gear shift is to precisely track the set angular displacement as well as keep the shift motor operating steadily. To this end, the overall architecture of the MSMADRC algorithm for angular displacement is organized, as shown in Figure 2.

Figure 2.

The overall architecture of MSMADRC algorithm.

MSMADRC controller consists of a sliding mode controller, expansion state observer (ESO), and disturbance compensator. ESO is the heart of the controller, it observes the uncertainties existing in the system as total disturbances and eliminates the influence of the disturbances in the form of disturbance compensation. The observed state variables are employed to design the sliding mode controller, which will greatly simplify the design of the sliding mode controller. What's more, to reduce the chattering phenomenon, the adaptive sliding controller is introduced by designing the sliding surface and sliding reaching law.

Active disturbance rejection controller design

ADRC aims to accommodate the disturbances in a way that the design can be executed without a detailed mathematical model. As in Figure 3, the idea of ADRC is to actively estimate the disturbance and cancel it by the control action. ²⁸ Using the disturbance before the process represents the total disturbance, From Figure 3, the response of the real motor system subjected to control action and external disturbance is

$$Y=P(U+W)=P_m(1+\mathrm{\Delta })(U+W)=P_m(U+\underset{D}{\underbrace{U\mathrm{\Delta }+W\mathrm{\Delta }+W}})=P_m(U+D)$$

(2)

Figure 3.

Active disturbance rejection control scheme.

Where Y is the output of the control system, P is the control process, U is the control action, W is the disturbance, $P_m$ is the nominal model of the control process, $\mathrm{\Delta }$ is model uncertainty, D is the total uncertainty. Disturbances and uncertainties are inevitable in the working situation for the shift motor. So the nominal shift motor model in equation (1) can be expressed in a state space form as:

$$\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=bu+a{x}_2+d(t)\end{array}$$

(3)

Where $x_1$ represents the angular displacement of the motor, $x_2$ represents the angular velocity of the motor. $b=1/J$ and $a=-B/J$ are the parameters of the plant, $d(t)=-D(t)/J$ represents the disturbance.

According to the basic idea of the ADRC, the second order motor system can be described as

$$\{\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=bu+f\end{array}$$

(4)

Where $f=a{x}_2+d(t)$ is a general nonlinear, time-varying dynamic representing the total disturbance including both the uncertainty of the system itself and the unknown disturbance of the outside the system. The key idea is to extend the state definition of f and estimate it in real time using the state observer. So regard f as an extended state $x_3$ and is augmented to the original system, the state vector is defined as

$$X=[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}{]}^T=[\begin{array}{ccc}{x}_1& x_2& f\end{array}{]}^T$$

(5)

Subsequently, equation (4) can be transformed into third orders extended state space, which is described as

$$\{\begin{array}{c}\dot{X}=AX+Bu+E\dot{f}\\ y=CX\end{array}$$

(6)

$$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{c}0\\ b_{}\\ 0\end{array}\right],C={\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]}^T,E=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$$

ESO is the core of linear active disturbance rejection control technology, it is used to estimate the state variables and the total disturbance of the system without giving an accurate mathematical model.

Assuming that the total disturbance f of the system is derivable, according to equation (5), a linear extended state observer (ESO) is constructed as:

$$\{\begin{array}{c}{\dot{z}}_1=z_2+l_1(y-\hat{y})\\ {\dot{z}}_2=z_3+b_{0}u+l_2(y-\hat{y})\\ {\dot{z}}_3=l_3(y-\hat{y})\\ \hat{y}=z_1\end{array}$$

(7)

Where $l_1,l_2,l_3$ are the ESO parameters to be determined later. In the above model, the state vector Z and the total disturbance can be estimated and the outputs of the LESO will converge to $x_1,x_2,x_3$ by appropriate choosing the parameters. Equation (6) is transformed into state space as

$$\{\begin{array}{c}\dot{Z}=AZ+Bu+L(y-\hat{y})\\ \hat{y}=CZ\end{array}$$

(8)

Where the matrix A, B, C in the above Equation are the same as in equation (5). Z is the state variable matrix of the extended state observer and $z_1$ , $z_2$ , $z_3$ are the outputs of the ESO, $z_1$ , $z_2$ represents the observation values of $x_1$ , $x_2$ respectively, that are the estimated values of motor angle and motor speed, $z_3$ represents the observation of $x_3$ , that is the estimated value of the total disturbance, $L={\left[\begin{array}{ccc}{l}_1& l_2& l_3\end{array}\right]}^T$ is the observer gain vector. $\hat{y}$ is the estimation of output. The closed-loop observer that is the correction term $L(y-\hat{y})$ is responsible for accommodating the unknown initial states, the uncertainties in parameters and disturbances. According to the idea of ADRC, the total disturbance can be actively estimated and canceled by the control action.

The standard procedure of ADRC is firstly to design the observer gain vector as

$$L=[\begin{array}{ccc}{\beta }_1{\omega }_0& {\beta }_2{\omega }_0^2& {\beta }_3{\omega }_0^3\end{array}{]}^T$$

(9)

Where ${\omega }_0$ is the observer bandwidth, ${\beta }_i$ $(i=1,2,3)$ are the gain of the ESO parameters and selected to ensure the polynomial $s^3+l_1{s}^2+l_{2}s+l_3$ is Hurwitz. To make ESO easy to tune, the ESO gains can be chosen so that all its eigenvalues are placed at ${\omega }_0$ , that is $s^3+l_1{s}^2+l_{2}s+l_3=(s+{\omega }_0^3)$ , and then the gain of the ESO parameters can be determined by

$${\beta }_i={\displaystyle \frac{(n+1)!}{i!(n+1-i)!},1\le i\le n+1}$$

(10)

Where $n=2$ represents the order of the subsystem, so the gains are obtained as

$${\beta }_1=3,{\beta }_2=3,{\beta }_3=1$$

With the state observer properly designed, the conventional ADRC control law that is the output of the controller $C_a$ in Figure 3 can be got as:

$$u=(k_1(r-z_1)+k_2(\dot{r}-z_2)-z_3)/b_0$$

(11)

where $k_1$ and $k_2$ are controller gains, r is the reference signal, and $b_0$ is the estimation of b.

As can be seen from equation (5), disturbance is added as an extended state and it is a real time status variable. What's more, the disturbance is estimated according to equation (7) and the estimated disturbance play a role during the controlling process as part of the controller output which can be referenced as equation (11).

Remark 1:

It can be seen from the design procedure for the ESO that the ${\omega }_0$ is the only parameter that can be adjusted. Therefore ${\omega }_0$ determines the whole performance of ESO, because the ability to meet the control requirements is largely dependent on the speed at which the observer can track the states. So the primary concern is to select the bandwidth. In general, observer should be made to work as fast as the measurement noise allowed. ²⁹ A larger value for bandwidth ${\omega }_0$ gives higher tracking accuracy of ESO and the sooner the disturbance is observed and cancelled by the controller, but it will also amplify the noise signal, which will affect the control quality of the controller. As for the engineering application, the bandwidth should be selected considering the limited sampling rate and noises so that there is no significant oscillation in its states.

Sliding mode control law design

As we know, sliding mode control also is a powerful tool to handle uncertainties and disturbances. Its dynamics of the system is determined by the selected sliding surface, by which excellent robust performance can be achieved as well as high tracking accuracy. So through the fusion of sliding mode control and ADRC, not only the control performance of the system is improved but also the parameter setting for the controller is simplified.

Based on the theory of sliding mode control, the deviation between the actual measured motor angle and desired motor angle is selected as the system tracking error

$$\hat{e}=z_1-{\theta }_d$$

(12)

Where ${\theta }_d$ is the desired angel. Differential angle error and angle error are selected as the variables of the sliding mode function in order to obtain good effect. The conventional sliding surface is introduced as

$$s=\lambda \hat{e}+\dot{\hat{e}}$$

(13)

where λ is the slope of the switching line, it is a positive constant and can be switched as a function of the state and time. Moreover, take differentiation on both sides of Equation (14) at the same time to get as

$$\dot{s}=\lambda (z_2-{\dot{\theta }}_d)-{\ddot{\theta }}_d+b_{0}u+z_3$$

(14)

According to the conventional sliding model design method, exponential reaching law is used to design the sliding mode controller.

$$\dot{s}=-ks-\epsilon \mathrm{sgn}(s)$$

(15)

where k is the control law factor, it is a positive constant, $\epsilon$ is the forward gain value, it is switched by the $\mathrm{sgn}(s)$ function

$$\mathrm{sgn}(s)=\{\begin{array}{c}\begin{array}{cc}1& \begin{array}{cc}\mathrm{if}& s>0\end{array}\end{array}\\ \begin{array}{cc}-1& \begin{array}{cc}\mathrm{if}& s<0\end{array}\end{array}\end{array}$$

(16)

The conventional sliding mode based SMADRC control law can be obtained from equations (15) and (16)

$$u={\displaystyle \frac{-\lambda (z_2-{\dot{\theta }}_d)+{\ddot{\theta }}_d-ks-\epsilon \mathrm{sgn}(s)-z_3}{b_0}}$$

(17)

The control law is composed of sliding mode control law $b_0^{-1}(-\lambda (z_2-{\dot{\theta }}_d)+{\ddot{\theta }}_d-ks-\epsilon \mathrm{sgn}(s))$ and disturbance compensator $-z_3/b_0$ . Sliding mode control action plays an active role when there is a large difference between the actual angle and desired angle, for example, in the case of system start-up. When the system is suddenly disturbed, the disturbance compensator will improve the dynamic quality according to the observed disturbance by ESO.

Modified sliding mode control law design

As can be seen from equation (17) that $\mathrm{sgn}(s)$ function is used in conventional sliding mode control design procedure, it will cause chattering when the state reaches the sliding surface. For the shift actuator controller by motor, the chattering will cause the motor to rotate forward and reverse continuously, which not only can not control the position precisely but also shortens the lifetime of the shift motor even damages the motor and gears. So continuity method based on relay characteristic is adopted to reduce the chatting. The continuous function $\theta (s)$ is used to replace the $\mathrm{sgn}(s)$ function as

$$\theta (s)={\displaystyle \frac{s}{|s|+\delta }}$$

(18)

Where $\delta$ is a positive constant, and the control law based on the modified sliding control is

$$u={\displaystyle \frac{-\lambda (z_2-{\dot{\theta }}_d)+{\ddot{\theta }}_d-ks-\epsilon \theta (s)-z_3}{b_0}}$$

(19)

Remark 2:

As can be seen from the above equation, there are four parameters that can be adjusted to obtain a good control performance. As for the sliding mode control, a smaller $\lambda$ value results in slower response, and a bigger one will cause overshoot. But the value of $\lambda$ has little effect on the MSMADRC. A bigger k will decrease the time from the initial position of the system state variable to the sliding surface, but it will reduce the existing area of the sliding mode. $\epsilon$ determines the speed of the system state variable to the sliding surface. A smaller value causes a long time to converge to the sliding surface but a too big value will lead to obvious jitter around the switching surface. To achieve a good control effect, increasing the $\epsilon$ value should decrease the k value at the same time. After conducting many simulation studies on different $\delta$ , we conclude that too little value will cause oscillation both in system output and control action, however a bigger $\delta$ has little effect on the system performance.

Stability analysis

Lemma 1 ³⁰ : As for $V(t):[0,\mathrm{\infty })\in R$ , if $\dot{V}(t)\le -\alpha V+f$ , $\mathrm{\forall }t\ge t_0\ge 0$ , then $V(t)$ satisfies

$$V(t)\le e^{-\alpha (t-t_0)}V(t_0)+{\int }_{t_0}^t{e}^{-\alpha (t-\tau )}f(\tau )d\tau$$

According to the designed sliding mode surface, the Lyapunov function $V(t)=1/2s^2$ is selected for the stability analysis

$$\dot{V}(t)=s\dot{s}=s(\lambda \dot{\hat{e}}+\ddot{\hat{e}})=s(\lambda \dot{\hat{e}}+b_{0}u+z_3-{\ddot{\theta }}_d+l_2{e}_1)$$

(20)

$e_1=y-\hat{y}$ is determined according to the observation error of the ESO. Combining equation (13) and (20), we can get

$$\dot{V}(t)=s(\lambda \dot{\hat{e}}-\lambda \dot{\hat{e}}+{\ddot{\theta }}_d-ks-\epsilon s/(|s|+\delta )-z_3+z_3-{\ddot{\theta }}_d+l_2{e}_1)$$

(21)

Simplifying it furtherly, we get as

$$\dot{V}(t)=s\dot{s}=-\left(k+{\displaystyle \frac{\epsilon }{|s|+\delta }}\right)s^2+l_2{e}_1$$

(22)

where $l_2$ , k, $\epsilon$ and are positive constants, supposing ${\mathrm{\Delta }}_{max}\ge |l_2{e}_1|$ , it can be obtained as

$$\begin{array}{rl}\dot{V}(t)\le & -k_t{s}^2+{\displaystyle \frac{1}{2}(s^2+{\mathrm{\Delta }}_{max}^2)=-\left(k_t-{\displaystyle \frac{1}{2}}\right)s^2+{\displaystyle \frac{1}{2}{\mathrm{\Delta }}_{max}^2}}\\ =& -(2k_t-1)V(t)+{\displaystyle \frac{1}{2}{\mathrm{\Delta }}_{max}^2}\end{array}$$

(23)

where $k_t=k+\epsilon /(|s|+\delta )$ . According to Lemma 1, the solution to the inequality

$$\stackrel{\cdot }{V(t)}\le -(2k_t-1)V(t)+{\mathrm{\Delta }}_{max}^2/2$$

with definition $\alpha =2k_t-1$ , $f={\mathrm{\Delta }}_{max}^2/2$ as

$$\begin{array}{rl}V(t)\le & e^{-\alpha (t-t_0)}V(t_0)+{\displaystyle \frac{1}{2}{\mathrm{\Delta }}_{max}^2{\int }_{t_0}^t{e}^{-\alpha (t-\tau )}d\tau }\\ =& e^{-(2k_t-1)(t-t_0)}V(t_0)+{\displaystyle \frac{1}{2(2k_t-1)}{\mathrm{\Delta }}_{max}^2(1-e^{-\alpha (t-t_0)})}\end{array}$$

When $k_t>1/2$

$$\underset{t\to \mathrm{\infty }}{lim}V(t)\le {\displaystyle \frac{1}{2(2k_t-1)}{\mathrm{\Delta }}_{max}^2}$$

(24)

When $t\to \mathrm{\infty }$ , $V(t)={\mathrm{\Delta }}_{max}^2/(2(2k_t-1))$ , and it can be concluded from equation (23) and (24) that $\stackrel{\cdot }{V}\le 0$ will be satisfied. The stability and convergence rate of the system depend on the SMC parameters k, $\epsilon$ and $\delta$ as well as the bandwidth of ESO.

Simulation results

Simulations are implemented to verify the performance of proposed MSMADRC algorithm for motor position control. On the basis of the model and controller proposed above, motor model and controllers are developed on MATLAB/Simulink platform. The parameters of the selected motor can be refered in Table 1. Based on many simulation tests, the MSMADRC controller parameters are set as $b_0=6250$ , ${\omega }_0=50$ , $\lambda =8$ , $k=30$ , $\delta =5$ . At the same time, to show more clearly the superiority of the designed MSMADRC, the conventional SMC and ADRC are compared with the MSMADRC, they are using the control action as equations (17) and (11) respectively. What's more, in order to ensure the reliability of the simulation results, the controller parameters of ADRC and SMC are set to the same as those of the MSMADRC.

Table 1.

Parameters of the motor.

Parameter	Value
Supply voltage	12 V
Armature inductance	0.0003 H
Armature resistance	0.2 Ω
Back emf coefficient	0.0405 V/(rad/s)
Rotational inertia	0.00016 Kg·m·m
Coefficient of viscous friction	0.001 N·m/(rad/s)
Torque coefficient	0.0405 N·m/A

With these controller settings, the performances of the three methods are simulated. Firstly, a sinusoidal signal input is used to evaluate the tracking response of the three control methods and the simulation results are shown in Figures 4 and 5. It can be seen from the figures that the three control methods can realize the tracking to the sinusoidal signal, and the tracking error is less than 0.01 rad, which indicates that the three control methods have good dynamic tracking performance. However, the local enlarged image shows that the tracking error of traditional ADRC exceeds 0.005 rad and the tracking error of SMC and MSMADRC are both close to 0, which also shows that the combination of ADRC and SMC can effectively reduce the error and improve the control accuracy.

Figure 4.

Tracking response to sinusoidal signal.

Figure 5.

Tracking error to sinusoidal signal.

Secondly, in order to demonstrate the response speed and stability of the system, three control methods are carried out to track the r pulse signal without disturbance. The simulation results are shown in Figures 6 and 7. It can be seen from the local enlarged image that ADRC has the slowest response speed with 1.2 s response time. There is no steady state error but overshoot. For SMC response, it takes about 1 s to the steady state but there is a steady state error. As for the performance of MSMADRC, it only take 0.6 to the steady state as well as there is no steady state error. At the same time, control actions that are the control input torques to the shift motor of SMC and MSMADRC are shown in Figure 8. As mentioned above, for the traditional SMC, the control action will be inevitably chattering due to the presence of $\mathrm{sgn}(s)$ . On the contrary, in the improved MSMADRC method, $\mathrm{sgn}(s)$ is replaced with $\theta (s)$ , which greatly weakens the chattering phenomenon and stabilizes the output of the controller. Therefore, compared with ADRC and SMC, MSMADRC has the advantage of fast response speed and no tracking error.

Figure 6.

Tracking response to square signal.

Figure 7.

Tracking error to square signal.

Figure 8.

Control action to square signal.

Thirdly, in order to verify the disturbance rejection performance of SMADRC, three methods are carried out to track the pulse signal with disturbance. The disturbance is randomly set between [0,1]. The tracking error and control actions are shown in Figures 9 and 10. From the simulation results it can be seen that when there is disturbance in the system, the control error of SMC is the largest And the ability of disturbance rejection of MSMADRC is obviously superior than ADRC and SMC. And the control actions are basically both smooth for ADRC and SMC.

Figure 9.

Perturbed system response to square signal.

Figure 10.

Control action to square signal.

Fourthly, to analyze the methods robustness, now suppose that there exists 100% increment for rotational inertia and coefficient of viscous friction respectively. The corresponding system outputs are shown in Figures 11–14. The model mismatched system parameters used in simulation are shown in Table 2. It can be seen from the results that for the traditional SMC method, there are always oscillations because of the existence of $\mathrm{sgn}(s)$ . Compared with ADRC, the system responses of MSMADRC are precise and stable during the tracking process, which reflects the low dependence on the model for MSMADRC, and also shows the strong robustness of MSMADRC in the model parameters error case.

Figure 11.

Model mismatched system I tracking response to step signal.

Figure 12.

Model mismatched system I tracking error to step signal.

Figure 13.

Model mismatched system II tracking response to step signal.

Figure 14.

Model mismatched system II tracking error to step signal.

Table 2.

Model parameters used in simulations.

Figure	J(Kg·m2)	B(N·m·s)
Figures 11 and 12	$3.2\times {10}^{-4}$	0.001
Figures 13 and 14	$1.6\times {10}^{-4}$	0.0005

It can be seen from the above results that the designed MSMADRC system can accurately and quickly track the setpoint and the performances of disturbance rejection and robustness are superior to the other two methods, which can meet the requirements of rapidity and accuracy for the position control of AMT shift actuator.

Conclusion

1. To improve the accuracy and speed of the shifting gear for agricultural tractor AMT, a gear shift motor control system based on MSMADRC method was presented.

2. To cope with the unknown disturbances, extended state observer was designed by utilizing the advantage of the ADRC. The estimated disturbances and uncertainties were combined with the nonlinear state error feedback and suppressed by designing the sliding mode controller. What's more, smooth control action was obtained by adopting the continuity method based on relay characteristic to reduce chattering.

3. The simulation results showed that the MSMADRC method demonstrated not only higher precision, faster response but also superior robustness and disturbance rejection compared with the traditional ADRC and SMC. The superior performance indicated that MSMADRC algorithm has the promise of reducing the shift time, eliminating the power interruption and improving the shift quality of the agriculture tractor.