High-efficiency aircraft antiskid brake control algorithm via runway condition identification based on an on-offvalve array

Dong SUN, Zongxi JIAO,b, Yoxing SHANG,b,*, Shui WU,b,Xiocho LIU,b

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

b Ningbo Institute of Technology, Beihang University, Ningbo 315800, China

KEYWORDS

Abstract The aircraft antiskid braking system is an important hydraulic system for preventing tire bursts and ensuring safe take-off and landing. The brake system adjusts the force applied on the brake discs by controlling the brake pressure.Traditional aircraft antiskid braking systems achieve antiskid performance by controlling the braking pressure with an electrohydraulic servo valve.Because the pilot stage of an electrohydraulic servo valve is easily blocked by carbonized hydraulic oil,the servo valve would become a dangerous weak point for aircraft safety.This paper proposes a new approach that uses an on-off valve array to replace the servo valve for pressure control.Based on this new pressure control component,an efficient antiskid control algorithm that can utilize this discontinuous feature is proposed.Furthermore,the algorithm has the ability to identify the runway circumstances. To overcome the discontinuity in the process of using an on-off valve array, the Filippov framework is introduced.The conditions of convergence of the system are also discussed.The results of the digital simulations and the hardware-in-the-loop (HIL) braking experiments are used to verify the efficiency and stability of the proposed control algorithm.The method also proves that the on-off valve array can replace the servo valve perfectly as a new type of antiskid braking pressure control component.

1. Introduction

The aircraft antiskid brake system ensures that the tires will not wear excessively or burst while braking by reducing the brake pressure when the wheel is locked, thereby ensuring the safety of aircraft take-off and landing. An antiskid brake system requires high adaptability to the runway environment and high dynamic response speed. Although with the development of more electric aircraft,the electrical drive brake system appears, but due to technical maturity and reliability,the hydraulic brake system is still the mainstream in the field of aircraft brake.1Hydraulic brake system obtains hydraulic energy from the engine driven pump (EDP) through a long pipeline.2The traditional aircraft brake system uses servo valves as brake pressure regulating components, as shown in Fig.1, Psis the supply pressure, Pbis the brake pressure, Pbris the reference brake pressure, PTis the reservoir pressure or tank pressure. Because of the vibration and high temperatures in the braking process, the hydraulic oil can easily carbonize, and carbonized hydraulic oil often blocks the pilot stage of a servo valve.3This blockage will lead to the loss of the pressure regulating function of the aircraft brake system, which substantially affects aircraft safety.

Compared with a traditional brake servo valve, an on-off valve array has great advantages in response speed and oil pollution resistance. The port of an on-off valve can be fully opened and closed in a few microseconds. Moreover, a comparison of these two kinds of valves with the same maximum flow shows that the valve opening area of an on-off valve is hundreds of times larger than that of the pilot stage of a servo valve, and the reliability of an on-off valve is higher than that of a servo valve in the case of hydraulic oil contamination.

Considering the difficulties of high-precision control caused by large-flow on-off valves and the low-pressure response speed caused by small flow valves, the control accuracy and response speed of the system can be guaranteed by using different flow-size on-off valves to form an array. The structure of the on-off valve array is shown in Fig.2.

During the process of increasing pressure and decreasing pressure, many upstream or downstream on-off valves are opened at the same time to realize pressure control. In the stage of pressure maintenance, the on-off valves are closed,and there is no flow loss. This structure can also increase the redundancy and reliability of the system.

The purpose of using an on-off valve array is to improve the antipollution ability of the system while realizing efficient antiskid braking control. Therefore, the primary goal is to achieve high-efficiency antiskid brake control.

Fig.1 Structure of a traditional aircraft hydraulic brake system.

The performance of the antiskid brake system is seriously affected by the runway conditions(e.g.,dry,wet or icy),which have strong nonlinear and time-varying characteristics. In the field of traditional aircraft antiskid brake control, many effective control methods have been proposed to solve this problem,and some methods have been applied in aircraft for actual use.

The pressure-bias-modulated(PBM)control method developed by Stubbs et al. has been applied in the Boeing 707 and some other kinds of aircraft; this method was presented as early as 1979.4By setting different thresholds of wheel speed tracking error, the braking pressure is adjusted according to the wheel skidding depth.In recent decades,many new control algorithms have been proposed. In 1995, neural networks and fuzzy logic were used in aircraft antiskid brake control by Tseng et al.5Tseng’s method uses these two tools to learn runway friction characteristics.Robust control was introduced by Tunay and Fengyu Li et al. These previous studies6-8completed the control for the nonlinear characteristics of the hydraulic part of the brake system. Li and Jiao8used the‘‘Lugre” friction model and parameter adaptive law to estimate runway friction characteristic parameters and calculate the optimal slip ratio. In 2017, Jiao et al. proposed a method to detect the runway friction coefficient by the slip factor, in which the calculation of the slip factor requires the installation of a brake torque sensor to obtain a brake torque signal.9In 2017, D’Avico et al. presented a method based on mixedslip-deceleration control, and they calculated ground friction with a detailed model of landing gear for antiskid braking control.10In 2018, Jia et al. proposed an active interference suppression control based on the optimal slip ratio, which controlled the working point of the system near the optimal slip ratio and restrained the interference in the braking process to ensure the high efficiency and stability of the brake.11These methods can achieve good control effects under different runway conditions,but almost all of these methods need to introduce other sensor signals, such as aircraft speed or braking torque. However, in a traditional braking system, the wheel speed signal and the brake pressure are the only inputs, and it is very difficult to add other sensors to an aircraft.In general,these traditional algorithms provide many ideas for the key problems of runway identification in antiskid control.

Furthermore, considering the discontinuity of on-off valve controls, on-off valve antiskid braking control is seldom used in aircraft brakes. In contrast, on-off valve antiskid braking control is usually used in an automobile antilock brake system(ABS).

Fig.2 Structure of the on-off valve array.

In the field of aircraft braking, Huang et al. proposed a switching control method that achieved aircraft antiskid brakes using two on-off valves in 2013.12,13They defined a switching surface to determine the control strategy of the onoff valves.However,this method needed to determine the optimal slip ratio of the runway in advance.In the field of automobile brakes,a four-phase control scheme was presented by Kuo and Yeh in 1992.14By adding a high-pressure holding mode and a low-pressure holding mode, the system achieved predetermined deceleration in a few cycles, and the road condition was also detected by wheel angular deceleration.In 1999,Choi and Cho proposed an antiskid braking method using pulsewidth modulation (PWM) to control on-off valves to achieve slip ratio control.This control method achieved a good control effect. However, the PWM control diminished the response speed and service life of the on-off valves.15In 2003,the effects of ABS antiskid control using different sliding surfaces were compared by Johansen et al. A new type of sliding surface was proposed that greatly reduced the system vibration caused by using on-off valves.16A sliding mode control (SMC) controller was designed for the antilock control system, and the effects of the sliding surface design were analyzed by Shim et al.in 2008.17In 2011,Jing et al.proposed a switched control strategy, and the algorithm was sufficiently robust to guarantee good performance in consideration of the uncertainties of the actuator characteristics and road friction coefficient.18In this method,the controller needs to use an acceleration sensor to estimate the vehicle speed as an input. These control methods in the field of automotive braking provide many good ideas for the design of aircraft antiskid braking systems based on on-off valve arrays.

This article is organized as follows: A dynamic model of aircraft brake is built in part 2; The algorithm structure and details are designed in part 3; Simulation and analysis of the algorithm are shown in part 4; In part 5, the hardware-inthe-loop (HIL) brake experiments are carried out, and the results are analyzed; In part 6, the characteristics of the method are summarized.

2. Modeling for an aircraft brake system

The process of aircraft antiskid braking is complex,and many nonlinear factors exist within this process. To simplify the modeling and analysis processes, the secondary factors that influence the system, such as the residual thrust of engines,are neglected. First, the kinematic model of the aircraft and the main wheels of the aircraft are built,the force balance diagram of the wheel is shown in Fig.3. This model considers only the longitudinal motion of the aircraft, and the rolling resistance acting on the wheel is neglected. The equations of the model are shown as follows:

The forces acting on the aircraft mainly include the friction between the two main wheels and the runway Ff. In addition,unlike automotive dynamics,aerodynamic drag Faexists in the aircraft model because of the fast speed.M0is the mass of the aircraft, Vpis the velocity of the plane, ρ is the density of air,Cais the aerodynamic drag coefficient,S is the windward area of the aircraft,r is the wheel radius,ω is the angular velocity of the wheel,FNrepresents the support force acting on the wheel,μ is the friction coefficient, Tbis the brake Torque acting on the wheel, J is the wheel inertia, and G in Fig.3 represents the gravity acting on the fuselage.

Fig.3 Force balance diagram of the wheel.

ΔM represents an equivalent change in aircraft mass.In the high-speed stage of aircraft landing, engine thrust reverser or deceleration parachute are usually used to decelerate, and brake is usually intervened in the middle-speed stage. With the existence of aerodynamic drag, the aircraft would decelerate more easily,which means that the equivalent mass M of the aircraft will be reduced,as shown in Eq.(2).It should be noted that due to the aircraft aerodynamic shape design and the relatively low brake start speed, the aerodynamic drag has a low impact on the equivalent inertia of the aircraft, and changes slowly compared with wheel speed. It can be considered that the equivalent inertia does not change in a short period of time(e.g. within 1 s). This will facilitate our further analysis.

μ in Eq. (1) represents the friction coefficient between the runway and the tires. The change of λ affects the value of μ.λ is the slip ratio, which can be obtained by the following equation:

The relationship between the friction coefficient μ and the slip ratio λ can be described by the ‘‘Magic Equation” which is expressed as follows19-22:

where B,C,D and E are the friction characteristic description coefficients.

For a hydraulic brake actuator, without considering heat transfer, temperature changes, time-varying oil properties and the dynamic piston process (including oil filling process),the brake actuator model can be expressed by the following equation12:

where kbis the brake disc friction coefficient and Kerepresents the equivalent pressure-volume coefficient of the brake actuator, which can be obtained from Eq. (6):

where Vba0is the initial volume of the brake actuator; Kbdis the comprehensive stiffness of the brake device; Q represents the flow of the system, which is affected by the brake pressure and the state of the on-off valve array; Abais the area of the brake actuator; and Ehis the elastic modulus of hydraulic oil.

3. Controller design

3.1. Runway circumstance identification

From the relationship between the ground friction coefficient and the slip ratio,we can conclude that under different runway conditions, the maximum friction coefficient and the optimal slip ratio of runway are different,23-25as shown in Fig.4,Ffmaxrepresents the maximum friction and λoptrepresents the optimal slip ratio that maximizes the friction.

A large number of runway friction tests indicate that because there are too many factors affecting the runway friction, most of these factors have nonlinear characteristics. It is difficult to obtain an empirical database of runway friction from which we can match the current runway condition.Moreover, the shape of these friction-slip ratio curves under different runway conditions indicates that we can judge whether the friction is close to the maximum friction. Here,we use the slope of the curves dFf/dλ as an indicator. Obviously, dFf/dλ ＞0 suggests that the friction increases with the increase of slip ratio. When dFf/dλ=0, the system achieves the optimal working point, which means that the maximum friction is achieved. When wheel begins to slip, the friction decreases with increasing slip ratio, and dFf/dλ ＜0. According to these conclusions, we only need to monitor the system status until dFf/dλ is close to 0, which can make the working point of the system close to the optimal point but located on the left side of the optimal point,which ensures that the wheels will not easily slip. To achieve this goal, a friction point determination threshold η (η ＞0) should be set. When 0 ＜dFf/dλ ＜η is satisfied, the system status should be maintained.

To obtain dFf/dλ, the friction force Ffand the aircraft velocity Vpshould be calculated in advance. In the process of aircraft braking,unlike the antiskid control of automobiles,the input to the controller is only the wheel speed and braking pressure, so the aircraft speed and friction should be observed first.Many studies have investigated vehicle and aircraft brake control.26It can be concluded that the friction Ffcan be observed by some methods, such as the Kalman filter. However,to ensure the observability of the system and the accuracy of the observation results, a relatively accurate aircraft speed Vpis needed as the input of the observer. A simple method of aircraft speed estimation is presented here.

According to Eq. (2), the following expression can be obtained:

During the initial stage after landing, there is no braking pressure,so the linear speed of the wheel is equal to the aircraft speed. When the pressure begins to increase without slippage,Eq. (8) can be used to calculate the aircraft speed, and a relatively accurate aircraft speed can be obtained.

Fig.4 Relation between slip ratio and friction.

The Kalman filter is used to observe and estimate the runway friction force.27-30Usually,after doing several differential operations on friction Ff,it can be assumed that the result is a random value. The state vector can be defined as follows:

The state equations can be expressed as follows:

Eq. (10) can be divided into the following expressions:

The measurement equation can be expressed as follows:

where w(t) is the Gaussian white noise produced by the state estimating process, n(t) is the measurement of the Gaussian white noise.

The equations for estimating the state and the error of the covariance matrix are as follows:

where ^ indicates the estimated value of the variable, P is the estimated error covariance matrix, F is the Jacobian matrix fkf(X), Qkfand Rkfare covariance matrixes, the subscripts of letters represent Kalman filter; K is the Kalman filter gain,which is determined by Eq. (14):

The change rate of friction dFf/dt and the friction Ffcan be obtained with the Kalman filter.Using the aircraft velocity Vp,the slip ratio can be obtained, and the differential of the slip ratio can also be calculated. The key parameter dFf/dλ can be obtained. During the initial stage of braking, the braking pressure is controlled by the on-off valve array, which causes the pressure to increase. The runway condition is identified by the condition 0 ＜dFf/dλ ＜η. Once this condition is met,the current braking pressure is recorded as Pbmax,which represents the maximum brake pressure that does not slip on the current runway condition. According to Pbmaxand the condition above, we can obtain the maximum deceleration that the runway can provide to the aircraft. The calculation method is as follows:

It should be noted that the whole optimization process depends on a stable pressurization process. To make the pressure increase process as smooth as possible,the minimum flow on-off valve group is used for controlling pressure increases,and the appropriate error band is set to avoid the data fluctuation caused by frequent switching of valves, as shown in Fig.5. Because of the gap between the brake actuators and brake discs, all the valves are opened at the same time at first.When the pressure increases to a lower pressure Pbm,the largeflow on-off valve is closed and the valve with minimum flow is opened. The pressure Pbmshould be lower than Pbmaxunder all runway conditions.

3.2. State feedback based wheel speed control

3.2.1. System phase plane analysis

To use the on-off valve array to control the wheel deceleration according to the maximum deceleration,the phase plane analysis of the system is carried out. The dynamic of the wheel speed is determined by the braking pressure, and the braking pressure is determined by the flow Q through the on-off valve array. The influence of Q on the tracking error of the wheel speed and wheel deceleration is analyzed in Fig.6.

The flow characteristic of an on-off valve array control is discontinuous. To simplify the problem, three kinds of flow input (Q ＞0, Q=0 and Q ＜0) are analyzed. It is easy to determine from Fig.6 above that to reduce the tracking error of wheel speed and deceleration, it is necessary to switch the flow Q at the appropriate time.

Fig.5 Control process of pressure growth with a constant slope.

Fig.6 Phase plane analysis of wheel speed control.

By using this state feedback for switching control of the onoff valve array,the phase trajectory of the system can be gradually approached to the set of equilibrium points of the system,as shown in Fig.8. In the set of equilibrium points, the tracking error of the wheel deceleration can be guaranteed to be 0,which is also the purpose of our wheel speed control.

Note that when the system reaches the set of equilibrium points,the tracking error of the wheel speed is not zero,which does not affect the tracking of wheel deceleration.Experiments show that in the actual system,as long as the tracking error of the wheel speed is limited and sufficiently small,stable tracking control of wheel deceleration can be guaranteed. Hence, the system stability can be guaranteed by choosing appropriate parameters ε1and ε2.

3.2.2. State feedback function design

The key point of the control is to design an appropriate state feedback function f(x).First,the state equation of the system is constructed.

For the state variable x2, according to the previous modeling process, we can obtain the following result:

In the last section, we used dFf/dλ as a criterion to determine whether the peak value of friction was found. With Kλ=dFf/dλ,Kλrepresents the slope of the friction curve near the optimal point, as shown in Fig.9.

Fig.7 Phase diagram after introducing f(x).

Fig.8 Phase trajectory after introducing f(x).

After finding the optimal working point of the system, the system controls the wheel speed. The actual working point of the system swings back and forth in a very small range near the optimal working point. During this process, the following linearization process is performed.

Note that x2can be written as follows:

Since the slip ratio λ=1-ωr/Vp, the differential of the slip ratio is as follows:

If the system does not slip, the slip ratio is very small.Moreover, it can be stated that the aircraft speed is approximately equal to the wheel speed.

The validity of this approximation has been verified by simulation. Fig.10 shows that the differential values of the slip ratio before and after approximation are very close during the process of braking pressure variation.

Combining Eq. (1), Eq. (15), Eq. (16), and Eq. (21) produces the following expression:

Bringing the above results into the system state equation produces the following expressions:

Fig.9 Practical significance of Kλ.

Fig.10 Real and approximate dλ/dt.

The inertia of the wheel and the mass of the aircraft differ enormously in magnitude, usually up to thousands of times.Hence,compared with the dynamic change in the wheel speed,the dynamic change in the aircraft speed is very small. By letting a1=-Kλr2/JVPand a2=-kbKe/J,the following expression can be obtained:

Let f(x1, x2)=f(x) be the state feedback equation. The input of the system is selected according to the state feedback:

Note that the system is a right-hand discontinuous system.To analyze the characteristics of this discontinuous system,the Filippov framework is introduced here. This framework is introduced in detail in the literature.31The existence and uniqueness of the right-hand discontinuous system solution are analyzed in Appendix A.

In addition, according to the previous analysis, we define the equilibrium point set of the system:

The method of system stability is used to analyze and calculate the system state feedback f(x). Because the system is discontinuous on the right-hand side, we choose the Lur’e Lyapunov function to analyze the stability of the system. This method is widely used for similar problems.32

To ensure that the generalized derivative of V(x)is less than or equal to 0, the state feedback function f(x) needs to satisfy the following conditions. The detailed proof and derivation of the system stability are given in Appendix B.

According to the above restrictions,it is easy to find a state feedback function f(x) that can ensure the stability of the system. As shown below, the feedback control law of the system Q=φ(f(x)) is designed.

The obtained state feedback function f(x) can be directly applied to wheel speed control using a single group of on-off valves. Note that the wheel deceleration is used in the state feedback function. To obtain this variable, the wheel speed must be differentiated.The tracking differentiator(TD)is used for the differential filter32and achieves good results.

3.2.3. Runway change detection based on state feedback

In Section 3.1,an optimization method for runway recognition is proposed. This method is effective in the initial stage of braking. However, the environment of the runway is not constant throughout the braking process. Water may accumulate in some sections of the runway. In other words, the road condition will switch among dry,wet or other runway conditions.If the optimization result of the initial braking stage is used during the whole braking process, frequent wheel skidding or low efficiency braking would occur. This realization also means that the controller needs to detect the changes in the runway condition.

First, the impact of changing runway conditions is analyzed.

A. Low friction coefficient runway to high friction coefficient runway.

Eq. (1) in the modeling section shows that the dynamics of the wheel are related only to the braking torque and the friction torque. When the aircraft enters the high friction coefficient runway, the change in the working point of the system is shown by the green arrow in Fig.11, wherein only the final stable state of the system is considered. Because the braking torque does not change in this process and there exists the relationship that rFf≈Tb,we can conclude that the friction remains approximately constant in this process, which means that the slip ratio of the system will decrease from λ1to λ2.In this process, because the deceleration of the aircraft is approximately constant, the wheel deceleration and wheel speed will rapidly increase.

The changes in the above process are reflected in Fig.12.When this process occurs, the system working point originally located in the shaded area will enter the first quadrant due to the rapid increase in the wheel speed and wheel deceleration.

Using this feature, runway change detection can be achieved by using the state feedback function and setting the error limit εc(εc＞ε1) when Q ≥0 (pressure maintenance or increase). If the working point suddenly enters the area of f(x)＞εc, then we can determine that the maximum friction coefficient of the runway increases. After this judgment, the following work involves using the optimization method in Section 3.1 to find the maximum friction point of the high coefficient runway and controlling the wheel speed using the state feedback function.

Fig.11 Changes during runway switching (low to high).

Fig.12 Runway state switching detection (low to high).

B. High friction coefficient runway to low friction coefficient runway.

Unlike the first case, when the μmaxof the runway the runway condition becomes worse. Because the braking torque does not change in a short period of time and the friction torque decreases rapidly,the wheel will be locked quickly and the slip ratio will change very rapidly from λ2to λ1(λ1=1), as shown in Fig.13.

In the system diagram analysis shown in Fig.14, in the presence of wheel speed control, the controller will actively reduce the pressure and stabilize the working point in the shaded part.However,because the preset deceleration of wheel speed control is determined by the runway with a high friction coefficient, this deceleration is too large for the runway with a low friction coefficient, which will eventually lead to wheel skidding. The working point enters the third quadrant from the shaded area.

Fig.13 Changes during runway switching (high to low).

Using this feature, runway change detection can be achieved by using the state feedback function and setting the error limit-εc(-εc＜-ε2)when Q ≤0(pressure maintenance or decrease) and Pb＜γpPbmax. If the working point suddenly enters the area of f(x)＜-εc, then we can determine that the maximum friction coefficient of the runway decreases. Note that γpis a threshold coefficient used to determine the difference between the actual braking pressure Pband Pbmax. This kind of detection needs to be maintained for a period of time to eliminate disturbances, such as local water accumulations on the runway. After this judgment, the following work involves decreasing the pressure to a pressure that does not slip under any runway conditions and using the optimization method in Section 3.1 to find the maximum friction point of the low coefficient runway.

3.2.4. Multivalve array control designs

To further explore the characteristics of this control algorithm and derive control strategies of multiple groups of on-off valves, simulations of on-off valve control with different flow Q are carried out.During the simulation process,the threshold ε1and ε2of the state feedback function remain constant.

The inlet flow of the switch valve in the simulations is q1and 4q1, and the outlet flow is -q2and -4q2, i.e., the flow of the high flow on-off valve group is four times that of the other group.

Figs.15 and 16 show the speed control capability of the system at a constant deceleration under two kinds of on-off valve flow. The preset deceleration and the wheel reference speed curve are the same in the simulations. The only difference is the flow of the on-off valve. The control signal of the on-off valve in the simulation is displayed.When the value of the signal is+1,the upstream on-off valve is controlled to open and the downstream on-off valve is controlled to close, which will cause the brake pressure to increase.When the value of the signal is -1, the downstream on-off valve is controlled to open and the upstream on-off valve is controlled to close,which will cause the brake pressure to decrease. When the value of the signal is 0,both valves are closed,which will cause brake pressure to maintain.

Fig.14 Runway state switching detection (high to low).

Fig.15 Wheel speed control effect (Q={q1,-q2}).

Fig.16 Wheel speed control effect (Q={4q1,-4q2}).

Note that to reflect the actual situation as accurately as possible, the gap between the initial position of the brake actuators and the brake discs is added in the simulation, which means that the on-off valve controlling the increase pressure needs to be kept open for a period of time to fill the chamber so that the actuators contact the brake discs. After this process,the brake pressure will increase rapidly.This is the reason why the pressure is approximately constant while the upstream valves remain open at the beginning of the simulation curves.

The results and phase diagrams in Fig.17 show that a small flow will cause a slow convergence process; however, the control process is smoother under a small flow than a large flow.The system would converge faster if the flow of the brake valve is increased, but it will lead to the fluctuations in the system that are difficult to suppress.Similar conclusions can be drawn from the system phase trajectories of the two simulation processes. Under a large flow, the state feedback function can restrict the error within a small range, but it will lead to a larger control step, which means that the system trajectory is more likely to cross the system equilibrium point set and oscillate near the equilibrium point set.These oscillations will result in many unnecessary actions of the valve during the control process. When the flow is small, the system converges slowly due to the smaller control step,and the error range may be larger,but it is easier to converge to the set of equilibrium points.

Through the above simulations and analyses, we hope to combine the control advantages of the two kinds of on-off valves so that the whole wheel speed control process can be fast and smooth. In addition, the number of valve actions in the whole control process should be minimized to prolong the service life of the on-off valve.

The basic idea of wheel speed control using an on-off valve array is to select the valve with appropriate flow according to the value of the state feedback function.The greater the difference is between the value of the state feedback function and the state feedback threshold(ε1,ε2),the greater the flow of the onoff valve array will be applied.

Take the on-off valve array composed of three groups of on-off valve arrays as an example. The flow of upstream onoff valves is Quv={q1u,q2u, q3u}(q1u＜q2u＜q3u), and the flow of downstream on-off valves is Qdv={-q1d, -q2d,-q3d}(-q1d＞-q2d＞-q3d), the subscript u represents upstream, d represents downstream. Flow combination can be achieved by controlling multiple on-off valves to open and close simultaneously. Therefore, in the process of wheel speed control, the upstream flow of the system is Qu={q1u,q2u,q3u, q1u+q2u,q1u+q3u,q2u+q3u,q1u+q2u+q3u} and the downstream flow is Qd={-q1d,-q2d,-q3d,-q1d-q2d,-q1d-q3d,-q2d-q3d, -q1d-q2d-q3d}. The positive threshold of the state feedback function is defined as εp={εp1,εp2,εp3,... ,εp8}, and the negative threshold is defined as εn={-εn1,εn2,εn3, ... ,εn8}, the subscript p represents positive, n represents negative. The control law can be designed as follows:

Fig.17 Wheel speed control phase diagram under different flow.

3.3. Overall structure design of the controller

The design in the above two parts mainly solves the problem of runway condition identification and wheel speed control by using the on-off valve array. The antiskid control law of the aircraft braking system is composed of two parts. The overall structure of the controller is shown in Fig.18.

The whole control algorithm consists of two parts: runway identification and wheel speed control based on state feedback.The two parts are controlled at different times,and the control law needs to be switched.

The two control laws should be switched depending on the system status, and the function block ‘‘Switching of control laws” in Fig.18 is a function designed to achieve this goal.The switching of the control laws can be triggered by two conditions. One condition is that the maximum friction is found by the optimization control in the initial stage of braking;the control law is converted from optimization control to wheel speed control. The other condition is that the change in the μmaxis detected;the wheel speed control will be switched to optimization control.Note that when the latter condition is satisfied, it is necessary to determine the baseline value of brake pressure increase for optimization control according to the change direction of μmax.

Fig.18 Overall control structure diagram.

In the process of wheel speed control, the system samples the wheel speed ω and uses the TD to calculate the state feedback f(x),and the error is judged by the state feedback threshold ε.The output flow Q is determined according to the error.Then, the flow Q is converted to the array on-off signal and output to the on-off valve array.Moreover,Q and Pbare combined with the state feedback function f(x) to monitor the change in the maximum friction coefficient of the runway.

Through the above processes, aircraft antiskid brake control based on an on-off valve array is achieved.

4. Simulation and analysis of the control algorithm

To verify the effectiveness of the antiskid control of the aircraft by using the switch valve array,the control algorithm is simulated and analyzed first. In this paper, simulations are carried out on the MATLAB/Simulink platform, and the step size is fixed at 1×10-4s. The sampling period and control calculation period are both 5 ms.The main parameters of the simulation model are listed in Table 1.

This algorithm is designed to achieve antiskid control. To evaluate the effectiveness of the control algorithm,the concept of the friction coefficient efficiency of the brake is introduced to evaluate the effectiveness of the control algorithm in the antiskid process.

Table 1 Main parameters list of the simulation.

where ημrepresents the friction coefficient efficiency, μbis the actual ground friction coefficient used in braking, μmaxis the maximum friction coefficient of runway, and L is the braking distance.The efficiency of the friction coefficient is obtained by integrating the utilization ratio of the friction coefficient in the whole braking process with respect to the braking distance.The ‘‘PBM” algorithm, which has no aircraft speed signal as an input,is used for comparison.Many simulations and experiments show that the friction coefficient efficiency is 50%-80%under wet runway conditions and 80%-90% under dry runway conditions when using the PBM control method.10

Using the above parameters and evaluation methods, we simulated four different working conditions. To simulate the braking process, the braking instructions are given after the aircraft landed for 1 s. In simulations, the pressure increase slope of the optimization process is set to 8 MPa/s. Since the flow of the on-off valve has been defined in the above parameter table, we quantify the on-off signal of the on-off valve array as {-7,-6,...,6,7} in the simulation process to open or close the on-off valve or a combination of on-off valves to produce a particular flow.

I Dry runway condition. The maximum friction coefficient of the runway is 0.5, and the runway state does not change during the whole process. The simulation results are shown in Fig.19.

Fig.19 Brake simulation on dry runway.

The simulation results above show that the algorithm is very effective under dry runway conditions.Fig.19(a)and(b)show that the whole braking process is smooth without skidding.The braking pressure remains high and stable after finding the maximum friction point. During the optimizing process, the pressure fluctuates slightly while the slope increases. Fig.19(c)shows that although there is pressure fluctuation, the differential tracker takes into account both signal smoothness and tracking speed.Although the estimated value of dFf/dλ fluctuates slightly,the overall trend is consistent with the actual value of dFf/dλ, and the maximum friction point is successfully detected at the threshold η without slipping. Fig.19(d) shows that the maximum friction coefficient of the runway is fully utilized during the braking process and that μbis close to μmax.Fig.19(e)shows that,except for pressure following in the optimization stage,there is little need for the on-off valve to act on the whole control process. The braking process is smooth and efficient, and the braking friction coefficient efficiency reaches 91.54%.The proposed control algorithm is more efficient than the ‘‘PBM”algorithm under this condition.

II. Wet runway condition. The maximum friction coefficient of the runway is 0.25, and the runway state does not change during the whole process. The simulation results are shown in Fig.20.

Fig.20 Brake simulation on wet runway.

In the simulation results of working condition II,Fig.20(a)and (b) show that the whole braking process is also smooth and steady. The braking pressure keeps high and stable after finding the maximum friction point. Brake pressure is maintained at a low level after the optimization process, but Fig.20(d)shows that μbis close to μmaxduring the whole process. Fig.20(c) shows that the estimated value of dFf/dλ can still follow the true value of dFf/dλ accurately, and the maximum friction point is successfully detected at the threshold η without slipping.Fig.20(e)shows that,the whole braking process hardly requires the action of the on-off valve.The braking friction coefficient efficiency reaches 85.77%. The proposed control algorithm is more efficient than ‘‘PBM” under this condition.

III. Wet-dry-wet runway condition. Conditions of dry and wet runways are consistent with those of the first two conditions.At 7 s,the runway changes from wet runway to dry runway,and at 15 s,from dry runway to wet runway.The runway state does not change in the rest of the time. The simulation results are shown in Fig.21.

Fig.21 Brake simulation on wet-dry-wet runway.

Fig.22 HIL braking test bench.

In the simulation results of working condition III,Fig.21(a)and(b)show that when the runway changes at 7 s and 15 s,the wheel speed fluctuates.At 15 s,due to the decrease in the runway friction coefficient,a slight slip occurs.The whole brakingprocess is smooth and steady.Fig.21(b)shows that the braking pressure can automatically change into a suitable value according to the maximum friction coefficient. It suggests that the wheel speed control can achieve accurately runway condition identification. According to the variation trend in the runway condition, the initial pressure of the first optimization is the pressure value of the wheel speed control at the last moment,while the initial pressure of the second optimization is 1.8 MPa.Fig.21(c)shows that the estimated value of dFf/dλ is still very close to the real value after the state feedback detects the road condition change at 7 s, and the maximum friction point with high friction coefficient is successfully detected without slipping.Fig.21(d)shows that μbis close to μmaxduring the whole process under different runway conditions.At 15 s,the runway friction coefficient suddenly decreases. To prevent excessive slipping, and because the value of the state feedback function exceeds the preset threshold,the controller gives an on-off signal that triggers a valve combination that produces a flow of 4 L/min, as shown in Fig.21(e). This signal rapidly reduces the pressure,causing the slip to become very small.Moreover,due to the existence of the state feedback function,the braking pressure does not decrease to 0 but maintains a relatively low level without slipping until condition Q ≤0, Pb＜γpPbmaxand f(x)＜-εcare satisfied.The braking friction coefficient efficiency reaches 86.83%.

Table 2 List of experimental parameters.

5. Experiments

The control algorithm proposed in this paper has achieved good results in simulations. In addition, the effectiveness and stability of the algorithm are further validated by HIL experiments. This method connects the hydraulic components of the brake system with the dynamic model of the wheel and the aircraft. In the field of braking, this kind of experimental method is often used to verify the effectiveness of hydraulic components and control algorithms in the brake system. In this paper, a new hydraulic component and a new algorithm are provided. It is difficult to accurately model the hydraulic part because of many uncertainty factors, such as leakage.The HIL experiment directly provides the physical object of this part, so the component and control algorithm can be validated effectively.

Fig.22 shows the experimental facility for the HIL experiments. Aside from a reduction in pipeline length, the experimental facility has a hydraulic component configuration that is exactly the same as that of the aircraft. At the beginning of the experiments, the dynamic model of the aircraft and the wheel starts to run on the computer. The brake pressure in the model is collected by the pressure sensor in the brake chamber. The brake control unit causes the pressure to increase and decrease by controlling the on-off valve array,and the brake pressure is transmitted directly to the wheel brake actuator. This process will continue until the aircraft speed in the computer model drops to 0.

Fig.23 HIL experiment on a dry runway.

The prototype of the designed on-off valve array is also shown in Fig.22. The upper three valves are used to control the pressure increase upstream, and the lower three valves are used to control the pressure decrease downstream. Six on-off valves are divided into three groups according to the flow. The flow of each group from left to right is 4 L/min,2 L/min, and 1 L/min (ΔP=10 MPa).

The experimental parameters are listed in Table 2. The experimental results are provided in Fig.23, Figs. 24, and 25.

Fig.23 shows the results of the dry runway test. Fig.23(c)shows that by setting the threshold and calculating the dFf/dλ,the controller successfully detected the maximum friction of the runway.During the tests,the motor of the hydraulic source produces electromagnetic interference to the sampling of pressure sensors, which also exists in the aircraft landing process.This problem is difficult to accurately repeat in simulations.This will lead to nonsmooth pressure sampling. These disturbances introduce a substantial amount of noise to the calculation of wheel deceleration, which increases the noise in the dFf/dλ signal.However,these disturbances do not affect the detection of the maximum friction point.The target deceleration is generated according to the Pbmax. In contrast to Fig.23(b) and (e), when both upstream and downstream onoff valves are closed, the brake chamber pressure still slowly decreases. Hence, it can be concluded that there exists a slight leakage in the brake chamber. The decrease in pressure will lead to a change in the state feedback function. The control algorithm opens the upstream on-off valves according to this change and increases the braking pressure to maintain the stable tracking of the reference wheel velocity. Leakage was not considered in the simulation,which explains why the number of actions of the on-off valve in the experiment is larger than that in the simulation. In addition, the braking process is smooth, and there is no slippage. Fig.23(d) shows that the maximum friction coefficient of the runway is 0.5, and the actual utilization of the friction coefficient during the braking process is very close to 0.5. According to the calculation method of braking efficiency mentioned above, the braking efficiency reaches 90.72%.

The wet runway HIL braking experiment results are displayed in Fig.24.The controller successfully detected the maximum friction of the runway. The number of actions in this experiment is larger than that in simulations because of the leakage of the system. In addition, the braking process is smooth and there is no slippage. Fig.24(d) shows that the maximum friction coefficient of the runway is 0.25, and the actual utilization of the friction coefficient during the braking process is very close to 0.25. According to the calculation method of braking efficiency mentioned above, the braking efficiency reaches 83.27%. Through the analysis of the two working conditions above, it can be concluded that the proposed algorithm can effectively adapt to the features of the actual hydraulic system.

Fig.24 HIL experiment on a wet runway.

Fig.25 HIL experiment on dry and wet runway.

Similar to the simulation above, the HIL braking experiment for dry-wet runway switching condition is also performed in Fig.25 When the wet runway is switched to the dry runway,the switch process is smooth and there is no slippage.The algorithm accurately finds the maximum friction points of the two runways in the optimization process. The maximum friction coefficients of the two runways are 0.5 and 0.25. The runway friction coefficient used in braking process is also very close to these two values. It should be noted that in the process of switching from dry runway to wet runway, there exists a deep slippage. From Fig.25(a) and (b), the reason is that although the controller gives the control signals of all downstream onoff valves opening at the same time, the pressure drop is not timely due to the relatively slow response of large flow onoff valves. This is also difficult to simulate accurately in the simulations. On the other hand, this process also verifies the effectiveness of the algorithm. When the slip occurs, the controller gives the maximum pressure relief instructions immediately to prevent deep slippage. If the features of the on-off valve are better, the algorithm can also achieve better performance. The braking efficiency reaches 80.69%.

6. Conclusion

In this paper, the on-off valve array is used to replace the traditional servo valve used in the aircraft braking process to overcome the oil contamination problem. Based on this new control component, an efficient aircraft antiskid braking control is proposed. Under the condition that the controller only has the wheel speed and the pressure signals as inputs, by introducing the Filippov framework,the discontinuous feature of the on-off valve is used to design the control algorithm. By observing the slope of the friction curve with Kalman filter,the maximum friction of the runway is detected, and the state change of the runway is detected by constructing the state feedback function.These two methods together constitute a precise and stable runway recognition algorithm.The stability and the disturbance resisting capacity of the system are also verified.The results of the simulation analyses and HIL braking experiments show that the proposed control algorithm can accurately identify the maximum friction of the runway without skidding and achieve higher braking efficiency than the‘‘PBM” method. It can be concluded from this paper that the on-off valve array can replace the servo valve perfectly as a new type of antiskid braking pressure control component,and achieve high braking efficiency.

Acknowledgments

The authors would like to thank the Science and Technology on Aircraft Control Laboratory and the National Nature Science Foundation of China (Nos. 51775014 and 51890882).

Appendix A. Analysis of the existence and uniqueness

According to Filippov theory,the definition of the Filippov solution of a differential equation with right-hand discontinuous can be understood intuitively as follows:if x(t)is the solution of a right-hand discontinuous differential equation g(x,u), its tangent vector at the discontinuous point must belong to the convex hull of the limit vector field in the infinitesimal neighborhood of discontinuous points.32

The existence and uniqueness of the system solution in Filippov framework are proved here. The system state vector x and the input Q are bounded, so the value of piecewise continuous function g(x,Q) is locally bounded. Through the mapping of functional K[·], the set K[g] is a non-empty locally bounded closed convex set in the solution space of the system.Moreover, the function K[g] is semi-continuous to x, that is,at any point x0.

The system satisfies the basic conditions of the existence theorem of solutions in reference.

For uniqueness, we mainly discuss the uniqueness of solutions at discontinuous switching surfaces. Using the f(x)=ε1as an example. The vector function g+is defined as the limiting value of the region (f(x)＞ε1). The same method defines g-, corresponding region (f(x)＜ε1).

The normal vector of the switching surface is defined as N.According to the sufficient condition of uniqueness of the solution, the projection of h to N hNshould satisfy the condition that hN≤0.

?f/?x2≥0 is a sufficient condition for our system to have an unique solution.

Appendix B. Analysis of system stability

The method of system stability is used to analyze and calculate the system state feedback f(x). Because the system is discontinuous at the right-hand side, we choose the Lur’e Lyapunov function to analyze the stability of the system, this method is widely used in the field of similar problems33-35.

We choose the Lyapunov function as shown above. Obviously, V(x) is regular whether it is on the switching surface or not. In addition, we find that V(x) is continuous, but not smooth. In the process of stability analysis, we need to derive V(x). For non-smooth V(x), Clark’s generalized time derivative method is used. In this method:

In order to ensure the stability of the system,i.e.dV/dt ＜0,Clark’s generalized derivative must be guaranteed to be less than or equal to 0.The stability analysis is needed in different regions.

Case (f(x)=ε1):

On the switching surface, the generalized differential of V(x) can be written as:

Similar to the previous analysis, ?V can be expressed as:

Owing to ζ∈?V, the ζ can be written as:

The generalized derivative of V(x) can be written as:

Because in this case, f(x) satisfies the condition f(x)=ε1:

In order to make the generalized derivative of V(x)less than or equal to 0, conditions ?f/?x1≥0 must be satisfied:

Case (-ε2＜f(x)＜ε1):

The generalized derivative calculation is performed directly.Obviously, this derivative satisfies the condition that it is less than or equal to 0 and no restriction is needed to be imposed on the state feedback function f (x).

Case (f(x)＞ε1):

In this case, the derivative of V(x) can also be calculated directly.

In order to make the generalized derivative of V(x)less than or equal to 0,the following restrictions need to be imposed on the state feedback function f(x):

Using the same analytical method to analyze the remaining two cases.

Case (f(x)=-ε2):

Case (f(x)＜-ε2):

In order to make the generalized derivative of V(x)less than or equal to 0, the state feedback function f(x) needs to satisfy the following conditions.