Journal of Sound and Vibration 509 (2021) 116241
Contents lists available at ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsv
Adaptive sliding mode control based nonlinear disturbance
observer for active suspension with pneumatic spring
Cong Minh Ho a, Duc Thien Tran a, Kyoung Kwan Ahn b,∗
a
b
Graduate School of Mechanical and Automotive Engineering, University of Ulsan, Daehakro 93, Nam-gu, Ulsan, 44610, Korea
School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan, 44610, Korea
a r t i c l e
i n f o
Article history:
Received 1 August 2020
Revised 20 February 2021
Accepted 25 May 2021
Available online 29 May 2021
Keywords:
Pneumatic spring
Nonlinear disturbance observer
Active suspension systems (ASSs)
Sliding mode control
a b s t r a c t
Pneumatic springs have been used for the automotive suspension to induce a flexible stiffness and create the control forces according to various uncertain masses of passengers
and exogenous inputs. However, the control of the vehicle suspension system using the
pneumatic spring is a very complicated task because it involves parametric uncertainties,
external disturbances, and system nonlinearities. In this paper, an adaptive sliding mode
control (SMC) based on nonlinear disturbance observer (NDOB) is designed to obtain passenger comfort and keep the driving safety of the pneumatic active suspension. The nonlinear disturbance observer is proposed to asymptotically reject the external disturbances
and overcome parametric uncertainties which exist in the suspension system such as road
profiles, different passenger masses, and actuator dynamics. Finally, comparative simulations and experimental results are given and compared by the pneumatic vehicle suspension test bench to demonstrate the efficiency of the proposed SMC - NDOB scheme for
different road conditions. Results show that the RMS acceleration value is decreased by
41.5% when the proposed control is used for the experiment with the bump road profile.
© 2021 Elsevier Ltd. All rights reserved.
1. Introduction
Nowadays, the active suspension has received more attention in the automotive industry to improve passenger comfort and driving safety capacity for vehicles, which plays an important role in automobile manufacturing [1–3]. There are
many types of actuators such as electromagnetic, hydraulic, or pneumatic devices which are increasingly used in modern
vehicles [4]. Electromagnetic actuators can generate the active control force to minimize the displacement of the chassis
during vibrations and can improve dynamic behavior with precise force control [5]. However, their masses are increased in
comparison with the other systems since their mechanisms include a lot of permanent magnet actuators and mechanical
structures. Hydraulic active suspension can meet the requirements of high force density, ease of control, and reliability [6,7].
Nonetheless, it needs to be kept at constant high pressure, so the system is a complex structure and requires high investment costs [8]. To increase the performance of the suspension, pneumatic actuators are proposed to improve the stiffness
and damping of the system [9–12]. Based on the important advantages of the variable stiffness coefficient and adaptive
force, the pneumatic suspension system can be utilized in practical applications [13,14]. However, the high nonlinearity is
one of the drawbacks of the pneumatic system, which will make it difficult and complicated in the suspension model and
∗
Corresponding author.
E-mail address: kkahn@ulsan.ac.kr (K.K. Ahn).
https://doi.org/10.1016/j.jsv.2021.116241
0022-460X/© 2021 Elsevier Ltd. All rights reserved.
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
controller design [15–17]. Therefore, many researchers have focused on dynamic simulation to analyze the characteristics of
pneumatic spring suspension [18]. Moreover, pneumatic springs are also used to regulate the displacement of the vehicle
chassis while driving under the different bump road profiles [19–21]. However, the driving safety and handling stability were
not considered as the main objectives in these studies.
On the other hand, many control schemes have been applied to enhance the performance of passenger comfort and
driving safety for vehicle suspension such as robust control [22–24], fuzzy control [25–27], backstepping control [28,29],
and so on. The robust state feedback control can solve the suspension system with parametric uncertainties [23], but it
cannot guarantee both the passenger comfort and the driving safety simultaneously [30]. W. Li et al. [25] applied fuzzy
output feedback control and considered the time delay and output constraints of the suspension system. In [29], an adaptive
backstepping control was proposed to handle nonlinear uncertainty parameters. This controller provided a good result of
passenger comfort, but the performance requirements of active suspension such as road holding capability and actuator
saturation were not considered. Also, they assumed the ideal cases that the active forces of actuators were measured while
the road input acted as a disturbance. Besides, it is not easy to demonstrate an exact system model of the real vehicle
suspension, so model-free strategies such as neural networks, genetic algorithms, and fuzzy logic controllers have been
proposed [31–33]. To overcome these drawbacks, the sliding mode control has been used to solve various requirements
of modern active suspension systems [34,35]. The ability of sliding mode control is its robust tracking performance even
with the uncertainty of parameters while real-time measurement of road input is not required [36–38]. Several studies have
modified the skyhook model and combined the skyhook model with sliding mode control strategy or inertial delay control
to compensate for the disturbances of different road profiles [39,40]. However, the objectives of improving the passenger
comfort and reducing the vertical displacement of the chassis to ensure handling stability will conflict with each other
[41]. To improve the performance of the active suspension, some researchers applied the sliding mode control scheme and
combined it with disturbance observer to reject the unknown parameters [42–44]. However, the effects of real actuator
dynamics were ignored, and those actuators were considered as ideal force sources [45]. Although these methods can solve
the requirements of active suspension, they request more flexible control actuators. Therefore, they are too difficult to be
applied in the real system, especially for pneumatic active suspension.
Basing on the above discussions, we firstly propose the real vehicle suspension system using a pneumatic spring actuator
in this paper. Because the stiffness of the pneumatic spring is caused by the contour of the airbag, it is more flexible
than pneumatic cylinders and can be used to enhance the efficiency of the automobile requirements. In this study, a new
model of the pneumatic active suspension is designed, using only one pneumatic spring as the active actuator. Since the
ride comfort always conflicts with the rattle space limit of the chassis; for example, reducing the chassis displacement
will require a larger suspension deflection, the adaptive sliding mode control is proposed by integrating NDOB to enhance
both the passenger comfort and the driving safety of the vehicle suspension system. The pneumatic suspension system
contains some uncertain nonlinear factors such as the time-varying mass of the passengers and pneumatic spring dynamics
which depend on different working conditions and external forces. Consequently, the effects of the actuator imperfection and
unknown parameters can be compensated by using a nonlinear disturbance observer in this work. Moreover, the stability of
the proposed control is guaranteed by using the Lyapunov theory. Finally, comparative simulations and experimental studies
based on a realistic test bench are executed to prove the robustness of the SMC - NDOB scheme.
The contributions of this paper can be listed as follows.
1 A new quarter car model which considers the flexible stiffness and hysteresis characteristics of pneumatic spring is
designed to investigate the behavior of the actual suspension system.
2 A nonlinear disturbance observer is used to overcome the parametric uncertainties and external disturbances of the
pneumatic spring. Then, the adaptive sliding mode control is applied to improve the performance of the pneumatic
active suspension, i.e., passenger comfort, driving safety, and handling stability.
The remainder of this paper is arranged as follows. The nonlinear mathematical of the quarter car model is established in
Section 2. SMC – NDOB control and system stability have been proposed and improved in Section 3. Besides, the comparative
simulations are analyzed in Section 4. Moreover, the experimental studies are implemented with two different cases of road
disturbance conditions in Section 5. Finally, the conclusion of pneumatic active suspension is taken place in Section 6.
2. System descriptions
2.1. Nonlinear quarter car active suspension model
The dynamics of the vehicle suspension system are conducted by the movement of the chassis which is called the sprung
mass. The unsprung mass denotes the weight of the wheel, tire body, and suspension structure. The chassis is affected by
external disturbances that cause continuous excitations to the passengers. The suspension system is designed to dissipate
this vibration for the passenger comfort. According to [46], the structure of the pneumatic suspension is described in Fig. 1.
The mechanical equations of the pneumatic active suspension are described as follows
ms z̈s = −Fs (zs , zu , t ) + Fu
(1)
mu z̈u = Fs (zs , zu , t ) − Ft (zu , zr t ) − Fu
(2)
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C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 1. Pneumatic active suspension.
where Fu is the force control of active suspension, ms and mu denote the sprung mass and unsprung mass, zs and zu determine the positions of the sprung mass and unsprung mass while the road profile will be represented byzr .
The forces Fs and Ft are generated by the pneumatic spring, mechanical springs, and tire, which can be calculated as
Fs (zs , zu , t ) = (ks + ka )(zs − zu )
(3)
Ft (zu , zr , t ) = kt (zu − zr )
(4)
where ks , kt , and ka represent the stiffness coefficients of the mechanical spring, tire, and pneumatic spring, respectively.
The tire force depends on safe driving conditions and is expressed by the following equation [36]
Ft =
kt (zu − zr )
0
if
if
kt (zu − zr ) < (ms + mu )g
kt (zu − zr ) ≥ (ms + mu )g
(5)
where g represents the gravitational acceleration.
2.2. Pneumatic spring modeling
Electro-pneumatic control can achieve suspension modulation by adjusting the flow rate of air mass into or out of the
pneumatic spring [21]. However, the controllers of the pneumatic active suspension system are so complicated because
we need to control both the pneumatic spring and the proportional pressure regulator simultaneously. Moreover, the air
pressure and airflow through proportional valves are nonlinear parameters. In this study, the stiffness of the pneumatic
spring is established by mathematical equations to describe the dynamic characteristics based on thermodynamic theory.
Then, the model of the pneumatic spring will be applied to study the behavior of the suspension system.
Compared with mechanical springs, the pneumatic stiffness of the air spring relates to the supply pressure source and
depends on the vertical displacement along the axis. To study the response of pneumatic active suspension, the different
stiffness of air spring is demonstrated depending on its working conditions. The pneumatic stiffness will comply with thermodynamic theory, which is calculated by [18].
ka = π P (
γ
γ P0V
28r
Dπ
+
) + γ +10 ×
90
8
V
28r Dπ π D2 2
4r 2
+ 2a
+
+
π −
9
90
8
4
(6)
where r is the meridian radius of pneumatic spring, D denotes the circumferential diameter of pneumatic spring, a represents the axial displacement of pneumatic spring, γ is the specific heat ratio, and P indicates the present pressure inside
the pneumatic spring while P0 and V0 are the initial values of pressure and volume, respectively.
The dynamic equation of the pneumatic spring in combination with active suspension can be demonstrated by [19].
P˙ Aas (zas0 + zs − zu ) = γ RT (qin − qout ) − γ P0 Aas (z˙ s − z˙ u )
(7)
where Aas represents the working area of the pneumatic spring, zas0 is the initial height of the pneumatic spring, R is the
ideal gas constant, T denotes the air temperature, qin is the airflow to the pneumatic spring, qout is the airflow out of the
pneumatic spring.
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C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
2.3. Problem formulation
The output force of this pneumatic actuator is computed by
Fu = Aas P
(8)
Assumption 1. Because the pressure inside the pneumatic spring is controlled by a proportional pressure regulator, the
relationship between the input voltage u and the pressure in the pneumatic springP is assumed as a linear function as
follows
P = kin u
(9)
wherekin is the pressure coefficient.
Assumption 2. Since the limitations of the vehicle structure and actual physical performance, the sprung mass is limited by
ms min < ms < ms max
(10)
The sprung mass is presented as
m s = m s 0 + m s 0
(11)
where ms0 is nominal sprung mass and ms0 is uncertain sprung mass.
In practice, the uncertain mass of the car-body part and noise measurement in sensors always exist, which can affect
the performance of the suspension system. To solve these problems, when we design the suspension controller, the system
dynamics are based on Assumption 1, Eqs. (1), (2), (8), (9), and (11). The definitions of the state variables are given as x1 = zs ,
x2 = z˙ s , x3 = zu , and x4 = z˙ u . Hence, the state space form of active suspension can be written as follows
x˙ 1 = x2 + d1 (t )
x˙ 2 =
(12)
1
[−ks x1 + Aas kin u] + d2 (t )
mso
(13)
x˙ 3 = x4 + d3 (t )
x˙ 4 =
(14)
1
[(ks + ka )(x1 − x3 ) − kt (x3 − zr ) − Aas kin u]
mu
(15)
m
s
where d2 (t ) = − msos0 x˙ 2 − mkaso (x1 − x3 ) + mkso
x3 is an unknown function and d1 (t ), d3 (t ) are noise measurements.
Assumption 3. The derivatives of the disturbance parameters d˙i (t ), (i = 1, 2, 3) are bounded and satisfy
lim d˙i (t ) = 0,
t→∞
i = 1, 2, 3
Remark 1. To ensure the ride comfort, some advanced control strategies have been adopted to reduce the chassis displacement. However, they often considered the suspension system with mechanical spring, so system nonlinearities are neglected
in the control design. In this study, the NDOB is used to estimate the uncertain nonlinear factors of pneumatic spring to
improve the suspension efficiency.
3. Adaptive Sliding mode control based Nonlinear Disturbance Observer
The block diagram of the proposed control is displayed in Fig. 2. The NDOB is utilized to approximate the uncertainties
d1 and d2 , which is then combined with SMC to guarantee the suspension requirements. The dynamic equations for control
design are simplified from Eqs. (12) and (13) as follows.
x˙ 1 = x2 + d1 (t )
(16)
x˙ 2 = f (x ) + g(x )u + d2 (t )
(17)
where f (x ) = m1so (−ks x1 ) and g(x ) = m1so Aas kin
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Journal of Sound and Vibration 509 (2021) 116241
Fig. 2. Diagram of Sliding mode control based NDOB.
3.1. Nonlinear Disturbance Observer control scheme
An adaptive SMC - NDOB is designed as the following procedure:
Step 1: An NDOB is applied to estimate the matched and mismatched uncertainties.
Step 2: An adaptive sliding mode control approach is then designed based on the disturbance observer.
First, we arrange Eqs. (16) and (17) as follows
x˙ = f1 (x ) + p(x )u + d (t )
y = x1
(18)
where x = [x1 , x2 ]T , f1 (x ) = [x2 , f (x )]T , p(x ) = [0, g(x )]T , and d (t ) = [d1 (t ), d2 (t )]T .
Consider the Nonlinear Disturbance Observer as [47]
h˙ = −vh − v[vx + f1 (x ) + p(x )u]
dˆ = h + vx
(19)
where dˆ is the disturbance estimation of d,h denotes the internal state of the nonlinear observer, and v is the observer
control gain.
We assume that the disturbance estimation errors are defined as
edi = di − dˆi , (i = 1, 2)
(20)
In addition, the error dynamics can be formulated from Eq. (20)
˙
e˙ d = d˙ − dˆ
(21)
Substituting Eq. (18) and Eq. (19) into Eq. (21), the error dynamics will be reduced to
e˙ d = d˙ − h˙ + vx˙
(22)
e˙ d = d˙ − {−vh − v[vx + f1 (x ) + p(x )u] + v[ f1 (x ) + p(x )u + d]}
(23)
e˙ d = d˙ − ved
(24)
According to Assumption 3, the disturbance d can be estimated by the disturbance estimation dˆ if the observer control
gain is ensured by v > 0. Hence, the Eq. (24) can be rewritten as
e˙ d + ved = 0
(25)
Therefore, the estimation error will converge to zero asymptotically according to Eq. (25).
The sliding mode surface based on the disturbance estimation for this system is chosen as
S = λe1 + e2 + dˆ1
(26)
where λ is the control parameter, and the errors e1 and e2 are defined as follows
e1 = x1 − xd
e2 = x2 − x˙ d
(27)
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Journal of Sound and Vibration 509 (2021) 116241
where xd is the reference trajectory.
The control law for SMC-NDOB is chosen as follows
u = ueq + un
(28)
˙
ueq = g−1 (x ) −λ(e2 + dˆ1 ) − f (x ) − dˆ2 − dˆ1 + ẍd
(29)
un = −g−1 (x )[ηsign(S )]
(30)
with:
The system (18) will be converged to the desired equilibrium point along the sliding surface asymptotically with the
control law (28) if the switching control gain η is selected to satisfy the following conditions
η > sup λed1 + ed2 and λ > 0
(31)
t>0
Proof. The demonstration will be presented in Section 3.2 below.
3.2. Stability
Choose the Lyapunov function
V =
1 2
S
2
(32)
The time derivative of V is
V˙ = SS˙
(33)
From Eq. (26), we have
˙
S˙ = λe˙ 1 + e˙ 2 + dˆ1
(34)
˙
S˙ = (x˙ 2 − ẍd ) + λ(x˙ 1 − x˙ d ) + dˆ1
(35)
Besides, substituting the control u in Eq. (28) into Eq. (35), we have
˙
˙
S˙ = −λ(e2 + dˆ1 ) − dˆ2 − dˆ1 + ẍd − ηsign(S ) + d2 − ẍd + λ(x2 + d1 − x˙ d ) + dˆ1
(36)
˙
˙
S˙ = −λ(x2 − x˙ d + dˆ1 ) − dˆ2 − dˆ1 + ẍd − ηsign(S ) + d2 − ẍd + λ(x2 + d1 − x˙ d ) + dˆ1
(37)
S˙ = −λdˆ1 − ηsign(S ) − dˆ2 + d2 + λd1
(38)
Applying Eq. (38), we can write the Eq. (33) as follows
V˙ = SS˙ = S −λdˆ1 − ηsign(S ) − dˆ2 + d2 + λd1
(39)
From Eq. (20), the disturbance estimation errors can be detailed by
ed1 = d1 − dˆ1
ed2 = d2 − dˆ2
We have
(40)
V˙ = SS˙ = S −ηsign(S ) + λed1 + ed2
Therefore
(41)
V˙ ≤ |S| −η + λed1 + ed2
(42)
Or, we have
√ V˙ ≤ − 2V η − λed1 − ed2
(43)
With the condition of Eq. (31), the Lyapunov function will gradually decrease to guarantee the stability of the control
system.
Based on the Eq. (43), the system states can approach the specified sliding surface S = 0 in a finite time as soon as the
condition in Eq. (31) is satisfied.
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Journal of Sound and Vibration 509 (2021) 116241
Table 1
Pneumatic active suspension parameters.
Parameter
Value
Unit
ms
mu
ks
kt
xR
r
zas0
R
Aas
P0
V0
30 ± 16.6
5.5
26,140
190,000
0.04
21×10−3
0.16
287.5
0.0047
101330
7.2×10−4
1.4
293.15
0.078
kg
kg
Nm−1
Nm−1
m
m
m
J.kg−1 .K−1
m2
Pa
m3
K
m
γ
T
D
The following condition can be obtained from Eq. (26)
e˙ 1 = −λe1 + ed1
(44)
Then, the system states will satisfy
lim ed1 (t ) = 0 and lim e1 (t ) = 0
t→∞
(45)
t→∞
Therefore, the system states will converge to the desired equilibrium point with the SMC - NDOB.
Remark 2. It can be seen from Eq. (45) that the estimation errors will converge to zero asymptotically; thus, the performance of the active suspension can be improved with the proposed control. Besides, the NDOB can estimate the external
disturbances precisely, so the amplitude of the estimation errors ed will be much smaller than the amplitude of the disturbances d. Hence, the switching gain of the proposed control can be reduced in comparison with that of the traditional SMC,
and the chattering phenomenon can be alleviated.
4. Simulation results
4.1. Simulation description
The comparative simulations of the proposed SMC-NDOB scheme are compared with passive suspension and traditional
SMC to verify the effectiveness of this method. The efficiency of the active suspension is evaluated by the human body’s
sensitivity to acceleration and quantified by the ride comfort. According to ISO 2631-5:2004, people are very sensitive to
acceleration in the frequency range of 4 – 8 Hz, so the active suspension systems must be kept to a minimum in this
frequency range. The ride comfort is investigated at the peak and the root mean square (RMS) values of chassis acceleration.
To find the weighted root mean square of acceleration, a filter is proposed in [48].
W (s ) =
81.89s3 + 796.6s2 + 1937s + 0.1446
s4 + 80s3 + 2264s2 + 7172s + 21196
(46)
To meet two objectives of limiting the magnitude of the chassis and ensuring the road holding of the tire, two parameters
are defined as relative suspension deflection (RSD) and relative tire force (RTF), which are presented in Appendix.
In a real active suspension system, a lot of parameters cannot be clearly defined because some information about passenger masses as well as the road profiles is random values. These factors are often ignored in other suspension systems,
so passenger comfort cannot be guaranteed. In this study, we consider all the main parameters of pneumatic suspension as
shown in Table 1.
Two types of road profiles are used to analyze the control scheme in this research
1 Sin road profile
2 Bump road profile
In this work, all the mathematical models and simulations are accomplished by using MATLAB with Simulink 2019a.
All the simulation programs are built by selecting automatic solver configuration and a sample time of 0.01 second. The
diagram block of the simulation for pneumatic active suspension is displayed in Fig. 3.
The control parameters for SMC-NDOB are selected as v = diag{2, 2}, λ = 10, and η = 20 throughout the simulation. The
disturbances are given by the sinusoidal with amplitude 0.015 m and frequency 0.5 Hz. The time-varying mass of the chassis
is also based on sinusoidal ms (t ) = 30 + 16.6 sin(t ) kg.
7
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 3. Structure of the simulation for pneumatic active suspension.
Table 2
Summary of simulation results with sin road profile.
Measure
Passive
SMC
SMC-NDOB
Peakx˙ 2
RMS
Peakx1
Peak RSD
Peak RTF
2.4204
0.8390
0.0237
0.0787
0.1916
1.8047
0.4572
0.0070
0.4583
0.3365
0.4057
0.0428
0.0046
0.4007
0.0772
4.2. Simulation example
4.2.1. Case 1: Simulation with sin road profile
The sin road profile is generated by a sinusoidal with an amplitude of 0.02 m and a frequency of 0.5 Hz
zr (t ) = 0.02 sin(π t )
(47)
The simulation results of sprung mass acceleration and displacement, relative suspension deflection, relative tire force,
and control signals of the passive suspension, conventional SMC, and SMC-NDOB with sin road profile are displayed in
Fig. 4 and Table 2. The passenger comfort, driving safety, and handling stability are strongly improved with the proposed
SMC-NDOB because the sprung mass acceleration and deflection, RSD, and RTF are all guaranteed. The SMC-NDOB can
enhance the ride comfort compared with the conventional SMC because the maximum value of sprung mass acceleration
is reduced as shown in Fig. 4(a). Compared with passive suspension, the conventional SMC can reduce the acceleration,
but there are some peak values due to the external disturbances. Besides, the proposed control can guarantee less sprung
mass deflection than the conventional SMC and passive suspension as shown in Fig. 4(b). The SMC-NDOB provides less
magnitude of the RSD than the conventional SMC as shown in Fig. 4(c), and this RSD is guaranteed to be smaller than 1.
Because enhancing the ride comfort will require a larger suspension stroke, the RSD of the passive suspension is minimal.
The SMC-NDOB scheme can guarantee passenger comfort by adjusting the sprung mass displacement under the exogenous
road disturbances while keeping the magnitude of RSD within the limit value. It is also possible to see the RTF values from
Fig. 4(d) that dynamic stroke constraints are guaranteed within the limits. The RTF value of SMC-NDOB is also smaller than
the conventional SMC to ensure the stability of the chassis. From Fig. 4(f), we can see that the shape of the control signal
is always against the impacts of the road profile in Fig. 4(e), so it is possible to reduce the displacement of the chassis to
obtain the ride comfort. Furthermore, the control signal of the conventional SMC is larger and more chattering than the
proposed control.
Fig. 5 shows the matched and mismatched disturbances and their estimation states. The mismatched disturbance can be
caused by the inaccuracy of the chassis sensor as established in Fig. 5(a). The matched disturbance comes from the different
masses of passengers and the flexible stiffness of the pneumatic spring is shown in Fig. 5(b). The results show that the
8
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 4. Simulation results with sin road profile (Amplitude 0.02 m, 0.5 Hz): (a) Sprung mass acceleration. (b) Sprung mass deflection. (c) Relative suspension
deflection (RSD). (d) Relative tire force (RTF). (e) Sin road profile. (f) Control input.
proposed control can provide an effective method for estimating the matched and mismatched disturbances in order to
obtain the objectives of the suspension system.
4.2.2. Case 2: Simulation with bump road profile
The bump road profile is selected for the simulation according to ISO 8608. The sinusoidal disturbances are employed to
consider the rough profile which is described by the following equation [49]
zr (t ) =
⎧
3
2
−0.0592(t − 3.5 ) + 0.1332(t − 3.5 ) + d (t );
⎪
⎪
3
2
⎪
⎨0.0592(t − 6.5 ) + 0.1332(t − 6.5 ) + d (t );
3
2
0.0592(t − 8.5 ) − 0.1332(t − 8.5 ) + d (t );
⎪
3
2
⎪
⎪
⎩−0.0592(t − 11.5 ) − 0.1332(t − 11.5 ) + d (t );
d (t );
where d (t ) = 0.0 02sin2π t + 0.0 02sin7.5π t
9
3.5 ≤ t < 5
5 ≤ t < 6.5
8.5 ≤ t < 10
10 ≤ t < 11.5
else
(48)
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 5. Response curves of disturbances and estimation results with sin road profile: (a) Mismatched disturbance. (b) Matched disturbance.
Table 3
Summary of simulation results with bump road
profile.
Measure
Passive
SMC
SMC-NDOB
Peakx˙ 2
RMSx˙ 2
Peakx1
Peak RSD
Peak RTF
4.6447
1.3528
0.0225
0.0936
0.3812
2.0378
0.4593
0.0063
0.4278
0.2918
0.4273
0.1598
0.0043
0.4146
0.0818
The simulation results with the bump road profile case are displayed in Fig. 6. Due to the nonlinear parameters, the
acceleration of passive suspension is very high, and it is impossible to guarantee the passenger comfort as shown in Fig. 6(a).
Although the conventional SMC can reduce the chassis acceleration to get the ride comfort of the vehicle, its vibration is
still greater than the proposed control. As the SMC-NDOB is used, the performance of active suspension is improved with
passenger comfort, road holding capacity, and handling stability. A summary of simulation results with the bump road
profile is given in Table 3. The RMS acceleration of the proposed control is reduced by 88.18% in comparison with that
of the passive suspension. Besides, due to the impact of the unspecified dynamics of the pneumatic actuator, the passive
suspension cannot guarantee the chassis deflection as shown in Fig. 6(b). When the vehicle crosses the peak of road profile,
the actuator of the active suspension is activated to pull the chassis down as shown in Fig. 6(e) and (f).
The matched and mismatched disturbances and their estimation states are shown in Fig. 7. Because the stiffness coefficient of pneumatic spring depends on the rubber behavior under vertical displacement, the matched disturbance will take
the profile of exogenous road as shown in Fig. 7(b). The disturbance observer can estimate the effects of passenger masses
and pneumatic spring stiffness to guarantee the suspension performance.
5. Comparative experimental studies
5.1. Experimental platform
The active suspension for the quarter car model driven by the pneumatic spring is set up to prove the effectiveness of the
proposed SMC - NDOB scheme as displayed in Fig. 8. The experimental test bench consists of three separate plates moving
along two sliding guide axes and connected by mechanical springs and a pneumatic spring. These plates represent chassis,
tire, and road, respectively. The top plate denotes the chassis of the suspension, and it will be connected to the middle
plate by two mechanical springs 1, 2 and one pneumatic spring. The middle plate simulates the tire and wheel assembly.
The pneumatic spring is assembled between the middle and top plate and acts as an active suspension to provide an active
force that dissipates the exogenous effect of the road disturbances. The middle plate connected to the bottom plate through
two mechanical springs 3, 4 simulates the tire of the suspension system. The bottom plate, which represents the road profile
simulation, is driven by a three-phase AC servo motor controlled to create different road profiles. The vertical displacements
of the chassis, tire, and road profile are recorded by three cable encoders. An accelerometer is mounted on the top plate to
determine the acceleration of the chassis during the experiment process. The controller is applied to record the data and to
send the feedback signal to the computer via a PCI card as shown in Fig. 9.
A pneumatic spring (FESTO GmbH, Model EB-145-100) is used for providing the active force of the suspension system.
The hardware contains an ADVANTECH compatible PC (Core 3.2G) that transmits the voltage control signal u(t ) to a proportional pressure valve, via a PCI Card (NI Corporation, Model 6229) to convert the control signal from a computer to analog
10
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 6. Simulation results with bump road profile: (a) Sprung mass acceleration. (b) Sprung mass deflection. (c) Relative suspension deflection (RSD). (d)
Relative tire force (RTF). (e) Bump road profile. (f) Control input.
voltage. To evaluate the ride comfort of active suspension, a piezoelectric accelerometer (PCB, 352C03) is used to measure
the acceleration of the chassis.
The experimental road profiles will be generated by the motion of the bottom plate, driven by the servo motor and
gearbox. The servo motor, which is controlled by PC via PCI card, can be used to simulate various road profiles. Both the sin
road and the bump road profiles are also used for experiments. The main objective is to validate the proposed sliding mode
control based nonlinear disturbance observer law to isolate the chassis from the disturbances of road profiles within a finite
time and minimize the acceleration for the passenger comfort.
5.2. Experimental results
5.2.1. Case 1: Sin road profile
The amplitude of the sin road profile is selected to be suitable for the limited displacement of the test bench, which
is constrained by the working stroke of the pneumatic spring. In these experiments, the maximum amplitude of the road
profile is chosen to be 0.01 m and the frequency is 0.5 Hz. The experimental results are displayed in Fig. 10 while the
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C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 7. Response curves of disturbances and estimation results with bump road profile: (a) Mismatched disturbance. (b) Matched disturbance.
Fig. 8. Photograph of pneumatic active suspension test bench.
Fig. 9. Diagram of the pneumatic active suspension test bench.
effectiveness of the proposed control is summarized in Table 4. Similar to simulation results, the SMC-NDOB can provide
better performance than the passive suspension and conventional SMC. The vertical displacement and root mean square
acceleration of sprung mass are decreased to obtain the ride comfort. When the proposed control is used, the RMS acceleration value is reduced by 43.6% compared with the passive suspension for the experiment with sin road profile. Despite a
difference between acceleration results in Figs. 10(a) and 4(a), both can clearly show that the passenger comfort has been
improved. This is due to the fact that the accelerometer cannot respond to the high frequency of the system. Furthermore,
the RSD and RTF parameters are also kept to less than 1 to meet the objective of active suspension in Fig. 10(c) and (d).
5.2.2. Case 2: Bump road profile
The amplitude of the pump road disturbances is compressed to 0.01m, which corresponds to the working constraints of
the experimental setup. The maximum acceleration value is significantly reduced when we use the SMC-NDOB as shown
12
C.M. Ho, D.T. Tran and K.K. Ahn
Journal of Sound and Vibration 509 (2021) 116241
Fig. 10. Experimental results with sin road profile (Amplitude 0.01 m, 0.5 Hz) (a) Sprung mass acceleration. (b) Sprung mass deflection. (c) Relative
suspension deflection (RSD). (d) Relative tire force (RTF). (e) Sin road profile. (f) Control input.
in Fig. 11(a) while the RMS value is decreased by 41.5% in Table 5. Since the active suspension enhances the ride comfort
resulting in a larger suspension stroke, the RSD value of SMC-NDOB is greater than the passive suspension and conventional
SMC as displayed in Fig. 11(c). Moreover, the sprung mass displacement is guaranteed within the limits of the maximum
displacement because the RSD value is always less than 1. In Fig. 11(d), the RTF value is also maintained to be less than 1 to
achieve the goal of driving safety. These results demonstrate the effectiveness of the proposed control under different road
conditions.
6. Conclusion
This article has proposed an adaptive sliding mode control based nonlinear disturbance observer for the pneumatic active
suspension to compensate for the uncertain parameters. In order to describe the vehicle suspension exactly, a mathematical
model of the pneumatic spring is applied to formulate the nonlinear system. The nonlinear disturbance observer is investigated to handle the unknown passenger masses and external disturbances to ensure the requirements of modern suspension.
Moreover, comparative simulations and experimental results prove that the proposed SMC – NDOB scheme is efficient and
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Journal of Sound and Vibration 509 (2021) 116241
Table 4
Summary of experimental results with sin road
profile.
Measure
Passive
SMC
SMC-NDOB
Peakx˙ 2
RMSx˙ 2
Peakx1
Peak RSD
Peak RTF
0.0572
0.0055
0.0106
0.0975
0.3584
0.0148
0.0037
0.0087
0.1352
0.5199
0.0073
0.0031
0.0040
0.2508
0.4628
Fig. 11. Experimental results with bump road profile (a) Sprung mass acceleration. (b) Sprung mass deflection. (c) Relative suspension deflection (RSD). (d)
Relative tire force (RTF). (e) Bump road profile. (f) Control input.
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Journal of Sound and Vibration 509 (2021) 116241
Table 5
Summary of experimental results with bump road
profile.
Measure
Passive
SMC
SMC-NDOB
Peakx˙ 2
RMSx˙ 2
Peakx1
Peak RSD
Peak RTF
0.0491
0.0053
0.0098
0.0416
0.2720
0.0100
0.0035
0.0086
0.1669
0.2786
0.0088
0.0031
0.0058
0.1568
0.3153
powerful and can reduce the vertical displacement and chassis acceleration. The RMS acceleration value is decreased by
41.5% when the proposed control is used for the experiment with the bump road profile. Besides, the objectives of driving
safety and handling stability are satisfied because the RSD and RTF values based on the simulations and experimental results
are smaller than 1. Therefore, this method is effective not only to improve the ride comfort but also to maintain the driving
safety, and it can be a promising method for the automotive industry.
Author Contributions
K.K.A. was the funding supervisor and project manager, and he reviewed and edited the manuscript. C.M.H. investigated,
analyzed, created MATLAB software, set up the experiment, and wrote the main content of the paper. D.T.T. supported the
control method and checked the introduction of the article.
Funding
This research was supported by Basic Science Program through the National Research Foundation of Korea (NRF) funded
by the Ministry of Science and ICT, South Korea (NRF-2020R1A2B5B03001480).
Declaration of Competing Interest
The author declares no conflict of interest.
Appendix
RSD is the constraint value that specifies the maximum vibration distance of the chassis and is defined by
RSD =
zs − zu
xR
(A.1)
wherexR called rattle space is defined as the distance between the tire and the chassis when the car stays at a rest position.
RTF which is called the safety driving factor is used to ensure the contact between the tire and road profile and can be
described by comparison of the dynamic tire force with the total weight of the frame, wheel, and tire as follows
RT F =
Ft
[ms (t ) + mu ]g
(A.2)
Road Holding and Handling stability Analysis: To analyze the road holding and handling stability of suspension systems,
we suppose that the system states will converge to the desired equilibrium point with the proposed control. In this section,
we consider the dynamic equations of the unsprung mass with x3 andx4 in the system (14) and (15). Then, substituting (28)
into (15), we obtain
X˙ = EX + Z
(A.3)
where
0
x
X = [ 3 ]; E = [ kt
x4
−m
u
1
d
m
m
˙
]; Z = [ 3 ];ψ = kmt zur + msu0 {− msos0 x˙ 2 − ed2 + λ(e2 + dˆ1 ) + dˆ1 − ẍd + ηsign(S )}
ψ
0
Besides, the estimation errors will converge to zero asymptotically according to (28). Based on Assumption 3, ψ is
bounded, i.e., there is a constant ψ̄ such that ψ ≤ ψ̄ .
Then, the Lyapunov function is selected as follows
Va = X T P X
(A.4)
where P is a positive definite symmetric matrix.
The time derivative of Va can be described by
V˙ a = X˙ T P X + X T P X˙
(A.5)
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Journal of Sound and Vibration 509 (2021) 116241
Substituting (A.3) into (A.5), we have
V˙ a = X T E T P + P E X + 2X T P Z
(A.6)
There is a positive definite symmetric matrix Q > 0 so that the Lyapunov equationE T P + P E = −Q holds. Basing on
2
2
Young’s inequality theorem for nonnegative real numbers a and b, we haveab ≤ a /2ς + b ς/2. Hence, we can write the
form of 2X T P Z as follows
1
2X T P Z ≤
ς
X T PT PX + ς ZT Z
(A.7)
where ς > 0 is the adjustable parameter.
Substituting (A.7) into (A.6), we have
−1
−1
P /2 Q P /2
V˙ a ≤ − λmin
1
−
ς
λmax (P ) V + ς ZT Z
(A.8)
where λmin and λmax represent the minimal and maximal eigenvalues of the matrix.
Choosing the appropriate parameter ς to satisfy
λmax (P )
ς >2
λmin
−1
−1
P /2 Q P /2
(A.9)
From (A.9), we can define χ and ν as follows
−1
−1
1
χ = λmin P /2 Q P /2 − λmax (P )
ς
(A.10)
ν = ς ZT Z
(A.11)
Then, the inequality equation (A.8) can be rewritten
V˙ a ≤ −χ Va + ν
(A.12)
Integrating both sides of (A.12), we acquired
Va ≤ Va (0 ) −
ν −χ t ν
ν
e
+
≤ Va (0 )e−χ t +
χ
χ
χ
(A.13)
Obviously, we can know that
|xi (t )| ≤
Va (0 )e−χ t +
ν
/λmin (P ) , i = 3, 4
χ
(A.14)
Now we can deal with the relative tire force condition in (A.2). The tire forces Ft in (4) can be calculated as
Ft (zu , zr , t ) = kt (zu − zr ) ≤ kt
ν
/λmin (P ) + kt zr ∞
χ
Va (0 )e−χ t +
(A.15)
From (A.15), the RTF constraint can be achieved by choosing the design parameters ς and matrix P so that |Ft | ≤
(ms (t ) + mu )gcan be guaranteed.
Similarly, the handling stability (A.1) can be expressed as
|zs − zu | = |x1 − x3 | ≤ |x1 | + |x3 | ≤ |x1 | +
Va (0 )e−χ t +
ν
/λmin (P )
χ
(A.16)
Then, the handling stability constraint which is denoted by the RSD value can be guaranteed by adjusting the control
parameters to ensure |zs − zu | ≤ xR .
From the above analysis, it can be demonstrated that all suspension performance requirements are satisfied by the selection of initial conditions and control parameters to meet the ride comfort, road holding, and handling stability, i.e., the
mechanical structure and safety of the pneumatic suspension can be guaranteed.
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