Adaptive backstepping control with grey signal predictor for nonlinear active suspension system matching mechanical elastic wheel

Mechanical Systems and Signal Processing 131 (2019) 97–111

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Adaptive backstepping control with grey signal predictor for nonlinear active suspension system matching mechanical elastic wheel Qiuwei Wang, Youqun Zhao ⇑, Han Xu, Yaoji Deng College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 25 January 2019 Received in revised form 1 April 2019 Accepted 21 May 2019 Available online 30 May 2019 Keywords: Mechanical elastic wheel Nonlinear active suspension Adaptive backstepping control Grey DGM (2,1) model Hybrid damping control

a b s t r a c t In order to stabilize the attitude and improve ride comfort of vehicles equipped with mechanical elastic wheel (MEW), an adaptive backstepping controller with grey signal predictor is presented for active suspension system, considering the uncertainty and nonlinearity of suspension and MEW. The MEW’s mathematical model is established through experiments firstly. Then the ideal suspension motions are generated according to a designed hybrid damping control. To track these ideal signals, Lyapunov theory and the thought of backstepping are used to estimate the nonlinearity of active suspension and obtain final control laws. Grey DGM (2,1) model is further applied to derive future motions of suspension, thus commands can be carried out in advance to realize control in time. The stability and convergence of whole scheme are also proved by Lyapunov-like lemma. Finally, simulations on two different road profiles (step and pulse) show that this strategy can effectively suppress the vertical motion of vehicle body by 19.7% and pitch motion by 9.5% respectively compared with that without control. And furthermore, the bounce of wheels is also decreased at the same time. The controller expands the application of MEW and establishes a good theoretical foundation for the novel wheels’ matching. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction With the rapid development of automotive industry, vehicle’s safety has been greatly improved, and more manufacturers focus on noise, vibration and harshness [1,2]. Much effort has been made to improve the ride comfort until now, including active steering, control of engine, etc. [3,4]. As one of the necessary components in vehicles, suspension plays a significant role in improving the ride comfort and handling [5–7], whose main function is to transmit the forces and torques between the vehicle body and road surface. Thus the suspension is often required to maintain a good balance among vehicle’s characteristics, not only to isolate the vibrations caused by different environments but also to keep a good contact with road surface [8]. Compared with traditional passive suspension and semi-active suspension, the active suspension is able to change stiffness and damping coefficient simultaneously. Although it needs a mass of external energy to operate, it’s the most effective and has great potential in improving the overall performance of vehicle [9–11], so researchers apply themselves to the control of active suspension. Nowadays, researches about active suspension control can be roughly divided into two categories: one is intelligent control that relies on accurate models [12–19], mainly including optimal control, robust control, adaptive control, and neural ⇑ Corresponding author. E-mail address: [email protected] (Y. Zhao). https://doi.org/10.1016/j.ymssp.2019.05.046 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

networks which can solve the nonlinear problem well and improve the vehicle’s performance. But it’s worth noting that fully known system models don’t exist, intelligent control strategies are challenged, stimulating the appearance of another control called free-model which doesn’t depend on system models. The free-model control is represented by fuzzy control, and the suspension is controlled and adjusted according to experts’ knowledge and experience [9,20–23]. This kind of strategy avoids complicated and cumbersome modeling work and realizes the application of human knowledge. However it ignores the useful information of systems and its precision is not always satisfying. Modern active suspensions are more likely to combine these controls’ advantages together like adaptive backstepping control does. Unlike traditional control methods, this kind of strategy uses converse thinking to solve problems. Adaptive methods are also applied to estimate some system’s unknown parameters. This makes control laws more understandable and adaptable. And the adaptive backstepping control strategy is first used in active suspensions to deal with hard constraints [24]. While the existing active suspension controls mostly cooperate with traditional pneumatic wheel and replace the wheel with simplified stiffness and damping, without considering the actual characteristics of wheels. The algorithm may be simple, but it’s far away from reality. At the same time, a new type of mechanical elastic wheel (MEW) exists. It combines the advantages of explosion-proof and lightweight, which has raised wide attention. Relevant literatures show that the wheel has specific structures and the characteristics of this MEW are extremely different from the ordinaries’ [25–28]. Traditional control strategies don’t adapt well to the suspension systems equipped with MEW. Even if these wheels are assembled on active suspension, the performance is difficult to be guaranteed owing to its higher radial stiffness. In general, there is limited work focusing on control strategies to match this kind of wheel to improve vehicle’s ride comfort, especially with active suspension. It’s necessary to do something on this area while considering the nonlinearity and uncertainty. The nonlinearity and uncertainty is necessary to be considered to reflect the actual movement of suspension system. Although many efforts have been made to deal with the nonlinear and uncertain systems, like sliding mode control, adaptive control, fuzzy control and H1 control. Calculation is still a big problem when taking these into account [29–32]. This makes control strategies too complicated to implement. According to the thought of model predictive control, if the future states can be obtained in advance, we can control accordingly. Traditional prediction methods include regression, ARMA model, Holte Winters method [33,34], but they are not easy enough to be applied in engineering. The grey system theory can fully utilize less data information to find the mathematical relationship among factors and then make predictions. The grey DGM (2,1) model is especially suitable for describing non-monotonic wobble sequence, and has been widely used in signal processing [35,36]. The computational burden is smaller and less information is needed. Therefore the grey DGM (2,1) model is used in this paper, providing predicted information so that controller can better optimize the suspension system. In short, the main contributions of this paper are to propose a new mechanical elastic wheel to prevent tire explosion and design an effective controller to match the novel wheel with active suspension. To reflect actual situation, we take the wheels’ and suspension’s nonlinearity and uncertainty into account, and use Taylor series to expand the nonlinear expressions in spring forces and damping forces with unknown parameters. Then Lyapunov theory and adaptive backsteping control are applied to estimate these unknown parameters and track ideal suspension motions to achieve better performance. These ideal suspension motions are further designed by an improved hybrid damping control. Considering timeliness, grey signal predictor is utilized to derive future suspension motions. Finally, simulation results validate the effectiveness and reliability of the proposed control method. The rest of this paper is organized as follows. The stiffness characteristic of MEW is firstly obtained through experiments in Section 2. Then Section 3 establishes the nonlinear semi-vehicle model including MEW. After that, active suspension controller is designed in Section 4, including: adaptive backstepping control laws, suspension ideal motions design and grey signal predictor. Section 5 simulates the response of active suspension with MEW under step and pulse road conditions to illustrate the effectiveness of designed controller. Finally, section 6 draws the conclusion that ride comfort of vehicle assembling MEW under the proposed control is improved. 2. Modeling of mechanical elastic wheel The mechanical elastic wheel (MEW) is mainly composed of an elastic wheel (rubber layer, wire ring), a hub and hinge groups, as shown in Fig. 1(a) and (b). The hub is suspended in center of the elastic wheel through hinge groups, and the hinge groups can be moved vertically within a certain range. The elastic wheel consists of rubber layer and wire ring. For simplification, Fig. 1(c) is given. Hinge group’s length l is slightly larger than the distance R between the hub and the elastic wheel and we define d = l  R, which is called free travel of hinge group. When unloaded, hinges are allowed a certain radial movement, but when MEW is loaded, the load is transmitted to the wheel through the hinge groups. The grounding part is obviously deformed, and the upper part has a radial contraction trend. Thanks to this kind of structure, the hinge groups are subjected to tension only. The MEW was tested to obtain its characteristics. The load increased from 0kN to 20kN with the step of 2kN and vertical deformation of the wheel was recorded. The test apparatus and results are shown in Fig. 2(a) and 2(b), respectively. Fig. 2(b) shows that the MEW has a vertical deformation (about 3 mm) when unloaded. The main reason is the existence of d. MEW is affected by gravity and the hub can be moved freely within a certain distance. After the hinge groups extend to the maximum length, the load is applied, the wheel begins to bear, and the stiffness characteristics of MEW are obviously nonlinear. The relationship between the vertical deformation and the load is as follows,

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

99

Fig. 1. Structure of MEW.

Fig. 2. Test apparatus and test results of MEW.


y¼

0

0 6 x 6 0:003

1E þ 07x þ 318439x  1064:6

0:003 < x

2

ð1Þ

When the wheel is in contact with the ground and the relative motion is not beyond d = 0.003 m, there is no tire force. Otherwise, there exists the secondary relationship between the tire force and vertical deformation of MEW. 3. Problem formulation Mechanical elastic wheels are equipped in active suspension, and a half-car model is used to describe the system, as shown in Fig. 3. The suspension spring and damping force are considered to be high-order nonlinear and unknown. System   parameters are shown in Table 1, and ½ x1 x2 x3 x4 x5 x6 x7 x8  ¼ z z_ h h_ z1 z_ 1 z2 z_ 2 are chosen as state

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Fig. 3. The model of half-car equipped with active suspension and MEW.

Table 1 Nomenclature. Symbol

Description

M m1 ,m2 z z1, z2 Iy h a ,b Fk1, Fk2 Fc1 , Fc2 u1 , u2 Ft1 , Ft2 v q1 , q2

Mass of car body Mass of front or rear MEW Vertical displacement of car body Vertical displacement of front or rear MEW Moment of inertia around the centroid Pitch angle of car body Distances between centroid and front or rear suspensions Spring force of front or rear suspension Damping force of front or rear suspension Forces generated by front or rear actuator Tire forces of front or rear MEW Speed of vehicle Road profiles

variables. The original state is defined at the position where there are no forces in the suspensions and tires. Axis direction is given in the figure. The dynamic can be described as follows.

8_ x1 > > >_ > > x2 > > > > > x_ 3 > > > > < x_ 4 > x_ 5 > > > > _6 > x > > > > > _ > x7 > > : x_ 8

¼ x2 ¼ M1 ½F k1  F c1 þ u1  F k2  F c2 þ u2  Mg ¼ x4 ¼ I1y ½F k1 a þ F c1 a  u1 a  F k2 b  F c2 b þ u2 b ¼ x6 ¼

1 m1

ð2Þ

½F k1 þ F c1  u1 þ F t1  m1 g

¼ x8 ¼ m12 ½F k2 þ F c2  u2 þ F t2  m2 g

Suspension’s spring and damping forces are unknown, which are nonlinear functions related to the relative displacement and velocity between the wheel and vehicle body. It’s reasonable for simplifying to assume that the forces generated in the positive and negative stroke are equal and the suspension’s characteristics don’t change with time. Then we expand Fki and Fci (i = 1, 2) with Taylor series, the first three items are retained as approximations according to experiments in [37].

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

  8 F k1 ¼ a11 ðz01  x5 Þ þ a12 ðz01  x5 Þz01  x5  þ a13 ðz01  x5 Þ3 > > > >   > < F c1 ¼ b11 ðz_ 0  x6 Þ þ b12 ðz_ 0  x6 Þz_ 0  x6  þ b13 ðz_ 0  x6 Þ3 1 1 1 1   > F k2 ¼ a21 ðz0  x7 Þ þ a22 ðz0  x7 Þz0  x7  þ a23 ðz0  x7 Þ3 > 2 2 2 2 > > >  0  : 3 0 0 0 F c2 ¼ b21 ðz_ 2  x8 Þ þ b22 ðz_ 2  x8 Þz_ 2  x8  þ b23 ðz_ 2  x8 Þ

101

ð3Þ

where:




(

z01 ¼ x1  ax3 z02 ¼ x1 þ bx3 a_ i1 ¼ 0; a_ i2 ¼ 0; a_ i3 ¼ 0 b_ i1 ¼ 0; b_ i2 ¼ 0; b_ i3 ¼ 0

ð4Þ

ði ¼ 1; 2Þ

ð5Þ

According to Newton’s second law and Eq. (1), there is the relationship between the tire forces of MEW and the state variables.

(

F t1 ¼ ( F t2 ¼

0

q 1  x5 6 d

1E þ 07ðq1  x5 Þ2 þ 318439ðq1  x5 Þ  1064:6 q1  x5 > d 0

q 2  x7 6 d

ð6Þ

1E þ 07ðq2  x7 Þ2 þ 318439ðq2  x7 Þ  1064:6 q2  x7 > d

So far, the active suspension system equipped with MEW has been modeled, considering the nonlinearity of the suspension and MEW, and the suspension forces and tire forces of MEW are expressed by Taylor series and experimental analysis respectively. Next section is the design of active suspension controller to match the MEW. 4. Active suspension controller The suspension controller includes adaptive backstepping control laws, ideal suspension motions generator and grey signal predictor. The suspension motions are controlled to track the ideal ones, and the signal predictor is used to predict the required states according to control laws to regulate in advance. The control structure is shown in Fig. 4. 4.1. Adaptive backstepping control law In order to stabilize the car body, we assume that there are ideal suspension motions x1d and x3d. If the actual vertical and pitch motion of vehicle (x1, x3) can track the desired ones, thus ride comfort can be guaranteed. Now we define tracking errors firstly.




e1 ¼ x1d  x1

ð7Þ

e3 ¼ x3d  x3

Fig. 4. The structure of controller.

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It’s necessary to introduce a positive definite (PD) Lyapunov function related to the system states so that the tracking errors gradually approach to zeros if its derivative is negative definite (ND). Here is the selected function.

Vðe1 ; e3 Þ ¼

1 2 1 2 e þ e 2 1 2 3

ð8Þ

Combining with Eqs. (2) and (7), the derivative of Eq. (8) can be obtained.

_ 1 ; e3 Þ ¼ e1 e_ 1 þ e3 e_ 3 Vðe ¼ e1 ðx_ 1d  x2 Þ þ e3 ðx_ 3d  x4 Þ

ð9Þ

And we define another two desired states.




x2d ¼ x_ 1d þ k1 e1 x4d ¼ x_ 3d þ k3 e3

ðk1 > 0Þ

ð10Þ

ðk3 > 0Þ

Obviously, tracking errors tend to zeros when x2 equals to x2d and x4 equals to x4d, because the derivative is ND. Therefore we introduce new tracking errors and Lyapunov function.




d2 ¼ x2d  x2

ð11Þ

d4 ¼ x4d  x4 Vðe1 ; e3 ; d2 ; d4 Þ ¼

1 2 1 2 1 2 1 2 e þ e þ d þ d 2 1 2 3 2 2 2 4

ð12Þ

The derivative of Eq. (12) is obtained according to Eqs. (2), (7), (10) and (11).

_ 1 ; e3 ; d2 ; d4 Þ ¼ e1 e_ 1 þ e3 e_ 3 þ d2 d_ 2 þ d4 d_ 4 Vðe ¼ e1 ðx_ 1d  x2 Þ þ e3 ðx_ 3d  x4 Þ þ d2 d_ 2 þ d4 d_ 4 ¼ e1 ½x_ 1d  ðx_ 1d þ k1 e1  d2 Þ þ e3 ½x_ 3d  ðx_ 3d þ k3 e3  d4 Þ þ d2 d_ 2 þ d4 d_ 4 ¼ k1 e2  k3 e2 þ d2 ðe1 þ d_ 2 Þ þ d4 ðe3 þ d_ 4 Þ 1

3

ð13Þ

Note that there are two parts in Eq. (13): one is k1e21  k3e23, which is ND. Another is the rest items which we don’t know whether it’s PD or ND. But it can be changed to zero just need to satisfy Eq. (14). Then Eq. (13) is semi-negative definite (NSD) that makes the system of suspension is bounded according to Lyapunov theory.

(

d_ 2 ¼ e1 d_ 4 ¼ e3

ð14Þ

Control signals then are derived by bringing the Eqs. (2) and (11) into (14).

(

b 1 b b 1 u1 ¼ aþb Mð€x1d þ k1 e_ 1 Þ  aþb Iy ð€x3d þ k3 e_ 3 Þ þ F k1 þ F c1 þ aþb Mg þ aþb Me1  aþb Iy e3 a 1 a a 1 u2 ¼ aþb Mð€x1d þ k1 e_ 1 Þ þ aþb Iy ð€x3d þ k3 e_ 3 Þ þ F k2 þ F c2 þ aþb Mg þ aþb Me1 þ aþb Iy e3

ð15Þ

However, there are unknown forces Fk1, Fk2, Fc1 and Fc2 in control signals. Eq. (3) indicates different suspension has different characteristics and their forces are not the same naturally. To adapt different active suspensions, parameters in (3) need to be estimated, and the estimated suspension forces denoted by F^k1 ; F^k2 ; F^c1 ; F^c2 . The actual and estimated parameters are listed.

(

r ¼ ½a11 ; a12 ; a13 ; b11 ; b12 ; b13 ; a21 ; a22 ; a23 ; b21 ; b22 ; b23 T

ð16Þ

T

r^ ¼ ½a^11 ; a^12 ; a^13 ; b^11 ; b^12 ; b^13 ; a^21 ; a^22 ; a^23 ; b^21 ; b^22 ; b^23  The estimation errors are also defined.





























r ¼ r  r^ ¼ a11 ; a12 ; a13 ; b11 ; b12 ; b13 ; a21 ; a22 ; a23 ; b21 ; b22 ; b23

T ð17Þ

And the derivatives of these estimated errors can be derived according to Eq. (5).  _



 _

 _

 _

 _

 _

 _

 _

 _

 _

 _

 _

 _

T

r ¼ a11 ; a12 ; a13 ; b11 ; b12 ; b13 ; a21 ; a22 ; a23 ; b21 ; b22 ; b23

 T ^_ 11 ; b ^_ 12 ; b ^_ 13 ; a ^_ 21 ; b ^_ 22 ; b ^_ 23 ^_ 11 ; a ^_ 12 ; a ^_ 13 ; b ^_ 21 ; a ^_ 22 ; a ^_ 23 ; b ¼ a

ð18Þ

Therefore the control signals change from Eqs. (15)–(19),

(

b 1 b b 1 ^ 1 ¼ aþb Mð€x1d þ k1 e_ 1 Þ  aþb Iy ð€x3d þ k3 e_ 3 Þ þ F^k1 þ F^c1 þ aþb Mg þ aþb Me1  aþb Iy e3 u ^ 2 ¼ a Mð€x1d þ k1 e_ 1 Þ þ 1 Iy ð€x3d þ k3 e_ 3 Þ þ F^k2 þ F^c2 þ a Mg þ a Me1 þ 1 Iy e3 u aþb

aþb

aþb

aþb

aþb

ð19Þ

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

103

Naturally, we hope the estimated values can keep up with the actuals after a certain period of time, so that the control signals approach to the ideal control signals as shown in Eq. (15). A new Lyapunov function is defined again. 

Vðe1 ; e3 ; d2 ; d4 ; rÞ ¼

1 2 1 2 1 2 1 2 1 T  e þ e þ d þ d þ r Ir 2 1 2 3 2 2 2 4 2

ð20Þ

where: I is a proper unit matrix. Combining with Eqs. (2), (7), (10), (11) and (20), we get the derivative of (20).  T _ 1 ; e3 ; d2 ; d4 ; r ^_ Vðe Þ ¼  k1 e21  k3 e23 þ d2 ½ð€x1d þ k1 e_ 1  x_ 2 Þ þ e1  þ d4 ½ð€x3d þ k3 e_ 3  x_ 4 Þ þ e3 Þ  r Ir h i ^ ^ ^ ^ k2 F k2 ÞþðF c2 F c2 Þ ¼  k1 e21  k3 e23 þ d2 ðF k1 F k1 ÞþðF c1 F c1 ÞþðF M h i T ^ ^ ^ ^ k2 F k2 ÞþbðF c2 F c2 Þ ^_ þ d4 aðF k1 F k1 ÞaðF c1 F c1 IÞþbðF  r Ir y   T d2 ad4 d2 bd4 ^_ ½ðF k1  F^k1 Þ þ ðF c1  F^c1 Þ þ ½ðF k2  F^k2 Þ þ ðF c2  F^c2 Þ  r Ir ¼  k1 e21  k3 e23 þ  þ M Iy M Iy

ð21Þ Further substitute Eq. (3) into (21).

   

3    d2 ad4  _ 1 ; e3 ; d2 ; d4 ; r ½a11 ðz01  x5 Þ þ a12 ðz01  x5 Þz01  x5  þ a12 z01  x5 þ b11 ðz_0 1  x6 Þ Vðe Þ ¼ k1 e21  k3 e23 þ  M Iy         d2 bd4  3 ½a21 ðz02  x7 Þ þ a22 ðz02  x7 Þz02  x7  þ þ b12 ðz_0 1  x6 Þz_0 1  x6  þ b13 ðz_0 1  x6 Þ  þ M Iy    

3  3 

T ^_ þ a23 z02  x7 þ b21 ðz_0 2  x8 Þ þ b22 ðz_0 2  x8 Þz_0 2  x8  þ b23 z_0 2  x8   r Ir

ð22Þ

where,             T r Ir^_ ¼ a11 a^_ 11 þ a12 a^_ 12 þ a13 a^_ 13 þ b11 b^_ 11 þ b12 b^_ 12 þ b13 b^_ 13 þ a21 a^_ 21 þ a22 a^_ 22 þ a23 a^_ 23 þ b21 b^_ 21 þ b22 b^_ 22 þ b23 b^_ 23

ð23Þ

According to Lyapunov theory, if Eq. (22) is ND, tracking errors and estimation errors all tend to zeros gradually. But it seems impossible, because Eq. (22) just can be NSD only when estimated parameters satisfy Eq. (24). Then the system is bounded but not asymptotic stable. Of course, we don’t care about whether the estimated parameters are accurate or not. The errors which are bounded are satisfying. But it is necessary to prove the tracking errors tend to zeros.







8   _  ^_ 12 ¼ dM2  adI 4 ðz01  x5 Þz01  x5  a ^13 ¼ dM2  adI 4 z01  x5 3 ^_ 11 ¼ dM2  adI 4 ðz01  x5 Þ a > a > y y y > > >





>   _ > ^_ 12 ¼ d2  ad4 ðz_0 1  x6 Þz_0 1  x6  b ^13 ¼ d2  ad4 z_0 1  x6 3 ^_ 11 ¼ d2  ad4 ðz_0 1  x6 Þ b >




  _  > > ^_ 22 ¼ dM2 þ bdI 4 ðz02  x7 Þz02  x7  a ^23 ¼ dM2 þ bdI 4 z02  x7 3 ^_ 21 ¼ dM2 þ bdI 4 ðz02  x7 Þ a >a > y y y > >





>   _ > :b ^_ 22 ¼ d2 þ bd4 ðz_0 2  x8 Þz_0 2  x8  b ^_ 21 ¼ d2 þ bd4 ðz_0 2  x8 Þ b ^23 ¼ d2 þ bd4 z_0 2  x8 3 M Iy M Iy M Iy

ð24Þ

Proof1: According to the Lyapunov-like lemma, if, (1) V P 0 (2) V_ 6 g ðt Þ where g ðtÞ P 0 (3) g_ ðtÞ 2 L1 Then lim gðtÞ ¼ 0. t!1

In the designed controller, 

Vðe1 ; e3 ; d2 ; d4 ; rÞ ¼

1 2 1 2 1 2 1 2 1 T  e þ e þ d þ d þ r Ir P 0 2 1 2 3 2 2 2 4 2

 _ 1 ; e3 ; d2 ; d4 ; r Vðe Þ ¼ k1 e21  k3 e23 6 gðtÞ ¼ ðk1 e21 þ k3 e23 Þ

ð25Þ ð26Þ

When Eq. (24) is satisfied, the system is bounded and state e1 ; e3 and e_ 1 ; e_ 3 are bounded, g_ ðtÞ ¼ 2k1 e1 e_ 1 þ 2k3 e3 e_ 3 is bounded, too. Finally, we draw the conclusion that the tracking errors are asymptotic stable and will tend to zeros with time.

limðk1 e21 þ k3 e23 Þ ¼ 0 ) e1 ! 0; e3 ! 0

t!1

ð27Þ

The feasibility of the above controller and the convergence of tracking errors are theoretically proved. The control signals and estimated parameters are shown by Eqs. (19) and (24) respectively, and the overall control laws for active suspension to match MEW are completed.

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Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

In order to generate the required control forces in vehicle actually, the hydraulic active suspension is chosen. The force generator of system consists of hydraulic source, actuator and electro-hydraulic servo valve. As is shown in Fig. 5, the pressure difference between the two sides of piston in the actuator is directly controlled by electro-hydraulic servo valve. The hydraulic oil flows into the actuator and generates the force u on the piston due to the pressure difference. Then the generated force can be calculated.

u ¼ As Pdif ¼ As ðp1  p2 Þ

ð28Þ

According to Eq. (19), the pressure difference is calculated to generate the required control forces.

(

^

P1dif ¼ uA1s

ð29Þ

^

P2dif ¼ uA2s

where, AS is the piston’s cross-sectional area; P1dif and P2dif is the pressure difference of front and rear suspension, which is controlled by electro-hydraulic servo valve directly. 4.2. Ideal motions The above mentioned strategy is given to make the suspension track the ideal motions to adapt different suspensions. Here is the ideal motion design. An improved hybrid damping control which integrates sky-hook and ground-hook damping control is presented to generate the required ideal motions as shown in Fig. 6. The ideal suspension motions can be described as follows.

8 €z ¼ 1 ½F k1  F c1  F k2  F c2  Mg  F sky1  F sky3  > > > d M > > < €hd ¼ I1 ½F k1 a þ F c1 a  F k2 b  F c2 b þ F sky1 a  F sky3 b y > €z1d ¼ m1 ½F k1 þ F c1 þ F t1  m1 g  F sky2  > > 1 > > : €z ¼ 1 ½F þ F þ F  m g  F  c2 t2 2 2d k2 sky4 m2

ð30Þ

where,

8

> < F sky1 ¼ csky1 z_ d  ah_ d

> : F sky3 ¼ csky3 z_ d þ bh_ d

ð31Þ

By selecting appropriate damping coefficient, the vibration can be directly suppressed. However, considering the special situation when the tire is separated from the ground, the wheel needs to contact with ground as quickly as possible under gravity. Here is an improved hybrid damping strategy used to deal with the problem when considering the novel structure of MEW.




F sky2 ¼
F sky4 ¼

csky2 jx6 j q1  x5 > d csky2 x6 q1  x5 6 d csky4 jx8 j q2  x7 > d csky4 x8

ð32Þ

q 2  x7 6 d

Fig. 5. The actuator of active suspension system.

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

105

Fig. 6. Sketch of hybrid damping control.




x1d ¼ zd

ð33Þ

x3d ¼ hd

The needed ideal motions are obtained through the improved hybrid damping control as shown in Eq. (33). Note that the hybrid damping controller is virtual, and it’s unnecessary to actually design that controller in real world, and the virtual damping coefficients are given as control signals. 4.3. Grey signal predictor Observing the control law, it includes the tracking errors and their rates. In other words, it’s related to the ideal and actual suspension displacement and velocity. The ideal signals are designed. Only the actual signals x1 ; x_ 1 ; x3 ; x_ 3 can be predicted in advance to further improve ride comfort. The DGM (2,1) model in grey system theory was used to design predictor based on less known data (at least 4), which is easy to implement on a microcomputer. Suppose there is a suspension motion sequence whose number is n.

  xð0Þ ¼ xð0Þ ð1Þ; xð0Þ ð2Þ;    ; xð0Þ ðnÞ ðx ¼ x1 ; x_ 1 ; x3 ; x_ 3 Þ

ð34Þ

We generate another two sequences based on Eq. (34) called 1-AGO and 1-IAGO respectively.

  xð1Þ ¼ xð1Þ ð1Þ; xð1Þ ð2Þ;    ; xð1Þ ðnÞ

að1Þ xð0Þ



 ¼ að1Þ xð0Þ ð2Þ;    ; að1Þ xð0Þ ðnÞ

ð35Þ ð36Þ

where,

xð1Þ ðkÞ ¼

Pk

i¼1 x

ð0Þ

ðiÞ ðk ¼ 1; 2;    ; nÞ

að1Þ xð0Þ ðkÞ ¼ xð0Þ ðkÞ  xð0Þ ðk  1Þ ðk ¼ 2; 3;    ; nÞ

ð37Þ ð38Þ

Grey DGM (2,1) model then can be described as follows.

að1Þ xð0Þ ðkÞ þ pxð0Þ ðkÞ ¼ q

ð39Þ

T

Define h = [p, q] , formula (39) can be changed into (40).

Bh ¼ Y where,

ð40Þ

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Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

2

xð0Þ ð2Þ 1

3

3

2

að1Þ xð0Þ ð2Þ 6 að1Þ xð0Þ ð3Þ 7 7 6

6 xð0Þ ð3Þ 1 7 7 6 7 6 B¼6 .. .. 7; Y ¼ 6 6 4 4 . .5 xð0Þ ðnÞ 1

.. .

að1Þ xð0Þ ðnÞ

2

6 6 7¼6 7 6 5 4

xð0Þ ð2Þ  xð0Þ ð1Þ xð0Þ ð3Þ  xð0Þ ð2Þ ...

3 7 7 7 7 5

ð41Þ

xð0Þ ðnÞ  xð0Þ ðn  1Þ

^ can be obtained, which is the estimation of h. Through the least square method, h 1 ^ ¼ ½p ^T ¼ ðBT BÞ BT Y ^; q h

ð42Þ

And there is the solution of DGM (2,1),

"

^xð1Þ ðk þ 1Þ ¼

#   ^ xð0Þ ð1Þ p^k q ^ ^ 1þp ^ q q ð0Þ ðk þ 1Þ þ x e  þ ð1Þ  ^ ^ ^ ^ p p p p ^2 p

ð43Þ

Finally, we get the prediction function,

^xð0Þ ðk þ 1Þ ¼ að1Þ ^xð1Þ ðk þ 1Þ ¼ ^xð1Þ ðk þ 1Þ  ^xð1Þ ðkÞ

ð44Þ

When the prediction completed, actual signal value is measured in time. Since the suspension motion is greatly affected by the random excitation, it is difficult to ensure the absolute accuracy. By comparing the predicted value with the actual value, if the error is allowable, the predicted value is output, otherwise it directly output actual signal. Considering that the short-term state signal will not change sharply. Therefore, 5 times absolute difference between the current signal and the previous one is set as error threshold, as shown in Eq. (45).

8       ð0Þ ð0Þ > < ^x ðn þ 1Þ ^x ðn þ 1Þ  xð0Þ ðnÞ 6 5xð0Þ ðnÞ  xð0Þ ðn  1Þ ð0Þ ^x ðn þ 1Þ ¼     ^ð0Þ  > : xð0Þ ðnÞ x ðn þ 1Þ  xð0Þ ðnÞ > 5xð0Þ ðnÞ  xð0Þ ðn  1Þ

ð45Þ

Next prediction of suspension motions x1 ; x_ 1 ; x3 ; x_ 3 add the measured value xð0Þ ðn þ 1Þ to the end of the sequence xð0Þ (the far right of the row vector) and deletes xð0Þ ð1Þ to generate a new sequence respectively, repeating the above steps to complete the metabolism of DGM (2,1) model. Proof2: According to proof1 we draw the conclusion that the system is Lyapunov stable and convergent if control laws use the actual state signals, and the derivative of Lyapunov function as shown in Eq. (22) is NSD. And we let,  _ 1 ; e3 ; d2 ; d4 ; r Vðe Þ ¼ rðr > 0Þ

When signals x1 ðkÞ; x_ 1 ðkÞ; x3 ðkÞ; x_ 3 ðkÞ are replaced by x1 ðk þ 1Þ; x_ 1 ðk þ 1Þ; x3 ðk þ 1Þ; x_ 3 ðk þ 1Þ. The Eq. (22) changes into,

ð46Þ the

signals

predicted

by

DGM

2

2  _ 1 ; e3 ; d2 ; d4 ; r Þ ¼ k1 e1 þ l1  k3 e3 þ l3 þ G½x1 ðk þ 1Þ; x_ 1 ðk þ 1Þ; x3 ðk þ 1Þ; x_ 3 ðk þ 1Þ Vðe

2

2   ¼ k1 e1 þ l1  k3 e3 þ l3 þ G x1 ðkÞ  l1 ; x3 ðkÞ  l3 ; x_ 1 ðkÞ  s1 ; x_ 3 ðkÞ  s3

(2,1)

model,

ð47Þ

where: G(*) is a complex continuous function, which can be obtained from Eqs. (21)–(24).

8 l1 ¼ x1 ðkÞ  x1 ðk þ 1Þ > > > < l3 ¼ x3 ðkÞ  x3 ðk þ 1Þ > s1 ¼ x_ 1 ðkÞ  x_ 1 ðk þ 1Þ > > : s3 ¼ x_ 3 ðkÞ  x_ 3 ðk þ 1Þ

ð48Þ

  Then the derivative of Lyapunov function becomes the continuous function of variables l1 ; l3 ; s1 ; s3 . It equals to r which is obviously negative if all variables equal to zeros. According to the characteristic of continuous function, if the change of variables is tiny enough, the sign of function will not change. Therefore we can always find a positive number f. When,

k





l1 ; l3 ; s1 ; s3 T k 6 f

ð49Þ

We still have,  _ 1 ; e3 ; d2 ; d4 ; r Vðe Þ<0

ð50Þ

And it’s the fact that all the system states are the continuous function of time, if the time step is tiny enough, we can also draw the conclusion that the states change little. The step time T step in DGM (2,1) needs to satisfy Eq. (51) to make the controller stable and bounded. When it comes to the system of e1 and e3, other variables are considered extraneous variables. The derivative of Lyapunov function of e1 and e3 is ND.

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Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

T step 2 argk





l1 ðtÞ; l3 ðtÞ; s1 ðtÞ; s3 ðtÞ T k 6 f

ð51Þ

Up to now, we have proved the tracking errors’ local asymptotic stability. 5. Simulation results and discussion The above designed controller receives the states from grey signal predictor and uses adaptive backstepping control to track the ideal motions generated by hybrid damping control to stabilize active suspension motions equipped with MEW. To verify the effect of controller further, the accuracy of DGM (2,1) model is checked by a set of random suspension’s motion data firstly, The number of signal sequence in predictor is set as 6 and step time equals to 0.001 s. Then two common road surfaces are designed in Simulink (step and pulse) to illustrate the effectiveness of the presented controller. The road excitation acting on the front and rear wheels has a certain time delay which is related to the wheelbase and vehicle speed, as shown in (52), and the simulation parameters are given in Table 2.

(

q2 ðtÞ ¼ q1 ðt  T 0 Þ T 0 ¼ aþb v

ð52Þ

It can be seen from Fig. 7 that the difference between accurate data and the prediction made by DGM (2,1) model is tiny, especially when the tendency is gentle. When the data changes sharply, tracking errors of DGM (2,1) model becomes a little larger, but it still can meet the request of engineering. Therefore the reliability of DGM (2,1) is ensured. 5.1. Simulation on step road surface The step road profile is shown in Fig. 8(a). A step signal with amplitude of 0.1 m appears at 1 s, and then the height of the road surface is constant. The response of active suspension equipped with MEW is observed during the whole process. It can be seen from Fig. 8(b) that the active suspension control forces appear at 1 s when the road excitation exists. The front suspension control force’s peak value is 11490 N, and the rear control force is generated after about 0.3 s which reaches to 14,790 N. After a period of time, the control forces of actuator both equal to zeros when vehicle is stabilized again. In Fig. 8

Table 2 Simulation parameters. Parameter

Value

Unite

Parameter

Value

Unite

M Iy m1 m2 g a b v a11 a12 a13 b11 b12 b13

730 1230 40 40 9.8 1.1 1.8 10 19,600 1 1 1290 1 1

kg kg*m2 kg kg m*s2 m m m*s1 N*m1 N*m2 N*m3 N*s*m1 N*s2*m2 N*s3*m3

a21 a22 a23 b21 b22 b23 d Csky1 Csky2 Csky3 Csky4 k1 k2 –

19,600 1 1 1290 1 1 0.003 2000 2000 2000 2000 10 10 –

N*m1 N*m2 N*m3 N*s*m1 N*s2*m2 N*s3*m3 M N*s*m1 N*s*m1 N*s*m1 N*s*m1 – – –

Fig. 7. Comparison between random data and prediction made by DGM (2,1) model.

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

0.1

1.5

0.08

1 Control force/N

Road profile/m

108

0.06 0.04 0.02

x 10

4

u1 u2

0.5 0

-0.5

0 -1 0

1

2

time/s

3

4

5

0

(a) Step road profile

Pitch motion x3/rad

Vertical displacement x1/m

3

4

5

-0.04 -0.06 -0.08 -0.1 -0.12 -0.14

-0.2

1

2

0.04 0.02

0.04 0.02

0

0

-0.02 1.5

0

with control and predictor with control but no predictor without control Ideal signal

0.06

-0.1

-0.15

3

-0.02

2 4

5

-0.04

time/s

1 0

1

0.2 0.25 0.2

0.1

0.15 0 1

0.1 0.05

1.1

1.2

with control and predictor with control but no predictor without control

0 0

1

2

3

4

time/s

(e) Vertical displacement of front wheel

2

time/s

3

1.5 4

5

(d) Pitch motion

0.3

5

Vertical displacement of rear wheel x7/m

(c) Vertical displacement of car Vertical displacement of front wheel x5/m

time/s

0.08 with control and predictor with control but no predictor without control Ideal signal

-0.05

-0.05

2

(b) Control signals u1 &u2

0

-0.25

1

0.3 0.25

0.2

0.2

0.1

0.15

0 1.2

0.1 0.05

1.6

with control and predictor with control but no predictor without control

0

-0.05

1.4

0

1

2

time/s

3

4

5

(f) Vertical displacement of rear wheel

Fig. 8. Suspension control under step input.

(c) and 8(d), the vertical displacement of car body and pitch motion under step input are described. The adaptive backstepping control strategy that with or without grey signal predictor both can make the active suspension system with MEW track the designed ideal motions well, and stabilize the vehicle more quickly (<1.5 s) compared to that without control (>3s). While the effectiveness of adaptive backstepping control with grey signal predictor is better than traditional one, the maximum amplitude of vertical motion of the car body is suppressed by 7.44% and the pitch motion is decreased by 19.7%. The motion amplitude under controller with grey signals predictor is smaller than that without. Fig. 8(e) and 8(f) reflect the vertical movements of front and rear wheel under different controllers. They show that the maximum movement of front wheel is about 0.23 m when there is no control, but it decrease to about 0.17 m under the traditional adaptive backstepping control strategy. And the front wheel’s movement is reduced to 0.13 m under the adaptive backstepping control with grey signal predictor. The vertical movement of the rear wheel is reduced by 0.04 m, but there is little difference between adaptive backstepping control with and without signal predictor. Under the step input, the control strategy can stabilize the car body and maintain the vehicle handling stability. In general, the overall performance when applying the grey signal predictor is better.

109

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

5.2. Simulation on pulse road surface

0.1

1.5

0.08

1 Control force/N

Road profile/m

The pulse road profile is shown in Fig. 9(a). There is no road excitation during 0–1 s and 2–5 s. The road height becomes 0.1 m during 1–2 s. The simulation results are shown in Fig. 8. Similarly, Fig. 9(b) shows that the control forces mainly act when road excitation exists and disappears, and the maximum control forces to maintain vehicle stability reach to 1.4kN. Fig. 9(c) and 9(d) indicate that the active suspension with adaptive backstepping control strategy can still stabilize the vehicle well. The vertical displacement of the vehicle body is reduced by 0.04 m when traditional adaptive backstepping control is applied. But when the grey signal predictor is further used, the effectiveness is better. The pitch angle is also reduced by about 0.01 rad and the reduction is almost the same no matter whether the grey signal predictor is applied or not. At the same time, the movement of front wheel decreases by 0.14 m under adaptive backstepping control without signal predictor, while it’s reduced by 0.1 m with grey signal predictor. And the maximum amplitude of the vertical motion of the rear wheel is also attenuated by 0.02 m, but the result of the predictor is similar, as shown in Fig. 9(e) and 9(f). That is to say,

0.06 0.04 0.02

x 10

4

u1 u2

0.5 0 -0.5

0 0

1

2

3

4

-1

5

0

1

2

time/s

(a) Pulse road profile

-0.1 -0.15

0.06 Pitch motion x3/rad

Vertical displacement x1/m

with control and predictor with control but no predictor without control Ideal signal

-0.2

0.04 0.02 with control and predictor with control but no predictor without control Ideal signal

0 -0.02

0

1

2

3

4

-0.04

5

0

1

2

time/s

0.2 0.2 0.15

0.1

0.1

0.05

1

0 0

1

2

1.1 3

4

5

Vertical displacement of rear wheel x7/m

with control and predictor with control but no predictor without control

0.15

4

5

(d) Pitch motion

0.3 0.25

3 time/s

(c) Vertical displacement of car Vertical displacement of front wheel x5/m

5

0.08

-0.05

-0.05

4

(b) Control signals u1 &u2

0

-0.25

3 time/s

0.3 with control and predictor with control but no predictor without control

0.25 0.2

0.25

0.15

0.2

0.1

0.15

0.05

0.1 1.4

0 -0.05

0

1

time/s

(e) Vertical displacement of front wheel

2

time/s

3

1.6

4

(f) Vertical displacement of rear wheel

Fig. 9. Suspension control under pulse input.

5

110

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

under the pulse input of the road surface, the suspension control proposed in this paper can ensure the stability of the vehicle and improve the ride comfort meanwhile. 6. Conclusion In this study, an active suspension control strategy for the MEW is put forward to improve the ride comfort of vehicle and suppress the body motions under the road excitation. And a theoretical foundation for the active suspension system to match the MEW is established. Firstly, through the experiment, the nonlinear mechanical model of MEW is built. In this condition, the nonlinear characteristics of the active suspension are considered simultaneously, and the half-car model is completed. Then based on Lyapunov theory, the control laws used to estimate the stiffness and damping force parameters and track ideal suspension motions are derived. After that, an improved hybrid damping control scheme is proposed to generate the ideal signals required by the control laws. The stability and feasibility of the system are proved by Lyapunov-like theorem. Furthermore, the grey DGM (2,1) model is implemented in the controller to predict suspension motions in advance. Finally, the suspension control under two typical road excitations is simulated, verifying the effectiveness of the control strategy and draws the following conclusions: (1) The stiffness characteristic of MEW can be expressed with a piecewise function related to the free travel of the wheel and there are no tire forces when relative movement of the road surface and the hub is within the free travel. But when it exceeds the free travel, the wheel forces can be described by a quadratic function related to displacement. (2) The proposed control strategy is verified by typical road surface simulation, results indicate that it can effectively suppress the vertical and pitch movement of vehicle body, which can be attenuated by 9.5% and 19.7%, improving the ride comfort and maintaining the handling meanwhile. (3) The adaptive backstepping control with grey signal predictor can further stabilize the vehicle body matching MEW, while suppressing the vertical bounce of the front wheel. But as for rear wheel, the improvement is limited.

Declaration of Competing Interest The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Acknowledgement Sincerely thank the support from the National Natural Science Foundation of China (11672127) and the Fundamental Research Funds for the Central Universities (NP2016412, NP2018403). And also thank editors and reviewers for their guidance and constructive comments. References [1] Q. Wang, K. Rajashekara, Y. Jia, J. Sun, Multiple quantile regression analysis of longitudinal data: heteroscedasticity and efficient estimation, IEEE Trans. Veh. Technol. 66 (9) (2017) 7722–7729. [2] J. Andert, K. Herold, R. Savelsberg, M. Pischinger, NVH optimization of range extender engines by electric torque profile shaping, IEEE Trans. Control Syst. Technol. 25 (4) (2017) 1465–1472. [3] C. Zhang, D. Zhang, J. Wen, Q. Gao, N. Han, Engine-start Control Strategy of P2 Parallel Hybrid Electric Vehicle Engine-start Control Strategy of P2 Parallel Hybrid Electric Vehicle, 2017, pp. 0–6. [4] H. Zhang, J. Wang, Active steering actuator fault detection for an automatically-steered electric ground vehicle, IEEE Trans. Veh. Technol. 66 (5) (2017) 3685–3702. [5] C. Moreno Ramírez, M. Tomás-Rodríguez, S.A. Evangelou, Dynamic analysis of double wishbone front suspension systems on sport motorcycles, Nonlinear Dyn. 91 (4) (2018) 2347–2368. [6] X. Sun, H. Zhang, W. Meng, R. Zhang, K. Li, T. Peng, Primary resonance analysis and vibration suppression for the harmonically excited nonlinear suspension system using a pair of symmetric viscoelastic buffers, Nonlinear Dyn. 94 (2) (2018) 1243–1265. [7] S. Kilicaslan, Control of active suspension system considering nonlinear actuator dynamics, Nonlinear Dyn. 91 (2) (2018) 1383–1394. [8] Y. Kawamoto, Y. Suda, H. Inoue, T. Kondo, Electro-mechanical suspension system considering energy consumption and vehicle manoeuvre, Veh. Syst. Dyn. 46 (Suppl. 1) (2008) 1053–1063. [9] H. Li, J. Yu, C. Hilton, H. Liu, Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach, IEEE Trans. Ind. Electron. 60 (8) (2013) 3328–3338. [10] K.J. Waldron, M.E. Abdallah, An optimal traction control scheme for off-road operation of robotic vehicles, IEEE/ASME Trans. Mech. 12 (2) (2007) 126– 133. [11] L. Pugi et al, Design and experimental results of an active suspension system for a high-speed pantograph, Mechatronix 13 (5) (2008) 548–557. [12] R. Wang, H. Jing, H.R. Karimi, N. Chen, Robust fault-tolerant H1 control of active suspension systems with finite-frequency constraint, Mech. Syst. Signal Process. 62 (2015) 341–355. [13] S.A. Chen, J.C. Wang, M. Yao, Y.B. Kim, Improved optimal sliding mode control for a non-linear vehicle active suspension system, J. Sound Vib. 395 (2017) 1–25. [14] G. Wang, C. Chen, S. Yu, Robust non-fragile finite-frequency H1static output-feedback control for active suspension systems, Mech. Syst. Signal Process. 91 (2017) 41–56. [15] S. Weichao, Z. Ye, L. Jinfu, Z. Lixian, G. Huijun, Active suspension control with frequency band constraints and actuator input delay, IEEE Trans. Ind. Electron. 59 (1) (2012) 530–537.

Q. Wang et al. / Mechanical Systems and Signal Processing 131 (2019) 97–111

111

[16] J.J. Rath, S. Member, M. Defoort, H.R. Karimi, S. Member, Output feedback active suspension control with higher order terminal sliding mode, IEEE Trans. Ind. Electron. 64 (2) (2017) 1392–1403. [17] V.S. Deshpande, P.D. Shendge, S.B. Phadke, Nonlinear control for dual objective active suspension systems, IEEE Trans. Intell. Transp. Syst. 18 (3) (2017) 656–665. [18] W. Sun, Z. Zhao, H. Gao, Saturated adaptive robust control for active suspension systems, IEEE Trans. Ind. Electron. 60 (9) (2013) 3889–3896. [19] P. Brezas, M.C. Smith, Linear quadratic optimal and risk-sensitive control for vehicle active suspensions, IEEE Trans. Control Syst. Technol. 22 (2) (2014) 543–556. [20] D. Ning, S. Sun, F. Zhang, H. Du, W. Li, B. Zhang, Disturbance observer based Takagi-Sugeno fuzzy control for an active seat suspension, Mech. Syst. Signal Process. 93 (2017) 515–530. [21] X. Sun, C. Yuan, Y. Cai, S. Wang, L. Chen, Model predictive control of an air suspension system with damping multi-mode switching damper based on hybrid model, Mech. Syst. Signal Process. 94 (2017) 94–110. [22] S.D. Nguyen, H.V. Ho, T.T. Nguyen, N.T. Truong, T. Il Seo, Novel fuzzy sliding controller for MRD suspensions subjected to uncertainty and disturbance, Eng. Appl. Artif. Intell. 61 (2017) 65–76. [23] R.J. Lian, Enhanced adaptive self-organizing fuzzy sliding-mode controller for active suspension systems, IEEE Trans. Ind. Electron. 60 (3) (2013) 958– 968. [24] W. Sun, H. Gao, O. Kaynak, Adaptive backstepping control for active suspension systems with hard constraints, IEEE/ASME Trans. Mech. 18 (3) (2013) 1072–1079. [25] Y. Deng, Y. Zhao, F. Lin, Z. Xiao, M. Zhu, H. Li, Simulation of steady-state rolling non-pneumatic mechanical elastic wheel using finite element method, Simul. Model. Pract. Theory 85 (2018) 60–79. [26] X. Du, Y. Zhao, F. Lin, H. Fu, Q. Wang, Numerical and experimental investigation on the camber performance of a non-pneumatic mechanical elastic wheel, J. Brazilian Soc. Mech. Sci. Eng. 39 (9) (2017) 3315–3327. [27] Y. Zhao, X. Du, F. Lin, Q. Wang, H. Fu, Static stiffness characteristics of a new non-pneumatic tire with different hinge structure and distribution, J. Mech. Sci. Technol. 32 (7) (2018) 3057–3064. [28] C. Chomchai, S. Sonjaipanich, S. Cheewaisrakul, Patient expectations for health supervision advice in continuity clinic: experience from a teaching hospital in Thailand, J. Med. Assoc. Thail. 96 (1) (2013) 26–32. [29] Y. Wang, Y. Xia, S. Member, H. Shen, A.S.S. Design, SMC design for robust stabilization of nonlinear markovian jump singular systems, IEEE Trans. Automat. Contr. 63 (1) (2018) 219–224. [30] Y. Li, S. Tong, T. Li, X. Jing, Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets Syst. 235 (2014) 1–24. [31] H. Li, X. Jing, H. Lam, P. Shi, Fuzzy sampled-data control for uncertain, IEEE Trans. Cybern. 44 (7) (2014) 1111–1126. [32] H. Li, X. Jing, H.R. Karimi, ‘‘Output-Feedback-Based H 1 Control for Vehicle Suspension Systems With, Control Delay” 61 (1) (2014) 436–446. [33] L. Ferbar Tratar, E. Strmcˇnik, The comparison of Holt-Winters method and Multiple regression method: a case study, Energy 109 (2016) 266–276. [34] Y. Shen, J. Guo, X. Liu, Q. Kong, L. Guo, W. Li, Long-term prediction of polar motion using a combined SSA and ARMA model, J. Geod. 92 (3) (2018) 333– 343. [35] Y. Shao, Y. Wei, H. Su, On approximating grey model GM(2,1), J. Grey Syst. 22 (4) (2010) 333–340. [36] Y.-F. Huang, C.-N. Wang, H.-S. Dang, S.-T. Lai, Evaluating performance of the DGM(2,1) model and its modified models, Appl. Sci. 6 (3) (2016) 73. [37] V.S. Deshpande, B. Mohan, P.D. Shendge, S.B. Phadke, Disturbance observer based sliding mode control of active suspension systems, J. Sound Vib. 333 (11) (2014) 2281–2296.