Research Article  Open Access
Asymmetric Barrier Lyapunov FunctionBased Wheel Slip Control for Antilock Braking System
Abstract
As an important device of the aircraft landing system, the antilock braking system (ABS) has a function to avoid aircraft wheels selflocking. To deal with the strong nonlinear characteristics, complex nonlinear control schemes are applied in ABS. However, none of existing control schemes focus on the braking operating status, which directly reflects wheels selflocking degree. In this paper, the braking operating status region is divided into three regions: the healthy region, the light slip region, and the deep slip region. An ABLFbased wheel slip controller is proposed for ABS to constrain the braking system operating status in the healthy region and the light slip region. Therefore the ABS will be prevented from operating in the deep slip region. Under the proposed control scheme, selflocking is avoided completely and zero steady state error tracking of the wheel optimal slip ratio is implemented. The HardwareInLoop (HIL) experiments have validated the effectiveness of the proposed controller.
1. Introduction
Antilock braking system (ABS), as an important safety guarantee of aircraft landing process, which keeps the braking direction stable and enhances braking performance, has a function to control aircraft landing dynamics. With the help of ABS, selflocking of main wheels and lost direction of front wheels, which will give rise to the sideslip and tail, are avoided. Meanwhile, under a good performance of ABS control, stopping distance is shortened, stopping time is saved, and tire wear is also reduced.
Taking advantage that the wheel speed is easy to obtain and the control scheme is simple to realize, the traditional ABS control is based on the wheel deceleration rate. Compared with the traditional control, ABS based on the wheel slip ratio control has a better control performance [1, 2]. The main objective of the slip control is to regulate wheel slip at its optimum value, which ensures that the aircraft braking system has a maximum tireroad friction. When the wheel slip of ABS operates at the optimum value, the wheel selflocking is avoided. Based on wheel slip control, some solutions have been proposed to deal with this complex nonlinear problem, such as neural network control [3], sliding mode control [1, 2, 4–6], integral feedback and back stepping control [7], combined control [8–10], iterative learning control [11], fuzzy control [12, 13], adaptive control [10, 14], nonlinear control [7, 15, 16], and extremum seeking control [17–19]. These studies always focus on how to track the optimum slip ratio with a good dynamic performance and steady state accuracy. However, none of the previously mentioned schemes take into account braking operating status, which directly reflects wheel slip degree.
In fact, in the slip ratio control, the braking operating region can be divided into three regions (the healthy region, the light slip region, and the deep slip region) dependent on the relationship between the slip ratio and the combination coefficient [20]. When the ABS operates in the deep slip region, not only is the system unable to obtain the maximum tireroad friction, but also the tires would be selflocking finally. Obviously, the ABS control should be designed to avoid braking operating status in the deep slip region.
In practice, outputconstrained tracking control is an important problem to deal with and some results have been achieved. Among them, [21–24] propose Barrier Lyapunov Function (BLF) based constrained control, which has a good tracking performance without violation of any constraint. Taking into account the initial condition of aircraft landing process, an Asymmetric Barrier Lyapunov Function (ABLF) based ABS wheel slip control scheme is proposed to constrain braking system operating status in the healthy region and the light slip region rather than the deep slip region. It prevents wheels from selflocking. In addition, the wheel slip ratio tracks optimum slip ratio with a zero steady state error.
The paper is organized as follows: Section 2 describes aircraft landing system. Section 3 formulates the Barrier Lyapunov Function problem. And Section 4 designs the controller based on ABLF for antilock braking. Section 5 analyzes HIL experiment results. Section 6 contains concluding remarks.
2. System Dynamics
Establishing the aircraft landing system model is the basis of the ABS controller design. Compared with common vehicles models, the aircraft landing system is affected by many aerodynamic forces, such as aerodynamic drag, aerodynamic lift, and parachute drag, which all are nonlinear parts with respect to aircraft speed. It makes the aircraft ABS model more complex and difficult to control. In order to simplify system model, some assumptions are as follows:(i)The aircraft is ideal rigid one.(ii)In the aircraft braking process, the engine is at idle state or reverse state.(iii)The system ignores the crosswind effect.(iv)The system ignores the tire deformation.(v)All wheels performances are the same, and they are controlled synchronously.
The aircraft force analysis is shown in Figure 1 and the parameters employed in the system are given in Table 1. The force balance equation of aircraft landing system is According to the aerodynamic theory, we have aerodynamic force functions as (2) and the definition of the coefficients is described in Table 2:The combination coefficient is defined as Substitute (3) into (1), and we obtain The model of wheel dynamics is given in Figure 2. Taking the main single wheel model into account (shown in Figure 2), the wheel dynamics are described as follows: where is angular velocity of the wheel; is inertia of the wheel; is support force of the wheel (the same as (1)); is radius of the wheel; is drag torque coefficient; is braking torque; the wheel slip ratio is defined as Since , we have that from (6); while , braking friction force is zero; and while , the wheel is in the selflocking state, which will cause sideslip and tail. There exists optimal slip ratio . When , the braking system has the maximum friction coefficient . and have a nonlinear relationship as follows [25] (shown in Figure 3):


(a) and
(b) Health region
(c) Slip region
As shown in Figure 3(a), braking operating status region is divided into three regions: the healthy region, where , the light slip region, where , and the deep slip region, where . If wheel slip ratio is (shown in Figure 3(b)), the braking operating status is in the healthy region. When the braking torque is applied to the main wheel, according to (5), the wheel angular velocity decreases. As a result, increases from to and increases to , which will give birth to the redaction of and increase of to . Accordingly, decreases from to and the system would be stable finally. If wheel slip is (shown in Figure 3(c)), the braking operating status is in the slip region (the light slip region or the deep slip region). When braking torque is applied to the main wheel, decreases and increases from to . Correspondingly, decreases to , which will give birth to the increase of and redaction of to . As a result, decreases from to . Finally, will increase up to one and the wheel will be selflocking. Compared with operating in the light slip region, the braking system operating in the deep slip region will cause more severe tire wear and even the wheel selflocking. It is the reason why the control law should keep ABS working in the healthy region and the light slip region, where .
According to (6), we obtain Substituting (4), (5), and (2) into (8), we have where
From (10), we find that the braking system has a strong nonlinear characteristic which is caused by the following factors:(i)The relationship of the wheel slip ratio with the wheel speed and the vehicle speed is nonlinear.(ii)The  relationship is nonlinear.(iii)There are multiplicative terms in the system state function.(iv)Equations (4) and (5) are nonlinear.
Taking into account the braking operating status region, ABS control problem is described as follows.
If possible, design a nonlinear controller for system (9) such that(1)the closed loop system is stable;(2);(3)the closed ABS operates in the healthy region and the light slip region. It implies that .
3. Barrier Lyapunov Function Preliminaries
In practice, it is an important problem for a fixed plant to track a desired trajectory while ensuring that all closed loop signals are bounded and that the output constraint is not violated. To prevent the output from violating the constraint, we employ a Barrier Lyapunov Function (BLF), which is defined as follows [24].
Definition 1. Barrier Lyapunov Function is a continuously differentiable and positive definite scalar function , defined with respect to the systemon an open region containing the origin. And it has the property as approaches the boundary of and satisfies along the solution of for and some positive constant .
A useful lemma [24] is as follows.
Lemma 2. An open set , where and are known constants. Consider the system where is piecewise continuous in and locally Lipschitz in , uniformly in , on . Suppose there exists a continuously differentiable function positive definite such thatIf the initial condition satisfies and derivative is negative definite, remains in the open set .
4. Asymmetric Barrier Lyapunov Function Controller
In the ABS design, the objective of control scheme is to ensure that the wheel slip ratio tracks the optimum wheel slip ratio . In the different road conditions, the optimum wheel slip ratios are not the same. And the optimum wheel slip ratio can be obtained by the identification technology. In this paper, the dry road condition is chosen and the optimum wheel slip ratio is 0.25. To ensure that the wheel slip ratio remains operating in the healthy region and the light slip region (it is ), an outputconstraint control is designed. Since the initial condition of the aircraft landing system is , which does not satisfy the initial condition of BLF function, an Asymmetric Barrier Lyapunov Function [24] (ABLF) is employed to relax the initial condition.
Considering system (9), the aircraft landing system can be rewritten as where the state variable . The input . The function is defined in (10). And the function . In the aircraft braking process, when the aircraft speed decreases to a fixed speed , which is a small positive constant such as 0.1 m/s, the ABS will be turned off and the aircraft stops by itself. So is positive, .
Define
Since is a constant, we have
Design the upper bound and the lower bound of state system as where is the upper bound of the light slip region. is the initial state of the system (14). And the initial condition for the aircraft landing system . Since , we have the initial condition . And the initial condition of Lemma 2 is satisfied.
Choose an ABLF as wherewhere denotes the natural logarithm of . And for any , . If and only if , we have . It implies that is positive definite, since is continuous within each of the two intervals and , respectively. Meanwhile, we have It implies that is continuously differentiable.
When , . And . The time derivative of is given by Design the control law as where is a positive constant. Substituting (22) into (21), we obtain Since , is negative definite for .
When , . The Lyapunov function . Its derivative is Design the control law as where is a positive constant. Substituting (25) into (24), we obtain Since , is negative definite for .
According to (22) and (25), the controller is designed as where is defined in (19).
According to (18) and (27), the derivative of the Lyapunov Function is
is negative definite. The system is asymptotically stable. And . Since the initial condition of Lemma 2 is satisfied, based on Lemma 2, remains in the open set . From , remains in the open set . Thus, the output constraint will never be violated and the ABS will operate in the healthy region and the light slip region.
According to (27), is continuous within each of the two intervals and , respectively. And there exists It implies that is continuous and there is no chatter in the switching process.
5. HardwareInLoop (HIL) Experiment
In this section, a HIL experiment platform is established and HIL experiment results are addressed.
5.1. HIL Experiment Platform Setup
To verify the performance of the controller, the aircraft braking HardwareInLoop (HIL) experimental platform is built and its structure is shown in Figure 4. The platform consisted of the host PC, the target CPU, the antilock controller, the braking actuator controller, the actuators, and the aircraft wheel. The host PC and the target CPU are the software simulation part, while the other four parts are hardware part.
Based on MATLAB/Simulink, the aircraft landing model, including the force balance model (described as (1) and (2)), the aircraft wheel model (described as (5)), and  model (described as (7)), is established in host PC. With the help of the discrete tool in MATLAB, the aircraft landing model is discretized with the step of 0.00002 s. With the tools of automatic code generation in MATLAB RTW, we rebuild the discrete aircraft landing model into the realtime code and download the realtime code into the target CPU through the JTAG. As the realtime simulation environment, the DSP TMS320F28335based target CPU (shown in Figure 5(a)) sends the display data to host PC through RS232 and provides the aircraft speed and the wheel speed to the antilock controller, which can be measured in practice by speed sensors. Based on DSP TMS320F28335, the antilock controller (shown in Figure 5(b)), which is the focus of this paper, is designed to give birth to the braking torque reference . The detailed control strategy of the antilock controller is described in the last section. The braking actuator controller, which consisted of a DSP TMS320F2812based CPU and a threephase fullbridge power circuit (shown in Figure 5(c)), has a function to control the brushless DC motors (BLDCMs) of the actuators on the aircraft wheel such that the braking torque generated by the wheel asymptotically tracked the braking torque reference provided by the antilock controller. In detail, with the help of the gears and ball screws, BLDCMs exert pressures on the brake pads of the wheel and the mutual sliding between the brake pads gives birth to the braking torque. Since the braking torque is the proportional function related to the pressures provided by the BLDCMs, the braking torque feedback is observed by the proportional function related to the pressures which are measured by the pressure sensors. The actuator and the aircraft wheel are shown in Figure 5(d), and the picture of the platform is shown in Figure 5(e).
(a) Target CPU
(b) Antilock controller
(c) Braking actuator controller
(d) Actuator and aircraft wheel
(e) HIL experimental platform picture
5.2. HIL Experiment Results
On the HIL experiment platform, we in this subsection conduct the ABLFbased antilock control experiment compared with the pressurebiasmodulated (PBM) technique [26], which is a wellknown traditional aircraft antilock approach. The parameters of the aircraft landing system model are described by Tables 1 and 2. And the parameters of the wheel model are , m, and kg·m^{2}. The limitation of the maximum braking torque is set at 20000 Nm. The initial conditions of the aircraft speed and the wheel speed are m/s and m/s, respectively. This implies that . And the lower bound of the aircraft speed is set as 0.1 m/s. Once the aircraft speed is lower than this lower bound speed, ABS will be turned off. Correspondingly, the aircraft stops by itself. In the  function, the optimum wheel slip ratio and the maximum friction coefficient . The parameters of the antilock controller are chosen as follows: , , and .
The experimental results of the proposed ABLFbased antilock control and PBM control are shown in Figures 6–11. From these experimental results in Figures 6–11, it can be clearly observed that under the proposed ABLFbased antilock control the wheel slip ratio tracks the optimal slip ratio with zero steady state error after a short time transient. And in the whole braking process, the ABS operates in the healthy region initially and then at the optimal braking status. This implies that the selflocking of the aircraft wheel is avoided completely. While under PBM control, the wheel slip ratio cannot track the optimal slip ratio well and it finally works in deep slip region where selflocking will occur. Correspondingly, compared with PBM control, the ABLFbased antilock control proposed has a better performance in wheel slip ratio, friction coefficient, wheel speed, stopping time, stopping distance, and braking torque.
(a) Antilock control based on ABLF
(b) PBM control
(a) Antilock control based on ABLF
(b) PBM control
(a) Antilock control based on ABLF
(b) PBM control
(a) Antilock control based on ABLF
(b) PBM control
(a) Antilock control based on ABLF
(b) PBM control
(a) Antilock control based on ABLF
(b) PBM control
For instance, Figure 6 shows the wheel slip ratio . Under ABLFbased antilock control, after a short time transient (less than 1 s), the wheel slip ratio tracks optimal slip ratio 0.25 without steady state error, while under PBM control, its slip ratio is around optimal slip ratio 0.25 when the aircraft speed is in the middle speed process. Yet the slip ratio has an obvious tracking error when the aircraft is in the high or low speed process. Figure 7 shows the friction coefficient . Under ABLFbased antilock control, after a short time transient process (also less than 1 s), the friction coefficient follows the maximum friction coefficient 0.6 with zero steady state error, which implies the ABS will provide the maximum braking force at its steady state, while under PBM control, only when the ABS works in the middle speed process, the friction coefficient is around the maximum friction coefficient 0.6; when the ABS works at the high or low speed process, the friction coefficient will deviate from the maximum friction coefficient. Figure 8 shows the aircraft speed and the wheel speed . Compared with the PBM control, the wheel speed of the ABLFbased antilock control has no oscillation. Figure 9 shows the stopping distance. The performance of the stopping time and the stopping distance under the ABLFbased antilock control (17 s and 625 m, resp.) is also better than the one under PBM control (24.4 s and 885 m, resp.). Figure 10 shows the braking torque . Compared with PBM control, the braking torque of the ABLFbased antilock control has no oscillation which is good at extending the life of actuator.
Figure 11 describes the contour of friction coefficient. It can be observed clearly that the braking operating region is divided into three regions (the healthy region, the light slip region, and the deep slip region) by the dash lines. Under the ABLFbased antilock control, the bold solid line is the actual braking status. It is in the healthy region firstly and then it is at the optical braking status. Compared with the control scheme proposed, the braking operating status of PBM control is in deep slip region at the braking low speed process, which will cause wheel selflocking. Based on the HIL experiment results, the performance comparison of the two control schemes is provided in Table 3.

6. Conclusion
In this paper, aircraft braking operating status is divided into three regions, which reflect the wheel slip degree directly. An ABLFbased wheel slip controller is proposed, which not only implements zero steady state error tracking of the wheel slip ratio but also constrains the braking system operating in the healthy region and the light slip region. And in contrast to existing antilock control method, it prevents wheels from selflocking completely.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the NSF of China under Grant 51407143 and by Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20136102120049.
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Copyright © 2015 Xiaolei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.