Vibration Control of Vehicle Active Suspension Using Sliding Mode ...

Journal of Traffic and Logistics Engineering Vol. 3, No. 2, December 2015

Vibration Control of Vehicle Active Suspension Using Sliding Mode Under Parameters Uncertainty H. Metered and Z. Šika Helwan University, Cairo, Egypt Czech Technical University in Prague, Prague, Czech Republic Email: [email protected]; [email protected]

force the suspension system to achieve the behavior of some optimized and reference systems. Active suspensions use electro-hydraulic actuators which can be commanded directly to generate a desired control force. Semi-active suspensions employ semi-active damper whose force is commanded indirectly through a controlled change in the dampers’ properties. Compared with the passive system, active suspensions can offer high quality performance over a varied frequency range [3], [4]. Simultaneously, the control algorithms of active suspension systems have been introduced in a wide range from primarily linear quadratic regulator (LQR) controllers to smart and intelligent controllers based on new findings of computational intelligence. In order to improve the performance of active suspension systems, numerous research investigations have been achieved on the design and control of active suspension system algorithm in the last three decades. For example, optimal control [5], adaptive control [6], [7], model reference adaptive control [8], H∞ [9], LQG control [10], fuzzy control [11], and sliding mode control strategy [12], [13], feedback controller [14] and the references therein, have been employed in active suspension systems. The main contribution of this paper is to enhance the ride comfort and vehicle stability through using the SMC control algorithm depends on the ideal sky-hook reference model to calculate the variable actuator force. The rest of this paper is organized as follows: an active vehicle suspension based on the quarter model and the dynamic equations of motion are explained in the next section while the description of the SMC control algorithm is provided in section III. Section VI introduces the effectiveness of the proposed controller that illustrated by simulation results. Finally, the conclusion is given in section V.

Abstract—This paper introduces a theoretical investigation of active vehicle suspension system using sliding mode control (SMC) algorithm to enhance the ride comfort and vehicle stability under parameters uncertainty. SMC algorithm is a nonlinear control technique that regulates the dynamics of linear and nonlinear systems using a discontinuous control signal. The proposed control algorithm forces the suspension system to follow the behavior of the ideal sky-hook system behavior. A mathematical model and the equations of motion of the quarter active vehicle suspension system are considered and simulated using Matlab/Simulink software. The proposed active suspension is compared with the passive suspension systems. Suspension performance is evaluated in both time and frequency domains, in order to verify the success of the proposed control technique. Also, uncertainty analysis due to the increased of sprung mass and depreciated of spring stiffness and damping coefficient is also investigated in this paper. The simulated results reveal that the proposed controller using SMC grants a significant enhancement of ride comfort and vehicle stability even in the existence of parameters uncertainty.  Index Terms—active vehicle suspension, sliding mode control, vibration control.

I.

INTRODUCTION

The development of good quality control techniques for vehicle active and semi-active suspension systems is a main issue for the automotive industry. A good quality suspension system should enhance the ride comfort and vehicle stability. To improve ride comfort, it should minimize the vertical body acceleration of the vehicle due to the unwanted disturbances of the road surface. In terms of vehicle stability, however, it should offer a sufficient tyre-terrain contact and minimize the dynamic deflection of the tyre. Therefore, good quality suspension systems are difficult to obtain because they involve a trade-off between ride comfort and vehicle stability [1]. There are three major classifications of suspension systems: passive, active, and semi-active [2]. Passive suspensions using oil dampers are simple, reliable and cheap. However, performance drawbacks are inevitable. Active and semi-active have control algorithms which

II.

The two-degree-of-freedom (2DOF) system that represents the quarter vehicle suspension model is illustrated in Fig. 1. It consists of an upper mass, mb , representing the body mass, as well as a lower mass, m w , representing the wheel mass and its associated parts. The

Manuscript received February 1, 2015; revised April 23, 2015. ©2015 Journal of Traffic and Logistics Engineering doi: 10.12720/jtle.3.2.136-142

QUARTER VEHICLE MODEL

136


vertical motion of the system is described by the displacements x b and x w while the excitation due to road disturbance is x r . The suspension spring constant is k s and the tyre spring constant is k t . Also, c s is the

III.

The SMC algorithm applied in this paper depends on the ideal sky-hook system, Fig. 2, as a reference model [16]. As can be seen from this figure, the tyre flexibility has been ignored for simplicity, since the tyre is much harder than the suspension spring. The displacement of the lower mass of the reference system is then taken to be similar to x w , the displacement of the unsprung mass of

damping coefficient of the passive damper whereas the tyre damping is neglected and f a represents the actuator force. The data used here for the quarter vehicle system is similar as ref. [15] and listed in Table I. Newton’s second law is applied to the quarter vehicle model and the equations of motion of mb and m w are: mb xb  k s ( xb  xw )  cs ( xb  x w )  f a  0 mw xw  k s ( xb  xw )  cs ( xb  x w )  kt ( xw  xr )  f a  0

the actual system. Hence, the equation of motion of the reference system is given by: (3) mb, ref xb, ref  cs , ref xb, ref  ks , ref xb, ref  xw  The major advantages of employing this control technique are that the system can be designed to be robust with respect to modeling imprecision, and it can be synthesized for the linear and nonlinear active system. In this study, the model reference design approach is chosen. Therefore, a good reference needs to be considered. In practice, the vehicle mass varies with the loading conditions such as the number of riding persons and payloads. Therefore, we consider that parameter perturbations of the sprung mass exist in the system. The possible bound of the mass can be assumed as follows:

(1)

The proposed active suspension system can be represented in the state space form as follows:

x  A x  B f a  D xr

(2)

where, x  [ xb xw xb xw ]T ,

 0  0  k s A    mb  ks   mw  B  0 

0

1  mb

0

1

0 ks mb k s  kt  mw

0 c  s mb cs mw

0  1  cs ,  mb  c   s  mw 

T  1  , and D  0 0 0   mw 

kt   mw 

mb  mbo  mb and mb  0.2mbo

T

where

(5)

e  xb  xb,ref

(6)

is a constant and

Further, the error between the estimated nominal value and the real value is assumed to be bounded by known  :

Parameter

Symbol

Value (Unit)

Mass of vehicle body

mb mw ks cs kt

240 (kg)

1

36 (kg)





1 / mbo mb 1 m 1     b     1  . 1 / mb  mbo  mbo

(7)

(1  mb / mbo ) is maximum when mb  0.2mbo while it is minimum when mb  0.2mbo .

16 (kN/m) 980 (Ns/m) 160 (kN/m)

So,   1.25.

Tyre stiffness ©2015 Journal of Traffic and Logistics Engineering



S  e  e

Figure 2. Ideal sky-hook reference model

TABLE I. QUARTER VEHICLE SUSPENSION P ARAMETERS [15].

Tyre stiffness Damping coefficient

(4)

where, mbo represents the nominal mass and ∆mb is the uncertain mass. The uncertainty ratio 0.2 is selected here for the purpose of application. The sliding surface is defined as:

Figure 1. Quarter-vehicle active suspension model.

Mass of vehicle wheel Mass of vehicle wheel Suspension stiffness Suspension stiffness Damping coefficient

SLIDING MODE CONTROL ALGORITHM

137


 m  mbo [ K  sgn( S )]  1  bo ( xb,ref  e). mb  mb  when S   , S  0 , Equation (14) becomes

The objective of the model reference approach is to develop a control algorithm which forces the plant to follow the dynamics of an ideal model. In order to ensure the state will move toward and reach the sliding surface, a sliding condition must be defined. The sliding condition is considered as in ref. [16] SS   S

 S  



(8)

 m m  K   1  b ( xb,ref  e)  b  mbo  mbo 

K 



(10)

K 

K  [

(11)

where sgn is the signum function. Equation (11) can be expressed as

f a  mbo[uo  K  sgn( S )]



(12)

 f  Kval( S ) fa   a0  f a 0  K sgn S 

(19)

S  S 

m mb  1  xb, ref  e  b  mbo mbo

 (   1)  xb, ref  e    (   1) uo 

 (   1) uo  (   1)

(13)

(20)

k ks xb  s xw   mso mso

k ks xb  (   1) s xw   (21) mbo mbo

K  should be bounded by Equation (21), so it can be expressed to be:

  1 k k K   (   1)  uo  1 xb  1 xw    (22) mbo  mbo  mbo The SMC described above is summarized in Fig. 3.

k m S   s ( xb  xw )  bo [uo  K  sgn( S )]  xb,ref  e. mb mb ks m k  ( xb  xw )  bo [( s ( xb  xw ) ms mb mbo

IV.

RESULTS AND DISCUSSION

Suspension working space (SWS), vertical body acceleration (BA), and tyre deformation (TD) are the three main performance criteria in vehicle suspension design that govern the ride comfort and vehicle stability. Ride comfort is closely related to the BA. To certify good vehicle stability, it is required that the tyre’s dynamic deformation ( xw  xr ) should be low [17]. The structural characteristics of the vehicle also constrain the amount of

 e  xb ,ref )  K  sgn( S )]  xb ,ref  e. m k ks ( xb  xw )  bo [ s ( xb  xw )] mb mbo mb

mbo m [ K  sgn( S )]  [1  bo ]( xb,ref  e). mb mb

©2015 Journal of Traffic and Logistics Engineering

m mb  1]( xb, ref  e)  b  mbo mbo

From equation (10), the right hand side of equation (20) becomes

Now, the range of the switching gain K  to make the system stable is to be found. Equation (8) can be interpreted as  S 0  S   (14)  when  S 0 S  



m  mb    b  1( xb,ref  e) mbo  mbo 

 (  1)  xb, ref  e  

K mbo

To avoid the chattering problem, a saturation function could be applied to equation (11). Then the equation (11) can be written as:



(18)

Combining two cases from Equations (17) and (19),

f a  uo  K sgn(S )

mbo

 m  mbo [ K  sgn( S )]  1  bo ( xb,ref  e)   mb  mb 

m  m K    b  1( xb,ref  e)  b  mbo  mbo 

force f a can be expressed as

and K   

(17)

multiply Eq.( 18) by mb it becomes, mbo

In order to satisfy the sliding condition, despite mass uncertainty, a term discontinuous across the surface is added to the expression u o . The desired control

where u    uo o

m  mb    b  1( xb,ref  e) mbo  mbo 

Similarly, when S  0, S   Equation (14) becomes

The best approximation u o of a control law that would achieve S  0 is thus:

ks ( xb  xw )  e  xb ,ref ] mbo

(16)

multiply Eq.(16) by  mb it becomes, mbo

where  is a positive constant. It constrains trajectories to point towards the sliding surface. In particular, once on the surface, the system trajectories remain on the surface, i.e. S  0. From equations (1) and (5),   e  xb  xb, ref  e S  e (9) k f   s ( xb  xw )  a  xb , ref  e. mb mb

uo  mbo [

 m  mbo [ K  sgn( S )]  1  bo ( xb,ref  e)   mb mb  

(15)

138


SWS within certain limits. The goal is to minimize SWS, BA, and TD in order to improve suspension performance.

0.04

A. Time Domain Analysis This section studies suspension performance for two cases of vibration control; passive suspension and active damped suspension using the proposed SMC. The abovementioned performance criteria are used to quantify the relative performance of these control methods. Since the passive suspension is used as a base-line for comparisons. The value of cs is selected depends on the median-sized automotive applications [15]. A well-known real-world road bump is used in this section to reflect the transient response characteristics which defined by [18] as:

0.02

0.5  t  0.5 

for

db Vc

Suspension Working Space (m)

0

-0.02

-0.04

0

1

2

3

4

5

Time (s)

3 Passive SMC

(c) 2

Body Acceleration (m/s2)

 a 1  cos(r (t  0.5)), xr    0, 

Passive SMC

(b)

(23)

otherwise

where a is the half of the bump amplitude, r  2Vc / db , d b is the bump width and Vc is the vehicle velocity. In this study a = 0.035 m, d b = 0.8 m, Vc = 0.856 m/s, as in [18]. The time history of the suspension system response under road bump disturbance excitation is shown in Fig. 4. The displacement of the road input signal is shown in Fig. 4(a) and the SWS, BA, and TD responses are given in Figs. 4 (b, c, and d) respectively. The latter figures show the comparison between the controlled active using SMC controller and the passive suspension systems. From these results it is clearly seen that the SMC controlled active suspension system can dissipate the energy due to bump excitation very well, cut down the settling time, and improve both the ride comfort and vehicle stability.

1

0

-1

-2

-3

0

1

2

3

4

5

Time (s)

4.5

x 10

-3

Passive SMC

(d)

3

Tyre Deflection (m)

1.5

0

-1.5

-3

-4.5

0

1

2

3

4

5

Time (s)

Figure 4. System response under road bump excitation. (a- Road Displacement b- SWS c- BA d- TD)

Also, Fig. 4 shows that the proposed active suspension controlled using the SMC have the lowest peaks for the SWS, BA, and TD, demonstrating their effectiveness at improving the ride comfort and vehicle stability. The controlled system using SMC controller can reduce maximum peak-to-peak of SWS, BA, and TD by 15.9 %, 46.4 % and 57.2 %, respectively, compared with the passive suspension system. Figure 5 shows the improvement percentage of PTP for the active suspension controlled using the SMC compared to the passive suspension system. The results confirm that the active vehicle suspension system controlled using SMC offers a superior performance.

Figure 3. Schematic diagram of the sliding mode control algorithm 0.08 (a)

Road Displacement (m)

0.06

B. Frequency Domain Analysis Road irregularities are the main source of disturbance that causes unwanted vehicle body vibrations. These irregularities are usually randomly distributed. The random nature of the road irregularities is due to construction tolerances, wear and environmental action. The road surface irregularities have naturally been

0.04

0.02

0

0

1

2

3

4

5

Time (s)


139


described as a white noise random road profile defined by [18] as: xr  Vxr  VWn

3.6

Passive SMC

(24) 2.4

Tyre Deflection (m)

road irregularity parameter, and  2 is the covariance of road irregularity. In random road excitation, (  = 0.45 m-1 and  2 = 300 mm 2 ) the values of road surface irregularity were selected assuming that the vehicle moves on the paved road with the constant speed V = 20 m/s , as in [18]. In order to improve the ride comfort, it is important to isolate the vehicle body from the road disturbances and to decrease the resonance peak of the body mass around 1 Hz which is known to be a sensitive frequency to the human body [19]. Moreover, in order to improve the vehicle stability, it is important to keep the tyre in contact with the road surface and therefore to decrease the resonance peak around 10 Hz, which is the resonance frequency of the wheel [19]. In view of these considerations, the results obtained for the excitation described by equation (24) are presented in the frequency domain.

1.2

0

(a)

2

4

6

8 Frequency (Hz)

10

12

14

16

Figure 6. System response under random road excitation. (a- SWS bBA c- TD)

Fig. 6 shows the modulus of the Fast Fourier Transform (FFT) of the SWS, BA, and TD responses over the range 0.5-16 Hz. The FFT was appropriately scaled and smoothed by curve fitting as done in [20]. It is evident that the lowest resonance peaks for body and wheel can be achieved using the proposed SMC controller. According to these figures, just like for the bump excitation, the controlled system using SMC controller can dissipate the energy due to road excitation very well and improve both the ride comfort and vehicle stability. In the case of random excitation, it is the root mean square (RMS) values of the SWS, BA, and TD, rather than their peak-to-peak values, that are relevant. The controlled system using SMC controller has the lowest levels of RMS values for the SWS, BA, and TD. SMC controller can reduce maximum RMS values of SWS, BA, and TD by 33.1 %, 27.9 and 44.5 %, respectively, compared with the passive suspension system. Figure 7 shows the improvement percentage of RMS for the active suspension controlled using the SMC compared to the passive suspension system. The results again confirm that the semi-active vehicle suspension system controlled using SMC controller can give a superior response in terms of ride comfort and vehicle stability.

Figure 5. % improvements of PTP values compared to passive system.


-3

(c)

where Wn is white noise with intensity 2 2 V ,  is the

0.03

x 10

Passive SMC

0.02

0.01

0

2

4

6

8 Frequency (Hz)

10

12

14

16

2.25 (b)

Passive SMC


Figure 7. % improvements of RMS values compared to passive system 1.5

C. Uncertainity Analysis In order to prove the robustness of the proposed SMC for vibration control of vehicle active suspension, the sprung mass is increased by 30%, the suspension spring constant is reduced by 20%, and also, the damping coefficient of the passive damper is reduced by 20%. In this test, the road displacement was simulated as a bandlimited Gaussian white-noise signal which was band

0.75

0

2

4

6

8 Frequency (Hz)

10


12

14

16

140


limited to the range 0–3 Hz; this frequency range is appropriate for automotive applications and previous published work used a similar range (0.4–3 Hz such as in reference [21]), with 0.02m amplitude, as in reference [21], this random road is shown in Fig. 8 (a). The zoomed responses of SWS, BA, and TD are shown in Fig. 8 (b, c, and d), respectively. Similar to the above results, the proposed SMC still offer a significant improvement under the existence of parameter uncertainty.

V.

In this paper, a sliding mode controller (SMC) is applied as an effective control technique for active vehicle suspension system to improve the ride comfort and road holding. A mathematical model of an active damped quarter-vehicle suspension system was derived and simulated using Matlab/Simulink software. The proposed controller is applied to force the system to emulate the performance of an ideal reference system depends on the ideal sky-hook system behavior. The system performance generated by the proposed SMC algorithm is compared with the passive suspension system. System performance criteria were assessed in time and frequency domains in order to prove the suspension efficiency under bump and random road excitations. Theoretical results showed that the SMC controller potentially offers a significantly superior ride comfort and road holding over the passive suspension system. Under the presence of parameter uncertainties due to the increased of the sprung mass and depreciated suspension stiffness and damping, desired performance is still achieved for the proposed SMC.

0.02 (a)

Road Displacement (m)

0.01

0

-0.01

-0.02

0

2

4 Time (s)

6

8

0.033 Passive SMC

(b)


0.022

ACKNOWLEDGMENT

0.011

This publication was supported by the European social fund within the frame work of realizing the project "Support of inter-sectoral mobility and quality enhancement of research teams at Czech Technical University in Prague", CZ.1.07/2.3.00/30.0034. Period of the project’s realization 1.12.2012 – 30.6.2015.

0

-0.011

-0.022

-0.033

CONCLUSION

3

3.5

4

4.5 Time (s)

5

5.5

6

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2 (c)

Passive SMC

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1.5


1

0.5

0

-0.5

-1

-1.5 3

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x 10

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4.5 Time (s)

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6

-3

(d)

Passive SMC

2

Tyre Deflection (m)

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Figure 8. System response under uncertain parameters. (a- Road Displacement b- SWS c- BA d- TD) ©2015 Journal of Traffic and Logistics Engineering

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[11] P. Gong, X. Teng, and C. Yun, “Fuzzy logic controller for truck active suspension and related optimal control method,” Applied Mechanics and Materials, vol. 441, pp. 821-824, 2013. [12] S. Vaijayanti, B. Mohan, P. Shendge, and S. Phadke, “Disturbance observer based sliding mode control of active suspension systems,” Journal of Sound and Vibration, vol. 333, pp. 22812296, 2014 [13] Q. Yun, Y Zhao, and H. Yang, “A dynamic sliding-mode controller with fuzzy adaptive tuning for an active suspension system,” Proc. IMechE Part D: Journal of Automobile Engineering, vol. 221, pp. 417-428, 2007 [14] R. Shamshiri and I. Wan, “Design and analysis of full-state feedback controller for a tractor active suspension: Implications for crop yield,” Int. J. Agric. Biol., vol. 15, pp. 909-914, 2013. [15] S. Tu, R. Hu, and S. Akc, “A study of random vibration characteristics of the quarter-car model,” Journal of Sound and Vibration, vol. 282, pp. 111-124, 2005. [16] A. Lam and H. Liao, “Semi-active control of automotive suspension systems with magnetorheological dampers,” International Journal of Vehicle Design, vol. 33, pp. 50-75, 2003. [17] R. Rajamani, Vehicle Dynamics and Control, Springer Science and Business Media, New York, 2006. [18] S. Choi and W. Kim, “Vibration control of a semi-active suspension featuring electrorheological fluid dampers,” Journal of Sound and Vibration, vol. 234, pp. 537-546, 2000. [19] D. Fischer and R. Isermann, “Mechatronic semi-active and active vehicle suspensions,” Control Engineering Practice, vol. 12, pp. 1353-1367, 2004. [20] H. Metered, “Application of nonparametric magnetorheological damper model in vehicle semi-active suspension system,” SAE International Journal of Passenger Cars, Mechanical Systems, vol. 5, pp. 715-726, 2012. [21] H. Metered, P. Bonello, and S. Oyadiji, “An investigation into the use of neural networks for the semi-active control of a magnetorheologically damped vehicle suspension,” Proceedings of the Institution of Mechanical Engineers, Part D: Automobile Engineering, vol. 224, pp. 829-848, 2010. Hassan Metered was born in Alexandria, Egypt. He obtained his B.Sc. and M.Sc. degrees in Automotive Engineering from Helwan University, Egypt, in 1998 and 2004, respectively. He has a Ph.D. degree in Mechanical Engineering from Manchester University, UK, in 2010. From 2010 to 2013 he was a Lecturer of vehicle dynamics and control at Helwan University. From Jan. 2014 until now, he is a Postdoctoral Senior


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Researcher at the Czech Technical University in Prague, Czech Republic. His interested research areas are active & semi-active vehicle suspension systems using smart fluid dampers controlled with advanced control strategies (e.g. Sliding Mode Control, Optimal Pole Placement and Linear Quadratic Gaussian, Fuzzy Logic Control, and optimized PID), Mechatronics systems, Real-time Hardware in the loop simulation (HILS) of Mechanical systems, modeling and identification of nonlinear systems, Artificial intelligence application in mechanical systems such as neural networks and ANFIS. Zbynek Šika: 1990 Ing., FME CTU in Prague, diploma thesis. 1999 Ph.D., FME CTU in Prague, doctoral thesis “Synthesis and Analysis of Redundant Parallel Robots”. 2005 Doc., FME CTU in Prague, inaugural dissertation “Active and Semi-active Suppression of Machine Vibration”. 2010 Prof., branch Applied mechanics, professor lecture “Optimization of Mechanical and Mechatronical Systems”. 1994~05 Assistant Professor at the Dept. of Mechanics, FME, CTU in Prague. 2005-10 Associated Professor at the Dept. of Mechanics, Biomechanics and Mechatronics, FME, CTU in Prague. 2010~today Full Professor at the Dept. of Mechanics, Biomechanics and Mechatronics, FME, CTU in Prague. Areas of the scientific activities: calibration and control of robots, active and semi-active vibration control of machines, synthesis and optimization of mechanical systems, redundantly actuated parallel kinematic machines, and vehicle system dynamics & control. Selected research projects participated by applicant within last 5 years: GAČR project 13-39057S, GAČR project P101/11/1627, TAČR project TE01020075, and EC FP7 projects. Multibody system dynamics and kinematics, Redundantly actuated parallel kinematic machines, and Vehicle system dynamics & control.