Barrier Lyapunov Function-Based Adaptive Back-Stepping Control for Electronic Throttle Control System

This paper presents an adaptive constraint control approach for Electronic Throttle Control System (ETCS) with asymmetric throttle angle constraints. The adaptive constraint control method, which is based on barrier Lyapunov function (BLF), is designed not only to track the desired throttle angle but also to guarantee no violation on the throttle angle constraints. An ETC mathematic model with complex non-linear system is considered and the asymmetric barrier Lyapunov function (ABLF) is introduced into the design of the controller. Based on Lyapunov stability theory, it can be concluded that the proposed controller can guarantee the stability of the whole system and uniformly converge the state error to track the desired throttle angle. The results of simulations show that the proposed controller can ensure that there is no violation on the throttle angle constraints.


Introduction
The electronic throttle control system (ETCS) is one of the main engine electronic control system (ECU) devices, which has a function to control the air intake of the engine cylinder by adjusting the position of the electronic throttle valve accurately. Therefore, the research of ETCS control algorithm has attracted extensive attention in the past few years.
Previously, the ETCS control algorithm usually used proportional-integral-derivative (PID) control to calculate the required throttle opening angle [1]. In [1], a control strategy of automotive ETCS based on PID control is proposed. However, this method relies on empirical or experimental adjustment parameters to a large extent, and it is easy to overshoot, affecting the stability. Therefore, improving the control performance of ETS based on throttle angle control algorithm has become a research focus in recent years. Such as sliding mode control methods (SMC) [2][3][4], backstepping control [5][6][7][8], fuzzy/neural network controls [9,10], and other control methods.
In [2], according to the variation of the angle of the electronic throttle (ET), a new extended state observer method is proposed, which takes into account the stick-slip friction, the nonlinear uncertainty of the spring, and the external disturbance of the system. In [3], for the robustness of the electronic throttle valve (ETV) system, a robust controller is designed based on the theory of integral sliding mode control. An adaptive sliding-mode tracking control strategy for an ET is applied in [4]. In this approach, input saturation constraints and parameter uncertainties are fully considered, and a parameter adaptive law is proposed to estimate unknown parameters with ET. An auxiliary system is designed to handle the input saturation. In [5], a finite-time convergence chattering-free backstepping to ensure that the system satisfies the constraints for strictly feedback nonlinear systems with time-varying output constraints, and a controller is designed. This design method has been successfully applied to many practical control problems [26,27]. An asymmetric BLF controller and an asymmetric time varying BLF controller are designed respectively for the problem of slip rate constraint in the ABS system in [28,29]. A nonlinear position controller is designed in [30] by a backstepping procedure using the barrier Lyapunov function to satisfy the constraints of position error and yaw. In [31], in order to deal with robust stabilization and output constraints of the system, a novel tan-type BLF and a power integrator technique have been proposed. In order to deal with system constraints, BLF is used in the backstepping process, and the control input is regarded as the extended state variable.
Based on the above methods, BLF is introduced into ETCS to solve the problem of desired throttle angle tracking with asymmetric throttle angle constraint. The main contributions of this paper can be summarized as follows: (1) In the ETCS, we consider a constrained control problem with asymmetric throttle angle constraint to ensure that it does not violate the throttle angle constraint; (2) The principle of throttle opening limitation is analyzed and a nonlinear ETC mathematic model with uncertainties is established. In the process of constraint controller design, BLF is introduced to deal with symmetric output constraints; (3) The simulation results show that our BLF control algorithm is more effective than the quadratic Lyapunov function (QLF) control algorithm. Notation 1. The notations used in this paper are standard. Superscript 'T' denotes matrix transposition; and X stands for the Euclidean norm for vectors and the spectral norm for matrices.

Electronic Throttle Mathematics Model
Electronic throttle is mainly composed of electronic control unit, drive motor, reduction gear, throttle body, return spring, and valve plate [6]. Its structure is shown in Figure 1.
Mathematics 2021, 9, x FOR PEER REVIEW 3 of 14 system is investigated. In this problem, when all states are constrained, the BLF is used to prevent the state from violating the constraint [24]. In [25], an asymmetric time-varying BLF is adopted to ensure that the system satisfies the constraints for strictly feedback nonlinear systems with time-varying output constraints, and a controller is designed. This design method has been successfully applied to many practical control problems [26,27]. An asymmetric BLF controller and an asymmetric time varying BLF controller are designed respectively for the problem of slip rate constraint in the ABS system in [28,29]. A nonlinear position controller is designed in [30] by a backstepping procedure using the barrier Lyapunov function to satisfy the constraints of position error and yaw. In [31], in order to deal with robust stabilization and output constraints of the system, a novel tan-type BLF and a power integrator technique have been proposed. In order to deal with system constraints, BLF is used in the backstepping process, and the control input is regarded as the extended state variable. Based on the above methods, BLF is introduced into ETCS to solve the problem of desired throttle angle tracking with asymmetric throttle angle constraint. The main contributions of this paper can be summarized as follows: (1) In the ETCS, we consider a constrained control problem with asymmetric throttle angle constraint to ensure that it does not violate the throttle angle constraint; (2) The principle of throttle opening limitation is analyzed and a nonlinear ETC mathematic model with uncertainties is established. In the process of constraint controller design, BLF is introduced to deal with symmetric output constraints; (3) The simulation results show that our BLF control algorithm is more effective than the quadratic Lyapunov function (QLF) control algorithm. Notation 1. The notations used in this paper are standard. Superscript 'T' denotes matrix transposition; and X stands for the Euclidean norm for vectors and the spectral norm for matrices.

Electronic Throttle Mathematics Model
Electronic throttle is mainly composed of electronic control unit, drive motor, reduction gear, throttle body, return spring, and valve plate [6]. Its structure is shown in Figure 1. The parameters in the structure are listed in Table 1.

Description
Symbol Coefficient of sliding friction The parameters in the structure are listed in Table 1. In the Reference [6], in order to improve the tracking performance of automotive electronic throttle, a control-oriented nonlinear electronic throttle model is proposed. The system model can be expressed as follows: ..
where J = n 2 J m + J g . However, in this paper, the throttle return spring torque T sp neglected the nonlinearity of the tightening torque coefficient of the spring and was given by However, the movement of the valve plate is limited between the maximum and minimum values. The limited travel is achieved by the return spring, and ideally there is only limited gain. The characteristic of the modeled spring can be described as where T LH is the spring offset and θ min and θ max are the lower and upper bounds of the throttle opening angle, respectively. So, we can have Therefore, the system model can be rewritten as Defining state variables as θ, u = E a and output variable y = θ, the system model can be rewritten as represents the sum of modeling error and unknown disturbance uncertainty. The |d(t)| is bounded. What we're trying to do is to design a constraint backstepping controller for a nonlinear dynamic system (6), which can make the output y and can track a desired trajectory y d (t) while ensuring the output constraint is not violated. Therefore, a controller based on BLF is designed to suppress the disturbances and handle the symmetric output constraints to improve the ETCS robustness.

Problem Statement
For an electronic throttle control system, the performance requirements to be considered in controller design include the following three aspects.
Firstly, one of the main tasks is to limit the throttle position, getting achievement of transient zero overshoot and avoiding the collision of limited position [5,12]. For the sake of controller performance analysis, we imply the objective is to specify another compact as large as desired for any initial compact set to which y(0) belongs. In other words, the output constraints are not violated.
Secondly, the goal of the nonlinear dynamical system controller is to control the output y can track desired trajectory y d (t).
Lastly, the strategy ensures that all the closed-loop signals are bounded.

BLF Preliminaries
To better establish constraint compensation and performance boundaries, we introduce the following definitions, assumptions, and lemma. A barrier Lyapunov function should be symmetric or asymmetric according to the boundary character, which is shown in Figure 2.
, such that the desired trajectory y d (t) and its time derivatives where η := ω S 1 T ∈ N, and h := R + × N → R l+1 is piecewise continuous with respect to t and locally Lipschitz with respect to η, uniformly with respect to t, on R + × N. Suppose that there exist functions U : R l → R + and V 1 : S 1 → R + continuously differentiable and positive definite in their respective domains, such that where γ 1 and γ 2 are class K ∞ functions. Let V(η) := V 1 (S 1 ) + U(ω) and S 1 (0) ∈ S 1 . If the inequality holds definite, has continuous first-order partial derivatives at every point of D , has the property ( ) V x → ∞ as x approaches the boundary of D , and satisfies ( ) , and some positive constant b .
A barrier Lyapunov function should be symmetric or asymmetric according to the boundary character, which is shown in Figure 2.

Controller Design
In the process of controller design in this section, fixed throttle angle constraints are directly considered to ensure that the constraint boundary is not exceeded. We employ the ABLF to design controller.
Defining the fixed constraints k a1 , k b1 as where Y l and Y h are the minimum and maximum value of the desired throttle angle θ * , respectively. k c and k c are the lower and upper bounds of opening angles of the throttle plate, respectively. The algorithm can be written as follows: Step1: Denote z 1 = x 1 − θ * as the tracking error, and z 2 = x 2 − α 1 as a virtual error, where α 1 is a virtual control function.
By selecting the asymmetric barrier Lyapunov function, where q(z 1 ) = 1, 0 < z 1 < k b1 0, − < k a1 z 1 ≤ 0 (9) k a1 and k b1 are the asymmetric constraints on z 1 , and each of these can be set independently, depending on the upper and lower bounds of θ * . To simplify the problem, we will denote q(z 1 ) by q.
By taking the derivative of Equation (8), we get the following expression: . Because Design the virtual controller α 1 as where k 1 is a positive constant.
z 1 z 2 can be canceled in a later step.
Step2: Define the Lyapunov function candidate as where η =η − η is the estimation of uncertainty upper bound η and η ≥ |d(t)|, and γ is a positive constant. Differentiating Equation (14), we can obtain The adaptive control law can be written as where k 2 is a positive constant. Equation (16) is substituted into (15) to obtain The adaptive law can be designed as follows: .η = γ|z 2 | By substituting (18) into (17), we can obtain As can be seen in Equation (19), . V is negative definite when the control law is (16) and the adaptive law is (18). Thus, the system is asymptotically stable. When t → ∞ , the z 1 → 0 . Based on Lemma 1, z 1 remains in (−k a1 , k b1 ), ∀t ∈ [0 , ∞]. Therefore, the throttle angle constraints will never be violated.

Discussion
In this part, simulation is carried out for three different conditions to verify the effectiveness of the proposed control strategy. Throttle Model Parameters are given in Table 2.   So, we can have k a = Y l − k c = 0.4 and k b = k c − Y h = 0.1. The simulation results can be seen in Figure 3. Figure 3a shows the tracking response curve of the system when the expected throttle opening is 0.4 rad, and Figure 3b shows the tracking error in this process. From Figure 3a,b, the ABLF controller can obtain good throttle opening tracking performance, the actual throttle opening is strictly kept within the throttle angle constraint boundary (0, 0.5), and the tracking error is within the constraint (−0.4, 0.1). This implies the ETCS will impossibly collide with the limit position and control transient overshoot effectively, even zero overshoot. The stability time of the step response is less than 150 milliseconds, and the stability error approaches zero gradually. However, QLF controller not only exceeds the constraint boundary of throttle opening angle, but also exceeds the constraint boundary of error, which means that the excessive transient overshoot of the desired throttle angle tracking can result in colliding limit position. At same time, the throttle valve fluctuates sharply, seriously impacting on the lifespan of the actuator. In addition, too much opening angle of the throttle plate will cause the car to stop when the manual car is in low gear, which will lead to accidents.
System uncertainty satisfies ( )=0.1cos( ) . The simulation results can be seen in Figure 3.

Case 2
In this sub-section, expected value of throttle opening is jump signal. In 0-0.6 s, expected value of throttle opening is 0.8 rad, the upper boundary of throttle opening is 0.9, the lower boundary of throttle opening is 0. System uncertainty satisfies d(t) = 0.1 cos(t). The controller parameters are k 1 = 190, k 2 = 150, γ = 1, Y l = Y h = 0.8, k c = 0, and k c = 0.9. States variable initial value: x(0) = 0.1 0 T . So, we can have In 0.6-0.7 s, the system is in the process of shifting. The expected value of throttle opening and the boundary of throttle opening are switched.
In 0.7-5 s, expected value of throttle opening is 0.4 rad, the upper boundary of throttle opening is 0.5, and the lower boundary of throttle opening is 0. System uncertainty satisfies d(t) = 0.1 cos(t). The controller parameters are chosen as follows: k 1 = 190, k 2 = 150, γ = 1, Y l = Y h = 0.4, k c = 0, and k c = 0.5. So, we can have k a = Y l − k c = 0.4 and Note: Since the derivative of the step function tends to infinity as it passes from one gear to another, Assumption 1 is not satisfied. In this paper, the transition part of the step function is replaced by a linear function corresponding to a line segment near the vertical line, as shown in Figure 4a. This satisfies both the actual moving process and the Assumption 1. . Note: Since the derivative of the step function tends to infinity as it passes from one gear to another, Assumption 1 is not satisfied. In this paper, the transition part of the step function is replaced by a linear function corresponding to a line segment near the vertical line, as shown in Figure 4a. This satisfies both the actual moving process and the Assumption 1.
The simulation results can be seen in Figure 4.  − ， . Therefore, the ABLF control method achieves good tracking performance. The actual throttle angle is always within the constraint range, and the error is always within the constraint range in the whole braking process. Furthermore, the tracking curve is smooth during the shifting condition.
However, the QLF controller not only exceeds the constraint boundary of throttle opening angle, but also exceeds the constraint boundary of error, which means that the excessive transient overshoot of the desired throttle angle tracking can result in colliding limit position. At same time, the throttle valve exist fluctuations during the shifting, impacting on the lifespan of the actuator seriously.

Case 3
In this sub-section, expected value of throttle opening is ( )  The simulation results can be seen in Figure 4. Note from Figure 4a,b that actual throttle opening angle is always within the constraint range (0, 0.9) and the tracking error is in the constrained range (−0.8, 0.1) before shift. After shift, the actual throttle opening stays strictly within the throttle angle constraint range (0, 0.5) and the tracking error is in the constrained range (−0.4, 0.1). Therefore, the ABLF control method achieves good tracking performance. The actual throttle angle is always within the constraint range, and the error is always within the constraint range in the whole braking process. Furthermore, the tracking curve is smooth during the shifting condition.
However, the QLF controller not only exceeds the constraint boundary of throttle opening angle, but also exceeds the constraint boundary of error, which means that the excessive transient overshoot of the desired throttle angle tracking can result in colliding limit position. At same time, the throttle valve exist fluctuations during the shifting, impacting on the lifespan of the actuator seriously.

Case 3
In this sub-section, expected value of throttle opening is (0.4 + 0.1 · sin(2t)) rad, the upper boundary of throttle opening is 0.65, and the lower boundary of throttle opening is 0. System uncertainty satisfies d(t) = 0.1 cos(t). The controller parameters are k 1 = 190, k 2 = 150, γ = 1, Y l = 0.3, Y h = 0.5, k c = 0, and k c = 0.65. States variable initial value: simulation results are shown in Figure 5.  It can be seen from Figure 5a,b that the ABLF controller can receive good tracking performance of expected throttle opening, actual throttle opening angle is always within the constraint range (0 0.65) ， , and the tracking error is strictly within the error constraint range ( 0.3 0.15) − ， . This implies the ETCS will impossibly collide with the limit position and control transient overshoot effectively, even zero overshoot. The stability time of the step response is less than 250 milliseconds, and the stability error approaches zero gradually. However, the QLF controller not only exceeds the constraint boundary of throttle opening angle, but also exceeds the constraint boundary of error, which means that the excessive transient overshoot of the desired throttle angle tracking can result in colliding limit position.
As can be seen in Table 3, the values of the performance criteria-over-adjustment, adjustment time, adjustment error-of the adjustment system in the studied cases are It can be seen from Figure 5a,b that the ABLF controller can receive good tracking performance of expected throttle opening, actual throttle opening angle is always within the constraint range (0, 0.65), and the tracking error is strictly within the error constraint range (−0.3, 0.15). This implies the ETCS will impossibly collide with the limit position and control transient overshoot effectively, even zero overshoot. The stability time of the step response is less than 250 milliseconds, and the stability error approaches zero gradually. However, the QLF controller not only exceeds the constraint boundary of throttle opening angle, but also exceeds the constraint boundary of error, which means that the excessive transient overshoot of the desired throttle angle tracking can result in colliding limit position.
As can be seen in Table 3, the values of the performance criteria-over-adjustment, adjustment time, adjustment error-of the adjustment system in the studied cases are clear. The BLF control strategies achieve the requirement: it has a shorter adjustment time, no over-adjustment, no violation of boundary, and smaller maximum absolute tracking error. However, with the QLF controller, the boundary of the throttle opening angle and tracking error are violated with serious over-adjustment.

Conclusions
In this paper, a second order mathematical model of ETCS is proposed, and a constraint controller based on ABLF is designed to constrain the throttle angle within an appropriate range to avoid collision with the limit position. The proposed controller is compared with the controller based on QLF via simulation of the analysis of different signal wave tracking response performance, such as step signal, jump signal, and sinusoidal signal. The simulation results show that the controller we designed can realize zero error tracking of throttle angle, and the constraint boundaries of throttle angle can not be violated under different response signals. In contrast, QLF-based controllers can result in constraint boundary violations. This means that the excessive transient overshoot of the desired throttle angletracking can result in colliding limit position. In addition, compared with the control effect of QLF controller, the controller we designed will not make the throttle angle oscillate. Because of the constraint boundary, the controller is more complex and the calculation is more complicated than the traditional PID or other methods. However, when the constraint boundary is taken into account, the motion angle range of the electronic throttle is limited to a reasonable range. The electronic throttle does not collide with the limit position due to frequent over-adjustment, which will increase the life of the actuator. Moreover, such precise control will make air intake more accurate, enable the engine airfuel ratio to be accurately controlled, save fuel, and protect the environment. It is therefore worth further research.

Conflicts of Interest:
The authors declare no conflict of interest.