Modeling and Backstepping Control of the Electronic Throttle System

Abstract

Electronic throttle is widely used in modern automotive engines. An electronic throttle system regulates the throttle plate angle by using a DC servo motor to adjust the inlet airflow rate of an internal combustion engine. Its application leads to improvements in vehicle drivability, fuel economy, and emissions. In this paper, by taking into account the dynamical behavior of the electronic throttle, the mechanism model is first built, and then the mechanism model is transformed into the state-space model. Based on the state-space model and using the backstepping design technique, a new backstepping controller is developed for the electronic throttle. The proposed controller can make the actual angle of the electronic throttle track its set point with the satisfactory performance. Finally, a computer simulation is performed, and simulation results verify that the proposed control system can achieve favorable tracking performance.

1. Introduction

In recent years, many functions of modern automobiles are shifting from a purely mechanical to an electromechanical implementation. These functions are implemented by using the so-called “x-by-wire” systems, including drive-by-wire and steer-by-wire systems [1]. “X-by-wire” systems act as an interface between the driver and the targeted mechanical subsystem of the vehicle. Now, advanced control strategies, including the data-driven control [2], fuzzy control [3, 4], and neural network control [5, 6], have been widely applied in the process industry and automobile industry, for example, the Tennessee Eastman process [7], the suspension control system [8, 9], the electronic throttle control system [10, 11], and so on. In this paper, we focus on the control strategy of the electronic throttle system, which is one of the important drive-by-wire systems in the automobile industry.

In automotive spark ignition engines, the air coming into the intake manifold and therefore the power generated strongly depend on the angular position of a throttle valve [12]. In traditional systems, the throttle position is actuated by a mechanical link with the accelerator pedal, directly operated by the driver. The traditional mechanical throttle is difficult to achieve the accurate control result. Therefore, the vehicle drivability, fuel economy, and emissions are not satisfactory by using the traditional mechanical throttle. In recent years, new and increasing requirements in terms of emissions control, drivability, and safety have led to the development of electronic throttle system. The electronic throttle is essentially a DC-motor-driven valve that regulates air inflow into the vehicle’s combustion system, and the mechanical linkage between accelerator pedal and the throttle is replaced by an electronic connection [13]. Recently, several control strategies for electronic throttle have been presented. In [10], a new intelligent fuzzy controller is proposed. It can handle the nonlinear hysteretic of electronic throttle. In [11], the controller synthesis is performed in discrete time by solving a constrained time-optimal control problem of the throttle. In [12], a robust position controller for motorized throttle body in automotive applications is presented. Complexity of the control problem is explained and control architecture is also presented. In [13], a process to design the control strategy is proposed for a vehicle with the electronic throttle control and the automatic transmission, and the dynamic programming (DP) technique is also used to obtain the optimal gear shift and throttle opening angular which maximizes fuel economy while satisfying the power demand. In [14], the nonlinear hysteretic characteristic of the electronic throttle is described and the variable structure control method is proposed to control the electronic throttle. In [15], an adaptive control strategy for the electronic throttle is introduced. In [16], an integrated control strategy is proposed, which consists of a proportional-integral-derivative (PID) controller and a feedback compensator for friction and limp-home effects. In [17], a novel nonlinear controller for the electronic throttle valve is presented, which uses the approximate model method and support vector machine (SVM) modeling. Although the abovementioned control methods can achieve the acceptable control performance, these control methods have complex structure and algorithm. As we know, the controller complex design process often leads to the difficulty of its realization in the actual automotive manufacturing industry. Therefore, more attention has been paid to backstepping design technique because of its systematic design and the excellent transient performance of the closed-loop system. Backstepping design technique is a recursive and systematic design scheme first presented by Kanellakopoulos et al. in 1991 [18]. The main idea is to decompose a complex system into multiple small-scale subsystems, then to design recursively control Lyapunov function and virtual controller for each subsystem, and finally obtain the original control law and realize the global regulation and tracking for the controlled system [19–21]. For the systematic design process, backstepping control scheme is easy to be realized, and it has been applied in many cases, such as induction motor [22], chemical process [23, 24], ship course [25], and robot manipulator [26].

Motivated by the advantage of the backstepping design method, this paper investigates the backstepping control problem of the electronic throttle. Since the backstepping design technique is a typical model-based design method, the dynamical model of the electronic throttle is first built in this paper. Based on the proposed dynamical model, the backstepping control design method for the electronic throttle is presented. The proposed backstepping controller can achieve the satisfactory performance; that is, the actual angular of the electronic throttle can track its set point. Finally, a computer simulation is performed, and simulation results verify the effectiveness of the proposed control method.

This paper is organized as follows. Section 2 describes the mathematical model of the electronic throttle. Section 3 designs the electronic throttle controller by using the backstepping method. Section 4 illustrates the simulation results and finally Section 5 shows the conclusion of this paper.

2. Mathematic Model of the Electronic Throttle

There are some symbols in this section. At first, definitions of these symbols are described as follows: 𝜃^∗: Set point of the valve plate angular 𝜃(𝑡): Actual angular of the valve plate 𝜃₀: Static angular of the valve plate 𝜔(𝑡): Angular speed of the valve plate 𝑖_𝑎(𝑡): Armature current 𝑅_𝑎: Armature resistance 𝑈_𝑎(𝑡): Input voltage of the motor 𝑈_𝑏(𝑡): Electromotive force 𝑈_bat: Supply voltage 𝐷(𝑡): Duty cycle of the bipolar chopper 𝑇_𝑒(𝑡): Electromagnetism torque 𝑇_𝑠(𝑡): Return spring torque  𝑇_𝑓(𝑡): Friction torque 𝐾_𝑡: Torque constant 𝐾_𝑠: Elastic coefficient 𝐾_𝑚: Torque compensation coefficient 𝐾_𝑑: Friction coefficient 𝐽: Moment of inertia 𝑗: Gear ratio.

The schematic of a typical electronic throttle control system is shown in Figure 1.

Figure 1 

Electronic throttle control system.

There are a controller, a bipolar chopper, and an electronic throttle body (ETB) in Figure 1. ETB consists of a DC drive powered by the bipolar chopper, a gearbox, a valve plate, a return spring, and a position sensor. When the valve plate angular is regulated, the air inflow into the vehicle’s combustion system can also be regulated. The control objective of the electronic throttle is to control the valve plate angular tracking its set point with the satisfactory performance.

At first, we build the motion equation for the electronic throttle system. The motion equation is 𝑗𝑇_𝑒(𝑡)−𝑇_𝑠(𝑡)−𝑇_𝑓(𝑡)=𝑗²𝐽𝑑𝜔(𝑡).𝑑𝑡 (1)

The relation between current 𝑖_𝑎(𝑡) and input voltage 𝑈_𝑎(𝑡) in the armature circuit is described as 𝑖_𝑎(𝑡)𝑅_𝑎=𝑈_𝑎(𝑡)−𝑈_𝑏(𝑡), (2) where 𝑈_𝑎(𝑡)=𝑈_bat(𝑡),𝑈×𝐷_𝑏(𝑡)=𝐾_𝑡×𝑗×𝜔(𝑡). (3)

By substituting (3) into (2), we have 𝑖_𝑎(𝑡)=𝑈_bat×𝐷(𝑡)−𝐾_𝑡×𝑗×𝜔(𝑡)𝑅_𝑎. (4)

Computation formula of 𝑇_𝑒(𝑡) is 𝑇_𝑒(𝑡)=𝐾_𝑡𝑖_𝑎(𝑡). (5)

By substituting (4) into (5), we get 𝑇_𝑒(𝑡)=𝐾_𝑡𝑈_bat×𝐷(𝑡)−𝐾_𝑡×𝑗×𝜔(𝑡)𝑅_𝑎. (6)

Return spring torque 𝑇_𝑠(𝑡) and friction torque 𝑇_𝑓(𝑡) are 𝑇_𝑠(𝑡)=𝐾_𝑠𝜃(𝑡)−𝜃₀+𝐾_𝑚,𝑇_𝑓(𝑡)=𝐾_𝑑𝜔(𝑡). (7)

By substituting (6) and (7) into (1), we get 𝑑𝜔(𝑡)𝐾𝑑𝑡=−^𝑠𝑗²(𝑡)−𝐾×𝐽×𝜃^2𝑡𝐽𝑅_𝑎+𝐾_𝑑𝑗²𝜔(𝑡)+𝐾×𝐽_𝑡×𝑈_bat𝑗×𝐽×𝑅_𝑎𝐷(𝑡)+𝐾_𝑠𝜃₀−𝐾_𝑚𝑗².×𝐽 (8) Equation (8) is the mechanism model of the electronic throttle.

Defining state variables 𝑥₁(𝑡)=𝜃(𝑡), 𝑥₂(𝑡)=𝜔(𝑡), input variable 𝑢(𝑡)=𝐷(𝑡), and the output variable 𝑦(𝑡)=𝜃(𝑡), (8) can be rewritten as 𝑥̇₁(𝑡)=𝑥₂(𝑡),𝑥̇₂(𝑡)𝐾=−^𝑠𝑗²×𝐽×𝑥₁(𝑡)−𝐾_𝑡²𝐽𝑅_𝑎+𝐾_𝑑𝑗²𝑥×𝐽²(𝑡)+𝐾_𝑡×𝑈_𝑏𝑎𝑡𝑗×𝐽×𝑅_𝑎𝑢(𝑡)+𝐾_𝑠𝜃₀−𝐾_𝑚𝑗²,𝑦(𝑡)×𝐽=𝑥₁(𝑡). (9)(10)(11)

Equations (9)–(11) are the state-space model of the electronic throttle.

3. Backstepping Control Design and Stability Analysis

The control objective of this paper is to design a backstepping control system such that the output 𝑦(𝑡) of the system shown in (11) to track its set point 𝑥_𝑑 asymptotically. The proposed backstepping control procedure is described step by step as follows.

Step 1. For the position-tracking objective, define the tracking error as 𝑧₁(𝑡)=𝑥₁(𝑡)−𝑥_𝑑. (12)
Taking 𝛼(𝑡) as a virtual control and defining 𝑧₂(𝑡)=𝑥₂(𝑡)−𝛼(𝑡), (13)
consider the following Lyapunov function candidate: 𝑉₁(𝑡)=12𝑧²₁(𝑡). (14)
The time derivative of 𝑉₁(𝑡) is 𝑉̇₁(𝑡)=𝑧₁(𝑡)𝑧̇₁(𝑡). (15)
From (12) and (13), we obtain 𝑧̇₁(𝑡)=𝑥̇₁(𝑡)=𝑥₂(𝑡)=−𝑧₁(𝑡)+𝑧₁(𝑡)+𝑥₂(𝑡)−𝛼(𝑡)+𝛼(𝑡)=−𝑧₁(𝑡)+𝑥₂(𝑡)−𝛼(𝑡)+𝑧₁(𝑡)+𝛼(𝑡)=−𝑧₁(𝑡)+𝑧₂(𝑡)+𝑧₁(𝑡)+𝛼(𝑡). (16)
Choosing the virtual control function 𝛼(𝑡)𝛼(𝑡)=−𝑧₁(𝑡). (17)
By substituting (17) into (16), we have 𝑧̇₁(𝑡)=−𝑧₁(𝑡)+𝑧₂(𝑡). (18)
By using (18) and (15), we get 𝑉̇₁(𝑡)=𝑧₁(𝑡)𝑧̇₁(𝑡)=𝑧₁(𝑡)−𝑧₁(𝑡)+𝑧₂(𝑡)=−𝑧²₁(𝑡)+𝑧₁(𝑡)𝑧₂(𝑡). (19) From (19), we know if 𝑧₂(𝑡) is equal to zero, the time derivative of 𝑉₁(𝑡) will be smaller than or equal to zero. If 𝑉̇₁(𝑡)≤0, we know that 𝑧₁(𝑡) will converge to zero, and 𝑥₁(𝑡) will converge to the set point 𝑥_𝑑. Therefore, in the next step, we will design a controller 𝑢(𝑡) to make 𝑧₂(𝑡) converge to zero.

Step 2. Consider the following Lyapunov function candidate 𝑉₂(𝑡): 𝑉₂(𝑡)=12𝑧²₂(𝑡)+𝑉₁(𝑡). (20) The time derivative of 𝑉₂(𝑡) is 𝑉̇₂(𝑡)=𝑧₂(𝑡)𝑧̇₂(𝑡)+𝑉̇₁(𝑡)=𝑧₂(𝑡)𝑧̇₂(𝑡)−𝑧²₁(𝑡)+𝑧₁(𝑡)𝑧₂(𝑡) (21)
From (10), (13), and (17), we have 𝑧̇₂(𝑡)=𝑥̇₂(𝑡)−𝛼̇(𝑡)=𝜇₀𝑢(𝑡)−𝜇₁𝑥₁(𝑡)−𝜇₂𝑥₂(𝑡)+𝐹−𝛼̇(𝑡), (22) where 𝜇₀=𝐾_𝑡/(𝑗×𝐽×𝑅_𝑎), 𝜇₁=𝐾_𝑠/(𝑗²×𝐽), and 𝜇₂=𝐾_𝑏×𝐾_𝑡/𝐽𝑅_𝑎+𝐾_𝑑/𝑗²×𝐽 and 𝐹=(𝐾_𝑠𝜃₀−𝐾_𝑚)/(𝑗²×𝐽).
Note that 𝛼̇(𝑡)=−𝑧̇₁(𝑡)=−𝑥̇₁(𝑡)=−𝑥₂(𝑡). (23)
By substituting (23) into (22), we have 𝑧̇₂(𝑡)=𝜇₀𝑢(𝑡)−𝜇₁𝑥₁(𝑡)−𝜇₂𝑥₂(𝑡)+𝐹+𝑥₂(𝑡)=𝜇₀𝑢(𝑡)−𝜇₁𝑥₁(𝑡)−𝑥_𝑑+𝑥_𝑑−𝜇₂𝑥₂(𝑡)−𝛼(𝑡)+𝛼(𝑡)+𝐹+𝑥₂(𝑡)=𝜇₀𝑢(𝑡)−𝜇₁𝑧₁(𝑡)−𝜇₁𝑥_𝑑−𝜇₂𝑧₂(𝑡)−𝜇₂𝛼(𝑡)+𝐹+𝑥₂(𝑡)=𝜇₀𝑢(𝑡)−𝜇₁𝑧₁(𝑡)−𝜇₁𝑥_𝑑−𝜇₂𝑧₂(𝑡)+𝜇₂𝑧₁(𝑡)+𝐹+𝑥₂(𝑡). (24)
Choosing the control function 𝑢(𝑡)𝑢(𝑡)=1𝜇₀𝜇₁−1𝑧₁(𝑡)+𝜇₂−1𝑧₂(𝑡)+𝜇₁𝑥_𝑑−𝜇₂𝑧₁(𝑡)−𝐹−𝑥₂(𝑡). (25)
From (25) and (24), we have 𝑧̇₂(𝑡)=−𝑧₁(𝑡)−𝑧₂(𝑡) (26)
Substituting (26) into (21) results in 𝑉̇₂(𝑡)=𝑧₂(𝑡)𝑧̇₂(𝑡)−𝑧²₁(𝑡)+𝑧₁(𝑡)𝑧₂(𝑡)=𝑧₂(𝑡)−𝑧₁(𝑡)−𝑧₂(𝑡)−𝑧²₁(𝑡)+𝑧₁(𝑡)𝑧₂(𝑡)=−𝑧²₁(𝑡)−𝑧²₂(𝑡)≤0. (27) Equation (27) means that 𝑉̇₂(𝑡)≤0. Therefore, it is obtained that the variables 𝑧₁(𝑡) and 𝑧₂(𝑡) converge to zero; that is, the output 𝑦(𝑡)=𝑥₁(𝑡) of the system shown in (11) can track its set point 𝑥_𝑑 asymptotically.

4. Simulation Experiments

In this section, we perform simulation experiment to confirm the effectiveness of the proposed backstepping control. The values of the parameters in the electronic throttle system are given in Table 1. All these parameters are obtained from the experiment platform of the electronic throttle in our laboratory.

Table 1 

Parameter values.

Simulation results are shown in Figures 2–5. Figure 2 shows the set point of the electronic throttle angular, that is, 𝑥_𝑑. Figure 3 shows the input voltage of the DC servo motor. Figure 4 shows the actual angular of the electronic throttle, that is, 𝑥₁(𝑡). Figure 5 shows the actual angular speed of the electronic throttle, that is, 𝑥₂(𝑡). In Figure 2, set point 𝑥_𝑑 is 20 degrees during 0 to 200 seconds. After 200 seconds, 𝑥_𝑑 is increased from 20 to 50 degrees, and after 400 seconds, 𝑥_𝑑 is decreased from 50 to 40 degrees.

Figure 2 

Set point of the valve plate angular.

Figure 3 

Input voltage of the motor.

Figure 4 

Actual angular of the valve plate.

Figure 5 

Angular speed of the valve plate.

At 200 seconds, 𝑥_𝑑 is increased. In order to increase the actual angular 𝑥₁(𝑡), the input voltage should be increased. From Figure 3, at first, the input voltage is increased when time is 200 seconds. Increase of the input voltage 𝑢(𝑡) leads to the increase of the angular speed 𝑥₂(𝑡), which is shown in Figure 5. When the angular speed 𝑥₂(𝑡) is increased, the actual angular of the electronic throttle 𝑥₁(𝑡) will be also increased, which is shown in Figure 4. Therefore, the actual angular 𝑥₁(𝑡) is regulated to track its set point. When the dynamical regulation process is finished, the input voltage 𝑢(𝑡) is a new stable value, and 𝑥₂(𝑡) is controlled to zero.

At 400 seconds, 𝑥_𝑑 is decreased. When 𝑥_𝑑 is decreased, in order to decrease the actual angular 𝑥₁(𝑡), the input voltage should be decreased. From Figure 3, at first, the input voltage is decreased when time is 400 seconds. For the decrease of the input voltage, the angular speed 𝑥₂(𝑡) is also decreased, which is shown in Figure 5. When the 𝑥₂(𝑡) is decreased, the actual angular of the electronic throttle 𝑥₁(𝑡) will be decreased, which is shown in Figure 4. Therefore, the actual angular 𝑥₁(𝑡) is regulated to track its set point. When the dynamical regulation process is finished, the input voltage is a new stable value, and 𝑥₂(𝑡) is controlled to zero.

From Figures 2–5, we know that the dynamical process of the simulation experiment is right for the electronic throttle, and the tracking performance is also satisfactory.

5. Conclusions

In this paper, the model and control method on the electronic throttle is considered. The dynamical mechanism model and state-space model of the electronic throttle are presented. Based on the state-space model, a backstepping controller is developed. The proposed controller can make the actual angular of the throttle plate track its set point with the satisfactory performance. Simulation experiment is implemented, and the simulation results confirm the effectiveness of the proposed control method.

Conflict of Interests

None of the authors of the paper has declared any conflict of interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61074014), Natural Science Fundamental of Liaoning Province (201102089), Program for Liaoning Excellent Talents in University (LJQ2011062), and the State Key Laboratory of Synthetic Automation for Process Industries.