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Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear
Applied Mathematical Modelling xxx (2014) xxx–xxx
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear Fu Li a,⇑, Guan Wei a, Wang Qi a, Xu Xinhe b a b
Shenyang Aerospace University, Liaoning Shenyang 110136, China Northeastern University, Liaoning Shenyang 110104, China
a r t i c l e
i n f o
Article history: Received 27 December 2013 Received in revised form 9 August 2014 Accepted 17 October 2014 Available online xxxx Keywords: Aircraft landing gear Magneto-rheological buffer system State equation Adaptive control Semi-active control
a b s t r a c t The main advantages of magneto-rheological buffer system for aircraft landing gear are adjustable damping force, simple structure and independent of external energy. A kind of magneto-rheological damper structure for aircraft landing gear was presented in this paper. The state equation was established and the landing buffer system control strategy was designed based on self-made magneto-rheological landing impact platform. Integrating with model predictive control, a semi-active control method was studied and used it in realizing the adaptive magneto-rheological buffer system landing process. It has been proved in practice that the proposed control method fully considered the boundless of output damping force. The problem of balancing between landing and taxiing, which is caused by un-adjustable passive damping force in the working process of oleo-pneumatic buffer system, is solved. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The passive oleo-pneumatic buffer has been mainly used in airplane landing gear buffer system at present. It has advantages such as simple structure, high reliability, and so on. But some important parameters like damping force cannot be adjusted in the process of landing and taxiing, which would lead to passive response to the ground impact. Because that the design of Oleo-pneumatic Orifice for the impact process of aircraft landing will arise ‘‘too hard’’ phenomenon during taxiing, and on the contrary, the design of Oleo-pneumatic Orifice for the shock process of aircraft taxing will be ‘‘too soft’’ during landing, the passive landing buffer system is difficult to balance the impact force between landing and taxiing. However, the active control landing gear needs an external larger power source. In recent years, the newly magneto-rheological fluid (MRF) has been developed, which can control the damping force with the change of coil current adjusted by the magnetic field intensity. This provides a new solution for the impact load buffering device. MRF is a kind of functional materials, invented and developed by Rabinow [1] who worked in National Bureau of Standards of the United States. Magneto-rheological fluid is mainly composed of the non-conductor magnetic fluid and the tiny particles dispersed uniformly with low hysteresis and high permeability. Under the effect of magnetic field, MRF can be rapidly and reversibly changed from the Newtonian fluidity liquid to Bingham plastic solid in milliseconds, which is high viscosity and low fluidity [1,2]. Magneto-rheological Impact absorption of the helicopter undercarriage can be seen as a typical application in the antiimpact load. Werely [3] uses the composite material made of rubber and magnetic-rheological fluid as the damping medium ⇑ Corresponding author. http://dx.doi.org/10.1016/j.apm.2014.10.043 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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F. Li et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
of helicopter undercarriage. The results show that the damper under magnetic field can provide continuous and controllable damping force. Meanwhile, it can overcome the damping amplitude loss which is caused by single frequency excitation. Batterbee et al. proposed a novel solution to this problem, that is to implement semi-active damping using MRF. They enabled the geometry of a flow mode magneto-rheological valve to be optimized with the constraints of an existing passive landing gear [4]. Choi and Wereley theoretically evaluated the electrorheological and magnetorheological fluid based landing gear system during touchdown of the aircraft. The feasibility and effectiveness of the response on attenuating dynamic load and vibration due to the landing impact were demonstrated [5]. Andrzej and Mikolaj presented investigation results of a semi-active industrial shock absorber with MRF, which is capable of controlling the stopping process of moving objects. The proposed solution makes it possible to adjust the braking force according to the kinetic energy of moving object [6]. Magneto-rheological damper was applied in soft recoil gun by Ahmadian et al. also the shooting range test was carried out and satisfactory results were obtained [7–9]. Studies have been a useful reference in the application of impact buffer control. Magneto-rheological intelligent buffer damper based on semi-active control technology can make up for the shortcomings of passive buffer damper effectively. It greatly improves the dynamic performance of traditional oleo-pneumatic buffer and provides high reliability. Compared to the active control, semi-active control buffer has advantages like low cost, simple structure, no need of large power source. But at present, the research around buffer technology is mainly at the stage of theoretical research. In this paper, according to the characteristics of impact load, we propose a control strategy and control algorithm, it could adaptively track ideal buffer force and velocity of impact load with magneto-rheological intelligent buffer damper in the self-made impact test platform. 2. Structure and mechanical model 2.1. Magneto-rheological buffer The variable-structure oil holes of traditional oleo-pneumatic buffer were designed as constant geometries shown in Fig. 1. The damping force of the flowing oleo is controlled by the intensity of applied magnetic field, which is controlled by the impressed current. The gas chamber filled with nitrogen, which is mainly used to provide the need of transformable displacement when impact condition. The filling amount and the initial pressure are determined by the change of displacement and the initial position of piston rod. 2.2. Characteristics and parameter identification Accurate damper model is the foundation for designing control strategy and obtaining good control effect. Because that it is extremely complex and difficult to analyze the damping characteristics based on rheological theory for magneto-rheological
Gas chamber Return oil hole
Excitation coil
Side oil hole
Oil chamber
Fig. 1. Magneto-rheological intelligent buffer structure for impact.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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damper, we establish dynamic model by integrating theoretical model with system identification. Through the characteristic test, magneto-rheological damper mechanical model can be obtained [10].
_ F d I; d; d_ ¼ cd d_ þ sd ðIÞsgnðdÞ;
ð1Þ
where, Fd is the damping force of magneto-rheological damper, sd(I) is the yield force of magneto-rheological damper fluid, which is controlled by the magnetic field intensity being relative to the current I, cd is the viscosity coefficient of magnetorheological damper fluid, kd is the elastic coefficient, d is the length of damper piston rod, d_ is the velocity of piston rod, O is the Origin of Coordinates. These parameters could be obtained by parameter identification algorithm. 2.3. Equation of the motion Two displacement sensors are separately put on the counterweight and the magneto-rheological damper in the self-made impact test platform, just as shown in Fig. 2. The dynamic model of magneto-rheological damper is built facing impact load, shown as Fig. 3. According to mechanical model and flexible impacting characteristics, the discrete state equation of controlled object is constructed.
x_ 1 ¼ x2 ; m1 x_ 2 þ k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ þ F z m1 g ¼ 0; x_ 3 ¼ x4 ;
ð2Þ
m2 x_ 4 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ m2 g F z þ kt ½x3 x3 ð0Þ ¼ 0; where x2, x4 are velocity, x1, x3 are displacement. Then get the formula (3).
x_ 1 ¼ x2 ; x_ 2 ¼ g
k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ 1 Fz; m1 m1
x_ 3 ¼ x4 ; x_ 4 ¼
ð3Þ
k½x3 ð0Þ x1 ð0Þ ðx3 x1 Þ kt 1 þg ½x3 x3 ð0Þ þ Fz: m2 m2 m2
The formula (3) are discretized,
x1 ðk þ 1Þ ¼ x1 ðkÞ þ sx2 ðkÞ;
Fig. 2. Self-made impact test platform.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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Fig. 3. Magneto-rheological damper mechanics model facing impact load.
k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ 1 x2 ðk þ 1Þ ¼ x2 ðkÞ þ s g Fz m1 m1 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ s Fz; ¼ x2 ðkÞ þ sg s m1 m1 x3 ðk þ 1Þ ¼ x3 ðkÞ þ sx4 ðkÞ; k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ kt 1 x4 ðk þ 1Þ ¼ x4 ðkÞ þ s þg ðx3 ðkÞ x3 ð0ÞÞ þ Fz m2 m2 m2 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ kt s ¼ x4 ðkÞ þ sg þ s s ðx3 ðkÞ x3 ð0ÞÞ þ Fz: m2 m2 m2
ð4Þ
Then we can get the system model as follows.
xðk þ 1Þ ¼ f ðxðkÞÞ þ gðxðkÞÞuðkÞ;
ð5Þ
yðkÞ ¼ AxðkÞ; where,
2
x1 ðkÞ
3
6 x ðkÞ 7 7 6 2 xðkÞ ¼ 6 7; 4 x3 ðkÞ 5 x4 ðkÞ 2 6 6 f ðxðkÞÞ ¼ 6 6 4
2
0
3
6 s 7 6 m1 7 7 gðxðkÞÞ ¼ 6 6 0 7; 5 4
uðkÞ ¼ F z ;
A ¼ ½ 0 1 0 1 ;
s
m2
x1 ðkÞ þ sx2 ðkÞ x2 ðkÞ þ sg s
k½ðx3 ð0Þx1 ð0ÞÞðx3 ðkÞx1 ðkÞÞ m1
x3 ðkÞ þ sx4 ðkÞ 3 ðkÞx1 ðkÞÞ s mkt2 ½x3 ðkÞ x3 ð0Þ x4 ðkÞ þ sg þ s k½ðx3 ð0Þx1 ð0ÞÞðx m2
3 7 7 7: 7 5
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
F. Li et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
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3. Control strategy The adaptive model predictive control algorithm is able to predict the future dynamic model, in which the control action and the error feedback correction are implemented through repeated online optimized calculation. Control algorithm is designed for above dynamic model as Eq. (5). A characteristic of model predictive control method is that the input and output constraints can be considered in the design process of controller, in other words, this control strategy can make sure that the control signal would not exceed the boundary of actual execution mechanism, and ensure that the system state does not exceed the preset bounded region. This feature is consistent with actual control requirements of magneto-rheological damper. The output of damping force is a bounded value, and the damper may work in the saturated state, that requires enough consideration on the problem of limited execution ability in the process of designing control algorithm. Otherwise, the control command signal will not be effectively implemented by actuator (damper), and this will lead to system performance degradation, perhaps collapse. In Fig. 2, two displacement sensors are separately put on the counterweight and the magneto-rheological damper, so that four parameters, which are displacement and velocity of the counterweight and the magneto-rheological damper, can be obtained in real time. Then the four parameters will be used as input signal of the control algorithm program. Thereupon, after each sampling, an online optimization process will be performed base on the designed control strategy and the feedback of sensor, thus, the control output (damping force) could be gotten. According to formula (1) or comparison table among damping force, compression speed and current intensity, we will get current intensity by interpolation operation. Finally, the current intensity will be presented to the MR damper to complete the sampling, calculation and control in one cycle. In this way, the automatic control scheme for the magneto-rheological damper based on model predictive control strategy facing impact load could be obtained. However, it is not enough, because we need to set control target of model predictive control algorithm. When the velocity which the impact load reaches the ground is given, the time history curves of ideal compression velocity can be gotten through the active control of the oleo-pneumatic buffer system. That is to say, if the magneto-rheological damper system can compress according to the ideal compression velocity curve in the first stroke, the magneto-rheological buffer system will most effectively absorb the impact energy in the process of Drop-test shock. The corresponding control target is to make the velocity error between actual compression and ideal compression minimal for magneto-rheological damper. The control target can be achieved through the design of objective function in model predictive control algorithm. In a word, the model predictive control implementation is proposed in this paper. We introduced design process of the magneto-rheological damper model predictive control algorithm, which is suitable for impact situation. The detailed calculations on the output (magneto-rheological damper force) are given, based on four parameters, which are displacements and velocities of the impact load and the damper unit. 4. Model predictive control 4.1. Control algorithm Based on the dynamic equations (5) of impact load system for aircraft landing gear, the controller is designed. x(k) 2 Rn is the system state, u(k) 2 Rm is the controller input, y(k) 2 Rp is the controller output, f ðÞ and gðÞ are known nonlinear functions, A is a constant matrix. The system meets the constraints as follows:
umin 6 uðkÞ 6 umax ;
jDuðkÞj 6 Dumax ;
ymin 6 yðkÞ 6 ymax ;
where
Duðk þ jjkÞ ¼ uðk þ jjkÞ uðk þ j 1jkÞ: It is the increment of control input. The main research is to design controller for the impact model equation (5), which is able to meet the stability of closedloop system, and the adjusting output can rapidly track the reference signal without steady-state error, that is
lim eðkÞ ¼ 0;
k!1
eðkÞ ¼ rðkÞ yðkÞ:
The control algorithm should ensure not only the zero steady state error but also the rapid performance, only in this way the working requirements of drop buffer can be reached. The system state is estimated at each sampling point, and the optimal control vector obtained from the optimization process is solved. According to above model predictive control method, the structure of target function can be constructed as follows.
JðkÞ ¼
NX N u 1 X jjrðk þ jjkÞ yðk þ jjkÞjj2Q þ jjDuðk þ jjkÞjj2R ; j¼1
ð6Þ
j¼0
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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where, r(k + j|k) is the reference signal, y(k + j|k) is the predicted output, N and N u are the prediction length and control mm length, Q 2 Rpp are the weighted matrices with appropriate dimensions, the norm ||||Q and ||||R are defined as pffiffiffiffiffiffiffiffiffiffiffi ffi , R2R T jjzjjW ¼ z Wz. In order to predict the current state of the system, we need to get the former prediction value, the expression is as follows:
xðk þ 1jkÞ ¼ f ðxðkÞÞ þ gðxðkÞÞðuðk 1Þ þ DuðkjkÞ; xðk þ 2jkÞ ¼ f ðxðk þ 1jk 1ÞÞ þ gðxðk þ 1jk 1ÞÞðuðk 1Þ þ DuðkjkÞ þ Duðk þ 1jkÞÞ; ... xðk þ NjkÞ ¼ f ðxðk þ N 1jk 1ÞÞ þ gðxðk þ N 1jk 1ÞÞ; ðuðk 1Þ þ DuðkjkÞ þ Duðk þ N 1jkÞÞ: Define the following vector
ðkÞ ¼ ½yðk þ 1jkÞ yðk þ NjkÞT 2 RNp ; y xðkÞ ¼ ½xðk þ 1jkÞ xðk þ NjkÞT 2 RNn ; T
ðkÞ ¼ ½DuðkjkÞ Duðk þ Nu 1jkÞ 2 RNn m ; Du ~ xðkÞ ¼ AðGðk ~ ðkÞ þ Kðk 1Þ þ Fðk 1ÞÞ; ðkÞ ¼ A 1ÞDu y where
2
3 A 0 . . . 6 N N ~¼4. . . .. 7 A 5 2 R p n; . 0 A 2 6 6 Gðk 1Þ ¼ 6 6 4
gðxðk 1Þ
0
3
gðxðk þ 1jk 1ÞÞ .. .
gðxðk þ 1jk 1ÞÞ .. .
0 .. .
7 7 7 2 RNn Nm ; 7 5
gðxðk þ N 1jk 1ÞÞ gðxðk þ N 1jk 1ÞÞ gðxðk þ N 1jk 1ÞÞ 2
gðxðkÞÞuðk 1Þ
6 6 Kðk 1Þ ¼ 6 6 4
gðxðk þ 1jk 1ÞÞuðk 1Þ .. .
3 7 7 7 2 RN n ; 7 5
gðxðk þ N 1jk 1ÞÞuðk 1Þ 2 6 6 Fðk 1Þ ¼ 6 6 4
3
f ðxðkÞÞ
f ðxðk þ 1jk 1ÞÞ 7 7 7 2 RNn : .. 7 5 . f ðxðk þ N 1jk 1ÞÞ
Then the controller design problem of model prediction can be transformed into the optimization problems solved constantly as follows,
~ ~ ~ ðkÞjj2Q þ jjDu ðkÞjj2R ; 1Þ AKðk 1Þ AGðk 1ÞDu min jjrðkÞ AFðk max 6 Du ðkÞ 6 Du max ; s:t: Du max ; umin 6 uðk 1Þ þ HDuðkÞ 6 u ~ ðkÞÞ 6 y max ; min 6 AðFðk 1Þ þ Kðk 1Þ þ Gðk 1ÞDu y
ð7Þ
where
ðkÞ ¼ ½ uðkÞ uðkÞ 2 RNu m ; u r ðkÞ ¼ ½ rðk þ 1jkÞ rðk þ NjkÞ T 2 RNp ; max ¼ ½ Dumax Du
Dumax T 2 RNu m ; T
min ¼ ½ umin u
umin 2 RNu m ;
max ¼ ½ umax u
umax T 2 RNu m ;
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
F. Li et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
min ¼ ½ ymin y
ymin T 2 RNp ;
max ¼ ½ ymax y 2 I 0 6I I 6 H¼6 6 .. .. 4. .
ymax T 2 RNp ; 3 0 07 7 N u mN u m : .. .. 7 72R . .5
I
I
I
7
I
ðkÞ 2 RNu m , the optimization problem can be transformed into the quadratic programming probDefining variable v ¼ Du lem as following
1 T v W v þ cT v ; 2 s:t: lmin 6 Ev 6 lmax ; min
ð8Þ
where
min T 2 R3Nu mþ2Np ; lmin ¼ ½ 1 Du T
max 2 R3Nu mþ2Np ; lmax ¼ ½ b Du T ~ ~ W ¼ 2 AGðk 1Þ Q AGðk 1Þ þ R 2 RNu mNu m ; T ~ 1Þ Q ðrðkÞ Kðk 1Þ Fðk 1ÞÞ 2 RNu m ; c ¼ 2AGðk T ð3N mþ2N p ÞNu m ~ ~ ; AGðk 1Þ AGðk 1Þ I 2 R u 3 min þ u ðk 1Þ u 6 7 ðk 1Þ max u u 6 7 7 2 R2Nu mþ2Np : b¼6 6 y 7 ~ ~ þ AFðk 1Þ þ AKðk 1Þ 4 min 5 ~ ~ max AFðk 1Þ AKðk 1Þ y
E ¼ ½ H 2
H
In conclusion, the detailed calculations from four inputting parameters, including the counterweight displacement, the counterweight velocity, the damper displacement and the damper velocity, to the magneto-rheological damper force output are given, namely the quadratic programming problem as (8) is solved. The control strategy is totally given for landing gear system. 4.2. Results In the dynamic equation (5) of impact load magneto-rheological damper, on the basis of the self-made impact drop stand, let the reference signal r(k) take the ideal velocity and damping force for the existing active control oleo-pneumatic buffer system of aircraft landing gear. The drop height is controlled to be 0.45 meters, so the grounding velocity of counterweight is approximate to 3 m/s. In Fig. 4, yr(t) represents the ideal sinking velocity, x2(t) is the time-history curve of the actual sinking
Fig. 4. The curves of yr ðtÞ and x2 ðtÞ.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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F. Li et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
Fig. 5. Curves of the ideal damping force ur ðtÞ and the magneto-rheological damper force uðtÞ designed by the control algorithm.
velocity from the piston rod of magneto-rheological damper. ur(t) is the ideal damping force of falling shock process, u(t) is the actual curve of damping force under the control algorithm designed in this paper. The control effect is shown in Fig. 5. From Figs. 4 and 5, under the control strategy and control algorithm based on model prediction in this paper, the two groups of parameters are all coincidence, that is the time-history curve of sinking velocity for MR buffer control system x2(t) (solid line in Fig. 4), the actual damping force provided by the MR buffer control system u(t) (solid line in Fig. 5), and the sinking velocity time-history curve yr(t) (dashed line in Fig. 4) and the time-history curve of ideal damping force ur(t) (dashed line in Fig. 5) Good control effect has been achieved using the control strategy and algorithm in the process of impact buffering. It also reflects high control performance compared with the active control. By adjusting the time-history curve of ideal control velocity, the ground impact energy can be maximally absorbed. Fig. 5 shows the change curve of the ideal damping force and the change curve of control output damper force with MPC algorithm. From Figs. 4 and 5, you can see that the control strategy and the proposed method effectively complete the controlling on the landing buffer force and speed of impact load in this paper. 5. Conclusion The main conclusions of this paper are as following. (1) A kind of new magneto-rheological buffer structure is presented for impact load. It could effectively absorb the impact energy produced in aircraft landing gears, and the oil-return hole can be used to prevent the tire from jumping off the ground when landing. (2) The discrete state equations of magneto-rheological damper buffer system are constructed under impact effect, on the basis of the self-made magneto-rheological buffer control platform facing impact load. (3) The control strategy of magneto-rheological damper buffer is proposed, and buffer control algorithm of magneto-rheological damper is designed based on model predictive control. The complex solving problem caused by nonlinear magneto-rheological fluid is avoided in this paper, and the control effect is also verified. (4) The output boundness of damping force is taken consideration in the proposed control method. It is coincide with the actual control system. (5) The boundness of the damper force variation is also taken consideration in the control. It makes the control algorithm more realizable. Compared with the traditional oleo-pneumatic buffer, damper structure is given in this paper and the corresponding control strategy and control algorithm can be implemented in the damper compression process. Because that the output damping force could be rapidly and effectively adjusted, the optimal buffering effect is obtained. Acknowledgments This research is partially supported by the National Natural Science Foundation of China (No. 61074090), the Innovation Funds of Aviation Industry Corporation of China (No. cxy2013SH16), and the Shenyang Science and Technology Plan Projects (F13-095-2-00). Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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References [1] J. Rabinow, The magnetic fluid clutch, AIEE Trans. 67 (1948) 1308. [2] H.B. Cheng, W.L. Qu, J. Zhang, J.W. Tu, Preparation and properties of MR fluids and MR damper, Funct. Mater. 37 (5) (2006) 811–813. [3] W. Hu, N.M. Werely, Magnetorheological fluid and elastomeric lag damper for helicopter stability augmentation, Int. J. Mod. Phys. B 7–9 (2005) 1471–1477. [4] D.C. Batterbee, N.D. Sims, N.D. Stanway, M. Rennison, Magnetorheological landing gear: 2. Validation using experimental data, Smart Mater. Struct. 16 (6) (2007) 2441–2452. [5] Y.T. Choi, N.M. Wereley, Vibration control of a landing gear system featuring electrorheological/magnetorheological fluids, J. AIRCRAFT 40 (3) (2003) 432–439. [6] M. Andrzej, H. Mikolaj, Application of magnetorheological fluid in industrial shock absorbers, Mech. Syst. Signal Process. 28 (2012) 528–541. [7] F.D. Goncalves, M. Ahmadian, J.D. Carlson, Behavior of MR fluids at high velocities and high shear rates, Int. J. Mod. Phys. B 19 (7–9) (2005) 1395–1401. [8] M. Ahmadian, R.J. Appleton, J.A. Norris, Designing magneto-rheological dampers in a fire out-of-battery recoil system, IEEE Trans. Magn. 39 (1) (2003) 480–485. [9] M. Ahmadian, R.J. Appleton, J.A. Norris, An analytical study of fire out of battery using magnetorheological dampers, Shock Vib. 9 (2002) 129–142. [10] L. Fu, L.P. Lin, X.H. Xu, Research on modeling and fuzzy of magneto-rheological intelligent buffer system for impact load, J. Shanghai Jiaotong Univ. Sci. 17 (5) (2012) 567–572.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear Fu Li a,⇑, Guan Wei a, Wang Qi a, Xu Xinhe b a b
Shenyang Aerospace University, Liaoning Shenyang 110136, China Northeastern University, Liaoning Shenyang 110104, China
a r t i c l e
i n f o
Article history: Received 27 December 2013 Received in revised form 9 August 2014 Accepted 17 October 2014 Available online xxxx Keywords: Aircraft landing gear Magneto-rheological buffer system State equation Adaptive control Semi-active control
a b s t r a c t The main advantages of magneto-rheological buffer system for aircraft landing gear are adjustable damping force, simple structure and independent of external energy. A kind of magneto-rheological damper structure for aircraft landing gear was presented in this paper. The state equation was established and the landing buffer system control strategy was designed based on self-made magneto-rheological landing impact platform. Integrating with model predictive control, a semi-active control method was studied and used it in realizing the adaptive magneto-rheological buffer system landing process. It has been proved in practice that the proposed control method fully considered the boundless of output damping force. The problem of balancing between landing and taxiing, which is caused by un-adjustable passive damping force in the working process of oleo-pneumatic buffer system, is solved. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The passive oleo-pneumatic buffer has been mainly used in airplane landing gear buffer system at present. It has advantages such as simple structure, high reliability, and so on. But some important parameters like damping force cannot be adjusted in the process of landing and taxiing, which would lead to passive response to the ground impact. Because that the design of Oleo-pneumatic Orifice for the impact process of aircraft landing will arise ‘‘too hard’’ phenomenon during taxiing, and on the contrary, the design of Oleo-pneumatic Orifice for the shock process of aircraft taxing will be ‘‘too soft’’ during landing, the passive landing buffer system is difficult to balance the impact force between landing and taxiing. However, the active control landing gear needs an external larger power source. In recent years, the newly magneto-rheological fluid (MRF) has been developed, which can control the damping force with the change of coil current adjusted by the magnetic field intensity. This provides a new solution for the impact load buffering device. MRF is a kind of functional materials, invented and developed by Rabinow [1] who worked in National Bureau of Standards of the United States. Magneto-rheological fluid is mainly composed of the non-conductor magnetic fluid and the tiny particles dispersed uniformly with low hysteresis and high permeability. Under the effect of magnetic field, MRF can be rapidly and reversibly changed from the Newtonian fluidity liquid to Bingham plastic solid in milliseconds, which is high viscosity and low fluidity [1,2]. Magneto-rheological Impact absorption of the helicopter undercarriage can be seen as a typical application in the antiimpact load. Werely [3] uses the composite material made of rubber and magnetic-rheological fluid as the damping medium ⇑ Corresponding author. http://dx.doi.org/10.1016/j.apm.2014.10.043 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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of helicopter undercarriage. The results show that the damper under magnetic field can provide continuous and controllable damping force. Meanwhile, it can overcome the damping amplitude loss which is caused by single frequency excitation. Batterbee et al. proposed a novel solution to this problem, that is to implement semi-active damping using MRF. They enabled the geometry of a flow mode magneto-rheological valve to be optimized with the constraints of an existing passive landing gear [4]. Choi and Wereley theoretically evaluated the electrorheological and magnetorheological fluid based landing gear system during touchdown of the aircraft. The feasibility and effectiveness of the response on attenuating dynamic load and vibration due to the landing impact were demonstrated [5]. Andrzej and Mikolaj presented investigation results of a semi-active industrial shock absorber with MRF, which is capable of controlling the stopping process of moving objects. The proposed solution makes it possible to adjust the braking force according to the kinetic energy of moving object [6]. Magneto-rheological damper was applied in soft recoil gun by Ahmadian et al. also the shooting range test was carried out and satisfactory results were obtained [7–9]. Studies have been a useful reference in the application of impact buffer control. Magneto-rheological intelligent buffer damper based on semi-active control technology can make up for the shortcomings of passive buffer damper effectively. It greatly improves the dynamic performance of traditional oleo-pneumatic buffer and provides high reliability. Compared to the active control, semi-active control buffer has advantages like low cost, simple structure, no need of large power source. But at present, the research around buffer technology is mainly at the stage of theoretical research. In this paper, according to the characteristics of impact load, we propose a control strategy and control algorithm, it could adaptively track ideal buffer force and velocity of impact load with magneto-rheological intelligent buffer damper in the self-made impact test platform. 2. Structure and mechanical model 2.1. Magneto-rheological buffer The variable-structure oil holes of traditional oleo-pneumatic buffer were designed as constant geometries shown in Fig. 1. The damping force of the flowing oleo is controlled by the intensity of applied magnetic field, which is controlled by the impressed current. The gas chamber filled with nitrogen, which is mainly used to provide the need of transformable displacement when impact condition. The filling amount and the initial pressure are determined by the change of displacement and the initial position of piston rod. 2.2. Characteristics and parameter identification Accurate damper model is the foundation for designing control strategy and obtaining good control effect. Because that it is extremely complex and difficult to analyze the damping characteristics based on rheological theory for magneto-rheological
Gas chamber Return oil hole
Excitation coil
Side oil hole
Oil chamber
Fig. 1. Magneto-rheological intelligent buffer structure for impact.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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damper, we establish dynamic model by integrating theoretical model with system identification. Through the characteristic test, magneto-rheological damper mechanical model can be obtained [10].
_ F d I; d; d_ ¼ cd d_ þ sd ðIÞsgnðdÞ;
ð1Þ
where, Fd is the damping force of magneto-rheological damper, sd(I) is the yield force of magneto-rheological damper fluid, which is controlled by the magnetic field intensity being relative to the current I, cd is the viscosity coefficient of magnetorheological damper fluid, kd is the elastic coefficient, d is the length of damper piston rod, d_ is the velocity of piston rod, O is the Origin of Coordinates. These parameters could be obtained by parameter identification algorithm. 2.3. Equation of the motion Two displacement sensors are separately put on the counterweight and the magneto-rheological damper in the self-made impact test platform, just as shown in Fig. 2. The dynamic model of magneto-rheological damper is built facing impact load, shown as Fig. 3. According to mechanical model and flexible impacting characteristics, the discrete state equation of controlled object is constructed.
x_ 1 ¼ x2 ; m1 x_ 2 þ k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ þ F z m1 g ¼ 0; x_ 3 ¼ x4 ;
ð2Þ
m2 x_ 4 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ m2 g F z þ kt ½x3 x3 ð0Þ ¼ 0; where x2, x4 are velocity, x1, x3 are displacement. Then get the formula (3).
x_ 1 ¼ x2 ; x_ 2 ¼ g
k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 x1 Þ 1 Fz; m1 m1
x_ 3 ¼ x4 ; x_ 4 ¼
ð3Þ
k½x3 ð0Þ x1 ð0Þ ðx3 x1 Þ kt 1 þg ½x3 x3 ð0Þ þ Fz: m2 m2 m2
The formula (3) are discretized,
x1 ðk þ 1Þ ¼ x1 ðkÞ þ sx2 ðkÞ;
Fig. 2. Self-made impact test platform.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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Fig. 3. Magneto-rheological damper mechanics model facing impact load.
k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ 1 x2 ðk þ 1Þ ¼ x2 ðkÞ þ s g Fz m1 m1 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ s Fz; ¼ x2 ðkÞ þ sg s m1 m1 x3 ðk þ 1Þ ¼ x3 ðkÞ þ sx4 ðkÞ; k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ kt 1 x4 ðk þ 1Þ ¼ x4 ðkÞ þ s þg ðx3 ðkÞ x3 ð0ÞÞ þ Fz m2 m2 m2 k½ðx3 ð0Þ x1 ð0ÞÞ ðx3 ðkÞ x1 ðkÞÞ kt s ¼ x4 ðkÞ þ sg þ s s ðx3 ðkÞ x3 ð0ÞÞ þ Fz: m2 m2 m2
ð4Þ
Then we can get the system model as follows.
xðk þ 1Þ ¼ f ðxðkÞÞ þ gðxðkÞÞuðkÞ;
ð5Þ
yðkÞ ¼ AxðkÞ; where,
2
x1 ðkÞ
3
6 x ðkÞ 7 7 6 2 xðkÞ ¼ 6 7; 4 x3 ðkÞ 5 x4 ðkÞ 2 6 6 f ðxðkÞÞ ¼ 6 6 4
2
0
3
6 s 7 6 m1 7 7 gðxðkÞÞ ¼ 6 6 0 7; 5 4
uðkÞ ¼ F z ;
A ¼ ½ 0 1 0 1 ;
s
m2
x1 ðkÞ þ sx2 ðkÞ x2 ðkÞ þ sg s
k½ðx3 ð0Þx1 ð0ÞÞðx3 ðkÞx1 ðkÞÞ m1
x3 ðkÞ þ sx4 ðkÞ 3 ðkÞx1 ðkÞÞ s mkt2 ½x3 ðkÞ x3 ð0Þ x4 ðkÞ þ sg þ s k½ðx3 ð0Þx1 ð0ÞÞðx m2
3 7 7 7: 7 5
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3. Control strategy The adaptive model predictive control algorithm is able to predict the future dynamic model, in which the control action and the error feedback correction are implemented through repeated online optimized calculation. Control algorithm is designed for above dynamic model as Eq. (5). A characteristic of model predictive control method is that the input and output constraints can be considered in the design process of controller, in other words, this control strategy can make sure that the control signal would not exceed the boundary of actual execution mechanism, and ensure that the system state does not exceed the preset bounded region. This feature is consistent with actual control requirements of magneto-rheological damper. The output of damping force is a bounded value, and the damper may work in the saturated state, that requires enough consideration on the problem of limited execution ability in the process of designing control algorithm. Otherwise, the control command signal will not be effectively implemented by actuator (damper), and this will lead to system performance degradation, perhaps collapse. In Fig. 2, two displacement sensors are separately put on the counterweight and the magneto-rheological damper, so that four parameters, which are displacement and velocity of the counterweight and the magneto-rheological damper, can be obtained in real time. Then the four parameters will be used as input signal of the control algorithm program. Thereupon, after each sampling, an online optimization process will be performed base on the designed control strategy and the feedback of sensor, thus, the control output (damping force) could be gotten. According to formula (1) or comparison table among damping force, compression speed and current intensity, we will get current intensity by interpolation operation. Finally, the current intensity will be presented to the MR damper to complete the sampling, calculation and control in one cycle. In this way, the automatic control scheme for the magneto-rheological damper based on model predictive control strategy facing impact load could be obtained. However, it is not enough, because we need to set control target of model predictive control algorithm. When the velocity which the impact load reaches the ground is given, the time history curves of ideal compression velocity can be gotten through the active control of the oleo-pneumatic buffer system. That is to say, if the magneto-rheological damper system can compress according to the ideal compression velocity curve in the first stroke, the magneto-rheological buffer system will most effectively absorb the impact energy in the process of Drop-test shock. The corresponding control target is to make the velocity error between actual compression and ideal compression minimal for magneto-rheological damper. The control target can be achieved through the design of objective function in model predictive control algorithm. In a word, the model predictive control implementation is proposed in this paper. We introduced design process of the magneto-rheological damper model predictive control algorithm, which is suitable for impact situation. The detailed calculations on the output (magneto-rheological damper force) are given, based on four parameters, which are displacements and velocities of the impact load and the damper unit. 4. Model predictive control 4.1. Control algorithm Based on the dynamic equations (5) of impact load system for aircraft landing gear, the controller is designed. x(k) 2 Rn is the system state, u(k) 2 Rm is the controller input, y(k) 2 Rp is the controller output, f ðÞ and gðÞ are known nonlinear functions, A is a constant matrix. The system meets the constraints as follows:
umin 6 uðkÞ 6 umax ;
jDuðkÞj 6 Dumax ;
ymin 6 yðkÞ 6 ymax ;
where
Duðk þ jjkÞ ¼ uðk þ jjkÞ uðk þ j 1jkÞ: It is the increment of control input. The main research is to design controller for the impact model equation (5), which is able to meet the stability of closedloop system, and the adjusting output can rapidly track the reference signal without steady-state error, that is
lim eðkÞ ¼ 0;
k!1
eðkÞ ¼ rðkÞ yðkÞ:
The control algorithm should ensure not only the zero steady state error but also the rapid performance, only in this way the working requirements of drop buffer can be reached. The system state is estimated at each sampling point, and the optimal control vector obtained from the optimization process is solved. According to above model predictive control method, the structure of target function can be constructed as follows.
JðkÞ ¼
NX N u 1 X jjrðk þ jjkÞ yðk þ jjkÞjj2Q þ jjDuðk þ jjkÞjj2R ; j¼1
ð6Þ
j¼0
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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where, r(k + j|k) is the reference signal, y(k + j|k) is the predicted output, N and N u are the prediction length and control mm length, Q 2 Rpp are the weighted matrices with appropriate dimensions, the norm ||||Q and ||||R are defined as pffiffiffiffiffiffiffiffiffiffiffi ffi , R2R T jjzjjW ¼ z Wz. In order to predict the current state of the system, we need to get the former prediction value, the expression is as follows:
xðk þ 1jkÞ ¼ f ðxðkÞÞ þ gðxðkÞÞðuðk 1Þ þ DuðkjkÞ; xðk þ 2jkÞ ¼ f ðxðk þ 1jk 1ÞÞ þ gðxðk þ 1jk 1ÞÞðuðk 1Þ þ DuðkjkÞ þ Duðk þ 1jkÞÞ; ... xðk þ NjkÞ ¼ f ðxðk þ N 1jk 1ÞÞ þ gðxðk þ N 1jk 1ÞÞ; ðuðk 1Þ þ DuðkjkÞ þ Duðk þ N 1jkÞÞ: Define the following vector
ðkÞ ¼ ½yðk þ 1jkÞ yðk þ NjkÞT 2 RNp ; y xðkÞ ¼ ½xðk þ 1jkÞ xðk þ NjkÞT 2 RNn ; T
ðkÞ ¼ ½DuðkjkÞ Duðk þ Nu 1jkÞ 2 RNn m ; Du ~ xðkÞ ¼ AðGðk ~ ðkÞ þ Kðk 1Þ þ Fðk 1ÞÞ; ðkÞ ¼ A 1ÞDu y where
2
3 A 0 . . . 6 N N ~¼4. . . .. 7 A 5 2 R p n; . 0 A 2 6 6 Gðk 1Þ ¼ 6 6 4
gðxðk 1Þ
0
3
gðxðk þ 1jk 1ÞÞ .. .
gðxðk þ 1jk 1ÞÞ .. .
0 .. .
7 7 7 2 RNn Nm ; 7 5
gðxðk þ N 1jk 1ÞÞ gðxðk þ N 1jk 1ÞÞ gðxðk þ N 1jk 1ÞÞ 2
gðxðkÞÞuðk 1Þ
6 6 Kðk 1Þ ¼ 6 6 4
gðxðk þ 1jk 1ÞÞuðk 1Þ .. .
3 7 7 7 2 RN n ; 7 5
gðxðk þ N 1jk 1ÞÞuðk 1Þ 2 6 6 Fðk 1Þ ¼ 6 6 4
3
f ðxðkÞÞ
f ðxðk þ 1jk 1ÞÞ 7 7 7 2 RNn : .. 7 5 . f ðxðk þ N 1jk 1ÞÞ
Then the controller design problem of model prediction can be transformed into the optimization problems solved constantly as follows,
~ ~ ~ ðkÞjj2Q þ jjDu ðkÞjj2R ; 1Þ AKðk 1Þ AGðk 1ÞDu min jjrðkÞ AFðk max 6 Du ðkÞ 6 Du max ; s:t: Du max ; umin 6 uðk 1Þ þ HDuðkÞ 6 u ~ ðkÞÞ 6 y max ; min 6 AðFðk 1Þ þ Kðk 1Þ þ Gðk 1ÞDu y
ð7Þ
where
ðkÞ ¼ ½ uðkÞ uðkÞ 2 RNu m ; u r ðkÞ ¼ ½ rðk þ 1jkÞ rðk þ NjkÞ T 2 RNp ; max ¼ ½ Dumax Du
Dumax T 2 RNu m ; T
min ¼ ½ umin u
umin 2 RNu m ;
max ¼ ½ umax u
umax T 2 RNu m ;
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F. Li et al. / Applied Mathematical Modelling xxx (2014) xxx–xxx
min ¼ ½ ymin y
ymin T 2 RNp ;
max ¼ ½ ymax y 2 I 0 6I I 6 H¼6 6 .. .. 4. .
ymax T 2 RNp ; 3 0 07 7 N u mN u m : .. .. 7 72R . .5
I
I
I
7
I
ðkÞ 2 RNu m , the optimization problem can be transformed into the quadratic programming probDefining variable v ¼ Du lem as following
1 T v W v þ cT v ; 2 s:t: lmin 6 Ev 6 lmax ; min
ð8Þ
where
min T 2 R3Nu mþ2Np ; lmin ¼ ½ 1 Du T
max 2 R3Nu mþ2Np ; lmax ¼ ½ b Du T ~ ~ W ¼ 2 AGðk 1Þ Q AGðk 1Þ þ R 2 RNu mNu m ; T ~ 1Þ Q ðrðkÞ Kðk 1Þ Fðk 1ÞÞ 2 RNu m ; c ¼ 2AGðk T ð3N mþ2N p ÞNu m ~ ~ ; AGðk 1Þ AGðk 1Þ I 2 R u 3 min þ u ðk 1Þ u 6 7 ðk 1Þ max u u 6 7 7 2 R2Nu mþ2Np : b¼6 6 y 7 ~ ~ þ AFðk 1Þ þ AKðk 1Þ 4 min 5 ~ ~ max AFðk 1Þ AKðk 1Þ y
E ¼ ½ H 2
H
In conclusion, the detailed calculations from four inputting parameters, including the counterweight displacement, the counterweight velocity, the damper displacement and the damper velocity, to the magneto-rheological damper force output are given, namely the quadratic programming problem as (8) is solved. The control strategy is totally given for landing gear system. 4.2. Results In the dynamic equation (5) of impact load magneto-rheological damper, on the basis of the self-made impact drop stand, let the reference signal r(k) take the ideal velocity and damping force for the existing active control oleo-pneumatic buffer system of aircraft landing gear. The drop height is controlled to be 0.45 meters, so the grounding velocity of counterweight is approximate to 3 m/s. In Fig. 4, yr(t) represents the ideal sinking velocity, x2(t) is the time-history curve of the actual sinking
Fig. 4. The curves of yr ðtÞ and x2 ðtÞ.
Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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Fig. 5. Curves of the ideal damping force ur ðtÞ and the magneto-rheological damper force uðtÞ designed by the control algorithm.
velocity from the piston rod of magneto-rheological damper. ur(t) is the ideal damping force of falling shock process, u(t) is the actual curve of damping force under the control algorithm designed in this paper. The control effect is shown in Fig. 5. From Figs. 4 and 5, under the control strategy and control algorithm based on model prediction in this paper, the two groups of parameters are all coincidence, that is the time-history curve of sinking velocity for MR buffer control system x2(t) (solid line in Fig. 4), the actual damping force provided by the MR buffer control system u(t) (solid line in Fig. 5), and the sinking velocity time-history curve yr(t) (dashed line in Fig. 4) and the time-history curve of ideal damping force ur(t) (dashed line in Fig. 5) Good control effect has been achieved using the control strategy and algorithm in the process of impact buffering. It also reflects high control performance compared with the active control. By adjusting the time-history curve of ideal control velocity, the ground impact energy can be maximally absorbed. Fig. 5 shows the change curve of the ideal damping force and the change curve of control output damper force with MPC algorithm. From Figs. 4 and 5, you can see that the control strategy and the proposed method effectively complete the controlling on the landing buffer force and speed of impact load in this paper. 5. Conclusion The main conclusions of this paper are as following. (1) A kind of new magneto-rheological buffer structure is presented for impact load. It could effectively absorb the impact energy produced in aircraft landing gears, and the oil-return hole can be used to prevent the tire from jumping off the ground when landing. (2) The discrete state equations of magneto-rheological damper buffer system are constructed under impact effect, on the basis of the self-made magneto-rheological buffer control platform facing impact load. (3) The control strategy of magneto-rheological damper buffer is proposed, and buffer control algorithm of magneto-rheological damper is designed based on model predictive control. The complex solving problem caused by nonlinear magneto-rheological fluid is avoided in this paper, and the control effect is also verified. (4) The output boundness of damping force is taken consideration in the proposed control method. It is coincide with the actual control system. (5) The boundness of the damper force variation is also taken consideration in the control. It makes the control algorithm more realizable. Compared with the traditional oleo-pneumatic buffer, damper structure is given in this paper and the corresponding control strategy and control algorithm can be implemented in the damper compression process. Because that the output damping force could be rapidly and effectively adjusted, the optimal buffering effect is obtained. Acknowledgments This research is partially supported by the National Natural Science Foundation of China (No. 61074090), the Innovation Funds of Aviation Industry Corporation of China (No. cxy2013SH16), and the Shenyang Science and Technology Plan Projects (F13-095-2-00). Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043
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Please cite this article in press as: F. Li et al., Modeling and adaptive control of magneto-rheological buffer system for aircraft landing gear, Appl. Math. Modell. (2014), http://dx.doi.org/10.1016/j.apm.2014.10.043