Adaptive Study

8/18/2019 Adaptive Study

1/12

K. El MajdoubUniversity of Caen,

GREYC UMR CNRS,

Caen 14032, France

D. GhaniUniversite Mohamed V,

Electric Engineering Department,Rabat, Agdal 10000, Morocco

F. Giri1

University of Caen,

GREYC UMR CNRS,

Caen 14032, France

e-mail: [email protected]

F. Z. ChaouiENSET,

University Mohamed V,

Electric Engineering Department,

Rabat, Suissi 10100, Morocco

Adaptive Semi-Active Suspensionof Quarter-Vehicle WithMagnetorheological Damper

This paper addresses the problem of controlling quarter-vehicle semi-active suspensionsystems. Presently, the suspension system involves a magnetorheological (MR) damper featuring hysteretic behavior captured through the Bouc–Wen model. The control objec-tive is to regulate well the chassis vertical position despite the road irregularities. Thedifficulty of the control problem lies in the nonlinearity of the system model, the uncer-tainty of some parameters and the inaccessibility to measurements of the hysteresis inter-nal state variable. The control design is performed using Lyapunov control design tools;it includes an observer providing online estimates of the hysteresis internal state and anadaptive state-feedback regulator. The adaptive controller, obtained by combining thestate observer and the state-feedback regulator, is formally shown to meet the desired control objectives. This theoretical result is confirmed by several simulations. The latter illustrate the performances of the present adaptive controller and compare them withthose of earlier control approaches and those of the passive suspension.[DOI: 10.1115/1.4028314]

Keywords: semi-active suspension, MR damper, Bouc–Wen model, adaptive backstep-

ping control, state observation

1 Introduction

Vehicle suspension system control aims at improving passen-ger’s comfort by reducing the vibration caused by the onboardengine and the road irregularities [1 – 4]. A few years ago, MRdampers have become an effective mean toward the achievementof this objective. These are versatile low-cost small-size devicesgenerating high damping forces. They contain MR fluids thatreversibly change their rheological properties in the presence of avarying magnetic field. Then, the fluid force can be acted on bychanging the magnetic field amplitude. This action only necessi-tates low energy that usual size batteries can provide. Theseappealing features explain why MR dampers have found, onrecent years, several applications in so various domains as vibra-tion isolation and damping, earthquake and civil engineering,automotive and vehicle industry [5 – 8].

The complexity of MR dampers dynamics lies in the nonlinear-ity and the hysteretic nature of the force–velocity relationship [ 9].Different, more or less complex, dynamic models have been pro-posed to capture this relationship [10 – 12]. Most of them are infact variants of the Bouc–Wen hysteresis model and the LuGrefriction model. In these models, the hysteretic behavior of MRdamper is captured by a first-order nonlinear differential equationwhose state variable is not accessible to measurements. Other

types of models have also been considered, e.g., Bingham modeland polynomial model.

For the MR dampers to be used as actuators in semi-active sys-tems for vibration isolation, they need to be appropriately con-trolled. To this end, several control approaches have beenproposed, ranging from simple On–Off control techniques tomuch more advanced linear and nonlinear control techniques

including LQG/ H 1

techniques [13], LPV techniques [14], back-stepping and quantitative feedback theory (QFT) [15].

In Ahmadian and Blanchard [16], a skyhook damper controlalgorithm has been proposed for vehicle suspension and theobtained performances have been shown to be better than thoseobtained with passive suspension systems. The weakness of theskyhook control in depressing the vibration of the unsprung massof a suspension has been coped with using Groundhook and

Hybrid control approaches [17]. The conclusions of the last workis that the skyhook control is much more effective in improvingthe ride comfort, the Groundhook control is effective in achievingbetter road holding ability and improving vehicle stability, and thehybrid control is a tradeoff between the skyhook control and theGroundhook control. More recently, control solutions have beendeveloped based on the H

1 control [18], neural network control

[19], and fuzzy logic control [20]. Semi-active suspension controlbased on linear parameter-varying model design has been consid-ered in Zin et al. [14]. Mixed skyhook–ADD control is a comfort-oriented strategy involving switching between control laws [21].In Giorgetti et al. [22], optimal and model predictive controllerswere combined leading to semi-active hybrid controllers switch-ing between different control laws. In Zapateiro et al. [15], thebackstepping design technique and QFT have been combined to

control MR damper-based vehicle suspension.One common shortcoming of most previous control approachesis that all system states, including the MR damper internal state,are supposed to be accessible to measurements. Furthermore,some of them involve switching between different control laws,making complex the formal analysis of the closed-loop perform-ances. Others use fully active suspension actuators which are notyet available on most vehicles because of their high cost and lowperformances. As a matter of fact, semi-active actuators are gener-ally preferred in industrial applications.

This paper addresses the problem of semi-active suspensioncontrol for quarter vehicles where the involved semi-active actua-tors are based on MR dampers. The aim is to improve the vehicleride comfort while maintaining the suspension working space.

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedSeptember 2, 2013; final manuscript received August 11, 2014; published onlineSeptember 24, 2014. Assoc. Editor: Shankar Coimbatore Subramanian.

Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2015, Vol. 137 / 021010-1CopyrightVC 2015 by ASME

wnloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 09/20/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use


2/12

The complexity of the considered (semi-active suspension) con-trol problem is threefold: (i) the involved MR damper dynamicsare nonlinear and hysteretic; (ii) the state variable associated toMR damper hysteresis is not accessible to measurements; and (iii)some parameters of the overall controlled system are uncertain.The control problem is dealt with using the backstepping designtechnique and Lyapunov analysis tools [23,24]. The controldesign is based on the complete system model that accounts notonly for the linear vehicle vertical motion but also for the nonlin-ear and hysteretic behavior of the semi-active MR damper. Pres-

ently, the Bouc–Wen model is resorted to capture that hystereticbehavior. The control synthesis includes three main parts: (i) astate observer that estimates the damper internal state variable;(ii) a state-feedback control law designed for vertical motion sta-bility and regulation; and (iii) a parameter adaptive law that esti-mates the uncertain parameters. The adaptive controller thusconstructed is formally shown to meets its objectives. This theo-retical result is confirmed by simulations.

The paper is organized as follows: Sec. 2 is devoted to the mod-eling of the vertical quarter and the semi-active vehicle suspen-sion system; the controller design and analysis are presented inSec. 3; and the controller performances are illustrated by numeri-cal simulation in Sec. 4.

2 Vertical Quarter Model and Semi–Active

Suspension Modeling

2.1 Quarter Car Model. Vehicle suspension systems aremainly composed of a spring and a damper that filters effort trans-mission between the vehicle body and the road. The task of thespring is to carry the body mass and isolate it by absorbing roaddisturbances making the drive more comfortable. In fact, thedamper contributes to both driving safety and comfort.

Considering the vehicle’s symmetry, the study of the verticalsuspension can be dealt with based on the so-called quarter car model shown in Fig. 1 where the tire is modeled by a singlespring. The remaining elements represent the Coulomb frictionand active or semi-active components. The system under study isa one degree-of-freedom (1DOF) to make the emphasis on theMR damper control problem. Drive safety is the consequence of a

harmonious suspension design in terms of wheel suspension,springing, steering, and braking. The drive comfort is closelyrelated to the vibration level: the lower of the vehicle vibrationlevel cause the better of the drive comfort quality.

The quarter vehicle model depicted by Fig. 1 is characterizedby the sprung mass ms and the unsprung mass mus, the vertical

positions of zs and zus. As its damping coefficient is negligible, thetire is simply modeled by a spring linked to the road contact pointzr . The passive suspension between ms and mus is modeled by adamper and a spring. Analytically, the nonlinear passive modelbased upon later (for simulation and performance evaluation), isdefined by

ms€zs ¼ k szdef Cs _zdef Fc Fmr (1a)mus _zus ¼ k usðzus zr Þ þ k szdef þ Cs _zdef þ Fmr (1b)

with zdef ¼ zs zus is the damper deflection assumed to be meas-ured and _zdef ¼ _zs _zus is the deflection velocity, it can be directlycomputed from zdef . The coefficient Cs is the linearized viscousdamping coefficient, k s is the spring stiffness, and Fc is the exter-nal force applied on the mass, k us is the tire stiffness. The MRdamper control force Fmr is electrically generated through anexternal control voltage, denoted v . For convenience, the follow-ing phase coordinates are introduced:

X ¼ x1 x2 x3 x4½ T ¼ zs _zs zus _zus½ T (2)

where x1 and x2 are the vertical position and velocity of the sprungmass, respectively. On the other hand, x3 and x4 denote the verti-cal position and velocity of the unsprung mass, respectively.Then, the system (1) rewrites as follows:

ms _ x2 ¼ k sð x1 x3Þ Csð x2 x4Þ Fc Fmr (3a)mus _ x4 ¼ k usð x3 zr Þ þ k sð x1 x3Þ þ Csð x2 x4Þ þ Fmr (3b)

The involved model parameters depend on the vehicle type andsize. For instance, the numerical values corresponding to a“Renault Megane Coupe” are shown in Table 1 [25].

2.2 MR Damper Modeling. The nonlinear behavior of MRdampers can be described by several hysteresis models includingsimple models (e.g., Bingham) and more complex ones (e.g.,Bouc–Wen model). As a matter of fact, model accuracy growswith model complexity. Of course, too complex models entaillaborious control design and yield more complex controllers. Pres-

ently, Bouc–Wen’s model is retained because it represents a goodaccuracy/complexity ratio. Figure 2 illustrates how the hysteresis

Fig. 1 Model of quarter vehicle with MR damper

Table 1 Parameter numerical values of the model

Symbol Value Description

ms 315 kg Sprung massmsmax 2 ms Maximum sprung massmus 37.5 kg Unsprung massk s 29,500N/m Suspension linearized stiffnessCs 1500N/m/s Suspension linearized dampingk us 210,000 N/m Tire stiffness[zdefmin, zdefmax] [8, 6] cm Suspension’s deflection limits

Fig. 2 Bouc–Wen model for MR damper

021010-2 / Vol. 137, FEBRUARY 2015 Transactions of the ASME



3/12

effect in MR dampers is captured using this model. Analytically,the Bouc–Wen model consists of the following couple of equations

Fmr ¼ ðCoa þ CobvÞ _zdef þ k ozdef þ ðaa þabvÞx (4)_x ¼ q _zdef j jx b _zdef xj j þ k _zdef (5)

where the different notations are defined as follows: x is the inter-nal state variable (m), v is the input voltage (V), k o is the linear spring stiffness (N/m), aa is the stiffness of x (N/m), ab is the

stiffness of x influenced by v (N/m V), Coa is the viscous dampingcoefficient (N s/m), Cob is the viscous damping coefficient influ-enced by v (N s/m V), and q;b; k is the positive parameters char-acterizing the shape and size of hysteresis loop.

The Bouc–Wen model internal state variable x is not accessibleto measurements but is formally proved to be bounded under some conditions on the model parameters. In effect, it is shown inmany places that if b þ q > 0 and q b 0, then the signal xremains bounded as follows [26]:

xj j < max kb þ q ; xð0Þj j

(6)

Using the notations (2-4-5), the above model also rewrites asfollows:

Fmr ¼ ðCoa þ CobvÞð x2 x4Þ þ k oð x1 x3Þ þ ðaa þabvÞx (7)

_x ¼ q x2 x4j jx bð x2 x4Þ xj j þ kð x2 x4Þ (8)

3 Adaptive Backstepping Controller Design

The control objective is to enforce the following tracking error to asymptotically vanish, whatever its initial value:

e1 ¼def x1 x1ref (9)

In Eq. (9), x1ref denotes the position reference value (supposed tobe constant, i.e., _ x1ref ¼ 0). To meet this objective, the controldesign will mainly be based on the vertical motion model (3a),considering there the force Fmr as the control input. A verticalposition regulator can thus be obtained. The point is that Fmr isthe output of the MR damper (described by Eqs. (7) and (8)) andso it turns out to be a virtual control input in Eq. (3a). Then, theonline control action denoted Fmr , to be generated by the verticalposition regulator, should only be viewed as the desired trajectoryof the real force Fmr . To make the latter coincide with its desiredvalue, one needs to apply an adequate control voltage v computedfrom the algebraic equation (7). But the latter involves the damper state variable x which is not accessible to measurements. That is,we will design in the present section the vertical position control-ler that includes a position regulator and an observer for thedamper internal state. Owing to the observer, Eqs. (7) and (8) sug-gest the following structure:

^ Fmr ¼ ðCoa þ CobvÞð x2 x4Þ þ k oð x1 x3Þ þ ðaa þ abvÞx̂(10a)

_̂x ¼ q x2 x4j jx̂ bð x2 x4Þ x̂j j þ kð x2 x4Þ þ K oðvÞ ~ Fmr (10b)

with ~ Fmr ¼ Fmr ^ Fmr is the observation error and K oðvÞ is theobserver gain.

Note that Eqs. (10a) and (10b) are a copy of Eqs. (7) and (8)

with Eq. (10b) augmented by the term in the error ~ Fmr . This termprovides the observer ((10a) and (10b)) with a feedback structure.The observer analysis will be performed later as a part of thewhole adaptive controller analysis. In particular, the way the

observer gain should be online tuned will be determined bearingin mind the achievement of the output-reference tracking objec-tive. To prepare that analysis, introduce the observation error

~x ¼ x x̂ (11)

Then, it is readily seen, subtracting side-to-side Eq. (10a) fromEq. (7) and Eq. (10b) from Eq. (8), that the observation error undergoes the following equation:

_~x ¼ q x2 x4j j þ K oðvÞðaa þ abvÞð Þ ~x bð x2 x4Þ xj j x̂j jð Þ

(12)

~ Fmr ¼ ðaa þabvÞ ~x (13)

The damper observer ((10a) and (10b)) constitutes the first ingre-dient of the controller under construction. The control design willnow be completed by determining the vertical position regulator that will generate the desired trajectory Fmr so that the trackingerror e1 and ~x vanishes asymptotically. From Eqs. (10b) and (3a),it is readily seen that the system vertical movement is alsodescribed by the following state-space representation:

_e1 ¼ x2 (14a)_ x2

¼ k 1

ð x1

x3

Þ k 2

ð x2

x4

Þ k 3

ð Fc

þ ^ Fmr

Þ þk 3 ~ Fmr (14b)

with k 1 ¼ ðk s=msÞ, k 2 ¼ ðCs=msÞ, and k 3 ¼ ð1=msÞ.Interestingly, the parameters ðk 1; k 2; k 3Þ are not required to be

exactly known. This is coherent with the fact that the physicalquantities ðms; k s; CsÞ are likely to change with the operation con-ditions. It is only required that

k 3min k 3 k 3max (15)

where the lower and upper bounds ðk 3min; k 3maxÞ are supposed tobe a priori known. This assumption is a realistic one since k 3 isrelated to the chassis mass which variation range is a priori knownfor each given car. From Table 1, one gets that k 3min ¼ 1=ms max.To compensate for parameter uncertainty, the controller will beprovided with a parameter adaptation capability. To this end, intro-

duce the following notations: k ¼ k 1 k 2 k 3½ T is the unknownparameter vector and its estimate is k̂ ¼ k̂ 1 k̂ 2 k̂ 3

T. The

parameter vector estimation error is ~k ¼ k̂k ¼ ~k 1 ~k 2 ~k 3 T

.

The adaptive control law, the parameter adaptation law and theobserver gain will now be designed on the basis of ((14a) and(14b)), using the backstepping technique [24]. This is performedin three steps:

Design Step 1. Stabilization function design for subsystem(14a).

Consider the following Lyapunov function candidate for thesubsystem (13):

V 1 ¼ 12

e21 (16a)

The corresponding time-derivative is

_V 1 ¼ e1 _e1 ¼ e1 x2 (16b)

In Eq. (14a), the variable x2 stands as a virtual control signal.Let a1ð x1Þ denotes the corresponding stabilization function. Equa-tion (16b) suggests that

a1ð x1Þ ¼ c1e1 (17a)

where c1 > 0. Indeed, x2 ¼ a1ð x1Þ yields _V 1 ¼ c1e21 implyingthe asymptotic stability of the subsystem (14a) around the

Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2015, Vol. 137 / 021010-3



4/12

equilibrium e1 ¼ 0. As x2 is just a virtual control signal, we can-not set x2 ¼ a1ð x1Þ. Nevertheless, we retain the previous expres-sion of a1ð x1Þ and introduce the error

e2 ¼ x2 a1ð x1Þ ¼ x2 þ c1e1 (17b)

with this notation, Eqs. (14a) and (16b) are rewritten as

_e1 ¼ c1e1 þ e2 (18a)_V 1

¼ c1e

21

þe1e2 (18b)

Design Step 2. Control law design for the system ((14a)and (14b)).

From Eq. (16b) it follows, using Eqs. (17b) and (14b), that theerror e2 undergoes the following equation:

_e2 ¼ _ x2 þ c1 _e1¼ k 1ð x1 x3Þ k 2ð x2 x4Þ k 3ð Fc þ ^ Fmr Þ

þ k 3 ~ Fmr c21e1 þ c1e2 (19)

Consider the Lyapunov function candidate

V ¼ V 1 þ 12

e22 þ1

2~x2 þ 1

2c~k

T~k (20a)

where the positive real number c is a design parameter, the choiceof which will be discussed later. The time-derivation of Eq. (20a)yields, using Eqs. (19), (18b), and (12)

_V ¼ _V 1 þ e2 _e2 þ ~x _~x þ 1c~k

T _~k

¼ c1e21 þ e2e1 þ e2ðk 1ð x1 x3Þ k 2ð x2 x4Þ k 3ð Fc þ ^ Fmr Þ þ k 3 ~ Fmr c21e1 þ c1e2Þ ~x2 q x2 x4j j þ K oðvÞ aa þabvð Þð Þ b~xð x2 x4Þ xj j x̂j jð Þ þ 1

c~k

T _~k

¼ c1e2

1 þ e2e1 ^k 1ð x1 x3Þ ^k 2ð x2 x4Þ k̂ 3ð Fc þ ^ Fmr Þ c21e1 þ c1e2 þ k 3e2 ~ Fmr ~x2 q x2 x4j j þ K oðvÞ aa þabvð Þð Þ b~xð x2 x4Þ xj j x̂j jð Þ þ 1

c~k

T _~k þ ~kTwe2 (20b)

where ~k i ¼ k̂ i k iði ¼ 1; …; 3Þ and

w ¼ x1 x3 x2 x4 Fc þ ^ Fmr T

(20c)

Equation (20b) suggests that ^ Fmr should be chosen so that

e1 k̂ 1ð x1 x3Þ k̂ 2ð x2 x4Þ k̂ 3ð Fc þ ^ Fmr Þ c21e1 þ c1e2¼ c2e2 (20d )

where c2 > 0 is a new design parameter. This suggests that thedamper force ^ Fmr should be set to F

mr with

^ Fmr ¼ ð1 c21Þe1 þ ðc1 þ c2Þe2 k̂ 1ð x1 x3Þ k̂ 2ð x2 x4Þ k̂ 3 Fc

k̂ 3(21)

if ^ Fmr was the actual control input, it would be sufficient to let^ Fmr ¼ Fmr . Since this is not the case, we substitute Fmr to ^ Fmr inEq. (7) and solve this with respect to the control voltage v. Doingso, one gets the following damper control voltage v

v ¼ Fmr ðCoað x2 x4Þ þ aa x̂þ k oð x1 x3ÞÞ

Cobð x2 x4Þ þ ab x̂ (22)

The control law thus designed is defined by Eqs. (21) and (22). Design Step 3. Observer gain and parameter adaptation

algorithm.Using Eq. (21), Eq. (20b) boils down to

_V ¼ c1e21 c2e22 þ k 3e2 ~ Fmr ~x2 q x2 x4j j þ K oðvÞ aa þabvð Þð Þ

b~xð x2 x4Þ xj j x̂j jð Þ þ1

c~k

T _~k þ ~k

T

we2

¼ c1e21 c2e22 þ k 3e2ðaa þabvÞ ~x ~x2 q x2 x4j j þ K oðvÞ aa þ abvð Þð Þ b~xð x2 x4Þ xj j x̂j jð Þ þ 1

c~k

T _~k þ ~kTwe2 (23)

where the last equality is obtained using Eq. (13). On the other hand, applying the inequality abj j ða2 þ b2Þ=2, one gets

k 3ðaa þabvÞ ~xe2j j 12ðaa þabvÞ2 ~x2 þ k

23

2 e22 (24)

Using Eq. (24), it follows from Eq. (23) that

_V 0, k̂ 2ð0Þ > 0 and




5/12

Pðwe2Þ ¼ c2e2ð x1 x3Þ c2e2ð x2 x4Þ p c2e2ð Fc þ ^ Fmr Þ T

(29b)

where

p c2e2ð Fc Fmr Þð Þ

¼c2e2ð Fc þ ^ Fmr Þ if k̂ 3 > k 3minc2e2ð Fc þ ^ Fmr Þ if k̂ 3 ¼ k 3min and c2e2ð Fc þ ^ Fmr Þ 0

0 if ^

k 3 ¼ k 3min and c2e2ð Fc þ ^ Fmr Þ > 0

8>>>:

(29c)

The adaptive controller thus designed is analyzed in the followingtheorem.

THEOREM 1. Consider the closed-loop control system composed of :

(i) The quarter-vehicle semi-active suspension, described bythe motion Eqs. (3a) and (3b) and the hysteresis equations(7) and (8).

(ii) The adaptive controller including the damper stateobserver defined by Eqs. (10a), (10b), and (28a), the adapt-ive control law defined by Eqs. (21) and (22), and the parameter adaptation algorithm (29a) – (29c).

Let the control design parameters are chosen such that

c1 > 0; c2 > 1

2k 23max; 0 < d 1 and ko > 0

Then, one has the following properties:

(1) The closed-loop control system is described in terms of theerrors ðe1; e2; ~x; ~kÞ by the equations

_e1 ¼ c1e1 þ e2 (30a)

_e2 ¼ e1 c2e2 k 3ðaa þ abvÞ ~xþ ~kw (30b)

_~x ¼ ðq x2 x4j j þ K oðvÞðaa þabvÞÞ ~x bð x2 x4Þ xj j x̂j jð Þ (30c)

_~k ¼ c Pðwe2Þ (30d )

The above equations are referred to closed-loop error system.

(2) The point ðe1; e2; ~x; ~kÞ ¼ ð0;0; 0;0Þ is a stable equilibriumof the closed-loop error system and the tracking error e1 isasymptotically vanishing, whatever its initial value.

Proof of Part 1. It is readily seen that Eq. (30a) is a simplecopy of Eq. (18a). Similarly, Eqs. (30c) and (30d ) are copies of Eqs. (12) and (29a). It remains to establish Eq. (30b). To this end,note that as k̂ 3

ð0

Þ > k 3min the projection (29c) ensures that

k̂ 3ðt Þ k 3min, for all t > 0. Consequently, the control law (21) issingularity-free and the right-side of Eq. (21) can be substituted to Fmr in Eq. (19). This yields Eq. (30a) due to Eqs. (13), (20c), and(21). Part 1 is established.

Proof of Part 2. It is seen from Eq. (29c) that

pðc2e2ð Fc þ ^ Fmr ÞÞ ¼ c2e2ð Fc þ ^ Fmr Þ in all cases except for thesituation where k̂ 3 ¼ k 3min and c2e2ð Fc þ ^ Fmr Þ > 0. In thiscase, one has ~k 3 ¼ k̂ 3 k 3 ¼ k 3min k 3 < 0 because of Eq. (15).Then, one has ~k 3 c2e2ð Fc þ ^ Fmr Þ < 0 and ~k 3 c2e2ð Fc þ ^ Fmr Þ¼ ~k 3 pðc2e2ð Fc þ ^ Fmr ÞÞ. From these remarks, it follows that~k

Tw ~k T pðwÞ (all time) which in turn implies that ð1=cÞ~kT _~k

þ~kTw ð1=cÞ~kT _~k þ ~kT pðwÞ. Then, Eq. (25) simplifies to

_V 0, it readily follows from Eq. (32) that _V is seminegative definite with respective to ðe1; e2; ~x; ~kÞ. Thisproves the stability of the equilibrium ðe1; e2; ~x; ~kÞ ¼ ð0; 0; 0; 0Þ.Furthermore, the fact that _V 0 implies that V is bounded andtime decreasing. Then, it follows from Eq. (20a) that all errors inthe vector ðe1; e2; ~x; ~kÞ are bounded time functions. Then, it fol-lows using Eq. (30a) that the derivative _e1 is bounded. As e1 isbounded, one gets that

_e21 ¼ 2e1 _e1 2 L1 (33)

where L1 denotes the subspace of bounded signals. On the other hand, one gets from Eq. (29b) that c1e

21 _V . Integrating both

sides, over any interval ½0; t , yields

c1

ð t 0

e21ðsÞd s V ð0Þ V ðt Þ V ð0Þ; 8t > 0 (34)

Since V ð0Þ is independent on t , it follows from Eq. (34) thate21 2 L1 where L1 denotes the space of absolutely integrable sig-nals. This together with Eq. (33) implies, using Barbalat’s lemma,that e21 ! 0 as t ! 1 (e.g., [23]). Part 2 is established

Fig. 3 Vehicle suspension adaptive

Table 2 The numerical values of the MR damper parameters

Parameters Bouc–Wen model Dahl model

k o 0 N/m 0 N/mCoa 2100 N s/m 2100 N s/mCob 3500 N s/m V 3500 N s/m Vaa 1400 N/m 1400 N/mab 69,500 N/m V 69,500 N/m V

k 4 300q 48,000b 48,000




6/12

Remarks 1. (1) The adaptive controller analyzed in Theorem 1is illustrated by Fig. 3.

(2) The damper control voltage (22) may involve singularity intransient periods, due to a nonsuitable initialization of theobserver. Therefore, the following singularity-free version can beused in practice

v ¼ Fmr ðCoað x2 x4Þ þ aa x̂þ k oð x1 x3ÞÞ

d ðCobð x2 x4Þ þ ab x̂Þ (35)

where d ð:Þ designates the following preload dead zonefunction:

Fig. 4 Bump road inputs whereT 151 s and T 252 s

Fig. 5 (a ) The vertical displacement motion of the sprung mass for the bump road input. Solid:semi-active suspension with MR damper; Dotted: passive suspension. (b ) The vertical accelera-tion of the sprung mass for the bump road input. Solid: semi-active suspension with MRdamper; Dotted: passive suspension.




7/12

d ðCobð x2 x4Þ þ ab x̂Þ¼ Cobð x2 x4Þ þ ab x̂ if Cobð x2 x4Þ þ ab x̂j j r

rsgnðCobð x2 x4Þ þ ab x̂Þ if Cobð x2 x4Þ þ ab x̂j j 0.

4 Simulation Results

The performances of the adaptive controller analyzed in Theo-rem 1, and depicted by Fig. 3, will now be illustrated by numericalsimulations. The parameters of the quarter car vehicle model,equipped with a MR damper represented by the Bouc–Wen modeldescribed by Eqs. (3a) and (3b) and Eqs. (7) and (8), are given the

numerical values of Tables 1 and 2. These values correspond to a“Renault Megane Coupe” car. The latter is supposed to beunloaded, i.e., Fc ¼ 0 N. Throughout this section, a comparisonwill be performed between the considered semi-active suspensioncontrol system and a passive suspension control. The latter is asimpler variant of the former as the damping force expression(10a) then boils down to

Fp ¼ Coað x2 x4Þ þ k oð x1 x3Þ (37)

where the index “ p” refers to “passive” and the parameters Coaand k o are given the values of Table 2.

The RMS (root mean square) value of the vehicle body acceler-ation is often used as a measure of passenger drive comfort [28].

That is, the vertical acceleration RMS is considered in this studyto evaluate the quality of ride comfort. Recall that the RMS valueof an n-dimensional vector x is defined as follows:

xRMS ¼ xk k ffiffiffin

p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

n

Xni¼1

x2i

s ; i ¼ 1; …; n (38)

where k k denotes the usual Euclidian norm. The accelerationpeak value is also of some interest. It refers to the maximal magni-tude of vehicle body or passenger acceleration. The peak value is

xk k1¼ maxi¼1;…;n

xij jð Þ (39)

where k k1 denotes the infinity norm.The vehicle speed produces an effect on the performancesof the quarter vehicle suspension model. Indeed, the speeddetermines the frequency of the disturbances caused by roadirregularities which in turn affect the suspension frequencybehavior.

Based on these observations, the speed vehicle effect is indi-rectly accounted for, in the simulation part, through the depth, thewidth and the steepness of the road profiles.

The control design parameters c1; c2, ko, and c are given thevalues c1 ¼ 1, c2 ¼ 0:01, 1 ¼ 1, ko ¼ 10, and c ¼ 105 whichproved to be suitable.

Three considered road profiles are referred to bump road input,road input with limited ramp, and sinusoidal road input.

Fig. 6 MR damper actuator force in the case of bump road input

Fig. 7 MR damper actuator control voltage in the case of bump road input




8/12

4.1 Bump Road Input. This profile is the most encounteredin practice; it is analytically described by

zr ðt Þ ¼1

2hb 1 cos 2p t T 1

T 2 T 1

; T 1 t T 2

0 otherwise

8<: (40)

where hb ¼ 4 cm represents the height of the bump road input.This profile is shown by Fig. 4 and the system responses, obtainedwith the proposed semi-active control and with the passive sus-pension, are plotted in Fig. 5(a) (chassis vertical displacement)and Fig. 5(b) (acceleration). The variation of the actuator force isshown in Fig. 6. It is seen from Figs. 5(a) and 5(b) that the ampli-tude for the vertical position and acceleration vanish much faster

Fig. 9 (a ) The vertical displacement motion of the sprung mass for the limited ramp input.Solid: semi-active suspension with MR damper; Dotted: passive suspension. (b ) The verticalacceleration of the sprung mass for the limited ramp input. Solid: semi-active suspension withMR damper; Dotted: passive suspension.

Fig. 8 Road inputs with limited ramp where T 151s and T 251:1 s




9/12

with the semi-active suspension control than with passive suspen-sion control. In particular, the rapid chassis vertical acceleration,achieved with semi-active suspension control, entails a better ridecomfort. To quantify the ride comfort improvement, the RMS val-ues for the vertical acceleration of the vehicle body are computedusing Eq. (38). With the semi-active suspension one gets a nullRMS, while this equals 0.463 with passive suspension. That is,improvement ratio is 100% when passing from passive to semi-active suspension.

The comparison between both suspension techniques can also

be checked by analyzing the peak values of the chassis verticalacceleration. With passive suspension, the observed accelerationpeak value is 2.599 m/s2. In the case of semi-active suspension,the recorded acceleration peak value is zero. Again, the accelera-tion decrease improvement is 100% in favor of the semi-activesuspension. The MR damper actuator force in the case of bumproad input is shown in Fig 6. The input control voltage signal of the MR damper is shown in Fig.7.

4.2 Limited Ramp. Presently, a road profile with limitedramp, depicted by Fig. 8, is considered. This profile simulates asudden change in the road surface elevation. Analytically, theroad shape is defined as follows, where hr ¼ 0:02 m is the finalroad surface elevation:

zr ðt Þ ¼

0 0 t < T 1hr

ðt T 1ÞðT 2 T 1Þ T 1 t < T 2

h t T 2

8>>><>>>:

(41)

The obtained control performances are illustrated by Figs. 9(a),9(b), and 10. These show the quarter-vehicle body vertical dis-placement (Fig. 9(a)), the vertical acceleration (Fig. 9(b)) and the

MR damper force (Fig. 10). It is seen that the vehicle settlessmoothly to the final height value of the road input and the verti-cal displacement magnitude and acceleration are not large. Let usevaluate the RMS value of the vertical motion acceleration of thesprung mass. With passive suspension, the RMS value is 0.2917,while it is only equal to 0.0011 with the semi-active suspension.That is, the ride comfort is significantly improved with the latter as the RMS decrease is approximately 99.62%. The peak value of the vertical motion acceleration equals 2270 m/s2, with the passivesuspension, and only equals 0.01 m/s2, with the semi-active sus-pension. That is, the decrease is quite significant. The MR damper actuator force in the case of limited ramp road is shown inFig. 10. The input control voltage signal of the MR damper in thecase of bump road input is shown in Fig. 11.

Fig. 10 MR damper actuator force in the case of limited ramp input

Fig. 11 MR damper actuator control voltage in the case of limited ramp input




10/12

4.3 Sinusoidal Road Profile. The considered sinusoidal roadprofile, depicted by Fig. 12, is analytically defined by

zr ðt Þ ¼ hs2

1 þ sin pt p2

(42)

with hs ¼ 2 cm. The resulting closed-loop system responses areplotted in Figs. 13(a) and 13(b), which, respectively, show thesprung mass vertical displacement and acceleration. Clearly,the sprung mass vertical displacement and acceleration are much

better with the semi-active suspension than with the passive sus-pension. Numerically, the RMS value of the vertical accelerationis 0.0745 with the passive suspension and 0.5 103 with thesemi-active suspension. The latter ensures a RMS decrease of 99.32%. It is also seen that the peak value of the vertical accelera-tion is 0.1589 m/s2 when the passive suspension is used and is

only 0.005 m/s2 when semi-active suspension is used. The peakvalue decrease is 96.85%. Finally, the MR damper actuator forcevariation is shown in Fig. 14 and the input control voltage signalis shown in Fig. 15.

Fig. 12 Road inputs with sinusoidal road input

Fig. 13 (a ) The vertical displacement motion of the sprung mass for the sinusoidal road input.Solid: semi-active suspension with MR damper; Dotted: passive suspension. (b ) The verticalacceleration of the sprung mass for the sinusoidal road input. Solid: semi-active suspensionwith MR damper; Dotted: passive suspension.




11/12

5 Conclusion

In this paper, a new semi-active quarter-vehicle suspensionbased on MR damper is developed. The main component of theproposed suspension system is an adaptive controller designed bythe backstepping technique on the basis of a model that accountsfor the hysteresis effect in the MR damper. The controller consistsof an observer estimating the damper hysteresis internal state, aparameter adaptive algorithm estimating the system uncertainparameters and an adaptive state-feedback control law stabilizingthe suspension system. The adaptive controller is formally shownto meet its control objectives. This theoretical result is confirmedby several simulations that also illustrate the high supremacy of the semi-active suspension over the passive suspension. Com-pared to previous works on semi-active suspension the present

solution presents several features: (i) it does not assume all systemstates and all system parameters to be known; (ii) it does notinvolve switching between different control laws; and (iii) itenjoys a formal closed-loop stability analysis. Finally, it ischecked with the simulation study of Subsection 4.2 that the back-stepping adaptive controller, compared to others, is the only onethat substantially improves ride comfort.

References[1] Yagiz, N., and Hacioglu, Y., 2008, “Backstepping Control of a Vehicle With

Active Suspensions,” Control Eng. Pract., 16(12), pp. 1457–1467.[2] Du, H., and Zhang, N., 2007, “H

1 Control of Active Vehicle Suspensions With

Actuator Time Delay,” J. Sound Vib., 301(1–2), pp. 236–252.

[3] Bohn, C., Cortabarria, A., H€artel, V., and Kowalczyk, K., 2004, “ActiveControl of Engine Induced Vibrations in Automotive Vehicles Using Dis-turbance Observer Gain Scheduling,” Control Eng. Pract., 12(8), pp.

1029–1039.[4] Poussot-Vassal, C., Sename, O., Dugard, L., Gaspar, P., Szabo, Z., and Bokor,

J., 2008, “A New Semi-Active Suspension Control Strategy Through LPVTechnique,” Control Eng. Pract., 16(12), pp. 1519–1534.

[5] Kim, Y., Langari, R., and Hurlebaus, S., 2009, “Semi-Active Nonlinear Controlof a Building With a Magnetorheological Damper System,” Mech. Syst. SignalProcess., 23(2), pp. 300–315.

[6] Liu, Y., Matsuhisa, H., and Utsuno, H., 2008, “Semi-Active Vibration IsolationSystem With Variable Stiffness and Damping Control,” J. Sound Vib.,313(1–2), pp. 16–28.

[7] Spelta, C., Previdi, F., Savaresi, S. M., Fraternale, G., and Gaudiano, N., 2009,“Control of Magnetorheological Dampers for Vibration Reduction in a WashingMachine,” Mechatronics, 19(3), pp. 410–421.

[8] Yao, G. Z., Yap, F. F., Chen, G., Li, W. H., and Yeo, S. H., 2002, “MR Damper and its Application for Semi-Active Control of Vehicle Suspension System,”

Mechatronics, 12(7), pp. 963–973.[9] Carlson, J. D., 1999, “Magnetorheological Fluid Actuators,” Adaptronics and

Smart Structures: Basics, Materials and Applications, H. Janocha, ed.,

Springer, Berlin, Germany.[10] Çeşmeci, Ş. and Engin , T., 2010, “Mo deling and T esting of a Fiel d-Controllable

Magnetorheological Fluid Damper,” International Journal of Mechanical Scien-ces, 52(8), pp. 1036–1046.

[11] Choi, S. B., and Lee, S. K., 2001, “A Hysteresis Model for the Field-DependentDamping Force of a Magnetorheological Damper,” J. Sound Vib., 245(2), pp.375–383.

[12] Jimenez, R., and Alvarez, L., 2002, “Real Time Identification of StructuresWith Magnetorheological Dampers,” Proceedings of the 41st IEEE Conferenceon Decision and Control, Las Vegas, NV, pp. 1017–1022.

[13] Horvat, D., 1997, “Survey of Advanced Suspension Developments and RelatedOptimal Control Application,” Automatica, 33(10), pp. 1781–1817.

[14] Zin, A., Sename, O., Gaspar, P., Dugard, L., and Bokor, J., 2006, “An LPV,Active Suspension Control for Global Chassis Technology: Design and

Fig. 14 MR damper actuator force in the case of the sinusoidal road input

Fig. 15 MR damper actuator control voltage in the case of the sinusoidal road input



http://dx.doi.org/10.1016/j.conengprac.2008.04.003http://dx.doi.org/10.1016/j.jsv.2006.09.022http://dx.doi.org/10.1016/j.conengprac.2003.09.008http://dx.doi.org/10.1016/j.conengprac.2008.05.002http://dx.doi.org/10.1016/j.ymssp.2008.06.006http://dx.doi.org/10.1016/j.ymssp.2008.06.006http://dx.doi.org/10.1016/j.jsv.2007.11.045http://dx.doi.org/10.1016/j.mechatronics.2008.09.006http://dx.doi.org/10.1016/S0957-4158(01)00032-0http://dx.doi.org/10.1006/jsvi.2000.3539http://dx.doi.org/10.1016/S0005-1098(97)00101-5http://dx.doi.org/10.1016/S0005-1098(97)00101-5http://dx.doi.org/10.1006/jsvi.2000.3539http://dx.doi.org/10.1016/S0957-4158(01)00032-0http://dx.doi.org/10.1016/j.mechatronics.2008.09.006http://dx.doi.org/10.1016/j.jsv.2007.11.045http://dx.doi.org/10.1016/j.ymssp.2008.06.006http://dx.doi.org/10.1016/j.ymssp.2008.06.006http://dx.doi.org/10.1016/j.conengprac.2008.05.002http://dx.doi.org/10.1016/j.conengprac.2003.09.008http://dx.doi.org/10.1016/j.jsv.2006.09.022http://dx.doi.org/10.1016/j.conengprac.2008.04.003


12/12

Performance Analysis,” Proceedings of the IEEE American Control Conference(ACC), Minneapolis, MN, pp. 2945–2950.

[15] Zapateiro, M., Pozo, F., Karimi, H. R., and Luo, N., 2012, “Semi-Active Con-trol Methodologies for Suspension Control With Magnetorheological Damp-ers,” IEEE/ASME Trans. Mechatronics, 17(2), pp. 370–380.

[16] Ahmadian, M., and Blanchard, E., 2011, “Non-dimensionalised Closed-FormParametric Analysis of Semi-Active Vehicle Suspensions Using a Quarter-Car Model,” J. Veh. Syst. Dyn., 49(1–2), pp. 219–235.

[17] Ahmadian, M., and Vahdati, N., 2006, “Transient Dynamics of Semi-ActiveSuspensions With Hybrid Control,” J. Intell. Mater. Syst. Struct., 17(2), pp.145–153.

[18] Choi, S. B., and Han, S. S., 2003, “H1

Control of Electrorheological Suspen-sion System Subjected to Parameter Uncertaintie,” Mechatronics, 13(7), pp.

639–657.[19] Guo, D. L., Hu, H. Y., and Yi, J. Q., 2004, “Neural Network Control for a

Semi-Active Vehicle Suspension With a Magnetorheological Damper,” J. Vib.Control, 10(3), pp. 461–471.

[20] Haiping, D., James, L., Cheung, K. C., Weihua, L., and Zhang, N., 2013,“Direct Voltage Control of MR Damper for Vehicle Suspensions,” J. SmartStruct., 22(10), p. 105016.

[21] Savaresi, S. M., and Spelta, C., 2007, “Mixed Skyhook and ADD: Approachingthe Filtering Limits of Semi-Active Suspension,” J. Dyn. Syst. Meas. Control,129(4), pp. 382–392.

[22] Giorgetti, N., Benporad, A., Tseng, H. E., and Hrovat, D., 2006, “Hybrid ModelPredictive Control Application Toward Optimal Semi-Active Suspension,” Int.J. Control, 79(5), pp. 521–533.

[23] Khalil, H., 2003, Nonlinear Systems, 3rd ed., Prentice-Hall, Upper Saddle River, NJ.[24] Krstić , M., Kanellakopoulos, I., and Kokotovic, P., 1995, Nonlinear and Adapt-

ive Control Design, Wiley, Hoboken, NJ.[25] Zin, A., Sename, O., Basset, M., Dugard, L., and Gissinger, G., 2004, “A Non-

linear Vehicle Bicycle Model for Suspension and Handling Control Studies,”Proceedings of the IFAC Conference on Advances in Vehicle Control andSafety AVCS, Genova, Italy, pp. 638–643.

[26] Ikhouane, F., and Rodellar, J., 2007, Systems With Hysteresis Analysis, Identificationand Control Using the Bouc–Wen Model, Wiley, Hoboken, NJ.

[27] Ioannou, P. A., and Fidan, B., 2006, Adaptive Control Tutorial, SIAM, Phila-delphia, PA.

[28] Ahmadian, M., and Pare, C. A., 2000, “A Quarter-Car Experimental Analysisof Alternative Semi-Active Control Methods,” J. Intell. Mater. Syst. Struct.,11(8), pp. 604–612.

http://dx.doi.org/10.1109/TMECH.2011.2107331http://dx.doi.org/10.1080/00423114.2010.482671http://dx.doi.org/10.1177/1045389X06056458http://dx.doi.org/10.1177/1077546304038968http://dx.doi.org/10.1177/1077546304038968http://dx.doi.org/10.1088/0964-1726/22/10/105016http://dx.doi.org/10.1088/0964-1726/22/10/105016http://dx.doi.org/10.1115/1.2745846http://dx.doi.org/10.1080/00207170600593901http://dx.doi.org/10.1080/00207170600593901http://dx.doi.org/10.1106/MR3W-5D8W-0LPL-WGUQhttp://dx.doi.org/10.1106/MR3W-5D8W-0LPL-WGUQhttp://dx.doi.org/10.1080/00207170600593901http://dx.doi.org/10.1080/00207170600593901http://dx.doi.org/10.1115/1.2745846http://dx.doi.org/10.1088/0964-1726/22/10/105016http://dx.doi.org/10.1088/0964-1726/22/10/105016http://dx.doi.org/10.1177/1077546304038968http://dx.doi.org/10.1177/1077546304038968http://dx.doi.org/10.1177/1045389X06056458http://dx.doi.org/10.1080/00423114.2010.482671http://dx.doi.org/10.1109/TMECH.2011.2107331

Adaptive Study

Documents

Transcript of Adaptive Study