Self-Tuning Backstepping Control with Kalman-like Filter for High-Precision Control of Automotive Electronic Throttle

Abstract

The automotive electronic throttle (AET) control system has been widely applied in modern automotive engines, and accurate control of AET can improve engine performance as well as reduce pollution emissions. However, the noise in the sensor circuit and the variation in automotive driving conditions seriously affect the control performance of the AET system, making controller designing challenging. This paper proposes a self-tuning backstepping control with a Kalman-like filter (SBCKLF) strategy. First, the noise affecting the position sensor measurement is verified to be non-Gaussian by acquiring and processing the noise signal. To eliminate its influence on control precision, a Kalman-like filter is introduced to estimate the real position of the valve. Then, a self-tuning backstepping controller is designed to automatically adapt to the variation in vehicle driving conditions. A self-tuning algorithm based on fuzzy control is used to tune the parameters of the backstepping controller online, so as to optimize the controller performance. Finally, an experimental platform based on dSPACE for the AET control system is built to perform the controller comprehensive test in a real-time environment. Experimental results and performance analysis demonstrate the effectiveness of the proposed SBCKLF strategy. Compared to the best results of other methods, the proposed method reduces the maximum and root mean square tracking errors by 21.65% and the average error by 12.89%. The steady-state and tracking error bounds are controlled to 0.9° and 2.3°, respectively. It also shows that the SBCKLF strategy has the strongest robustness as well as the best response speed.

1. Introduction

With the ongoing global energy crisis and the strict emission regulations issued by countries worldwide, the emission and fuel economy of fuel vehicles and hybrid electric vehicles (HEVs) have attracted increasing research attention [1,2,3]. Whether it is a fuel vehicle or an HEV with gasoline, diesel, natural gas, or other fuels as the main power source, engine performance has a decisive influence on vehicle emission and fuel consumption [4]. The automotive electronic throttle (AET) is a key piece of equipment, which can control air flow into the engine cylinder. Engine efficiency is closely related to the performance of the AET. An advanced AET control system can guarantee the accuracy of fuel/air ratio and economy by optimizing the throttle opening and adjusting the amount of injected fuel. The AET system not only saves the running cost of the engine and reduces pollution emissions but also improves the engine’s ability to adapt to road conditions [5].

An AET system generally consists of a DC motor, a position sensor, a limp-home (LH) spring, a gearbox, and a valve plate. However, its performance is normally affected by its parameter uncertainty, disturbances in the external environment, the strong LH spring nonlinearity, the friction force of the throttle valve, and gear backlash. Therefore, accurately and rapidly tracking the reference angle of the valve opening requires dealing with the measurement noise and the nonlinearity of the system. At present, there is no controller that can perfectly solve this problem. It is a significant challenge for researchers and automotive engineers.

Eliminating the measurement noise and improving the control strategy are two approaches for improving the control precision of an AET system. To eliminate the influence of measurement noise on the control accuracy, a polyhedral approximation algorithm of set-valued estimation has been used in [6]. Additionally, a Kalman filter-based control algorithm has been proposed in [7] to reduce the effect of measurement noise on electronic throttle control. To further improve the estimation accuracy, an unscented Kalman filter has been employed to estimate the position of the valve plate in [8]. However, these existing studies assume that the disturbance noise is Gaussian white noise. In fact, the sensor noise is a superposition of multiple random noises that are non-Gaussian white noises [9]. Therefore, the above methods are no longer applicable.

Some researchers are trying to improve the control strategy to improve the control precision of AET. For example, in [10,11], a proportional integral derivative (PID) controller with a feedforward compensator has been introduced. To solve the nonlinearity of AET systems, variable structure control [12,13], optimal control [14,15], and nonlinear control [16] have been proposed. Moreover, several sliding mode control algorithms [17,18,19] have been applied to AET control systems, and a closed-loop robust control system has been designed under the influence of parameter uncertainty and external disturbance. In recent years, neural network control and fuzzy control methods have been proposed [20,21,22,23]. A self-learning control strategy based on neural networks has been proposed in [20], wherein a fuzzy neural network controller and a recurrent neural network identifier have been used to generate the control scheme. To adjust the fuzzy output membership functions, an intelligent fuzzy logic controller has been designed for a nonlinear throttle and a back-propagation algorithm has been proposed [21]. Furthermore, the fuzzy dynamics model of an AET system has been established in [22], together with a performance index of a fuzzy system. In addition, adaptive control methods have been used to improve their control accuracy and robustness of AET systems [24,25,26].

Backstepping control is a common and effective method to solve system nonlinearity. Generally, the backstepping design method uses regression approach for uncertain systems, and it combines the selection of the Lyapunov function with the controller design. Further, in this method, a complex nonlinear system can be decomposed into several subsystems, and a virtual controller can be developed for any subsystem so as to obtain the proper control law. Such system design advantages simplify the implementation of the backstepping technology; therefore, this backstepping design method has many applications in the field of industrial control. To address the influence of the nonlinearity and parameter uncertainty of AET systems, a backstepping controller that considers input saturation and unknown disturbance has been designed in [27]. Additionally, a finite-time backstepping method has been developed in [28], using new virtual control signals and modified error compensation signals. Furthermore, an adaptive backstepping controller based on the rigid graph theory and neural network has been designed in [29]. However, in the abovementioned backstepping control method, the controller parameters cannot be adjusted in real-time. As the working conditions of electronic throttle valves are changeable, thus the traditional backstepping control algorithm with fixed parameters cannot meet the requirements of the complicated working conditions of electronic throttle.

In summary, the non-Gaussian sensor noise and the variation in automotive driving conditions brings a challenge to the controller design of the AET system. The key problem is to design a non-Gaussian filter and automatically adjust the parameters of the backstepping controller. This paper proposes a self-tuning backstepping control strategy with a Kalman-like filter (SBCKLF) for the AET system. A Kalman-like filter is designed to attenuate the effect of the non-Gaussian white noise, which is superimposed by noises such as electrical and mechanical noise, on the measured values of a position sensor. A self-tuning backstepping controller is designed that considers the variation in automotive driving conditions. Different from the traditional backstepping controller, the fuzzy control approach is used to tune the parameters of the backstepping controller in real-time, thus improving control accuracy. A physical simulation platform is built to verify the superiority of the proposed method.

The rest of this paper is organized as follows. In Section 2, the mathematical model of the electronic throttle is constructed, and the major factors affecting the accuracy of the AET control are summarized. In Section 3, the SBCKLF strategy is proposed to improve control accuracy. Section 4 presents the experimental results and comparison of the related methods. Discussion and conclusion are drawn in Section 5 and Section 6, respectively.

2. Mathematical Model

The AET system consists of a drive motor, a position sensor, a gear reduction mechanism, a throttle valve plate, and an LH spring. Figure 1 shows the AET system structure. When the driver presses down the accelerator pedal, the pedal position will be converted into a voltage signal by the pedal sensor. Meanwhile, the electronic throttle position sensor also converts the real opening of its valve plate into another voltage signal. These two voltage signals are simultaneously transmitted to the electronic control unit (ECU). The ECU then generates an armature voltage that can be controlled by using the pulse width modulation (PWM) to control the drive motor. It can realize an increase or a decrease in the throttle opening based on the pedal’s position. The dynamic behavior of the AET body can be expressed using a DC motor model and a throttle dynamic model. The system parameters are listed in

Table 1. AET system parameters.

Symbol	Description	Nominal Values
$k_t$	Torque constant of motor	0.0185 $\mathrm{Nm}/\mathrm{A}$
$k_b$	Electromotive force constant of motor	0.0285 $\mathrm{V}\cdot \mathrm{s}/\mathrm{rad}$
$R_a$	Internal resistance of motor	1.15 $\Omega$
$k_{sp}$	Elastic coefficient of LH spring	0.087 $\mathrm{Nm}\cdot \mathrm{s}/\mathrm{rad}$
${\theta }_0$	Default throttle opening angle	12 $\mathrm{deg}$
$T_{LH}$	Offset of LH spring	0.396 $\mathrm{Nm}/\mathrm{A}$
$F_s$	Coulomb friction coefficient	0.284 $\mathrm{Nm}$
$n$	Transmission ratio of reduction gear	20.68
$J$	Equivalent total inertia	0.0021 $\mathrm{kg}\cdot {\mathrm{m}}^2$
$B$	Damping constant	0.0088 $\mathrm{Nm}\cdot \mathrm{s}/\mathrm{rad}$

[26,30].

As the armature inductance is very small and the dynamic characteristics of the armature current are ignored, the dynamic model of the AET system can be expressed as follows [26]:

$$J_m{\ddot{\theta }}_m=T_m-B_m{\dot{\theta }}_m-T_L,$$

(1)

$$T_m=k_t\frac{u-k_b{\dot{\theta }}_m}{R_a},$$

(2)

$$J_{\mathrm{g}}\ddot{\theta }=T_g-B_t\dot{\theta }-T_{sp}-T_f-T_d,$$

(3)

where $J_m$ and ${\theta }_m$ are the moment inertia and the rotation angle of the motor, respectively; $T_m$ and $T_L$ denote the motor torque and load torque, respectively; $B_m$ is the damping coefficient; $u$ is the input voltage; $J_g$ is the inertia moment of the throttle valve shaft; $\theta$ is the rotation angle of the throttle valve; $T_g$ is the transmission ratio torque; $B_t$ is the damping coefficient, $T_{sp}$ and $T_f$ are the LH spring torque and the friction torque, respectively. As for $T_d$, it denotes the airflow load torque, which is considered as a superposition of the external disturbances in this paper.

Considering the nonlinear problem of the LH spring, the LH spring model can be described as follows:

$$\begin{array}{ll}{T}_{sp}& =\{\begin{array}{l}-k_{sp}(\theta -{\theta }_0)-T_{LH} {\theta }_0<\theta <{\theta }_{\max }\\ k_{sp}(\theta -{\theta }_0)+T_{LH} {\theta }_{\min }<\theta <{\theta }_0\end{array}\\ & =k_{sp}(\theta -{\theta }_0)+T_{LH}\mathrm{sgn}(\theta -{\theta }_0) {\theta }_{\min }<\theta <{\theta }_{\max }\end{array}.$$

(4)

The electronic throttle valve is affected by the nonlinear friction force during rotation. The friction torque for the same can be calculated as follows:

$$T_f=F_s\mathrm{sgn}\dot{\theta }.$$

(5)

The nonlinear gear backlash model is described as follows:

$$\{\begin{array}{l}{\theta }_m=n\theta +d(\theta )\\ T_g(t)=n{T}_L(t)+d(T_L(t))\end{array},$$

(6)

where n is the transmission ratio of the reduction gear, $d(\cdot )$ is the nonlinear function of $T_L(t)$ and ${\dot{T}}_L(t)$, and $\epsilon$ is an uncertain bound that can be processed by the control strategy.

According to Equations (1)–(6), the AET dynamics model can be simplified as:

$$\begin{array}{ll}\ddot{\theta }=& \frac{1}{{\mathrm{n}}^2{J}_m+J_g}[\frac{n{k}_{t}u}{R_a}-(\frac{n^2{k}_b{k}_t}{R_a}+n^2{B}_m+B_t)\dot{\theta }-k_{sp}(\theta -{\theta }_0)\\ & -T_{LH}\mathrm{sgn}(\theta -{\theta }_0)-F_s\mathrm{sgn}\dot{\theta }]+D\end{array}.$$

(7)

Some factors were ignored in the model building, such as the production deviations and the variation in the external conditions. In this paper, these factors, as well as the abovementioned uncertainties, such as the airflow load torque $T_d$ and the nonlinear function $d(\cdot )$, have been added to form the total disturbance of the system, namely $D=f(\theta ,\dot{\theta },T_d,d)$, where $\left|D\right|\le \eta$($\eta$ is an unknown constant). The state vector ${[x_1,x_2]}^T={[\theta ,\dot{\theta }]}^T$ has been selected to establish the state space expression.

The expression of the AET control system is summarized as follows:

$$\begin{gathered} \begin{array}{l}\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-\frac{k_{sp}}{J}{x}_1-\alpha x_2+\beta (x_1,x_2)+\frac{n{k}_t}{J{R}_a}u+D\end{array},\end{array} \\ y=x_1 \end{gathered}$$

(8)

where, $J=n^2{J}_m+J_g$, $B=n^2{B}_m+B_t$, $\alpha =\frac{1}{J}(\frac{n^2{k}_b{k}_t}{R_a}+B)$, and $\beta (x_1,x_2)=\frac{1}{J}[k_{sp}{\theta }_0-F_s\mathrm{sgn}{x}_2-T_{LH}\mathrm{sgn}(x_1-{\theta }_0)]$.

3. SBCKLF Strategy

In this section, a self-tuning backstepping controller is developed to improve control accuracy by introducing a Kalman-like filter to attenuate the effect of the non-Gaussian white noise. The proposed tracking controller is shown in Figure 2.

3.1. Design of Kalman-like Filter

The position sensor in the AET system converts the actual opening of the AET into a voltage signal and feeds it to the controller for realizing the closed-loop control. Owing to the poor driving environment and complexity of the engine’s internal structure, multiple noise sources exist in the engine. For example, the aerodynamic, mechanical, combustion, and electrical noise affect the measurement accuracy of the AET position sensor [31,32], thereby affecting the accuracy of the AET control.

The position sensor noise is a superposition of multiple random noises that are non-Gaussian white noises. The detected noise probability density curve in the laboratory environment is shown in Figure 3.

In order to deal with the disturbance of the non-Gaussian white noise, the system estimation tool employs the Kalman-like filter in this paper. The Kalman-like filter uses an approximate minimum mean square error point process to estimate the state of the Markov chain system, and basic steps of its filtering algorithm are as follows.

Remark 1. 

The noise signal collected in this study is not only the measurement error of the position sensor but also the electrical noise in the transmission circuit of the position sensor. This is because in the actual process of automobile operation, the superposition of various noises impacts the AET controller (in addition to the impact of the position sensor measurement error). This paper also presents the follow-up control experiment on this platform.

Remark 2. 

Herein, non-Gaussian white noise is defined as noise for which the probability density function does not obey the normal distribution while the power spectral density obeys the uniform distribution.

System state prediction:

$${\widehat{X}}_{k+1/k}={\varphi }_k{\widehat{X}}_{k/k}.$$

(9)

System status update:

$$y_{k+1/k}=H_k\ast {\widehat{X}}_{k+1/k}$$

(10)

$${\widehat{X}}_{k+1/k+1}={\widehat{X}}_{k+1/k}+G_{k+1}(y_{k+1}-y_{k+1/k}).$$

(11)

Gain matrix:

$$G_{k+1}=P_{k+1/k}{H}_k^T\ast {(H_k{P}_{k+1/k}{H}_k^T+{\tilde{R}}_k)}^{-1}.$$

(12)

Covariance matrix prediction:

$$P_{k+1/k}={\varphi }_k{P}_{k/k}{\varphi }_k^T+Q_k.$$

(13)

Covariance matrix update:

$$P_{k+1/k+1}={\varphi }_k{P}_{k+1/k}{\varphi }_k^T+diag({\widehat{X}}_{k+1/k+1})-{\varphi }_{k}diag({\widehat{X}}_{k/k}){\varphi }_k^T,$$

(14)

where ${\widehat{X}}_{k/k}$ is the state vector, ${\varphi }_k$ is the state transition matrix, $H_k$ is the observation matrix, $G_k$ and $y_k$ are the gain matrix and the observation vector, respectively, $P_k$ and ${\tilde{R}}_k$ are the prediction error covariance matrix and the measurement noise covariance matrix, respectively, $Q_k$ is the system noise covariance matrix, ${\omega }_k$ is the system noise matrix, and $v_k$ is the measurement noise matrix. The structure block diagram of the Kalman-like filter is shown in Figure 4.

As can be seen from Equations (12)–(14) and Figure 4, there are three differences between the two filters:

(1) In the standard Kalman filter algorithm, the control quantity will affect the state variation in the system, whereas in the Kalman-like filter algorithm, the control quantity only affects the precision of the control and will not affect the state variation in the entire estimation system.

(2) The gain of Kalman filter and Kalman-like filter does not depend on the measured value but is related to the state value of the system. Because the system state of the Kalman filtering algorithm requires a Gaussian distribution, the system state of the Kalman-like filter operates for a discrete time, and given the finite state of the Markov chain, the direct result of this correlation is that the Kalman-like filter used herein constitutes a nonlinear filter, unlike the standard Kalman filter.

(3) Under the standard Kalman filter, the conditional distribution of system states is proven to be Gaussian distribution, the system state estimation is conditional mean, and the filtering error covariance matrix is conditional covariance matrix. However, in our setup, this distribution is consistent with the system state estimate [33]. The electronic throttle valve position estimation steps can be described as follows.

Step (1) The actual electronic throttle opening angle $\theta$ is defined as the state vector ${\widehat{X}}_{k/k}$, and the observation vector $y_k$ is obtained.

Step (2) The state prediction value of the system ${\widehat{X}}_{k+1/k}$ and the covariance matrix prediction value $P_{k+1/k}$ are obtained.

Step (3) The Kalman-like filter gain matrix $G_{k+1}$ is calculated.

Step (4) The system status update value ${\widehat{X}}_{k+1/k+1}$ and the covariance matrix update value $P_{k+1/k+1}$ are obtained.

Step (5) $k=k+1$ is set, and the steps are repeated starting from Step 1.

3.2. Self-Tuning Backstepping Control Strategy

The control goal of the AET is to make the real opening angle $\theta$ follow the reference angle accurately and quickly. According to the nonlinear characteristics of the system, the backstepping control method is used to guarantee the closed-loop stability of the system, and the fuzzy control is used to select parameters of the backstepping controller.

The backstepping method is used to construct the Lyapunov function step-by-step to obtain the corresponding virtual control, and then backstep to the equation that contains the real control input to obtain the control law of the system. The basic steps of designing an AET controller based on the backstepping algorithm are as follows:

Define the first error variable:

$${\epsilon }_1=x_1-{\theta }_r,$$

(15)

where $x_1$ is the real opening of the electronic throttle valve plate, and ${\theta }_r$ is the reference value of the throttle angle.

The Lyapunov function is:

$$V_1=\frac{{\epsilon }_1^2}{2}.$$

(16)

Its derivative is:

$${\dot{V}}_1={\epsilon }_1{\dot{\epsilon }}_1={\epsilon }_1(x_2-{\dot{\theta }}_r).$$

(17)

where $x_2$ is the derivative of $x_1$.

To make $d{V}_1/dt$ a negative definite, select $x_2$ as the virtual control. The ideal virtual control value is:

$$x_{2d}=-k_1{\epsilon }_1+{\dot{\theta }}_r,$$

(18)

where $k_1$ is a positive constant.

Define the second error variable:

$${\epsilon }_2=x_2-x_{2d}=x_2+k_1{\epsilon }_1-{\dot{\theta }}_r.$$

(19)

The Lyapunov function is:

$$V_2=\frac{{\epsilon }_1^2}{2}+\frac{{\epsilon }_2^2}{2}.$$

(20)

The derivative of $V_2$ is:

$$\begin{array}{ll}{\dot{V}}_2& ={\epsilon }_1{\dot{\epsilon }}_1+{\epsilon }_2{\dot{\epsilon }}_2=-k_1{\epsilon }_1^2+{\epsilon }_2[(1-\frac{k_{sp}}{J})x_1\\ & +(k_1-\alpha )x_2-{\theta }_r-k_1{\dot{\theta }}_r-{\ddot{\theta }}_r+\beta (x_{1,}{x}_2)+\frac{n{k}_t}{J{R}_a}u]\end{array}.$$

(21)

To make $d{V}_2/dt$ a negative definite, let:

$$\begin{array}{ll}-k_2{\epsilon }_2& =(1-\frac{k_{sp}}{J})x_1+(k_1-\alpha )x_2-{\theta }_r-k_1{\dot{\theta }}_r-{\ddot{\theta }}_r\\ & +\beta (x_{1,}{x}_2)+\frac{n{k}_t}{J{R}_a}u\end{array},$$

(22)

where $k_2$ is a positive constant.

The control law is:

$$\begin{array}{ll}u& =\frac{J{R}_a}{n{k}_t}[(\frac{k_{sp}}{J}-1-k_1{k}_2)x_1+(\alpha -k_1-k_2)x_2-\beta (x_1,x_2)\\ & +(1+k_1{k}_2){\theta }_r+(k_1+k_2){\dot{\theta }}_r+{\ddot{\theta }}_r]\end{array}.$$

(23)

A tracking error system was established for the backstepping control method, and its robustness was analyzed in [34]. The derivative of the two error variables is given as follows:

$${\dot{\epsilon }}_1={\epsilon }_2-k_1{\epsilon }_1$$

(24)

$${\dot{\epsilon }}_2={\dot{x}}_2+k_1{\dot{\epsilon }}_1-{\ddot{\theta }}_r=-{\epsilon }_1-k_2{\epsilon }_2+D.$$

(25)

The state space expression of the AET tracking error is expressed as:

$$\left[\begin{array}{c}{\dot{\epsilon }}_1\\ {\dot{\epsilon }}_2\end{array}\right]=\left[\begin{array}{cc}-k_1& 1\\ -1& -k_2\end{array}\right]\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]D.$$

(26)

The Lyapunov function is:

$$V_e=\frac{{\epsilon }_1^2}{2}+\frac{{\epsilon }_2^2}{2}$$

(27)

The derivative is obtained as:

$${\dot{V}}_e=-k_1{\epsilon }_1^2-k_2{\epsilon }_2^2+{\epsilon }_{2}D$$

(28)

According to Young’s inequality:

$${\epsilon }_{2}D\le \frac{\lambda {\epsilon }_2{}^2}{2}+\frac{D^2}{2\lambda }$$

(29)

where $\lambda$ is a constant and $\lambda >0$.

Substitute inequality (29) into Equation (28):

$${\dot{V}}_e\le -k_1{\epsilon }_1^2-(k_2-a){\epsilon }_2^2+\frac{D^2}{2a}.$$

(30)

where $a=\frac{\lambda }{2}$, and $k_2\ge a>0$. Let:

$$b=\min \left\{2k_1,2k_2-2a\right\}.$$

(31)

Then,

$${\dot{V}}_e\le -b{V}_e+\frac{D^2}{2a}.$$

(32)

The following inequality can thus be obtained:

$$\underset{t\to \infty }{\lim }\left|{\dot{V}}_e\right|\le \frac{D^2}{2ab}.$$

(33)

According to Equations (27) and (33):

$$\frac{{\epsilon }_1^2}{2}\le {\dot{V}}_e\le \frac{D^2}{2ab}.$$

(34)

Thus, the following conclusion can be drawn:

$${\epsilon }_1\le \sqrt{\frac{D^2}{ab}}.$$

(35)

According to Equations (26), (31), and (35), it can be concluded that the decay rate of the tracking error is related to the selection of $k_1$, and the larger the value of $k_1$, the faster is the decay rate. In addition, the tracking static error is related to the selection of $k_2$, and the larger the value of $k_2$, the smaller is the tracking static error. However, excessive values of $k_1$ and $k_2$ would result in a high gain of the controller, which can cause problems in amplifying the disturbance signal and the overshoot oscillation. Hence, herein, the fuzzy control algorithm is chosen to adjust the values of the two parameters $k_1$ and $k_2$ of the backstepping control in real time, and the fuzzy rules are formulated to select reasonable parameter values.

Remark 3. 

According to the stability proof presented herein, the error converges when the parameters $k_1$ and $k_2$ are positive constants. The parameters adjusted by the fuzzy rule conform to the condition of error convergence and can produce a better tracking effect than the fixed gain. The effect of experimental tracking can also be demonstrated as described above.

The fuzzy logic system comprises four parts: fuzzy module, fuzzy rule base, fuzzy module inference engine, and defuzzification module, as shown in Figure 5. In the fuzzy logic system, the tracking error ${\epsilon }_1$ and the derivative of the tracking error ${\dot{\epsilon }}_1$ are set as the input of the fuzzy control. The two parameters $k_1$ and $k_2$ of the adaptive control are used as output variables, and the range of the input and output parameters is defined according to the actual system output.

The principle of parameter self-tuning is based on the relationship between the parameters $k_1$, $k_2$, the tracking error ${\epsilon }_1$, and its derivative ${\dot{\epsilon }}_1$. If the absolute value of the tracking error ${\epsilon }_1$ is large, the parameter $k_2$ should be given a large positive value, so that the static tracking error ${\epsilon }_1$ can be eliminated as little as possible. If the absolute value of the derivative of the tracking error ${\dot{\epsilon }}_1$ is large, the parameter $k_1$ should be given a large positive value to increase the error decay rate. If the absolute values of ${\epsilon }_1$ and ${\dot{\epsilon }}_1$ are relatively small, the parameters $k_2$ and $k_1$ should be given relatively small positive values, respectively. It ensures that the controller has appropriate gain parameter when approaching a stable state, thereby avoiding the occurrence of system output ripple and oscillation.

Based on the above principles, the fuzzy sets that define the tracking error ${\epsilon }_1$ and the derivatives of the tracking error ${\dot{\epsilon }}_1$ are {NB (negative big), NS (negative small), ZO (zero), PS (positive small), PB (positive big)}. The fuzzy sets of the two parameters $k_1$ and $k_2$ for the adaptive control are {PS (positive small), PM (positive medium), PB (positive big)}. In fuzzy control, membership functions of triangle, trapezoid, Gauss, and other shapes are usually used to describe the fuzzy degree of input variables and output variables. Considering that the input variables are bounded error parameters, the triangle membership function method is used to fuzzify the input and output variables. The performance of the control system is adjusted by appropriately adjusting the parameter value of the membership function. The inference type of fuzzy control adopts Mamdani inference algorithm in this study.

The fuzzy rule base $R={\cup }_{i=1}^N{R}_i$ is a set of fuzzy rules, where each IF-THEN fuzzy rule $R_i$ is in the form of:

$$R_i:\mathrm{IF} {\epsilon }_1 \mathrm{is} A_1^i, {\dot{\epsilon }}_1 \mathrm{is} A_2^i \mathrm{THEN} k_1 \mathrm{is} B_1^i, k_2 \mathrm{is} B_2^i,$$

(36)

where $N$ is the number of rules, $A_1^i$ is the input membership function, and $B_1^i$ is the output membership function. The specific fuzzy rules are listed in

Table 2. Fuzzy rule for parameter k₁.

	${\mathit{\epsilon }}_1$	NB	NS	ZO	PS	PB
${\dot{\mathit{\epsilon }}}_1$
NB		PB	PB	PB	PB	PB
NS		PM	PM	PM	PM	PM
ZO		PS	PS	PS	PS	PS
PS		PM	PM	PM	PM	PM
PB		PB	PB	PB	PB	PB

and

Table 3. Fuzzy rule for parameter k₂.

	${\mathit{\epsilon }}_1$	NB	NS	ZO	PS	PB
${\dot{\mathit{\epsilon }}}_1$
NB		PB	PM	PS	PM	PB
NS		PB	PM	PS	PM	PB
ZO		PB	PM	PS	PM	PB
PS		PB	PM	PS	PM	PB
PB		PB	PM	PS	PM	PB

below. Using the average weighting method for defuzzification, the output of the fuzzy controller can be designed as:

$${\rho }_{\mathrm{fuzzy}}=\frac{{\displaystyle \sum _{i=1}^n{x}_i\rho \left(x_i\right)}}{{\displaystyle \sum _{i=1}^n\rho \left(x_i\right)}},$$

(37)

where $\rho \left(x_i\right)$ is the value of the membership function at $x_i$.

After defuzzification, the two parameters $k_1$ and $k_2$ of the backstepping controller are the outputs. The specific value range is shown in Figure 6. The backstepping controller selects the parameters $k_1$ and $k_2$ output by the fuzzy controller and calculates the optimal control voltage $u$ according to the tracking error ${\epsilon }_1$, thereby outputting the ideal output angle $\theta$ of the AET system.

Remark 4. 

The selection criteria of parameters are as follows:

Multiply both sides of Equation (32) by $e^{bt}$:

$$\frac{d}{dt}(V_e{e}^{bt})\le \frac{D^2}{2a}{e}^{bt}.$$

(38)

Integrating Equation (38) on [0, t] can be obtained as follows:

$$V_e(t)e^{bt}-V_e(0)\le \frac{D^2}{2a}{\displaystyle {\int }_0^t{e}^{b\tau }}d\tau .$$

(39)

Multiply both sides by $e^{-bt}$ to obtain:

$$V_e(t)\le V_e(0)e^{-bt}+\frac{D^2}{2a}{\displaystyle {\int }_0^t{e}^{-b(t-\tau )}}d\tau .$$

(40)

Thus:

$${\left|z(t)\right|}^2\le {\left|z(0)\right|}^2{e}^{-bt}+\frac{D^2}{2a}{\displaystyle {\int }_0^t{e}^{-b(t-\tau )}}d\tau .$$

(41)

In this paper, the minimum control objective is set as, from full opening to full closing of the valve plate, the adjustment time with a steady-state error of 2% being allowed within 200 ms. The following formula can be obtained by substituting it from Equation (41): $b\ge 40$.

Combined with Equation (31), it can be obtained: $k_1\ge 20$. After the value of $k_2$ is fixed, the value of $k_1$ is gradually increased by trial and error, and a relatively satisfactory compromise can be reached between the response speed and control precision within the interval of [45, 50]. Then, $k_1$ is fixed within the interval of [45, 50], and the value of $k_2$ is gradually increased to obtain that of $k_2$, which achieves a satisfactory system performance within the interval of [65, 70].

4. Experimental Verification

In order to verify the proposed self-tuning backstepping control algorithm, a small experimental platform based on dSPACE fast controller is built. In this platform, a car throttle is used as the controlled object. The performance advantages and disadvantages of the proposed method are analyzed by comparing it with other algorithms. The selection of comparison methods depends on: whether to use Kalman-like filter; whether to use backstep control; whether to use self-tuning. Five other methods are Kalman filter + self-tuning backstepping control, self-tuning backstepping control, traditional backstepping control, fuzzy PID control with feedforward compensation, and PID control. Detailed comparison results are given below.

4.1. Experimental Configurations

In order to verify the control performance of the electronic throttle system, all programs have been written using MATLAB and compiled in Simulink. Figure 7 shows the experimental platform of the AET (produced by Bosch) system. On this platform, dSPACE obtains the voltage signal of the electronic throttle position sensor with a frequency of 1.5 kHz and after being selected by the low-pass filter whose cutoff frequency is 6 kHz. The control algorithm runs at a sampling interval of 0.1 ms. In this study, two experiments have been conducted to verify the performance of the proposed method. In test 1, a trapezium instruction has been adopted to observe the accuracy and rapidity of the controller, whereas in test 2, a sinusoidal instruction has been adopted to verify the robustness of the controller.

In order to achieve a performance comparison, five other methods, including Kalman filter + self-tuning backstepping control, self-tuning backstepping control, traditional backstepping control, fuzzy PID control with feedforward compensation, and PID control schemes, have also been realized. Considering the fairness of the comparison, all parameters of five other controllers under testing need to be adjusted through intensive testing to achieve an acceptable steady-state and transient performance.

4.2. Performance Comparison

For attaining further clarity, this study uses the maximum (MAX) and root mean square (RMS) values of the tracking error to compare the control performance of the six controllers [35]:

$$MAX\left(\epsilon \right)=\max \left(\left|\epsilon \left(i\right)\right|\right),i\in [1, N_d]$$

(42)

$$RMS\left(\epsilon \right)=\sqrt{{\displaystyle \sum _{i=1}^{N_d}\frac{{\epsilon }^2\left(i\right)}{N_d}}},$$

(43)

where $N_d$ is the number of samples.

In

Table 4. Controller performance comparison.

Controller	Test	MAX Error (deg)	RMS Error (deg)
SBCKLF	Test1	1.7446	0.4108
	Test2	2.3497	0.5158
Kalman filter + self-tuning backstepping control	Test1	2.9144	0.7303
	Test2	2.9990	0.5921
Self-tuning backstepping control	Test1	3.3230	0.7379
	Test2	3.8513	1.1401
Traditional backstepping control	Test1	4.6724	1.6385
	Test2	6.2302	2.8291
Fuzzy PID control with feedforward compensation	Test1	4.5629	1.4842
	Test2	3.6892	2.8442
PID control	Test1	5.0027	2.3035
	Test2	6.3145	2.8279

, the MAX and RMS values of the tracking error $\epsilon (i)$ are used as performance indicators to further compare the performance of the six controllers. The tracking error MAX and RMS values of the SBCKLF strategy are considerably smaller than those of the other five control methods. In the other five methods, the best result is obtained by the Kalman filter + self-tuning backstepping control method. Its tracking error MAX and RMS values are 2.9990° and 0.5921° in test 2, respectively. Compared to this result, the proposed method reduces the MAX and RMS tracking error by 21.65% and the average error by 12.89%. Therefore, the results clearly show that the proposed SBCKLF strategy has the strongest robustness as well as the best tracking accuracy and response speed. Moreover, this strategy can be effectively implemented in actual AET systems.

4.3. Performance of Trapezium Commands (Test 1)

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the performance of the six control methods under the trapezium tracking with different amplitude command. (1) The steady-state error bound (SSEB) of the SBCKLF strategy is 0.9°. In comparison, the SSEBs of the Kalman filter + self-tuning backstepping control, self-tuning backstepping control, and traditional backstepping control are 2.3°, 2.8°, and 3.2°, respectively. Additionally, the fuzzy PID control with feedforward compensation and PID control have the maximum SSEBs of 2.6° and 3.9° at 14–15 s, respectively. (2) For the response time, considering the rising stage after 15 s as an example, the SBCKLF strategy reaches a minimum response time of 0.02 s, whereas the response times of the Kalman filter + self-tuning backstepping control, self-tuning backstepping control, traditional backstepping control, and PID control are 0.03, 0.07, 0.09, and 0.16 s, respectively. Thus, it can be concluded that the SBCKLF strategy achieves the fastest response. (3) For the transient tracking error bound (TEB), the fuzzy PID control with feedforward compensation and PID control have transient TEBs of 4.6° and 5° at 25–26 s, respectively. In contrast, the SBCKLF strategy, the Kalman filter + self-tuning backstepping control, the self-tuning backstepping control, and the traditional backstepping control have transient TEBs of 1.7°, 2.9°, 3.3°, and 4.7°, respectively. (4) Compared with the Kalman filter + self-tuning backstepping control and the self-tuning backstepping control, the SSEB of the SBCKLF strategy is reduced by 1.4° and 1.9°, respectively, and the transient TEB is reduced by 1.2° and 1.6°, respectively. The response time is reduced by 0.01 and 0.05 s, respectively. The SBCKLF strategy can effectively solve the piecewise constant error component (typically due to friction or other mechanical effects) caused by the tracking angle change and achieve a more accurate tracking effect and a faster tracking response. These results show that the Kalman-like filter could effectively eliminate the influence of the non-Gaussian white noise in the sensor circuit when using the electronic throttle control strategy. (5) Compared with the traditional backstepping control, the SSEB and transient TEB of the self-tuning backstepping control are reduced by 0.4° and 1.4°, respectively, and the response time is reduced by 0.02 s, which implies that the fuzzy control algorithm could improve the control accuracy and accelerate the response by adjusting the backstepping control parameters in real-time.

Remark 5. 

When the reference value is the trapezium signal, since the values before and after the trapezium signal are constant and the trapezium moment is not differentiable, its derivative is always considered to be 0 in this paper.

4.4. Performance of Sinusoidal Commands (Test 2)

Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 show the control performance of the six methods under the sinusoidal command. (1) In the medium frequency sinusoidal command tracking performance test, the transient TEB of the SBCKLF strategy is 1.7°, whereas the transient TEBs of the Kalman filter + self-tuning backstepping control, self-tuning backstepping control, and traditional backstepping control are 3°, 2.8°, and 4.5°, respectively. The transient TEBs of the fuzzy PID control with feedforward compensation and PID control are 3.3° and 5.1°, respectively. (2) In the high-frequency sinusoidal command tracking performance test, the transient TEB of the SBCKLF strategy is 2.3°, whereas the transient TEBs of the Kalman filter + self-tuning backstepping control, self-tuning backstepping control, and traditional backstepping control are 2.6°, 3.9° and 6.2°, respectively. The transient TEBs of the fuzzy PID control with feedforward compensation and PID control are 3.7° and 6.3°, respectively. (3) Compared with the Kalman filter + self-tuning backstepping control, self-tuning backstepping control and traditional backstepping control, the transient TEB of the SBCKLF strategy is reduced by 0.7°, 1.6° and 3.9°, respectively. (4) The fluctuation of the error curve in Figure 8 and Figure 14 is obviously smaller than that of the other five control methods, suggesting that the SBCKLF strategy could effectively eliminate the influence of the nonlinear problems of the LH spring, friction force, and gear backlash on the control accuracy of the AET.

5. Discussion

The above experimental results have demonstrated that the proposed SBCKLF strategy has the best performance compared to five other methods, including Kalman filter + self-tuning backstepping control, self-tuning backstepping control, traditional backstepping control, fuzzy PID control with feedforward compensation, and PID control. The experimental results assume that the reference signal is a trapezium or sine wave, which is a common reference signal type for control performance testing. However, the reference signal is complex and changeable in the real operating environment of the AET systems. The SSEB and TEB are two indicators of important reference significance. The SSEBs of these methods with trapezium command are listed in

Table 5. SSEB comparison.

Controller	SSEB (deg)
SBCKLF	0.9
Kalman filter + self-tuning backstepping control	2.8
Self-tuning backstepping control	2.3
Traditional backstepping control	3.2
Fuzzy PID control with feedforward compensation	2.6
PID control	3.9

. The TEBs of these methods with trapezium and sinusoidal commands are listed in

Table 6. TEB comparison.

Controller	TEB with Trapezium (deg)	TEB with Medium Frequency Sine (deg)	TEB with High Frequency Sine (deg)
SBCKLF	1.7	1.7	2.3
Kalman filter + self-tuning backstepping control	2.9	3.0	2.6
Self-tuning backstepping control	3.3	2.8	3.9
Traditional backstepping control	4.7	4.5	6.2
Fuzzy PID control with feedforward compensation	4.6	3.3	3.7
PID control	5.0	5.1	6.3

. In these two comparisons, the SSEB and TEB of the proposed method are controlled to 0.9° and 2.3°, respectively. In general, methods with better experimental results have a high probability of better performance, but not absolute best.

6. Conclusions

In this paper, the SBCKLF strategy was proposed for AET systems. Its contributions can be summarized as follows: (1) the non-Gaussian white noise that negatively affects the measurement accuracy of the position sensor is collected and analyzed. (2) The Kalman-like filter can effectively eliminate the influence of non-Gaussian white noise and significantly improve the measurement accuracy of the position sensor. (3) The fuzzy control algorithm is employed to adjust the backstepping control parameters online, which can improve the control accuracy and response speed of the AET system with variable driving conditions. The experimental results have proved the effectiveness of the SBCKLF strategy. Compared to the best results of other methods, the proposed method can reduce the maximum and root mean square tracking errors by 21.65% and the average error by 12.89%. The steady-state and tracking error bounds are controlled to 0.9° and 2.3°, respectively. Experimental results also demonstrate that the SBCKLF method has the strongest robustness as well as the best response speed. Our future research is to explore the optimal control parameter range with the help of intelligent algorithms and integrate the control algorithm into the electronic control unit of the vehicle to track the desired signal given by the accelerator pedal in the real engine environment.