Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 1, pp. 3-11, Warsaw 2016
DOI: 10.15632/jtam-pl.54.1.3

CONTINUOUS AND DISCRETE SLIDING MODE CONTROL OF

AN ACTIVE CAR SUSPENSION SYSTEM

Jan Łuczko, Urszula Ferdek

Cracow University of Technology, Faculty of Mechanical Engineering, Kraków, Poland

e-mail: jluczko@mech.pk.edu.pl; uferdek@mech.pk.edu.pl

In this paper, sliding mode control (SMC) algorithms are tested for their use in an active
car suspension system. Using the quarter model of the car as an example, the comparison
of the efficiency of the algorithms is made. A continuous and two discrete versions of the
sliding mode control are taken into consideration. The study is limited to finding the rela-
tion between the control parameters and the comfort factor. This is done by analyzing the
response of the model to the harmonic and impulse excitation.

Keywords: sliding mode control, active damping, car suspension

1. Introduction

The most common types of suspension systems used in automobiles are passives ones. Their
construction is based on several spring and damping elements with constant, linear or nonlinear
characteristics. When designing a passive car suspension, it is necessary to choose the optimal
damping coefficient that would allow both the comfort and safety to be on a high enough level
(Łuczko and Ferdek, 2012). An alternative option is to use suspension which can modify its
properties during ride of the vehicle – namely an active or a semi-active system. In the case of
active suspension, the spring and damping elements are replaced with servomotors. This way
the force generated in the suspension can be used to influence both the comfort and safety of
the ride. Although active suspension systems are more complex and require more energy, they
are more andmore often considered and included in designs of modern vehicles.

One of the possible options for the active suspension control is to use a slidingmode control
(SMC) algorithm (Yoshimura et al., 2001; Sam et al., 2004; Lin et al., 2009). The idea of the
algorithm is to guide themodel in the state-space, to a specific plane called “the sliding plane”
and to keep it as close to this plane as possible. Therefore, two separate phases of this control
can be listed: the “approach” phase, which lasts until the point describing the dynamics of
the system reaches the switching hyperplane, and the “sliding” phase. In the second phase, the
system is forced to “slide” along a slinging plane up to the desired point byusing adiscontinuous
control signal. Most often, it is assumed that the sliding plane equation is linear and related to
the regulation error.

The application of the sliding mode control to a physical system is accompanied by the
so-called “chattering” effect (Lee and Utkin, 2007) – high-frequency oscillations which might
cause wear or damage to the actuator. It is therefore essential to eliminate this effect from the
regulation process.

As most of modern control systems are implemented as digital, below a procedure for de-
signing a discrete sliding mode controller (DSMC) for control of a car suspension is presented.
Themost important advantages of the digital control are reduction of device cost, possibility to
implement complicated control rules and higher control precision.



4 J. Łuczko, U. Ferdek

In this study, a quarter model of the car is used for the purpose of testing the efficiency
of damping the vibration by application of different control strategies. The performance factor
includes parameters related to comfort of the ride.

2. Model of the system

Theanalysis hasbeenperformedonaquartermodelof thecar (HuangandChen,2006;Rajeswari
and Lakshmi, 2008; Snamina et al., 2011) consisting of a non-spring-supportedmass mw (mass
of thewheel, axle and some elements of the drive transmission) and a spring-supportedmassmb
(1/4 of the other mass, mostly the car body). It has been assumed that the road influence on
the wheel can be descibed using a harmonic or impulse kinematic excitation w(t). The analysis
included the influence of the control and disturbance on the behavior of the system. Both the
control and disturbance forces have been assumed to act at the point connecting the car body
and the suspension.
The oscillation of the system around the static equilibrium point can be given by a set of

the two second-order differential equations

mwÿw =−cw(ẏw− ẇ)−kw(yw−w)+ cb(ẏb− ẏw)+kb(yb−yw)−u−z

mbÿb =−cb(ẏb− ẏw)−kb(yb−yw)+u+z
(2.1)

where yw and yb are the displacement of the non-spring-supported and spring-supportedmass,
respectively. The parameters kw and cw define the stiffness and damping of the wheel, while kb
and cb are the same for the passive vibro-isolation system.
By configuring the velocities and displacements of both masses to be included in the sta-

te vector x = [yw,yb, ẏw, ẏb]
T, Eqs. (2.1) can be written in the form of a first-order matrix

differential equation

ẋ=Ax+B(u+z)+Fw (2.2)

where

A=











0 0 1 0
0 0 0 1

−(kw+kb)/mw kb/mw −(cw+ cb)/mw cb/mw
kb/mb −kb/mb cb/mb −cb/mb











B=











0
0

−1/mw
1/mb











F=











0
0

kw/mw
0











(2.3)

3. SMC continuous regulation algorithm

Figure 1 shows a block diagram of the system with the sliding mode control (Sam and Osman,
2005; Chen and Huang, 2005; Sam et al., 2008).
In order to increase the ride comfort, the parameters of the regulator are adjusted using the

minimization condition for the variables that describe the movement of the spring-supported
mass. The reduction of the car body vibration can be accomplished by introducing the following
sliding plane

S =Dx (3.1)



Continuous and discrete sliding mode control... 5

Fig. 1. Block diagram of a continuous SMC

where the transposed vector D is defined as follows

D= [0,κ,0,1] (3.2)

This definition of the vector D should provide the minimal displacement x2 = yb and velocity
x4 = ẏb of the spring-supportedmass. From the dynamical condition of the ideal slidingmotion

Ṡ=Dẋ=D[Ax+B(u+z)+Fw] =0 (3.3)

the equivalent (compensation) control ueq(t) is obtained, which after the omission of the excita-
tion and disturbance influence can be written using the formula

ueq(t)=−(DB)−1DAx (3.4)

An additional switching control usw(t) is also included in order to guarantee the stability of the
system. The control usw is discontinuous with the sliding plane set as the switching line. This
control can be calculated from the equation

usw(t)=−Ksw(DB)−1 sgn(Dx) (3.5)

The slidingmode control is the sum of both of these components

u(t)=ueq(t)+usw(t) (3.6)

The sliding control realized in accordance to the algorithm presented above operates by
dynamical compensation of the suspension stiffness and the introduction of additional damping.
The switching component of control keeps the system close to the sliding plane. If the constant
Ksw is large enough, then the systembecomes resistant to inaccuracies of themodel. In practice,
in order to limit the excessive switching (”chattering” effect) the signum function present in Eq.
(3.5) is substitutedby its approximation, e.g. “saturation” function.For thepurposeof numerical
calculations performed in this study, an arctangent approximation function has been used.

4. SMC discrete regulation algorithm

Due to the fact that the modern control systems are more often realized using digital devices,
the sliding algorithm presented above needs to be modified. In order to choose the parameters
for the discrete slidingmode controller, the following discrete equations are used (Yu et al., 2004;
Yan and Fan, 2012)

xk+1 =Adxk+Bd(uk+zk)+Fdwk (4.1)

where

Ad =Φ(Ts) Bd =ΨB Fd =ΨF (4.2)



6 J. Łuczko, U. Ferdek

with Ts being the sampling time. Thematrix functionsΦ andΦ are defined as

Φ(τ)= exp(Aτ) Ψ= [Φ(Ts)− I]A
−1 (4.3)

The sliding plane can be chosen in the similar form as before (3.1)

Sk =Dxk (4.4)

By introducing

fk =Bdzk+Fdwk (4.5)

from the condition

Sk+1 =Dxk+1 =D(Adxk+Bduk+fk)= 0 (4.6)

the equivalent control can be found

u
eq
k
=−(DBd)

−1
D(Adxk+fk) (4.7)

Usually, fk component, whose value changes depending on the unknown excitation and di-
sturbance, can be omitted, and the equivalent control component can be calculated from the
equation

u
eq
k
=−(DBd)

−1
DAdxk (4.8)

In order to eliminate the component fk from Eq. (4.7), an assumption is made that its value
varies only slightly from the value in the previous step, giving fk ≈ fk−1. By using additionally
discrete state equation (4.1), it can be shown that

fk ≈ fk−1 =xk−Adxk−1−Bduk−1 (4.9)

After transformation, the final form of the equation describing the equivalent control can be
written

u
eq
k
=uk−1− (DBd)

−1[(DAd+D)xk−DAdxk−1] (4.10)

The switching component usw is defined by a similar equation as Eq. (3.5)

uswk =−K
sw(DBd)

−1 sgn(Ddxk) (4.11)

where the final control is the sum of both components.

Fig. 2. Block diagrams of the discrete system (AD – analog/digital converter, DA – digital/analog
converter): (a) DSMC, (b) DMSMC

Using numerical calculations, the efficiency of the DSMC regulator (Fig. 2a) with the equ-
ivalent control obtained using formula (4.8) is compared with the efficiency of the modified
DMSMC regulator (Fig. 2b) related to equation (4.10).



Continuous and discrete sliding mode control... 7

5. Results of numerical calculations

The emphasis of the numerical simulation is placed on finding the influence of the disturbance
on the behavior and efficiency of the continuous SMC, discrete DSMC and DMSMC control-
lers. The following parameter values are chosen for matrices (2.3): mw = 28kg, mb = 510kg,
kw = 180000N/m, kb = 20000N/m, cb = 1000Ns/m. The analysis is limited to the response
of the system under the harmonic excitation of amplitude a0 =0.005m and angular frequency
ω=8.396rad/s (first vibrationmode of the passive systemwithout control) and to the impulse
excitation described in detail later on.
Several dimensionless parameters are introduced: βx, βv, γ, τs and defined using equations:

βx =κxKx, βv =κvKv, γ=K
sw/ω0 and τs =Ts/T0, where:Kx = a0,Kv =ω0a0, T0 =2π/ω0,

and ω0 =
√

kb/mb correspond to the first vibration mode of the system. The function sgn(S)
present in Eqs. (3.5) and (4.11) is substituted by continuous 2/πarctan(ηS) with η=100 used
for the calculations.
It is assumed that the disturbance is in the form of a Gaussian white noise of zero average

value and variance of value 25N2 (standard deviation s0 = 5N). An exemplary realization of
this disturbance for the sampling time Ts =0.05T0 is shown in Fig. 3.

Fig. 3. An example of the disturbance signal

Figure 4 shows the displacement of the spring-supported mass for the continuous SMC,
discrete DSMC,Eq. (4.8), andmodified discreteDMSMC regulator, Eq. (4.10). For the discrete
control, the sampling time was Ts = 0.02T0 (τs = 0.02). By comparing the results, one can
notice that in the case of discrete control, the response looks periodic in contrast to the aperiodic
continuous control SMC.Ontheassumedcontrol parameters, the efficiencyofSMCandDMSMC
regulator is comparable.

Fig. 4. Displacement plot: SMC (β
x
=β

v
=1, γ=0.1), DSMC (β

x
=β

v
=1, γ=0.001),

DMSMC (β
x
=β

v
=1, γ=0.001)

The irregularity seen in the response to the SMC regulator is caused by the high sensitivity
of the system to the external disturbance. This fact is supported byFig. 5 inwhich the response
to three disturbance signals of different standard deviations s = 5, 10 and 15N is shown (for
s=ns0, n=1,2,3, s0 =5N). The change in the response to these disturbances is nearly linear
(with linear rise of the disturbance). Although the amplitude of vibration can be attenuated by
increasing the value of parameters βx and γ, setting them too high is undesirable due to the
effect of “chattering”, and also due to the time delay present between the regulator and the
actuator.
In the case of the modified DMSMC digital controller, the response to the disturbance

is definitely lower. When using the same values of n that define the disturbance as before,



8 J. Łuczko, U. Ferdek

Fig. 5. Influence of the disturbance on the displacements – SMC (β
x
=β

v
=1, γ=0.1)

the difference between the responses is practically undetectable. Therefore, Fig. 6 shows the
displacement for n= 1 and n=20 (with τs =0.02). Even with twenty times higher value of s
(s = 5N and 100N), the differences are still minor, which means that the DMSMC system is
resistant to the disturbance.

Fig. 6. Influence of the disturbance on the displacements – DMSMC (β
x
=β

v
=1, γ=0.001)

The efficiency of the discrete regulator is decreased when the sampling time increases
(Fig. 7).Thepresenteddisplacements are obtained for theDMSMCwithdifferent sampling times
τs = 0.01, 0.02, 0.03 and βx = βv = 1, γ = 0.001 and s = 5N. The relation of the amplitude
of vibration with the sampling time is nonlinear, e.g. a change in τs from 0.01 to 0.02 leads
to the amplitude rise by ten times. When rise of τs is continued, the amplitude of vibration is
changed by a smaller factor. On the other hand, too small sampling time is undesirable due to
computation time taken for the realization of digital control.

Fig. 7. Relation between the sampling time and displacements – DMSMC (β
x
=β

v
=1, γ=0.001)

Figure 8 shows the displacements (Fig. 8a – for βx =0.1, 1, 2) and velocities (Fig. 8b – for
βx =1only) of the spring-supportedmass in theDMSMCsystem (βv =1, γ=0.001, τs =0.02).

With the riseof valueβx, themaximaldisplacements aredecreasedbut themaximal velocities
and accelerations are increased. In addition to that, the character of vibration is changed, with
more influence of high-frequency component present in the system.For high enough values ofβx,
the chattering effect starts to occur.

In order to test the behavior of the vibroisolation system,when crossing an obstacle, analysis
of the response to the impulse excitation has been performed.

To describe the excitation in simulations, the bump“roundedpulse” function has been used.
It is defined as follows (Shekhar et al., 1999)

w(t) =
1

4
h[eη(t− t0)]

2exp[−η(t− t0)]H(t− t0) (5.1)



Continuous and discrete sliding mode control... 9

Fig. 8. Effect of the parameter β
x
– DMSMC system (β

v
=1, γ=0.001, τ

s
=0.02): (a) displacements,

(b) velocities (β
x
=1)

whereH(t−t0) is theHeaviside function.Equation (5.1) has continuousfirstand secondderivates
and themaximal value equal toh (Fig. 9). Theparameter η defines the sharpness of the impulse.
At η =ωn1 =8.396rad/s, the impulse duration is equal to the half of the first vibration mode
period. In numerical calculations, the value h has been chosen to be equal to 0.05m.

Fig. 9. Impulse excitation (γ=8.396rad/s, h=0.05m)

Figure 10 shows the response of the SMC (βx =βv =1, γ =0.1) andDMSMC(βx =βv =1,
γ=0.001) system to impulse excitation (5.1). In both cases, the rate of displacement reduction
is similar. In the case of the SMC, the damping of the disturbance (Fig. 10a) is slower when
considered in relation to DMSMC, but it is also less steep, see the velocity plot in Fig. 10b.

Fig. 10. Response to the impulse excitation – regulators: SMC (β
x
=β

v
=1, γ=0.1) and DMSMC

(β
x
=β

v
=1, γ=0.001): (a) displacements, (b) velocities

In the considered example, if thevalueofγ is toohigh (forγ > 0.01,βx =1), the“chattering”
effect occurs (Fig. 11). Apart from the additional high-frequency oscillations, the changes in the
value are twice as high as for the systemwith optimally chosen parameters of control (Fig. 10a).



10 J. Łuczko, U. Ferdek

In addition to that, an increase in the value of βx, causes the possibility of occurring of the
chatter to be higher (at βx > 10, the chattering can be observed for γ=0.001).

Fig. 11. Displacement plot with chattering (β
x
=β

v
=1, γ=0.05, 0.1, τ

s
=0.02)

6. Summary

The analysis of the results allows formulating of the following conclusions:

• In all the considered active systems, the reduction of vibration is well enough, at least
within the range that includes the first vibration mode. As this mode is significant, when
considering vibration of the car body, it should be noted thatmaking use of the presented
regulators greatly increases the driving comfort.

• Thecontinuous SMCregulator, for veryhighvalues of parametersβx andγ, is theoretically
more effective than discrete regulators. However, it is more susceptible to disturbances as
well as the time delay present during control.

• From the two presented discrete regulators, the better one is definitely DMSMC. It is
much more resistant to disturbances, and for small sampling times, its performance is
comparable to SMC.

• When adjusting the parameters of sliding mode regulators, the possibility of the “chatte-
ring” effect to occur should be taken into consideration.Values of the regulator parameters
should allow the optimal and regular operation of the regulator without undesirable high-
frequency oscillations.

References

1. Chen P.-C., Huang A.-C., 2005, Adaptive sliding control of non-autonomous active suspension
systems with time-varying loadings, Journal of Sound and Vibration, 282, 1119-1135

2. Huang S.-J., Chen H.-Y., 2006,Adaptive sliding controller with self-tuning fuzzy compensation
for vehicle suspension control,Mechatronics, 16, 607-622

3. Lee H., Utkin V.I., 2007, Chattering suppression methods in sliding mode control systems,
Annual Reviews in Control, 31, 179-188

4. Lin J., Lian R.-J., Huang C.-N., Sie W.-T., 2009, Enhanced fuzzy slidingmode controller for
active suspension systems,Mechatronics, 19, 1178-1190

5. Łuczko J., Ferdek U., 2012, Comparison of different control strategies in a semi-active vehicle
suspension system (in Polish),Czasopismo Techniczne, 11, 6-M, 81-92

6. Rajeswari K., Lakshmi P., 2008, GA tuned distance based fuzzy sliding mode controller for
vehicle suspension systems, International Journal of Engineering and Technology, 5, 1, 36-47

7. Sam Y.M., Osman J.H.S., 2005, Modeling and control of the active suspension system using
proportional integral sliding mode approach,Asian Journal of Control, 7, 2, 91-98

8. Sam Y.M., Osman J.H.S., Ghani M.R.A., 2004, A class of proportional-integral sliding mode
control with application to active suspension system, Systems and Control Letters, 51, 217-223



Continuous and discrete sliding mode control... 11

9. Sam Y.M., Suaib N.M., Osman J.H.S., 2008, Hydraulically actuated active suspension system
with proportional integral slidingmode control,WSEAS Transactions on Systems and Control, 9,
3, 859-868

10. ShekharN.C., HatwalH.,MallikA.K., 1999,Performance of non-linear isolators and absor-
bers to shock excitations, Journal of Sound and Vibration, 227, 2, 293-307

11. Snamina J., Kowal J., Wzorek T., 2011, Analysis of energy dissipation in vehicle suspensions
for selected control algorithms (in Polish),Czasopismo Techniczne, 2, 1-M, 233-240

12. Yan M., Fan L., 2012, Discrete sliding mode control for DC-DC converters with uncertainties,
Przegląd Elektrotechniczny, 88, 5b, 60-63

13. Yoshimura T., Kume A., Kurimoto M., Hino J., 2001, Construction of an active suspension
system of a quarter car model using the concept of sliding mode control, Journal of Sound and
Vibration, 239, 2, 187-199

14. Yu W.-C., Wang G.-J., Chang C.-C., 2004, Discrete sliding mode control with forgetting
dynamic sliding surface,Mechatronics, 14, 737-755

Manuscript received December 12, 2014; accepted for print March 31, 2015