Research on clutch position control based on backstepping method for small-sized tractors

Abstract

Tractors installed automated manual transmission can avoid discomfort for the drivers because of excessive limbs intervention as well as improve shift quality. Automatic clutch control is critical for the performance of the automated manual transmissions. For a good operation, precise and fast control of clutch position is mandatory. In order to meet these requirements, an enhanced strategy focused on the clutch is presented adopting a simple tracking control method based on the detailed models in this study. Clutch models including DC motor model and mechanical actuator model are established and transformed to a controllable model. On the basis of the control model, a clutch position tracking control scheme is proposed composed of motor control circuit and motor angle tracking controller designed according to the backstepping method. Simulations are carried out in comparation with the internal model control method, and the results demonstrate the effectiveness of the presented control scheme by superior rapidity and accuracy of the controller response for the clutch position tracking system.

Keywords

Clutch actuator position tracking backstepping method tractor

Introduction

Effective automotive transmission can not only alleviate driver work load but also reduce fuel consumption and emissions. In last few years, some new transmission systems were presented for vehicles or agricultural tractors. Clutch is an important component to carry out the function of the transmission, it is designed to engage and release the engine power to the wheels.¹ Clutches used for tractor automated manual transmissions have different forms and working modes, and the control scheme of the cluster position tracking has a significant effect on the performance of gear shift.² Because the accurate clutch position decides the coordination between the clutch and the shafts, which means that fast and smooth capability of clutch control is fundamental to realize the automotive transmission.

In the referenced literatures, different clutch architectures have been proposed and different driving sources for the clutch actuator are designed for automated manual transmissions.^3–5 Electro mechanically actuator taking motor as power source is widely adopted in commercial cars or small and mid-sized vehicles. Hydraulic manipulator or electro hydraulic manipulator is an alternative drive form for the automatic mechanical transmission system not only used on the wheeled agricultural tractors but also on the heavy-duty automobile vehicles.^6–9 Moreover, electro pneumatic driven clutch actuators are also used for the shift gear control in heavy-duty vehicles.^10,11 To track the desired clutch engagement point, clutch piston displacement, clutch chamber pressure and transmission shaft torque are the mostly used control modes. For the torque control mode, it is complex because the torque transmission capability changes with temperature and humidity, what's more, the torque sensor is expensive. As far as pressure control is concerned, it is adopted rarely because it is difficult to sense the internal pressure of the actuator.

As we know, clutch is responsible for connecting or releasing the propulsion sources from the wheels. The position tracking control of clutch is important for the performance of the automated manual transmissions. Therefore, obtaining a precise and steady clutch control has been the challenging issue for the engineers and researchers, and some valuable achievements have been acquired in recent years. As a simple control method, proportional integral derivative (PID) has been adopted for a wide range of fields including the clutch control system due to its convenient implementation. In Ref. 12, a PID controller was designed for an electric clutch actuator used on agricultural tractor, the actuator model was denoted as a first-order stable process with integrator and time delay. In Ref. 13, a clutch position tracking control system was proposed included a switchable controller and a neuron PID controller. The switchable controller was responsible for decreasing the tracking error, and the PID controller was in charge of controlling the valves. However, it is difficult to obtain satisfied performances because the dynamics of the clutch system is too complex. Model predictive control method is also a popular method used for the clutch position control.¹⁴ To realize precise clutch position control, model predictive control method was presented on the basis of the estimation of resistance torque as well as corrected clutch wear.¹⁵ Tamas¹⁶ established a nonlinear model of the electromechanical actuator and designed a gain-scheduled MPC and an LPV-H ∞ controller to realize the position control of the actuator. Sliding mode control is simple and insensitive to external noise and unmodeled dynamic parameter perturbation, it is suitable for controlling the nonlinear systems. But chattering phenomenon must be avoided by designing enhanced control law when it is used.¹⁷ In Ref. 18, an angle feedback-based adaptive sliding mode controller was designed to track the reference trajectories and unscented Kalman filter was adopted to estimate the rotational speed and angular acceleration of the drive motor. In Ref. 19, an adaptive sliding mode controller for the torque control was designed based on the estimated disk friction coefficient in order to track the desired actuator position. Li et al.²⁰ proposed a terminal sliding mode with a feed-forward compensation control scheme to enhance the tracking performance of the clutch position control system, and designed uncertainty observer to estimate the model uncertainties. Moreover, variable structure control is also a promising method used extensively in various fields. In this control structure, more than one controller is designed to improve system performances, such as set point tracking and disturbance rejection. For clutch position control systems, Gao et al.²¹ designed linear and nonlinear controllers on the basis of a two degree of freedom control structure respectively. For a hybrid dual-clutch transmission, Laukenmann et al.²² proposed a two degree of freedom control structure in which a position tracking controller as well as a pressure tracking controller were designed. To track position of automatic clutch, a current-position double control method was proposed based on Udwadia–Kalaba theory.²³ What's more, some other superior methods for clutch actuator control were proposed. For example, Wang and Liu²⁴ designed an adaptive fuzzy iterative control method to decrease tracking error between desired position and actual position of the clutch engagement. In Ref. 2, a robust control scheme including three controllers was proposed based on a high-order model to provide precise and fast response for the tractor clutch actuator. For an electromechanical automatic clutch actuator driven by brushless direct current motor, Shi et al.²⁵ proposed a robust position control system based on robust output feedback control method, which included a high gain observer and recursive controller. At the same time, some methods based on deep learning or optimizing theory were proposed for the clutch position control systems,^26,27 but these methods are not suitable because of the limited hardware resource and computation ability. In practice, a method that is simple and easy to implement on the hardware is favorite, after all, the control for the clutch position is repetitive and frequent.

These proposed control strategies significantly improved the tracking quality for automatic clutch position tracking control. And it is important that these results will be beneficial references to design the clutch control for the small-sized tractor although the research on control of clutch for tractor is less than that of other vehicles. It can be seen from above that there are different clutch architectures and driving sources designed for clutch actuator, and different methods have been presented for the clutch position control. Because shift is a repetitive action that may be along with disturbance and uncertainty. Whatever control method is adopted, accurate and rapid tracking performance with robustness is the essential need for the control. Efficient driving model as well as precise and fast position tracking method are the difficulties for the actuator design. The purpose of this study is to design a position tracking system with simple control strategy but excellent performance for electric actuator used for agricultural tractor. By adopting a mature electro mechanically clutch system, a detailed clutch model is established including drive motor and mechanical actuator. Based on the detailed models, a clutch position tracking control scheme is proposed composed of motor angle controller according to the backstepping method and motor control circuit. Utilizing the advantage of recursive design method from the front to the back of the back-stepping control,²⁸ the designed clutch position tracking controller can realize the global adjustment, especially the robust performance for the interference or uncertainty that does not meet the matching conditions. What's more, the method is more suitable for online control with little calculation time. The simulation is carried out and compared the presented method with the internal model control method, by which the advantages of accuracy, disturbance rejection, and accuracy of the proposed method are demonstrated. For clear illustration, organization of this paper is arranged as: Clutch actuator description is introduced in Section 2. Clutch position tracking control strategy is detailed in Section 3. Effectiveness of the presented control scheme is exhibited in Section 4 and some conclusions are drawn in the last section.

Clutch actuator description

Structure of clutch actuator

The electrically controlled clutch is a set of automatic control system adapted on the basis of the mechanical clutch. It can automatically accomplish the operations needed in the process of start, shift, and motion. Automatic clutch system is mainly composed of sensors, ECU control unit, and clutch actuator as shown in Figure 1. Motor drive is characterized by simple structure, cheap and convenient control, so motor is adopted as driving source.^29,30 Because the clutch driving mechanism is designed to convert the rotational motion of the DC motor to the linear motion of the actuator, the mode of worm gear and worm drive is employed in this paper. The transmission system of the actuator is composed of one-stage worm gear pair and two-stage planetary gear pair.¹⁵ The torque of the motor is delivered to the worm gear pair and to the planet gear pair through the worm gear pair sequentially and to the separation bearing through the two-stage of planetary gear pair finally, by which the clutch engagement or separation can be realized. For this transmission system, there are advantageous aspects of high transmission efficiency, fast response speed, and high reliability. To achieve accurate clutch engagement control, the position of the clutch must be obtained in real-time. Because electric actuator is adopted, motor rotation position can be used to express the engagement point of the clutch. The angle position of the worm gear is employed to represent the engagement point of the clutch because there is linear relationship between the clutch engagement position and angle of the worm gear. Angle sensor installed on the worm shaft is used to obtain the clutch engagement position. The ECU sends an execution signal to the clutch actuator after calculating the data obtained by the sensors. The motor drives the actuator to engage or release the clutch. The tracking of the engagement point controlled by ECU directly affects the life of the clutch as well as the smoothness of starting and gear shift.

Figure 1.

Schematic diagram for the clutch actuator.

Model of clutch system

As the driving source of the actuator, the permanent magnet DC brush motor is selected with rated voltage 12 V, rated power 100 W, and rated speed 1600 R/min. It is characterized by small size and good reliability. To establish the model of the motor, the diagram of DC motor is given in Figure 2.

Figure 2.

The schematic diagram of DC motor.

In the light of Kirchhoff’s voltage law and Newton’s law of mechanics, the voltage balance equation of the motor and motion equation is obtained as follows

{\begin{matrix} e_{i} = e_{m} + L_{m} {\dot{i}}_{l} + R_{l} i_{l} \\ T = T_{o} + I_{m m} {\dot{ω}}_{m o} + d_{m} ω_{m o} \end{matrix}

(1)

Where

e_{i}

denotes the input voltage to DC motor,

i_{l}

denotes the current through the circuit,

L_{m}

denotes the inductance;

R_{l}

denotes the armature resistance,

e_{m}

denotes the back electromotive force of the DC motor,

T

denotes the torque of the motor,

I_{m m}

denotes the moment inertia of the motor rotor,

d_{m}

denotes the damping coefficient of the motor rotor.

T_{o}

and

ω_{m o}

are the output torque and speed of the motor, respectively.

According to the principle of DC motor, the input voltage, back electromotive force and total torque of motor are expressed as

{\begin{matrix} e_{i} = D_{c} U_{b} \\ e_{m} = C_{e} ω_{m o} \\ T = C_{T} i_{l} \end{matrix}

(2)

Where

D_{c}

is the PWM duty cycle,

U_{b}

is the supply voltage,

C_{e}

is the coefficient of back electromotive force,

C_{T}

is the torque coefficient. Substituting Equation (2) into Equation (1), the mathematical model of the motor can be obtained as

{\begin{matrix} D_{c} U_{b} = c_{e} ω_{m o} + L_{m} {\dot{i}}_{l} + R_{l} i_{l} \\ T_{o} = C_{T} i_{l} - I_{m m} {\dot{ω}}_{m o} - d_{m} ω_{m o} \end{matrix}

(3)

What's more, according to the principle of torque transfer and the mechanical part of the actuator, we can get

T_{o} = I_{a} {\dot{ω}}_{m o} + d_{a} ω_{m o} + T_{c}

(4)

Where

I_{a}

denotes the equivalent moment of inertia obtained by integrating the moment of inertia of all the components.

d_{a}

denotes the equivalent damping coefficient obtained by integrating the damping coefficient of all the components.

I_{a} = {\begin{matrix} I_{2} + \frac{1}{η_{23}} [\frac{I_{3} + I_{4}}{i_{23}^{2}} + \frac{Q m_{5} r_{6}^{2}}{i_{23} i_{26}} + \frac{Q I_{5}}{i_{23} i_{25}} + \frac{1}{η_{46}} (\frac{I_{6} + I_{7}}{i_{23} i_{46} i_{27}} + \frac{Q I_{8}}{i_{23} i_{46} i_{28}} + \frac{Q m_{8} r_{9}^{2}}{i_{23} i_{46} i_{29}} + \frac{1}{η_{79}} \frac{I_{9} + \frac{m_{10}}{2} L^{2} + m_{11} L^{2}}{i_{23} i_{46} i_{79} i_{29}})], (ω_{m o} > 0) \\ I_{2} - η_{32} [\frac{I_{3} + I_{4}}{i_{23}^{2}} + \frac{Q m_{5} r_{6}^{2}}{i_{23} i_{26}} + \frac{Q I_{5}}{i_{23} i_{25}} + η_{64} (\frac{I_{6} + I_{7}}{i_{23} i_{46} i_{27}} + \frac{Q I_{8}}{i_{23} i_{46} i_{28}} + \frac{Q m_{8} r_{9}^{2}}{i_{23} i_{46} i_{29}} + η_{97} \frac{I_{9} + \frac{m_{10}}{2} L^{2} + m_{11} L^{2}}{i_{23} i_{46} i_{79} i_{29}})], (ω_{m o} < 0) \end{matrix}

(5)

I_{i}

deontes the moment of inertia of the component i,

η_{i j}

denotes the transfer efficiency from part i to part j, Q denotes the amount of planetary gears,

i_{j k}

denotes the transmission ratio between part j and part k,

m_{i}

denotes the mass of the component i, L denotes the length from the output shaft to the throw out bearing.

d_{a} = {\begin{matrix} d_{1} + \frac{d_{2}}{η_{23} i_{23}^{2}} + \frac{d_{b} \cdot L^{2} + d_{3}}{η_{23} η_{46} η_{79} i_{23} i_{46} i_{79} i_{29}}, (ω_{m o} > 0) \\ d_{1} - \frac{η_{32} d_{2}}{i_{23}^{2}} - \frac{η_{32} η_{64} η_{97} (d_{b} \cdot L^{2} + d_{3})}{i_{23} i_{46} i_{79} i_{29}}, (ω_{m o} < 0) \end{matrix}

(6)

Where

d_{1}

denotes the damping coefficient of the input shaft bearing,

d_{2}

denotes the damping coefficient of cochlear axle bearing,

d_{3}

denotes the damping coefficient of the output shaft bearing,

d_{b}

denotes the damping coefficient of the release bearing. The

T_{c}

in Equation (4) is expressed as

T_{c} = {\begin{matrix} F_{s} (x_{b}) p / (η_{23} η_{46} η_{79}), (ω_{m o} > 0) \\ η_{32} η_{64} η_{97} F_{s} (x_{b}) p, (ω_{m o} < 0) \end{matrix}

(7)

Where

F_{s}

is release force of the clutch, it is related to the displacement of the release bearing.

p = L / (i_{23} i_{46} i_{79})

denotes the transmission ratio of the actuator. The relation is

{\dot{x}}_{b} = p ω_{m o}

x_{b}

is the displacement of the release bearing.

Clutch position tracking control design

In general, hierarchical control is the mostly used method to control the clutch system. The upper control is to adjust the speed of the clutch. It is an optimal clutch engagement or release law obtained in the light of the operation intention, running state and so on. The lower control is to track the desired position obtained from the upper controller. The primary goal of the clutch position tracking system is to drive the mechanical displacement accurately to track the input displacement command. For the clutch position tracking, it is servo control problems in essence, most of which employ position, velocity and current three loop control scheme. In the proposed control scheme, the speed loop will not be constructed because it is not convenient to install photoelectric encoder in the clutch. But excellent control performance can also be obtained by designing an outstanding clutch position tracking controller. The proposed clutch position tracking control system structure is given in Figure 3.

Figure 3.

Structure of clutch position tracking control system.

Motor angle control circuit

The angle control of the motor can effectively adjust the engagement point of the clutch. The precision of the current control is the key to tracking the desired clutch engagement point accurately. In this paper, pulse width modulation (PWM) technology is employed to adjust the current through the DC motor. The voltage applied on the motor is the product of the supply voltage and PWM duty cycle. The supply voltage is provided from the battery on the tractor. Changing the duty cycle of PWM can control the current through the motor. The designed current control circuit is given in Figure 4. To control the current precisely, two essential modules are designed. One is the motor current control module which is composed of relay control circuits and PWM modulation circuits. Where Relay 4 is responsible for controlling the power for the H-bridge. It is composed of two triodes Q16 and Q17 and is controlled by the PTB3 pin of the microprocessor. Relay 5 plays a part in supplying power for the DC motor, it is controlled by the PTB4 pin of the microprocessor. The H-bridge circuit is composed of four N-channel enhanced MOSFETs Q19 to Q22. When the tubes Q19 and Q22 are on and Q20 and Q21 are off, the motor is driven to rotate forward. When the tubes Q19 and Q22 are off and Q20 and Q21 are on, the motor is driven to rotate backward. Driver chip HIP4082 is selected and four pins AHO, BHO, ALO and BLO are used to control the four MOSFETs. The other is the current sensing module. In this module, a current sampling resistance R39 with resistance value 5mR is adopted to obtain the current through the motor. After processed by the designed amplifier, the sampled current value is sent to the input pin ADP8 of microprocessor.

Figure 4.

Schematic diagram of current control.

Clutch position tracking controller design

The mathematical models of the actuator and driving motor have been established in Equation (4) and Equation (3), respectively. According to the two equations, the clutch position control model is expressed as follow

{\begin{matrix} D_{c} U_{b} = c_{e} ω_{m o} + L_{m} {\dot{i}}_{l} + R_{l} i_{l} \\ C_{T} i_{l} = I_{t} {\dot{ω}}_{m o} + c_{l} ω_{m o} + T_{c} \end{matrix}

(8)

Where

I_{t} = I_{a} + I_{m m}

，

c_{l} = d_{a} + d_{m}

. Duty cycle of PWM

D_{c}

is regarded as the controlled variable and the angle of the motor

θ

is regarded as the system output which is also treated indirectly as the displacement of the release bearing

x_{b}

At the same time, $x_{1} = θ$ ， $x_{2} = ω_{m o}$ ， $x_{3} = i_{l}$ are designed as the state variable, and the state equation of the clutch actuator used for the controller design may be described as follow

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = (- c_{l} / I_{t}) x_{2} + (C_{T} / I_{t}) x_{3} - T_{c} / I_{t} \\ {\dot{x}}_{3} = (- c_{e} / L_{m}) x_{2} - (R_{l} / L_{m}) x_{3} + (U_{b} / L_{M}) D_{c} \end{matrix}

(9)

{\begin{matrix} \dot{x} (t) = A x (t) + B u (t) + C T_{c} (x (t)) \\ y = D x (t) \end{matrix}

(10)

Where,

A = [\begin{matrix} 0 & 1 & 0 \\ 0 & - c_{l} / I_{t} & C_{T} / I_{t} \\ 0 & - C_{e} / L_{m} & - R_{l} / L_{m} \end{matrix}], B = [\begin{matrix} 0 & 0 & U_{b} / L_{m} \end{matrix}], C = [\begin{matrix} 0 & - / I_{t} & 0 \end{matrix}], D = [\begin{matrix} 1 & 0 & 0 \end{matrix}]

As designed in the control scheme, DC motor is adopted as driving source for the clutch mechanical actuator. As we know, the motor control is sensitive to external load and it is difficulty to be regulated accurately. But the requirements of automatic clutch control can be met by designing a good control scheme for the adopted clutch actuator. Backstepping method is insensitive to the system uncertainty and it exhibits good robustness in dealing with the external disturbances.³¹ Therefore, backstepping method is an effective method can be adopted in the design the position or velocity tracking system.³²

As can be seen from state equation described in Equation (9), the mathematical model for the automatic clutch is a third order system. According to the principle of backstepping control method, the design procedure of backstepping controller for the automatic clutch system may involve three steps. Supposing the final desired point of the clutch engagement is defined as $x_{1 d}$ and the control error is $e = x_{1} - x_{1 d}$ .

Step 1: Intermediate control variable $x_{2 d}$ is designed to make the error $e = x_{1} - x_{1 d}$ to zero.

\dot{e} = {\dot{x}}_{1} - {\dot{x}}_{1 d} = x_{2} - {\dot{x}}_{1 d}

(11)

The Lyapunov function is designed as follow

V_{e} = \frac{1}{2} e^{2}

(12)

The derivative of Equation (12) is

{\dot{V}}_{e} = e \dot{e} = e (x_{2} - {\dot{x}}_{1 d})

(13)

According to Lyapunov's theory, it is obvious that

V_{e} > 0

is true. To satisfy the condition

{\dot{V}}_{e} \leq 0

, the following expression can be selected

x_{2 d} = - k_{1} e + {\dot{x}}_{1 d}

(14)

So a new control objective is introduced and a new error is defined as

δ = x_{2} - x_{2 d}

. In the next step, design procedure is employed to make

δ

to be zero.

Step 2: The second intermediate control variable $x_{3 d}$ is introduced to make the error $δ = x_{2} - x_{2 d}$ to be zero.

\dot{δ} = {\dot{x}}_{2} - {\dot{x}}_{2 d}

(15)

Substituting Equation (9), Equation (14) into Equation (15), we can get

\dot{δ} = {\dot{x}}_{2} - {\dot{x}}_{2 d} = - \frac{c_{l}}{I_{t}} x_{2} + \frac{C_{T}}{I_{t}} x_{3} - \frac{T_{c}}{I_{t}} - {\ddot{x}}_{1 d} + k_{1} \dot{e}

(16)

Another Lyapunov function is again designed as follow

V_{e, δ} = \frac{1}{2} e^{2} + \frac{1}{2} δ^{2}

(17)

Similarly, we can obtain the following equations

{\dot{V}}_{e, δ} = e \dot{e} + δ \dot{δ}

(18)

{\dot{V}}_{e, δ} = e \dot{e} + δ \dot{δ} = - k_{1} e^{2} + δ (e + \dot{δ})

(19)

According to Lyapunov's theory,

V_{e, δ} > 0

and

- k_{1} e^{2} \leq 0

is obviously true. To satisfy the condition

{\dot{V}}_{e, δ} \leq 0

, the inequation

δ (e + \dot{δ}) \leq 0

must be true. Supposing

e + \dot{δ} = - k_{2} δ

, we can get the following equation

e + ({\dot{x}}_{2} - {\dot{x}}_{2 d}) = - k_{2} δ

(20)

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

e - \frac{c_{l}}{I_{t}} x_{2} + \frac{C_{T}}{I_{t}} x_{3} - \frac{T_{c}}{I_{t}} - {\ddot{x}}_{1 d} + k_{1} \dot{e} = - k_{2} δ

(21)

So the second intermediate control variable

x_{3 d}

can be got as

x_{3 d} = \frac{I_{t}}{C_{T}} (\frac{c_{l}}{I_{t}} x_{2} + \frac{T_{c}}{I_{t}} + {\ddot{x}}_{1 d} - e - k_{1} \dot{e} - k_{2} δ)

(22)

A new control objective is produced and the new error is defined as

ξ = x_{3} - x_{3 d}

. The following design procedure will be executed to make

ξ

to be zero.

Step 3: Control variable $D_{c}$ will be obtained by making the error $ξ = x_{3} - x_{3 d}$ to zero. So the last Lyapunov function may be designed

V_{e, δ, ξ} = \frac{1}{2} e^{2} + \frac{1}{2} δ^{2} + \frac{1}{2} ξ^{2}

(23)

And its derivative can be got as

{\dot{V}}_{e, δ, ξ} = e \dot{e} + δ \dot{δ} + ξ \dot{ξ} = - k_{1} e^{2} - k_{2} δ^{2} - ξ ({\dot{x}}_{3} - {\dot{x}}_{3 d})

(24)

In the light of Lyapunov's theory,

- k_{1} e^{2} - k_{2} δ^{2} \leq 0

is obviously true. To satisfy the condition

{\dot{V}}_{e, δ, ξ} \leq 0

, we design the procedure so that the inequation

- ξ ({\dot{x}}_{3} - {\dot{x}}_{3 d}) \leq 0

makes sense.

{\dot{x}}_{3} - {\dot{x}}_{3 d} = - k_{3} ξ

(25)

Substituting Equation (9) and Equation (22) into Equation (25), we can get the equation as

- \frac{C_{e}}{L_{m}} x_{2} - \frac{R_{l}}{L_{m}} x_{3} + \frac{U_{b}}{L_{m}} D_{c} - \frac{I_{t}}{C_{T}} (\frac{c_{l}}{I_{t}} {\dot{x}}_{2} + \frac{{\dot{T}}_{c}}{I_{t}} + {\overset{⃛}{x}}_{1 d} - \dot{e} - k_{1} \overset{\ddot{-}}{e} k_{2} \dot{δ}) = - k_{3} ξ

(26)

So the duty cycle of PWM can be calculated as

D_{c} = \frac{L_{m}}{U_{b}} [\frac{C_{e}}{L_{m}} x_{2} + \frac{R_{l}}{L_{m}} x_{3} + \frac{I_{t}}{C_{T}} (\frac{c_{l}}{I_{t}} {\dot{x}}_{2} + \frac{{\dot{T}}_{c}}{I_{t}} + {\overset{⃛}{x}}_{1 d} - \dot{e} - k_{1} \overset{\ddot{-}}{e} k_{2} \dot{δ}) - k_{3} ξ]

(27)

System stability analysis

Barbalat lemma³³：supposing $x : [0, \infty) \to R$ is first order derivative, and there is limit when $t \to \infty$ . If $\dot{x} (t)$ is uniform continuity as $t \to [0, \infty)$ ，then $\underset{t \to \infty}{l i m} \dot{x} (t) = 0$ .

According to the Equation (24) and Equation (25), the followings can be draw

{\dot{V}}_{e, δ, ξ} = e \dot{e} + δ \dot{δ} + ξ \dot{ξ} = - k_{1} e^{2} - k_{2} δ^{2} - k_{3} ξ^{2}

(28)

- {\dot{V}}_{e, δ, ξ} \leq k_{1} e^{2}

(29)

We can obtain the following inequation by integrating both sides of Equation (29)

\int_{0}^{\infty} k_{1} e^{2} \leq V_{e, δ, ξ} (0) - V_{e, δ, ξ} (\infty)

(30)

Because there is boundedness for the

V_{e, δ, ξ}

, the following conclusion can be draw according to the Barbalat lemma

\underset{t \to \infty}{l i m} e = 0

(31)

In the same way, the deductions can be draw as

\underset{t \to \infty}{l i m} δ = 0

\underset{t \to \infty}{l i m} ξ = 0

. It can be concluded that the error of each subsystem will reach to zero and the designed backstepping controller is asymptotically stable.

Simulation results

To verify the superiority performance obtained by the proposed control scheme for the clutch position tracking system, model of the automatic clutch system and controllers are established. The parameters for the clutch actuator model employed in the simulation are shown in Table 1.²⁴

Table 1.

Parameters for the clutch system.

Parameter	Value	Parameter	Value
$L_{m}$	$0.0003 H$	$m_{8}$	$0.34 k g$
$R_{l}$	$0.2 Ω$	$m_{10}$	$0.112 k g$
$I_{m m}$	$0.0001 k g \cdot m^{2}$	$m_{11}$	$0.312 k g$
$d_{m}$	$0.001 N m s / r a d$	$L$	$45 m m$
$U_{b}$	$12 V$	$Q$	$3$
$C_{e}$	$0.0405 V / (r a d / s)$	$η_{23}$	$0.44$
$C_{T}$	$0.0405 N m / A$	$η_{32}$	$- 0.16$
$I_{2}$	$2.714 * 10^{- 6} k g \cdot m^{2}$	$η_{46}$ ,singledollar\eta _{64}singledollar	$0.9$
$I_{3}$	$4.311 * 10^{- 5} k g \cdot m^{2}$	$η_{79}$ ,singledollar\eta _{97}singledollar	$0.9$
$I_{4}$	$2.251 * 10^{- 5} k g \cdot m^{2}$	$d_{1}$	$0.0007 N m s / r a d$
$I_{5}$	$1.19 * 10^{- 6} k g \cdot m^{2}$	$d_{2}$	$0.0004 N m s / r a d$
$I_{6}$	$3.105 * 10^{- 5} k g \cdot m^{2}$	$d_{3}$	$0.0004 N m s / r a d$
$I_{7}$	$2.216 * 10^{- 5} k g \cdot m^{2}$	$d_{b}$	$21.5 N s / m$
$I_{8}$	$2.224 * 10^{- 6} k g \cdot m^{2}$	$p$	$0.000308$
$I_{9}$	$6.613 * 10^{- 5} k g \cdot m^{2}$	$I_{t}$	$2.727 * 10^{- 6} k g \cdot m^{2}$
$m_{5}$	$0.183 k g$	$c_{l}$	$0.00081 N m s / r a d$

From the final form of the backstepping controller expressed in Equation (27), we can see that there are only three adjustable parameters for the controller $k_{1}$ , $k_{2}$ , and $k_{3}$ , and they all have an effect on the response speed and dynamic property of the clutch position tracking system. The response speed of the system is mainly related to the parameter $k_{1}$ , the larger $k_{1}$ , the faster response to the input of the system. The dynamic property of the system is mainly affected by $k_{2}$ , the larger $k_{2}$ ,the higher tracking accuracy will be obtained. The smoothness of the tracking process is influenced by $k_{3}$ , too small $k_{3}$ will degrade the smoothness of the performance even give rise to shock for the clutch position process, but too large $k_{3}$ is not able to improve the smoothness drastically, inversely, it will weaken the dynamic property of the tracking control system. After a lot of simulation tests, the parameters for the backstepping controller are determined as $k_{1} = 50$ , $k_{2} = 30$ , $k_{3} = 200$ .

With the established clutch position system expressed in Equations. (1) to (10) and the designed back stepping controller shown in Equation (27), simulations are performed by setting $P_{d}$ as a square signal with period 10 s and amplitude 1at time t = 1 s. The output of the backstepping controller is given in Figure 5. The tracking error between the desired angle and the actual angle of the motor is exhibited as Figure 6.

Figure 5.

Output angle of the motor.

Figure 6.

Error of angle tracking.

We can observe that the tracking response of the control system to the square signal demonstrates the rapidly tracking performance and excellent dynamic property. There is no overshoot as well as fast response. With the control action of backstepping controller, the response time of clutch actuator to the desired angle is only 0.3 s and the maximum tracking error is no more than 9 × 10⁻⁵ rad, which can absolutely meet the requirements of fast and accurate control of automatic clutch actuator.

The output of the clutch engagement point controller is exhibited in Figure 7. The responses of the output speed of the motor and the resistance of the actuator during are exhibited in Figure 8. From the two figures we can generally get an idea about the process of engagement and release of clutch. As for the clutch release process, the drive motor controlled by big voltage is increased instantly with high speed at 9 rad/s, which drives the clutch actuator rapidly to separate from the release bearing. According to the characteristics of the diaphragm spring in the clutch and waveform spring, shifting resistance of the actuator is increasing with the increase of release bearing displacement.

Figure 7.

PWM duty cycle.

Figure 8.

Motor velocity and separation resistance of clutch.

What's more, release resistance of the clutch reaches a maximum while the release process will be completed. In the engagement process, the power will be transferred through clutch engagement realized by the reverse rotation of the drive motor. At the same time, we can observe that the output of the controller is smooth which makes for the steady velocity of the motor demonstrated in the Figure 8. what's more, both the control action and the motor velocity vary rapidly with the changed system input. For comparison, internal model control method is introduced and a sinusoidal input is selected as the desired position signal to evaluate the servo performance of the two methods. The outputs of the two control systems are given in Figure 9. We can see from the simulation results that the tracking effect of the two methods to the sinusoidal angle input is acceptable. However, there is almost no tracking error for the proposed method but there is bigger tracking error because of oscillation in the beginning for the slide mode controller, which can be observed from the local enlarged image figure. For the motor velocity in the process exhibited in Figure 10, we can see that there is oscillation in the beginning for the internal model control which results in the angle tracking error in Figure 9. The drastic oscillation may give rise to the rotation back and forth for the drive motor, which degrades the tracking capability for the desired position drastically. More seriously, it will damage the motor and gears in the presence of external disturbance.

Figure 9.

Angle tracking effects of the two controllers.

Figure 10.

Motor velocities of the two controllers.

We can see from the results that the presented position tracking control for the clutch system is able to track the desired angle accurately and quickly compared with the internal model control, which can satisfy the requirements of rapidity and precision for the clutch actuator control system. The advantages of the backstepping method are clearly reflected by virtue of precision and stability obtained through designing virtual control variable for low-order subsystem and gradual backstepping. What's more, for the design principle of the controllers, the back-stepping controller with simple form realizes the global control rapidly compared with the sliding mode controller.

Conclusions

Manual shift for the tractor in the operation process is easy to cause driver fatigue and reduce working efficiency, and even endanger the safety of the operation. Automated manual transmission is the effective solution for this issue and control of the clutch is a critical technical issue for automatic gear shift. In this paper, an enhanced control scheme devoted to the clutch position tracking control was presented to improve the effect of the gear shift. Based on the developed automatic clutch system model, an angle tracking controller was designed according to the back-stepping idea. Concluded from the simulation results, the proposed method and internal model control method could both realize the angle tracking function for the adopted clutch actuator system. But the proposed controller exhibited superior performance with response time 0.3 s and maximum angle tracking error less than 9 × 10⁻⁵ rad. Experiments with the proposed system design scheme can be conducted on a real small-sized tractor in future.

Footnotes

Acknowledgements

This work was financially supported by the Shandong Province Key Technologies R&D Program of China (Grant No.2016GNC112014).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Shandong Province Key Technologies R&D Program of China (2016GNC112014)

ORCID iD

Chengqiang Yin

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