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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 2281–2296

0022-46http://d

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journal homepage: www.elsevier.com/locate/jsvi

Disturbance observer based sliding mode control of activesuspension systems

Vaijayanti S. Deshpande a, B. Mohan b, P.D. Shendge c, S.B. Phadke c,n

a NBN Sinhgad School of Engineering, Pune, Indiab Tata Consultancy Services, Switzerlandc College of Engineering, Pune, India

a r t i c l e i n f o

Article history:Received 10 July 2013Received in revised form23 December 2013Accepted 25 January 2014

Handling Editor: L.G. Tham

to measure the states of the unsprung mass. The ultimate boundedness of the overall

Available online 18 February 2014

0X/$ - see front matter & 2014 Elsevier Ltd.x.doi.org/10.1016/j.jsv.2014.01.023

esponding author.ail addresses: vaijudesh@gmail.com (V.S. Desru@coep.ac.in (S.B. Phadke).

a b s t r a c t

In this paper, a novel scheme to reduce the acceleration of the sprung mass, used incombination with sliding mode control, is proposed. The proposed scheme estimates theeffects of the uncertain, nonlinear spring and damper, load variation and the unknownroad disturbance. The controller needs the states of sprung mass only, obviating the need

suspension system is proved. The efficacy of the method is verified through simulationsfor three different types of road profiles and load variation and the scheme is validated onan experimental setup. The results are compared with passive suspension system.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

There is a great interest in active suspension systems because of the improvement in ride comfort made possible by suchsystems. The suspension system of a vehicle is an uncertain nonlinear system subjected to an unknown road disturbance. Thedesign of an active suspension system is a challenge because the ride comfort is to be improved in the presence of uncertainties andunknown road disturbance while maintaining good road holding and operating within the rattle space limitations [1,2].

During the last three decades, many diverse control strategies, such as optimal control [3], adaptive control [4–7], modelreference adaptive control [8–10], H1 [11], LQG control [12] among many others, have been employed in active suspensionsystems. Adaptive backstepping control for active suspension systems with hard constraints is proposed in [13] forstabilizing the attitude of the vehicle while improving the ride comfort. In [14] a backstepping control is proposed tocompensate for actuator lag while in [15] an adaptive robust control is proposed to cope up with actuator saturation.

In general, it is difficult to find an accurate model of the suspension system. For this reason, many researchers have proposedsolutions based on model free strategies like the fuzzy control [16], fuzzy control with genetic algorithms and neural network[17,18]. Another well known strategy for control of uncertain systems affected by unmeasurable disturbances is the sliding modecontrol (SMC) strategy. The sliding mode control is not model free but can guarantee invariance for matched uncertainties and dis-turbances having a known bound. The application of SMC and its combinations with other strategies are reported in [9,10,19–21].

The performance of an active suspension system can be improved if accurate information about road profile is obtained in realtime. In the literature, efforts to obtain this information directly and indirectly are reported. Some researchers have tried to obtainthis information directly using special sensors or by using lead vehicles as sensors. A system with preview control employing avehicle mounted preview sensor is proposed in [22,23] while the use of the lead vehicle in a convoy as preview sensor is proposedin [24].

All rights reserved.

hpande), mohanbhaskara@yahoo.co.in (B. Mohan), pds.instru@coep.ac.in (P.D. Shendge),

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962282

Some researchers have obtained the information about the uncertainties and the road disturbance indirectly, obviatingthe need for employing special sensors. In [25,26], a function approximation technique is used to approximate the unknownfunction containing uncertainty and the road disturbance with an adaptive sliding mode control. In [27] an inertial delaycontrol that estimates the road disturbance is proposed for a sky-hook active suspension system. A drawback of this schemeis that it requires the states of the unsprung mass.

The main motivation of this paper is to improve the ride comfort which largely depends on the acceleration of the sprung mass.The acceleration of the sprung mass results from an unknown road profile and therefore a control scheme that can estimate theeffects of the road profile is required. Further, the need to position sensors to measure the states of the unsprung mass is to beobviated and the drawback of conventional SMC that needs the knowledge of the bounds of uncertainty is to be overcome. In thispaper, an active suspension system that uses the sliding mode control along with a Disturbance Observer (DO) is proposed. In theconventional sliding mode control, the uncertainties and disturbances can be compensated if their bounds are known. The roaddisturbance can vary considerably even in a small journey, making it difficult to obtain the bounds of the uncertainty that are nottoo conservative. In order to remove this difficulty, a scheme that estimates the effect of uncertainty and the road disturbance isproposed in this paper. The scheme for estimation is a modified version of the disturbance observer proposed in [28]. A preliminaryversion of this paper appeared in [29]. Salient features of this paper are as follows:

�

The proposed scheme estimates the effect of the road disturbance, nonlinearities and uncertainties of the suspensionsystem and then negates them using control.

�

The scheme does not require the states of the unsprung mass thereby reducing the sensor requirement. � Unlike the conventional sliding mode control, no knowledge of bounds on uncertainties and disturbances is required. � A single controller works satisfactorily under various loading conditions and road profiles. � Stability of the overall system is proved.

The paper is organized as follows: a quarter car model and the dynamic equations are explained in Section 2 whileSection 3 explains the sliding mode control. The disturbance observer is explained in Section 4. In Section 5 the stability ofthe overall system is proved and the ultimate bounds on estimation error and the sliding variable are calculated. The efficacyof the proposed controller is illustrated by simulation results in Section 6 and the scheme is validated on an experimentalsetup in Section 7. The conclusion is given in Section 8.

2. Problem statement

Consider the quarter car model of suspension system shown in Fig. 1. The sprung mass ms(t) is the mass of the car body,passengers, frame, internal components and it may vary according to the passenger loading condition of the car. It issupported by the suspension system consisting of a spring ks and a damper cs. The spring is modelled by a linear stiffnesscoefficient k1s and nonlinear stiffness coefficient k2s and the damper is modelled by linear damping coefficient c1 andnonlinear damping coefficient c2. The mass of the wheel, tyre, brake and suspension linkage mass is referred to as unsprungmass mu which is supported by the tyre modelled as a combination of linear spring and linear damper with coefficients ktand ct respectively. In an active suspension system, in addition to these passive components, an actuator is connectedbetween the sprung mass and the unsprung mass. The actuator generates a control force u so as to improve the ride comfort.

It is assumed that no a priori information is available about the unknown road profile. The vertical road disturbanceacting on the unsprung mass is denoted by z. The vertical displacements of the sprung mass and unsprung mass withrespect to their static positions are denoted by xs and xu respectively. Next, the dynamic equations of the suspension systemare expressed in the state variable form.

Let the state x¼ ½x1 x2 x3 x4�T be defined as

x1 ¼ xs; x2 ¼ _xs; x3 ¼ xu; x4 ¼ _xu (1)

Fig. 1. Quarter car suspension model.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2283

The motion equations of the quarter car model shown in Fig. 1 can be described in state variable form as

_x1 ¼ x2 (2)

_x2 ¼1

msðtÞ� f s� f dþu

� �(3)

_x3 ¼ x4 (4)

_x4 ¼1mu

f sþ f d� f t�u� �

(5)

where

f s ¼ k1sðx1�x3Þþk2sðx1�x3Þ3 (6)

f d ¼ c1ðx2�x4Þþc2ðx2�x4Þ2 (7)

f ts ¼ ktðx3�zÞ (8)

f td ¼ ctðx4� _zÞ (9)

f t ¼ f tsþ f td (10)

fs and fd are spring and damper forces respectively. The tyre force ft is the addition of tyre spring force fts and tyre dampingforce ftd. The tyre force is calculated as

f t ¼f tsþ f td if x3�zð Þo ðmsðtÞþmuÞg

kt

0 if x3�zð ÞZ ðmsðtÞþmuÞgkt

8>><>>:

(11)

where g is acceleration due to gravity. In this paper, the parameters of the passive suspension are assumed to be unknownand nonlinear. Further, the sprung mass is assumed to be time varying. The tyre spring constant and tyre damper constantsare also assumed to be unknown. Further, we assume that even the bounds of these uncertainties are unknown.

The objective is to control the acceleration of the sprung mass in order to get good ride comfort using the measurementsof x1 and x2 only. The motivation for using only these measurements is to simplify the implementation by avoiding thedeployment of sensors on the wheel and tyre assembly. To fulfill the objective, a sliding mode control strategy is used. SMCstrategy is explained in the next section.

3. Sliding mode control

In SMC, a suitable sliding surface of required dynamics is selected and control is designed so that sliding condition isalways satisfied. This makes the system insensitive to uncertainties and behave as per the definition of sliding surface. Inthis section, sliding surface and the control design are discussed.

3.1. Sliding surface

Select the sliding surface s

s¼ Sx1þx2 (12)

where S is a user chosen constant. Differentiating (12) and using (2) and (3),

_s ¼ Sx2þ1

msðtÞ� f s� f dþu

� �(13)

_s ¼ Sx2þ f ðx; tÞþGmðΔGðtÞ�1ÞuþGmu (14)

where

f x; tð Þ ¼ � 1msðtÞ

f sþ f d� �

(15)

and

1msðtÞ

¼ G tð Þ ¼ GmΔG tð Þ (16)

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962284

where Gm is a nominal constant value and ΔGðtÞ is multiplicative uncertainty in G(t). The bounds of ΔGðtÞ are assumed to beunknown. Rewriting (14) as

_s ¼ Sx2þeðx; tÞþGmu (17)

where eðx; tÞ is the lumped uncertainty, given by the equation

eðx; tÞ ¼ f ðx; tÞþGmðΔGðtÞ�1Þu (18)

In the sequel, at times eðx; tÞwill be denoted simply by e for simplicity. Next section explains design of control to compensatefor uncertainty and to achieve sliding condition.

3.2. Design of control

Control u to be designed is split into three parts viz. ueq, un and us. The component ueq is used to compensate the knownterms and un is used to compensate the lumped uncertainty e. A discontinuous component us with its smoothapproximation is used for further improvement in performance. The key idea of the proposed scheme is to estimate theuncertainty by the Disturbance Observer and then to use opposite of it in un to negate the effect of uncertainty. In thissection it is assumed that the estimate of e, denoted by e, is available while the method to estimate e using DO is explainedin Section 4

u¼ ueqþunþus (19)

with

ueq ¼ � 1Gm

Sx2þksð Þ (20)

un ¼ � 1Gm

e (21)

us ¼ � 1Gm

kst sat sð Þ (22)

sat sð Þ ¼sgn s if jsj4εsε

if jsjrε

8<: (23)

where k, kst and ε are positive constants, chosen by designer. Using (19), (20), in (17)

_s ¼ �ksþeþGmunþGmus (24)

Using (21) and (22) in (24)

_s ¼ �ks�kst satðsÞþ ~e (25)

where

~e ¼ e� e (26)

~e is estimation error. If the estimate e is such that ~e goes close to zero, sliding variable swill go close to 0, thereby improvingthe ride comfort. Next, a Disturbance Observer is designed to estimate e such that ~e goes close to zero. It may be noted thatthe control obtained in this section does not use the states of the unsprung mass.

4. Disturbance observer

The Disturbance Observer used here is a modified version of that proposed in [28]. Let the estimate of the lumpeduncertainty e be given by

e ¼ dðtÞþpðsÞ (27)

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where pðsÞ is some linear or nonlinear scalar function of s. Now, dðtÞ is to be updated in such a way that the estimation error~e goes to zero. Differentiating (27) we get

_e ¼ _d tð Þþ ∂p

∂s_s (28)

Substituting _s from (24)

_e ¼ _d tð Þþ ∂p

∂s�ksþeþGmunþGmusð Þ (29)

This suggests an update law for dðtÞ as_d tð Þ ¼ � ∂p

∂s�ksþ eþGmunþGmus

� �(30)

giving

_e ¼ ∂p∂s

~e (31)

Subtracting both sides of (31) from _e, we get

_~e ¼ � ∂p∂s

~eþ _e (32)

This suggests that for stability of ~e the choice of pðsÞ be such that ∂p=∂s be a positive function. For estimation error to bebounded, the following assumption is necessary.

Assumption 1. The lumped uncertainty e is such that

dedt

��������oμ (33)

where μ is a positive number.

The boundedness of ~e and s is discussed in Section 5.

5. Stability

In this section the condition for stability and ultimate bounds on ~e and s is calculated.Consider a candidate Lyapunov function

V s; ~eð Þ ¼ 12 s

2þ12~e2 (34)

Taking the derivative of Vðs; ~eÞ along (25) and (32),

_V s; ~eð Þ ¼ �ks2�kst sat sð Þsþ ~es� ∂p∂s

~e2þ ~e _e (35)

Using Young's inequality [30], ~esr 12

~e2þs2� �

and Assumption 1 and simplifying

_V s; ~eð Þr� k� 12

� �s2� ∂p

∂s� 1

2

� �~e2þ ~e μ�kst sat sð Þs

���� (36)

The control parameters can always be chosen so that k� 12 40 and ∂p=∂s� 1

2 40. From (36) it can be proved that thedynamics of s and the estimation error ~e are not asymptotically stable but are ultimately bounded in the sense of Corlessand Leitmann [31]. Omitting the details of derivation, we give the bounds on j ~ej and jsj. Calling the bound on j ~ej as λ1

~e�� ��rλ1 ¼

μ∂p∂s �1

2

� � (37)

As per the two conditions on jsj (23), two bounds for jsj are obtained from (25) and (37)

sj jrλ1�kst

kfor jsj4ε

λ1ε

kεþkstfor jsjrε

8>><>>:

(38)

It is easy to see that the bounds on jsj are lowered on account of kst.Thus, it can be seen that s and ~e are ultimately bounded and their bounds can be made arbitrarily small by choosing the

control function ∂p=∂s and the control parameters k and kst.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962286

6. Illustrative example

The efficacy of the proposed control is verified through simulations of the suspension model shown in Fig. 1 for three differenttypes of road profiles shown in Fig. 2. The performance of the passive system, the active system without us and the active systemwith us was assessed for each road profile and load variation. In the simulation results, active system is referred to as DOSMCsystem. In the figures, the plot of relative suspension deflection is the plot of the ratio of suspension deflection ðx1�x3Þ to the rattlespace. The plot of relative tyre force is the plot of the ratio of the dynamic tyre force to static tyre force. The results obtained for theroad profile 1, shown in Fig. 2a, are shown graphically as well as in tabular form. In order to save space, the results of the other casesare shown only in tabular form. The parameters of suspension system are shown in Table 1.

The initial conditions of plant are taken as x0 ¼ ½0 0 0 0�T . The control parameters are chosen as: k ¼10, W¼100, kst ¼ 2,ε¼ 0:01 and S ¼4. The control parameters and the initial conditions are kept unchanged for all the cases considered.

6.1. Case 1: road profile 1

The road profile 1 is as shown in Fig. 2a which is same as considered in [25] and is given by

zrðtÞ ¼

�0:0592 t31þ0:1332 t21þdðtÞ; 3:5rto50:0592 t32þ0:1332 t22þdðtÞ; 5rto6:50:0592 t33�0:1332 t23þdðtÞ; 8:5rto10�0:0592 t34�0:1332 t24þdðtÞ; 10rto11:5dðtÞ else

8>>>>>><>>>>>>:

(39)

where dðtÞ ¼ 0:002 sin 2πtþ0:002 sin 7:5πt is the sinusoidal disturbance and the time intervals are defined as t1 ¼ t�3:5;t2 ¼ t�6:5; t3 ¼ t�8:5; and t4 ¼ t�11:5. Notice that the sinusoidal disturbance is superimposed on the disturbance thatdepends on time intervals.

The responses for this road profile with passive system are shown in Fig. 3. The responses with DOSMC with us¼0 areshown in Fig. 4. The responses with DOSMC system with us are shown in Fig. 5. The results of the three suspension systemsare summarized in Table 2.

From Figs. 4h and 5h where the uncertainty e is superimposed on its estimate e, it can be seen that the DO is successful inestimating the uncertainty with just a small error.

It can be seen from Fig. 4c and Table 2 that the system with DOSMC brings about a significant improvement in the ridecomfort. Compared to the passive system, the DOSMC system reduces the rms acceleration of the sprung mass by a factor ofroughly 5. When a small us is added to the control, the performance markedly improves further as seen in Fig. 5c. Comparedto DOSMC without us, the DOSMC system with us reduces the rms sprung mass acceleration further by a factor of 5.

Fig. 2. Road profiles: (a) road profile 1, (b) road profile 2, (c) road profile 3.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2287

From Figs. 4d and 5d, it can be noted that this improvement in the ride comfort is obtained with only a marginal change inthe control effort.

The plots of relative suspension deflection shown in Fig. 4e and relative dynamic tyre force shown in Fig. 4f for DOSMCwithout us and the plots of relative suspension deflection shown in Fig. 5e and relative dynamic tyre force shown in Fig. 5ffor DOSMC with us are well within limits.

Fig. 3. Responses with passive suspension system for road profile 1: (a) sprung mass deflection and road profile, (b) sprung mass deflection,(c) acceleration of sprung mass, (d) relative suspension deflection, (e) relative dynamic tyre force.

Table 1Parameters of the suspension system.

Parameter Value Units

msm 290 kgms(t) 290760 kgGm 0.00356 kg�1

ΔG 0.9 kg�1

mu 59 kgk1s 14 500 N m�1

k2s 160 000 N m�1

c1 1385.4 N m�1 sc2 524.28 N m�2 s2

ct 170 N m�1 skt 190 000 N m�1

Rattle space 0.12 m

Fig. 4. Responses with DOSMC without us for road profile 1: (a) sprung mass deflection and road profile, (b) sprung mass deflection, (c) acceleration ofsprung mass, (d) control force, (e) relative suspension deflection, (f) relative dynamic tyre force, (g) sliding variable, (h) uncertainty and its estimate.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962288

6.2. Case 2: road profile 2

The road profile shown in Fig. 2b is referred to as Case 2 road profile. This road profile is given by

z¼ 0:05 cos ð2πtÞ sin ð0:6πtÞ (40)

The passive system was compared with the two variants of DOSMC as in Case 1 without changing the control parameters.The results of the three types of suspension systems are summarized in Table 3. It is clear that the improvement over passivesystem is significant.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 5. Responses with DOSMC with us for road profile 1: (a) sprung mass deflection and road profile, (b) sprung mass deflection, (c) acceleration of sprungmass, (d) control force, (e) relative suspension deflection, (f) relative dynamic tyre force, (g) sliding variable, (h) uncertainty and its estimate.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2289

6.3. Case 3: road profile 3

The road profile shown in Fig. 2c is given by

z¼ 0:05 sin ð1:5πtÞ sin ð0:15πtÞþ0:05 cos ð0:6πtÞ sin ð0:3πtÞ (41)

The results of simulation for this case are summarized in Table 4. The results for this case also show marked improvementwith DOSMC. Like the other two cases, addition of us produces a significant improvement over DOSMC without us.

Table 2Performance for road profile 1: maximum and rms values.

Type of suspension Displacement in m Acceleration in m/s2

max rms max rms

Passive 0.1037 0.0399 0.6827 0.2629DOSMC without us 0.0014 5.92 �10�4 0.1322 0.0474DOSMC with us 7.72�10�5 3.06�10�5 0.0484 0.0099

Table 3Performance for road profile 2: maximum and rms values.

Type of suspension Displacement in m Acceleration in m/s2

max rms max rms

Passive 0.0782 0.0393 3.5302 1.9505DOSMC without us 0.0020 0.0010 0.1685 0.0573DOSMC with us 1.29�10�4 6.20�10�5 0.0642 0.0119

Table 4Performance for road profile 3: maximum and rms values.

Type of suspension Displacement in m Acceleration in m/s2

max rms max rms

Passive 0.1168 0.0479 2.1010 0.9457DOSMC without us 0.0024 0.0011 0.0965 0.0237DOSMC with us 1.27�10�4 5.83�10�5 0.0258 0.0057

Table 5Performance for sprung mass¼230 kg for road profile 1: maximum and rms values.

Type of suspension Displacement in m Acceleration in m/s2

max rms max rms

Passive 0.1034 0.0396 0.7877 0.3079DOSMC without us 0.0014 5.9 �10�4 0.1403 0.0468DOSMC with us 7.70�10�5 3.05�10�5 0.0470 0.0087

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962290

6.4. Case 4: variation in sprung mass

In this case, varying loading was considered for road profile 1. The varying load may come because of varying passengerload or due to cornering. The load variation was assumed to be in the range msðtÞ ¼msm760. The results for the case whenmsm ¼ 230 kg are tabulated in Table 5.

The results show that the passive system is relatively more sensitive to the change in the sprung mass compared to thetwo variants of DOSMC.

6.5. Discussion

The road profile 1, shown in Fig. 2a, is a general rough road surface with a bump around 5 s, followed by a trough around10 s. In the case of passive suspension, it can be seen from Fig. 3a and b that the sprung mass deflection x1 almost follows theroad profile z. The relative suspension deflection plot in Fig. 3d shows only a marginal activity which implies that theunsprung mass deflection x3 too follows the road profile. In passive system, the ride comfort directly depends on the roadsurface.

In contrast, in the case of active suspension system, the effect of road disturbance and uncertainties, denoted by e, isestimated using a disturbance observer and the estimate, denoted by e, is used in the control law to negate the effect of theroad disturbance. Thus the sprung mass system gets excited, not by the uncertainty e but by just the estimation error

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2291

~e ¼ e� e which is much smaller than the uncertainty e itself as can be seen from Fig. 6b. This is the main reason why theperformance of the active suspension system with DO is markedly improved compared to passive system.

From Fig. 4a and b, it can be clearly seen that sprung mass deflection x1 remains close to zero even when the vehicle isgoing over a bump or going through a trough. From Fig. 4d, it can be seen that the control force is shaped to oppose theeffect of bump or trough. In contrast to the passive case, the plot of relative suspension deflection in Fig. 4e showsconsiderable activity and is in a direction opposite to that of the road profile. Thus when the vehicle is going over a bump,the actuator in active suspension tries to pull the sprung mass down, preventing it from following the road profile and it isthe DO which calculates the amount by which to pull the sprung mass.

In fact, this is the strategy executed throughout this road profile and for all other road profiles. For a clear illustration,Fig. 6a shows z;u and e on a single plot. Since the magnitudes of z, u and e have a wide variation, they are appropriatelyscaled for clarity of illustration. It may be noted from the figure that the shapes of z and e are similar while that of u isexactly opposite. Fig. 6b shows the uncertainty e and ~e on the same plot. It can be seen that ~e is much smaller than e. Theanalysis for the other two profiles confirms the same findings.

The results with DOSMC with us are even better because the effects of small estimation error ~e are countered by the termus. Interestingly, the improvement is achieved without an appreciable change in the magnitude of total control u.

It may be noted from the tabulated results that for the three road profiles, the passive suspension system showsconsiderable variation in the maximum and RMS values of sprung mass acceleration while the variation is much smaller

Fig. 6. Responses with active suspension system for road profile 1 (a) scaled road profile (solid line), scaled control (dashed) and scaled e (dash dot) and(b) uncertainty e (solid line), estimation error ~e (dash dot).

Fig. 7. Laboratory setup.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962292

with the proposed control. Thus a single controller works satisfactorily for various loading conditions and road profiles andgives the same level of ride comfort. The results were obtained without using any knowledge of the bounds of uncertaintyand disturbances.

7. Experimental results

In this section, it is intended to validate the proposed control on an actual hardware setup in the laboratory with thesame type of road profiles used in Section 6. The experimental setup [32] consists of a bench-scale model to emulate a

Table 6Parameters of the experimental setup.

Parameter Value Units

ms 2.45 kgmu 1 kgks 900 N m�1

bs 7.5 N m�1 sbus 5 N m�1 skt 2500 N m�1

Rattle space 0.038 m

Fig. 8. Experimental results with passive suspension system for scaled road profile 1: (a) sprung mass deflection and road profile, (b) sprung massdeflection, (c) acceleration of sprung mass, (d) relative suspension deflection, (e) tyre deflection.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2293

quarter-car model controlled by an active suspension mechanism. The plant consists of three plates on top of each other. Thetop plate resembles the vehicle body and is suspended over the middle plate with two springs and two dampers. A DCmotor drive stands between the top and middle plates to emulate an active suspension mechanism. The top plate has anaccelerometer mounted to measure the acceleration of the vehicle body relative to the plant ground. The middle plate is incontact with the bottom plate, i.e. the road, through a spring and a damper and constitutes the tire in the quarter-car model.The bottom plate provides the road excitation in the system. It is connected to a fast response DC motor so that the user cangenerate various road profiles. The system is shown in Fig. 7. The positions of the sprung and unsprung masses are sensed

(h)

Fig. 9. Experimental results DOSMC without us for scaled road profile 1: (a) sprung mass deflection and road profile, (b) sprung mass deflection,(c) acceleration of sprung mass, (d) control force, (e) relative suspension deflection, (f) tyre deflection, (g) sliding variable, (h) estimate of uncertainty.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–22962294

by 10-bit optical encoders while their velocities are obtained by high pass filters. In the experimental work the states of theunsprung mass were not used.

The suspension system is modelled by the following equations:

_x1 ¼ x2 (42)

_x2 ¼1ms

� f s� f dþu� �

(43)

Fig. 10. Experimental results DOSMC with us for scaled road profile 1: (a) sprung mass deflection and road profile, (b) sprung mass deflection,(c) acceleration of sprung mass, (d) control force, (e) relative suspension deflection, (f) tyre deflection, (g) sliding variable, (h) estimate of uncertainty.

V.S. Deshpande et al. / Journal of Sound and Vibration 333 (2014) 2281–2296 2295

_x3 ¼ x4 (44)

_x4 ¼1mu

f sþ f d� f t�u� �

(45)

where

f s ¼ ksðx1�x3Þ; f d ¼ bsðx2�x4Þ (46)

f ts ¼ ktðx3�zÞ; f td ¼ busðx4� _zÞ; f t ¼ f tsþ f td (47)

fs and fd are spring and damper forces respectively. The tyre force ft is the addition of tyre spring force fts and tyre dampingforce ftd, where the nominal values of the parameters are given as given in Table 6. In the design of control no knowledge ofthese parameters was assumed. The initial conditions of plant are taken as x0 ¼ ½0 0 0 0�T . The control parameters arechosen as, k ¼5, W¼100, kst ¼ 0:05, ε¼ 0:0015 and S¼4. The experiment with the proposed control was performed withthe same types of the three road profiles, the magnitude of the road profile was scaled down because the rattle space of thesuspension setup was very small and the setup required that the control voltage be limited within 730 V.

The responses for scaled road profile 1 with passive system are shown in Fig. 8. The responses with DOSMC with us¼0are shown in Fig. 9.

The responses with DOSMC with us, with kst ¼ 0:05 are shown in Fig. 10.

7.1. Discussion

The observations of Section 6.5 qualitatively hold good for the experimental results. It may be noted that the results areobtained with sprung mass position measurement by an optical encoder and sprung mass velocity by high pass filteringwhich has given rise to effects of quantization and measurement noise.

8. Conclusion

A disturbance observer based sliding mode control scheme for active suspension is designed. The scheme is successful inestimating the effect of uncertainties and irregular road profiles with a good accuracy. The results of simulation for three roadprofiles and load variation showed that the proposed DOSMC scheme, with a single set of control parameters, improves theperformance in terms of sprung mass displacement and acceleration markedly while meeting the handling requirements. Thevariant of DOSMC with the inclusion of a smooth approximation of the discontinuous component us is capable of furtherimprovement. Implementation of the proposed control law on an experimental setup validates the scheme.

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