Sage Journals: Discover world-class research

Abstract

Motivated by achieving precise positioning of the throttle plate, an extended state observer–based intelligent double integral sliding mode control strategy is proposed for the electronic throttle control system. Specifically, an extended state observer is first designed to observe the opening angle change of the electronic throttle and compensate unexpected external disturbance. On this basis, an intelligent double integral sliding mode controller for electronic throttle valve is presented based on Lyapunov stability theory and sliding mode control theory, which includes stability analysis and parameter adaptation design. Finally, extensive simulations are conducted to demonstrate the superior performance of the proposed method.

Keywords

Electronic throttle valve extended state observer intelligent double integral sliding mode control backpropagation neural network

Introduction

In past decades, a variety of electronic throttle (ET) control methods, which have great significance on drivability of automobiles and help reduce fuel emissions, have been proposed. Deur et al.¹ proposed an refined proportional–integral–derivative (PID) control approach for the ET valve considering compensation of friction and limp-home effects. However, the guidance rules of how to choose parameters of the compensator were not given. Based on the aforementioned work, Yuan and Wang² put forward a neural network–based PID controller, realizing precision control of the ET valve with adaptive updating of the PID parameters. However, the effect of the torque of the return spring is ignored. Subsequently, Sheng and Bao³ carried out a fractional-order fuzzy-PID control scheme, wherein optimal control parameters are found through the fruit fly optimization algorithm (FOA). Yet, the gear backlash was ignored in this approach, degrading the control performance. Furthermore, Panzani et al.⁴ designed an ET controller including linear part and hybrid feedforward–feedback friction compensator to address the control of ET body for ride-by-wire application in sport motorcycles.

On the basis of the nonlinear control, considering that the angular velocity of ET is not measurable, Pan et al.⁵ investigated variable structure control based on a sliding mode observer to estimate the angular velocity. However, it only considered the friction torque caused by Coulomb friction, while the stick-slip friction was ignored. Lately, Hu and colleagues^6,7 proposed a reduced-order observer-based backstepping controller for ET, where a guideline for selection of control parameters was given through the input-to-state stability (ISS) analysis. Unfortunately, though the disturbance was taken into account to show the ISS property, it was not handled directly in the controller design. Therefore, the disturbance would result in the performance deterioration. Recently, Li et al.⁸ proposed an extended state observer (ESO)-based backstepping sliding mode control (SMC) approach for the ET valve. Especially, an ESO was designed based on the ET nonlinear model to estimate the throttle opening angle change and uncertainty of ET simultaneously and then the backstepping sliding mode controller was developed to improve the control performance. However, this method only considered the throttle opening angle error while ignoring the throttle opening change error in the feedback signals. Subsequently, Li et al.⁹ put forward an ESO-based double-loop integral SMC approach for ET, considering both of these two errors as feedback signals. Numerical experiments verified the effectiveness of the proposed approach with respect to the accuracy and transient performance. Moreover, Eski and Yıldırım¹⁰ put forward a robust adaptive neural-based fuzzy inference control system to control the speed of heavy duty vehicle, which used an ET system. Recently, Sun et al.¹¹ proposed a fuzzy approach for optimal robust control design of automotive ET system, where the fuzzy dynamical model of the ET system with parameter uncertainties, nonlinearities, and external disturbances were established. The results showed that this method achieved a good control performance for ET system. Wang et al.¹² proposed a robust adaptive SMC scheme for automotive ET valve to eliminate the effects of the parameter uncertainties and nonlinearities including friction, return spring limp-home, and gear backlash. The simulation and experimental results showed that the proposed method achieved a comparative performance. With respect to the discrete domain, Amini et al.¹³ presented a generic robust discrete sliding mode controller (DSMC) for ET system, where a novel method was used to predict and incorporate the sampling and quantization imprecisions into the DSMC structure.

From the viewpoint of optimal control, Vašak et al.¹⁴ formulated a constrained optimal control problem based on the piecewise affine model of the ET valve considering both the gearbox friction and the “limp-home” effect. However, this approach assumed that the model was very accurate; otherwise, it would result in obvious degrading of the performance. To address this problem, Vašak et al.¹⁵ further presented a time-optimal control method based on model predictive control (MPC), which would make an improvement of the control performance. In addition, Kim et al.¹⁶ put forward a dynamic programming-based optimal controller for the ET valve, wherein the optimal gear shift and throttle opening angle could be obtained, and throttle gear map which could meet the power demand of the driver was developed. However, in the context of the wear and tear of the transmission components, it could not provide enough power since the map was lack of self-learning ability. Recently, Yuan et al.^17,18 presented a self-learning controller for the ET valve based on neural network, for which strong robustness was achieved. Furthermore, to handle the nonlinear hysteretic behavior of ET, Wang and Huang¹⁹ proposed an intelligent fuzzy control approach with feedforward closed-loop structure, wherein the fuzzy membership function was elegantly tuned with a novel closed-loop backpropagation (BP) method to achieve better tracking performance. However, the feedforward controller could not compensate nonlinear hysteresis (NH) accurately as the designed fuzzy rules were too simple. Montanaro et al.²⁰ presented a novel discrete-time model reference adaptive control (MRAC) method to cope with the nonlinear and discontinuous dynamics of the ET, where a class of minimal control synthesis (MCS) algorithms was extended by adding an explicit discrete-time adaptive integral action and an adaptive robust term. Experimental investigation verified the effectiveness of this method. Subsequently, Vargas et al.²¹ presented an output feedback control strategy for Markov jump linear systems with no mode observation, which was then applied for ET control, and the experimental results demonstrated that this approach could achieve good tracking performance. Lately, Zhang et al.²² first modeled an ET system as a linear parameter varying (LPV) system in discrete-time domain. Then, they designed mixed constrained $H_{2} / H_{\infty}$ LPV controller for the LPV throttle control system through the linear matrix inequality convex optimization method. Furthermore, Li and Jiao²³ investigated a finite-time servo control strategy through incorporating the particle swarm optimization (PSO) identification technique with the finite-time stability theory to enhance the rapidity and accuracy of throttle opening trajectory tracking for ET control systems.

From the previous analysis, the main factor restricting the performance of ET control is that the system model includes a barrage of nonlinear characteristics (i.e. stick-slip friction, gear backlash, and nonlinear spring) and external disturbances, but not all of previous studies have already taken these factors into account. Inspired by this observation, this article proposes an ESO-based intelligent double integral sliding mode controller with improved transient performance and strong robustness. In particular, an ESO is first introduced to estimate the ET angular velocity and external disturbance simultaneously. Meanwhile, an intelligent double integral sliding mode controller (IDISMC) is designed based on the Lyapunov stability theory and SMC theory, which can reject uncertainties effectively and improve the dynamic performance to guarantee the reliability and accuracy of ET. Furthermore, the BP neural network is applied for online self-adaption of control parameters to further improve the control performance. Comparative simulation results are provided to show the superior performance of the proposed approach in terms of transient performance and robustness to parameter variations.

To sum up, the proposed approach successfully merges the advantages of both traditional PID and sliding mode controllers and achieves improved performance. The main contributions are as follows:

An ESO is developed to observe the opening angle change of the ET as well as external disturbance simultaneously, which is further used to suppress the influence of disturbance.

On the basis of the ESO, an intelligent double integral sliding mode controller is carried out based on Lyapunov stability theory and SMC theory. Subsequently, the BP neural network is used to design the parameters of PID-like term of the controller to realize online self-adaption of control parameters.

As verified by simulation results, the proposed approach is robust with respect to parameter variations and external disturbance, and it achieves better transient performance compared with traditional SMC.

The organization of this article is stated as follows. Section “Mathematic model of ET” gives the system model of the ET valve. Section “ET controller design” designs the controller and the ESO. Then, comparative simulations with respect to the SMC are conducted in section “Simulations.” The conclusion is given in the final section.

Mathematic model of ET

As shown in Figure 1, the ET valve consists of a DC motor, a pinion gear, an intermediate gear, a sector gear, a valve plate, and a nonlinear spring.⁵ In the light of Kirchhoff’s law, the motor winding circuit is expressed as^8,9

\begin{matrix} L_{a} \frac{d i_{a}}{dt} + R_{a} i_{a} = K_{ch} u - K_{v} {\overset{\cdot}{θ}}_{m} \end{matrix}

(1)

where $L_{a}$ and $R_{a}$ represent the armature inductance and the overall resistance of the armature circuit, respectively. $i_{a}$ denotes armature current of the motor, and $u$ represents the input control voltage. $k_{ch}$ is the chopper gain while $k_{v}$ denotes the electromotive force constant, with ${\overset{\cdot}{θ}}_{m}$ being angular velocity of the DC motor.

Figure 1.

Electronic throttle valve.

According to the torque balance theory, the system model of an ET valve is given considering the ET valve drag torque caused by airflow, frictional torque, and throttle return spring torque^8,9

\begin{matrix} {JK}_{l}^{2} \overset{\cdot\cdot}{θ} = K_{l} K_{t} i_{a} - T_{f} (\overset{\cdot}{θ}) - T_{sp} (θ) - T_{l} (θ) \end{matrix}

(2)

where $J$ denotes the overall moment of inertia referred to the motor side, $θ$ represents the position (opening angle) of ET, $K_{l}$ is the ideal gear ratio^6,24,25 which can be calculated by $K_{l} = θ_{m} / θ$ , $K_{t}$ is the motor torque constant, $T_{f}$ depicts the frictional torque, $T_{sp}$ denotes the return spring torque of the ET valve, and $T_{l}$ represents the ET valve drag torque caused by airflow. According to the work of Hu et al.,^6,24 $T_{f}$ can be described as

\begin{matrix} T_{f} (\overset{\cdot}{θ}) = F_{s} \overset{\cdot}{θ} + F_{c} sgn (\overset{\cdot}{θ}) \end{matrix}

(3)

where $F_{s}$ and $F_{c}$ represent the sliding friction coefficient and the coulomb friction coefficient, respectively. The return spring torque $T_{sp}$ is represented as follows

\begin{matrix} T_{sp} (θ) = K_{sp} (θ - θ_{0}) + K_{p} sgn (θ - θ_{0}) \end{matrix}

(4)

where $K_{sp}$ is the spring elastic coefficient, $K_{p}$ depicts the spring tightening torque coefficient, and $θ_{0}$ represents the default opening angle of the ET valve. The airflow over the throttle plate induces a small torque $T_{l}$ as²⁵

\begin{matrix} T_{l} (θ) = R_{af} F_{a} \cos θ \end{matrix}

(5)

where $R_{af}$ represents the distance between the focus of airflow force and the center of the ET valve, $F_{a}$ represents the air force acting on the plate parallel to the airflow direction, which is given as²⁵

\begin{matrix} F_{a} = Δ P A_{p} \cos θ \end{matrix}

(6)

where $Δ P = P_{atm} - P_{m}$ , $P_{atm}$ denotes the atmospheric pressure, $P_{m}$ is the manifold pressure, which can be measured with sensors, $A_{p} = π R_{p}^{2}$ is the throttle plate area, $R_{p}$ denotes the radius of throttle plate, then equation (5) can be rewritten as

\begin{matrix} T_{l} (θ) = R_{af} Δ P π R_{p}^{2} co s^{2} θ \end{matrix}

(7)

The sampling period is set as $T = 5 ms$ by considering the time constant of the ET model.¹ It is noted that, since $T_{a} = L_{a} / R_{a} \leq T$ , the armature current dynamics is not taken into account. Therefore, equation (1) is simplified as¹

\begin{matrix} i_{a} = \frac{1}{R_{a}} (K_{ch} u - K_{v} {\overset{\cdot}{θ}}_{m}) \end{matrix}

(8)

Based on equations (2)–(4), (7), and (8), the ET model can be given as

\begin{array}{l} \overset{\cdot\cdot}{θ} = - \frac{K_{s p}}{J K_{l}^{2}} (θ - θ_{0}) - \frac{K_{l}^{2} K_{t} K_{v} + F_{s}}{J K_{l}^{2}} \dot{θ} \\ + \frac{K_{t} K_{c h}}{J K_{l}} u - \frac{K_{p}}{J K_{l}^{2}} sgn (θ - θ_{0}) - \frac{F_{c}}{J K_{l}^{2}} sgn (\dot{θ}) \\ - \frac{π R_{a f} R_{p}}{J K_{l}^{2}} Δ P \cos^{2} (θ) \end{array}

(9)

Then, let $x_{1} = θ$ , $x_{2} = \overset{\cdot}{θ}$ , and adding a new external disturbance term $d (t)$ to the system dynamics, equation (9) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = a_{21} (x_{1} - θ_{0}) + a_{22} x_{2} + bu \\ - α_{2} sgn (x_{2}) - α_{1} sgn (x_{1} - θ_{0}) \\ - α_{3} Δ Pco s^{2} x_{1} + d \end{matrix}

(10)

where $a_{21} = - K_{sp} / ({JK}_{l}^{2})$ , $a_{22} = - (K_{l}^{2} K_{t} K_{v} + F_{s}) / ({JK}_{l}^{2})$ , $b = K_{t} K_{ch} / (J K_{l})$ , $α_{1} = K_{p} / ({JK}_{l}^{2})$ , $α_{2} = F_{c} / ({JK}_{l}^{2})$ , $α_{3} = π R_{af} R_{p}^{2} / ({JK}_{l}^{2})$ , and $d \leq | D |$ .

In addition to the nonlinear stick-slip friction and throttle nonlinear return spring, the ET model also considers the drag torque of the ET valve caused by airflow. The control objective is to make the throttle plate track the desired trajectory more accurate without overshoot.

ET controller design

The ET valve is a complicated nonlinear system, which includes many nonlinear characteristics and immeasurable signals. Thus, in order to control ET with high accuracy, the following control strategy is given, as shown in Figure 2.

Figure 2.

Control diagram of electronic throttle valve system.

ESO design

In the electronic-controlled throttle valve system, only the ET opening angle $x_{1}$ is measurable. Therefore, the angular velocity $x_{2}$ and the disturbance $d$ (i.e. $x_{3}$ ), which play an important role in the calculation of the control input, should be estimated by a state observer. Based on the ESO with equivalent control proposed in the literature,^26–28 a state observer is designed as follows

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = {\hat{x}}_{2} - \frac{b_{1}}{g' ({\hat{x}}_{1} - x_{1})} g ({\hat{x}}_{1} - x_{1}) \\ {\overset{\cdot}{\hat{x}}}_{2} = {\hat{x}}_{3} - \frac{b_{2}}{g' ({\hat{x}}_{1} - x_{1})} g ({\hat{x}}_{1} - x_{1}) + bu \\ {\overset{\cdot}{\hat{x}}}_{3} = - \frac{b_{3}}{g' ({\hat{x}}_{1} - x_{1})} g ({\hat{x}}_{1} - x_{1}) \end{matrix}

(11)

The appropriate parameters $b_{i} (i = 1, 2, 3)$ are chosen to guarantee the $s^{3} + b_{1} s^{2} + b_{2} s + b_{3}$ is Hurwitz; then according to Hurwitz theory, the closed-loop system is stable. Moreover, the function $g (z)$ should satisfy the following conditions^8,28

$g (z)$ is continuously differentiable;

$g (0) = 0$ ;

$\frac{dg (z)}{dz} \neq 0$ .

Intelligent double integral sliding mode controller (IDISMC) design

Double integral sliding mode controller

Let the desired ET angle be $θ_{d}$ , then the tacking error is described as

e = θ_{d} - x_{1}

(12)

And, the sliding mode surface is given as²⁹

\begin{matrix} s (t) = \overset{\cdot}{e} (t) + k_{1} e (t) + k_{2} \int_{0}^{t} e (τ) d τ \\ + k_{3} \int_{0}^{t} \int_{0}^{t} e (τ) d τ d τ \end{matrix}

(13)

where $k_{1}$ , $k_{2}$ , and $k_{3}$ are the positive constant, respectively. Then, the differential of $s (t)$ is expressed as

\begin{matrix} \overset{\cdot}{s} (t) = \overset{\cdot}{e} (t) + k_{1} e (t) + k_{2} \int_{0}^{t} e (τ) d τ + k_{3} \int_{0}^{t} \int_{0}^{t} e (τ) d τ d τ \\ = {\overset{\cdot\cdot}{θ}}_{d} - a_{21} (x_{1} - θ_{0}) - a_{22} x_{2} - bu \\ + α_{2} sgn (x_{2}) + α_{1} sgn (x_{1} - θ_{0}) \\ + α_{3} Δ P \cos^{2} x_{1} - d + k_{1} \overset{\cdot}{e} (t) + k_{2} e (t) \\ + k_{3} \int_{0}^{t} e (τ) d τ \end{matrix}

(14)

A candidate Lyapunov function is chosen as^8,9,30,31

\begin{matrix} V = \frac{1}{2} s^{2} (t) + \frac{1}{2 β} {\tilde{d}}^{2} \end{matrix}

(15)

where $β > 0$ , $\tilde{d} = d - \hat{d}$ , and $\hat{d}$ is the estimation of $d$ .

Assumption

$d (t)$ satisfies the condition that $\overset{\cdot}{d} = 0$ when $t \to \infty$ . The derivation of $V$ can be described as

\begin{matrix} \overset{\cdot}{V} = s (t) \overset{\cdot}{s} (t) - \frac{1}{β} \tilde{d} \overset{\cdot}{\hat{d}} \\ = s (t) [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{d} - a_{21} (x_{1} - θ_{0}) - a_{22} x_{2} \\ - bu + α_{1} sgn (x_{1} - θ_{0}) + α_{2} sgn (x_{2}) \\ - \hat{d} + α_{3} Δ P \cos^{2} x_{1} + k_{1} \overset{\cdot}{e} (t) \\ + k_{2} e (t) + k_{2} e (t) + k_{3} \int_{0}^{t} e (τ) d τ \end{matrix}] \\ - \frac{1}{β} \tilde{d} (\overset{\cdot}{\hat{d}} + β s (t)) \end{matrix}

(16)

In terms of equation (16), the double integral sliding mode controller (DISMC) can be designed as

\begin{matrix} u = - \frac{1}{b} [- {\overset{\cdot\cdot}{θ}}_{d} + a_{21} (x_{1} - θ_{0}) + a_{22} x_{2} \\ - α_{2} sgn (x_{2}) - α_{1} sgn (x_{1} - θ_{0}) \\ - α_{3} Δ Pco s^{2} x_{1} + \hat{d} - k_{1} \overset{\cdot}{e} (t) - k_{2} e (t) \\ - k_{3} \int_{0}^{t} e (τ) d τ - δ sgn (s (t))] \end{matrix}

(17)

where $δ > 0$ . According to equation (16), the adaptive law of the disturbance can be designed as

\begin{matrix} \overset{\cdot}{\hat{d}} = - β s (t) \end{matrix}

(18)

Submitting equations (17) and (18) into equation (16) yields

\begin{matrix} \overset{\cdot}{V} = s (t) \overset{\cdot}{s} (t) - \frac{1}{β} \tilde{d} \overset{\cdot}{\hat{d}} \\ = - δ s (t) sgn (s (t)) = - δ | s (t) | \leq 0 \end{matrix}

(19)

Let $f = k \overset{\cdot}{e} (t) + k_{2} e (t) + k_{3} \int_{0}^{t} e (τ) d τ$ , then equation (17) can be rewritten as

\begin{array}{l} u = - \frac{1}{b} [- {\overset{\cdot\cdot}{θ}}_{d} + a_{21} (x_{1} - θ_{0}) + a_{22} x_{2} - α_{2} s g n (x_{2}) \\ + \hat{d} - α_{1} s g n (x_{1} - θ_{0}) - α_{3} Δ P c o s^{2} x_{1} - δ s g n (s (t)) - f] \end{array}

(20)

By replacing the immeasurable variable $x_{2}$ with its estimation ${\hat{x}}_{2}$ , the following DISMC based on the designed observer is obtained

u = - \frac{1}{b} [- {\overset{\cdot\cdot}{θ}}_{d} + a_{21} (x_{1} - θ_{0}) + a_{22} {\hat{x}}_{2} - α_{2} sgn ({\hat{x}}_{2}) + \hat{d} - α_{1} sgn (x_{1} - θ_{0}) - α_{3} Δ Pco s^{2} x_{1} - δ sgn (s (t)) - f]

(21)

Implementation of laws for control parameters

Compared with traditional SMC, DISMC possesses good integral control characteristic to decrease the steady-state error of the closed-loop system. In addition, considering the symbol $f$ in equation (21) is similar to PID control law, so the control parameters $k_{1}$ , $k_{2}$ , and $s_{3}$ can be designed like P, I, and D parameters of the PID control law. In this article, the BP neural network is used for online self-adaption of the control parameters of DISMC. Let

\begin{matrix} f_{1} = - {\overset{\cdot\cdot}{θ}}_{d} + a_{21} (x_{1} - θ_{0}) + a_{22} {\hat{x}}_{2} \\ - α_{2} sgn ({\hat{x}}_{2}) - α_{1} sgn (x_{1} - θ_{0}) \\ - α_{3} Δ Pco s^{2} x_{1} - δ sgn (s (t)) + \hat{d} \end{matrix}

(22)

Then

\begin{matrix} u = \frac{1}{b} (f - f_{1}) \end{matrix}

(23)

According to the classical incremental digital PID control algorithm, it is obtained that

\begin{matrix} f (k) = f (k - 1) + Δ f (k) \end{matrix}

(24)

\begin{matrix} Δ f (k) = k_{1} (e (k) - 2 e (k - 1) + e (k - 2)) \\ + k_{2} (e (k) - e (k - 1)) + k_{3} e (k) \end{matrix}

(25)

In order to adaptively update the parameters $k_{1}$ , $k_{2}$ , and $k_{3}$ , the BP neural network with three layers is designed, as shown in Figure 3, where the structure of neural network is chosen as 2-5-3.

Figure 3.

Neural network structure of BP.

The input of neural network is expressed as

\begin{matrix} O_{j}^{(1)} = z (j) (j = 1, 2, \dots, N) \end{matrix}

(26)

where $N$ is the number of input variables, and $e$ and $\overset{\cdot}{e}$ are chosen as the input of BP neural network, then $N = 2$ . The input and output of network hidden layers are, respectively, expressed as

\begin{matrix} {net}_{i}^{(2)} (k) = \sum_{j = 0}^{2} ω_{ij}^{(2)} O_{j}^{(1)} \\ O_{i}^{(2)} (k) = h ({net}_{i}^{(2)} (k)) (i = 1, 2, \dots, M) \end{matrix}

(27)

where $ω_{ij}^{(2)}$ represents hidden layer weighting coefficient, superscripts (1), (2), and (3) are the input, hidden, and output, respectively; $k$ denotes the $k th$ iteration. In addition, the positive and negative symmetry Sigmoid function is selected as the activation function of hidden layer neurons

\begin{matrix} h (z) = \tanh (z) = \frac{e^{z} - e^{- z}}{e^{z} + e^{- z}} \end{matrix}

(28)

The input and output of output layer are, respectively, described as

\begin{matrix} {net}_{l}^{(3)} (k) = \sum_{i = 0}^{M} ω_{li}^{(3)} O_{i}^{(2)} (k) \\ O_{l}^{(3)} (k) = p ({net}_{l}^{(3)} (k)) (l = 1, 2, 3) \\ O_{1}^{(3)} (k) = k_{1} \\ O_{2}^{(3)} (k) = k_{2} \\ O_{3}^{(3)} (k) = k_{3} \end{matrix}

(29)

The nodes of the output layer denote the three adjustable parameters $k_{1}$ , $k_{2}$ , and $k_{3}$ , respectively. Furthermore, the nonnegative Sigmoid activation function of the output layer is chosen because the $k_{1}$ , $k_{2}$ , and $k_{3}$ are not negative

\begin{matrix} p (z) = \frac{1}{2} (1 + \tanh (z)) = \frac{e^{z}}{e^{z} + e^{- z}} \end{matrix}

(30)

Based on Figure 2, then the performance index function is chosen as

\begin{matrix} E (k) = \frac{1}{2} {((θ_{d} (k) - x_{1} (k)))}^{2} \end{matrix}

(31)

where $θ_{d} (k)$ and $x_{1} (k)$ are the desired input and actual output of ET, respectively. The weighting coefficients are updated through the gradient descent as

\begin{matrix} Δ ω_{li}^{(3)} (k) = - η \frac{\partial E (k)}{\partial ω_{li}^{(3)}} + α Δ ω_{li}^{(3)} (k - 1) \end{matrix}

(32)

where $η$ denotes learning rate and $α$ represents the inertia coefficient. In terms of equation (32), the following equation can be achieved

\begin{array}{l} \frac{\partial E (k)}{\partial ω_{l i}^{(3)}} = \frac{\partial E (k)}{\partial x_{1} (k)} \cdot \frac{\partial x_{1} (k)}{\partial u (k)} \cdot \frac{\partial u (k)}{\partial Δ f (k)} \\ \cdot \frac{\partial Δ f (k)}{\partial O_{l}^{(3) (k)}} \cdot \frac{\partial O_{l}^{(3) (k)}}{\partial n e t_{l}^{(3)} (k)} \cdot \frac{\partial n e t_{l}^{(3)} (k)}{\partial ω_{l i}^{(3)} (k)} \end{array}

(33)

and

{\begin{matrix} \frac{\partial u (k)}{\partial Δ f (k)} = \frac{1}{b} \\ \frac{\partial {net}_{l}^{(3)} (k)}{\partial ω_{li}^{(3)} (k)} = O_{i}^{(2)} (k) \end{matrix}

(34)

Since $\partial x_{1} (k) / \partial u (k)$ of equation (34) is unknown, so the sign function $sgn (\partial x_{1} (k) / \partial u (k))$ is used to replace $\partial x_{1} (k) / \partial u (k)$ , and the inaccuracy effect of approximate calculation can be compensated by adapting the learning rate $η$ . According to equations (25) and (29), it is obtained that

\frac{\partial Δ f (k)}{\partial O_{l}^{(3) (k)}} = e (k) - 2 e (k - 1) + e (k - 2)

(35)

\frac{\partial Δ f (k)}{\partial O_{2}^{(3) (k)}} = e (k) - e (k - 1)

(36)

\frac{\partial Δ f (k)}{\partial O_{3}^{(3) (k)}} = e (k)

(37)

Then, the weighting coefficient learning algorithm of the output layer is derived as follows

\begin{matrix} \begin{matrix} Δ ω_{li}^{(3)} (k) = α Δ ω_{li}^{(3)} (k - 1) \\ + η δ_{l}^{(3)} O_{i}^{(2)} (k) \end{matrix} \end{matrix}

(38)

δ_{l}^{(3)} = \frac{1}{b} \cdot e (k) \cdot sgn (\frac{\partial x_{1} (k)}{\partial u (k)}) \cdot \frac{\partial Δ f (k)}{\partial O_{l}^{(3) (k)}} \cdot p' ({net}_{l}^{(3)} (k))

(39)

where $h' (\cdot) = (1 - h^{2} (z)) / 2, i = 1, 2, \dots, M$ . Similarly, the weighting coefficient learning algorithm of the hidden layer is described as

\begin{matrix} Δ ω_{ij}^{(2)} (k) = α Δ ω_{ij}^{(2)} (k - 1) \\ + η δ_{i}^{(2)} O_{j}^{(1)} (k) \end{matrix}

(40)

\begin{matrix} δ_{i}^{(2)} = h' ({net}_{i}^{(2)} (k)) \sum_{l = 1}^{3} δ_{l}^{(3)} ω_{li}^{(3)} (k) \end{matrix}

(41)

where $h' (\cdot) = (1 - h^{2} (z)) / 2, i = 1, 2, \dots, M$ .

Simulations

Three groups of simulations are presented in this section to demonstrate the superior performance of the proposed approach. In the first simulation, online parameter self-adaption and observer performance is verified; in the second simulation, the robustness of the proposed approach is verified, and finally, the controller performance for challenging square wave signals is tested in terms of dynamic performance.

The system parameters are given as shown in Table 1.⁷ The function $g (z) = (1 - e^{- z}) / (1 + e^{- z})$ is selected as the nonlinear function of ESO. For the BP neural network, $η = 0.3$ , $α = 0.06$ , and the initiate values of weighting coefficients are random numbers in the interval $[- 0.6, 0.6]$ . Moreover, the desired ET angle $θ_{d}$ and its derivatives are known, which depends on engine operating conditions. The simulation is implemented in MATLAB/Simulink, and the comparison test is also conducted with SMC⁵ to show the superior performance of the proposed controller.

Table 1.

Parameters of electronic throttle valve.

Parameters	Value	Units
$θ_{0}$	2	°
$K_{l}$	16.95	−
$K_{t}$	0.016	$N m / A$
$K_{p}$	0.107	$N m$
$R_{a}$	2.8	$Ω$
$J$	$1.15 \times 10^{- 3}$	$kg m^{2}$
$F_{a}$	0.0048	$N m$
$K_{ch}$	2.4	−
$K_{v}$	0.016	$V s / rad$
$F_{s}$	$4 \times 10^{- 4}$	$V m s / rad$
$K_{s} p$	0.0247	$N m / rad$
$R_{af}$	0.0027	$m$
$R_{p}$	$0.0018$	$m$

Simulation 1: online parameter self-adaption and observer performance verification

In the article, control gains of DISMC are updated through a BP neural network. As shown in Figure 4, it can be observed that these control gains are bounded: $k_{1}$ belongs to the range [0.33, 0.38], $k_{2}$ belongs to the range [0.35, 0.69], and $k_{3}$ belongs to the range [0.54, 0.78].

Figure 4.

Online self-adaption of control parameters.

In this part, the control performance of tracking a sinusoidal reference signal has been presented in Figures 5 and 6. On one hand, from the viewpoint of the response time, as shown in Figure 5, the presented method is apparently better than SMC. On the other hand, from the viewpoint of system error, it is seen from Figure 6 that tracking errors of SMC fluctuate in the interval [−1.76, 2.56], while tracking errors of proposed controller fluctuate in the interval [−0.91, 1.32]. Obviously, the tracking errors of the proposed controller are smaller than that of SMC. Moreover, Figure 7 illustrates the estimation of the angular velocity. It can be seen from Figure 7 that the ESO is able to observe the angular velocity of the ET precisely (the absolute error is around 2 rad/s and the relative error is less than 5%). Meanwhile, Figure 8 shows the estimation of the external disturbances by the designed observer.

Figure 5.

Sine signal tracking response.

Figure 6.

Sine signal tracking error.

Figure 7.

Estimation of throttle opening change.

Figure 8.

Estimation of disturbance.

Simulation 2: controller robustness analysis

Figure 9 illustrates the step-tracking performance of IDISMC and SMC, and it can be observed that these two methods can achieve the desired throttle angle without overshot. However, from the aspect of the settling time (ST), the ST of IDISMC is 0.084 s, and the ST of SMC is 0.163 s. This means the ST of IDISMC is smaller than SMC, which indicates that the dynamic performance of the proposed controller is better than that of SMC.

Figure 9.

Step response.

As means to validate the robustness of the proposed control scheme against parameter variations, the following cases are examined, where the system parameters are set as changing parameters so that $K_{t}$ changes from 0.0160 to 0.0112, $K_{sp}$ changes from 0.0247 to 0.0321, and $F_{c}$ changes from 0.0048 to 0.0062. The simulation results are provided in Figure 10, and it shows that the step performance of IDISMC and SMC with parameter variations. One can see that the tracking errors with IDISMC and SMC are about 0.05 and 0.3, respectively. That is to say, the tracking errors of IDISMC is smaller than SMC after parameters variations, which means the accuracy of the proposed controller is higher than SMC, and the robustness of IDISMC is much better than SMC.

Figure 10.

Step response under parameter changes.

Simulation 3: controller performance for square wave signals

In this simulation, the performance of proposed control scheme is further validated in the context of square wave signals since the sudden changes of the signal from small to large opening angles or vice versa in practical applications, which are important to engine control, are possible. Moreover, there exists the phenomenon of go-and-stop in the highway when congestion happens. The vehicle speed is changing either fast or slow with the phenomenon of go-and-stop, which implies the ET valve should have the ability of making response timely with the phenomenon of go-and-stop. So, controller performance of square wave signal is discussed here. As a matter of fact, according to working range of the nonlinear spring in Hu et al.,⁶ the minimum and maximum values of square wave are, respectively, set as $10^{\circ}$ and $80^{\circ}$ . Figures 11 and 12 show the square wave signal tracking performance of IDISMC and SMC. It is shown in Figure 11 that the rising time of the IDISMC and SMC are about 0.0142 and 0.0446 s, respectively. This means that the rising time with the IDISMC is smaller than that of the SMC. Additionally, from the viewpoint of the falling time shown in Figure 12, the falling time of the IDISMC and SMC are about 0.0240 and 0.0440 s, respectively. From the results, it can be seen that the performance of the proposed controller is better than that of the SMC under square wave signals.

Figure 11.

Square wave tracking response.

Figure 12.

Square wave tracking error.

Conclusion

In this article, a new mathematic model of ET is proposed with consideration of the drag torque of the throttle plate. Then, an ESO-based IDISMC is carried out to improve the control accuracy of ET. First, the ESO is applied to estimate the throttle plate angular velocity and external disturbance. Meanwhile, the Lyapunov theory and sliding mode method are used to controller design. Particularly, the BP neural network is used for online self-adaption of the control parameters. Finally, the simulation is conducted. Simulation results show that (1) BP neural network has a good ability for tuning the control parameters; (2) the ESO has good performance to estimate the disturbance and the angular velocity of the throttle plate; and (3) compared with the SMC, the proposed controller has better dynamic performance and robustness with respect to different input signals and parameters variation.

Footnotes

Handling Editor: Fakher Chaari

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 Open Foundation of First Level Zhejiang Key in Key Discipline of Control Science and Engineering, National Natural Science Foundation of China (grant nos 61773082 and 61573195), and by the Key Project of Basic Science and Emerging Technology of Chongqing under grant cstc2017jcyjBX0018.

References

Deur

Pavković

Perić

et al . An electronic throttle control strategy including compensation of friction and limp-home effects. IEEE T Ind Appl 2004; 40: 821–834.

Yuan

Wang

. Neural networks based self-learning PID control of electronic throttle. Nonlinear Dynam 2009; 55: 385–393.

Sheng

Bao

. Fruit fly optimization algorithm based fractional order fuzzy-PID controller for electronic throttle. Nonlinear Dynam 2013; 73: 611–619.

Panzani

Corno

Savaresi

. On adaptive electronic throttle control for sport motorcycles. Control Eng Pract 2013; 21: 42–53.

Pan

Özgüner

Dağci

. Variable-structure control of electronic throttle valves. IEEE T Ind Electron 2008; 55: 3899–3907.

et al . Observer-based output feedback control of electronic throttle. Acta Autom Sin 2011; 37: 746–754.

Chen

Guo

et al . Control of electronic throttle based on backstepping approach. Control Theory Appl 2011; 28: 491–496.

Yang

Zheng

et al . Extended state observer based adaptive back-stepping sliding mode control of electronic throttle in transportation cyber-physical-systems. Math Probl Eng 2015; 2015: 301656.

Yang

Zheng

et al . Extended state observer based double loop integral sliding mode control of electronic throttle valve. IEEE T Intell Transp 2015; 16: 2501–2510.

10.

Eski

Yıldırım

. Neural network-based fuzzy inference system for speed control of heavy duty vehicles with electronic throttle control system. Neural Comput Appl. Epub ahead of print 9 June 2016. DOI: 10.1007/s00521-016-2362-0.

11.

Sun

Zhao

Huang

et al . A fuzzy approach for optimal robust control design of automotive electronic throttle system. IEEE T Fuzzy Syst. Epub ahead of print 28 March 2017. DOI: 10.1109/TFUZZ.2017.2688343.

12.

Wang

Liu

et al . Robust adaptive position control of automotive electronic throttle valve using PID-type sliding mode technique. Nonlinear Dynam 2016; 85: 1–14.

13.

Amini

Razmara

Shahbakhti

. Robust model-based discrete sliding mode control of an automotive electronic throttle body. SAE Int J Commer Veh 2017; 10: 317–330.

14.

Vašak

Baotić

Morari

et al . Constrained optimal control of an electronic throttle. Int J Control 2006; 9: 465–478.

15.

Vašak

Baotić

Petrović

et al . Hybrid theory-based time-optimal control of an electronic throttle. IEEE T Ind Electron 2007; 54: 1483–1494.

16.

Kim

Peng

Bai

et al . Control of integrated powertrain with electronic throttle and automatic transmission. IEEE T Contr Syst T 2007; 15: 474–482.

17.

Yuan

Wang

Sun

et al . RBF networks-based adaptive inverse model control system for electronic throttle. IEEE T Contr Syst T 2010; 18: 750–756.

18.

Yuan

Wang

et al . Neural network based self-Learning control strategy for electronic throttle valve. IEEE T Veh Technol 2010; 59: 3357–3765.

19.

Wang

Huang

. A new intelligent fuzzy controller for nonlinear hysteretic electronic throttle in modern intelligent automobiles. IEEE T Ind Electron 2013; 60: 2332–2345.

20.

Montanaro

Gaeta

Giglio

. Robust discrete-time MRAC with minimal controller synthesis of an electronic throttle body. IEEE-ASME T Mech 2014; 19: 524–537.

21.

Vargas

Acho

Pujol

et al . Output feedback of Markov jump linear systems with no mode observation: an automotive throttle application. Int J Robust Nonlinear Control 2016; 26: 1980–1993.

22.

Zhang

Yang

Zhu

. LPV modeling and mixed constrained H2/H∞ control of an electronic throttle. IEEE-ASME T Mech 2015; 20: 2120–2132.

23.

Jiao

. Synthesis and validation of finite time servo control with PSO identification for automotive electronic throttle. Nonlinear Dynam. Epub ahead of print 14 August 2017. DOI: 10.1007/s11071-017-3718-4.

24.

Liu

Sun

et al . Design of an ADRC-based electronic throttle controller. In: 2011 30th Chinese control conference, Yantai, China, 22–24 July 2011, pp.6340–6344. New York: IEEE.

25.

Yadav

Gaur

. Robust adaptive speed control of uncertain hybrid electric vehicle using electronic throttle control with varying road grad. Nonlinear Dynam 2014; 76: 305–321.

26.

Guo

Zhao

. On convergence of non-linear extended state observer for multi-input multi-output systems with uncertainty. IET Control Theory A 2012; 6: 2375–2386.

27.

Yao

Jiao

. Adaptive robust control of DC motors with extended state observer. IEEE T Ind Electron 2014; 61: 3630–3637.

28.

Kang

Chen

. A design method of nonlinear extension state observer. Electr Mach Control 2001; 5: 199–203.

29.

Lin

Chen

Huang

. Intelligent double integral sliding-mode control for five-degree-of-freedom active magnetic bearing system. IET Control Theory A 2011; 5: 1287–1303.

30.

Park

Chwa

Eom

. Adaptive sliding mode antisway control of uncertain overhead cranes with high-speed hoisting motion. IEEE T Fuzzy Syst 2014; 22: 1262–1271.

31.

Zhang

Han

Zhang

et al . Sliding mode control with mixed current and delayed states for offshore steel jacket platforms. IEEE T Contr Syst T 2014; 22: 1769–1783.

Extended state observer–based intelligent double integral sliding mode control of electronic throttle valve

Abstract

Keywords

Introduction

Mathematic model of ET

ET controller design

ESO design

Intelligent double integral sliding mode controller (IDISMC) design

Double integral sliding mode controller

Assumption

Implementation of laws for control parameters

Simulations

Simulation 1: online parameter self-adaption and observer performance verification

Simulation 2: controller robustness analysis

Simulation 3: controller performance for square wave signals

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References