Intelligent Semi-Active Vibration Control Suspension System

7/28/2019 Intelligent Semi-Active Vibration Control Suspension System

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Journal of Mechanical Science and Technology 26 (2) (2012) 323~334

www.springerlink.com/content/1738-494xDOI 10.1007/s12206-011-1007-6

12

Intelligent semi-active vibration control of eleven degrees of freedom suspensionsystem using magnetorheological dampers

Seiyed Hamid Zareh*, Atabak Sarrafan, Amir Ali Akbar Khayyat and Abolghassem Zabihollah

School of Science and Engineering, Sharif University of Technology, Iran

(Manuscript Received February 1, 2011; Revised August 8, 2011; Accepted September 13, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract

A novel intelligent semi-active control system for an eleven degrees of freedom passenger cars suspension system using magnetor-

heological (MR) damper with neuro-fuzzy (NF) control strategy to enhance desired suspension performance is proposed. In comparison

with earlier studies, an improvement in problem modeling is made. The proposed method consists of two parts: a fuzzy control strategyto establish an efficient controller to improve ride comfort and road handling (RCH) and an inverse mapping model to estimate the force

needed for a semi-active damper. The fuzzy logic rules are extracted based on Sugeno inference engine. The inverse mapping model is

based on an artificial neural network and incorporated into the fuzzy controller to enhance RCH. To verify the performance of the NF

controller (NFC), comparisons with existing semi-active techniques are made. The typical control strategy are linear quadratic regulator

(LQR) and linear quadratic Gaussian (LQG) controllers with clipped optimal control algorithm, while inherent time-delay and non-linear

properties of MR damper lie in these strategies. Simulation results demonstrated that the NFC has better control performance and less

control effort than the optimal in improving the service life of the suspension system and the ride comfort of a car.

Keywords: Clipped optimal control algorithm; Full car model; Linear quadraticGaussian; MR damper; Neuro-fuzzy strategy; Suspension system

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1. Introduction

Suspension systems have long been of great concern for car

industries. It performs multiple tasks such as maintaining con-

tact between vehicle tires and the road, addressing the stability

of the vehicle, and isolating the frame of the vehicle from

road-induced vibration and shocks. In general, ride comfort,

road handling, and stability are the most important factors in

evaluating suspension performance.

In present work, passenger car suspension system is modi-

fied to reduce the amplitude of the car vibration caused by

applied road profile. In the passive suspension system, the

stiffness and damping parameters are fixed and effective over

a certain range of frequencies.

To overcome this problem, the use of semi-active suspen-sion systems which have the capability of adapting to chang-

ing road conditions by the use of an actuator has been consid-

ered; therefore an MR damper is added to the usual suspen-

sion systems while the other parts of suspension system are

intact. The significance of MR damper is that its viscosity

changes as the magnetic field is changed. A schematic model

of MR damper is shown in Fig. 1.

Considered suspension model is controlled by applied Lin-

ear quadratic regulator (LQR) and linear quadratic Gaussian(LQG). By using employed controller results, the amount of

viscosity of MR damper can be calculated incorporated with

clipped optimal strategy. Unfortunately, due to the inherent

nonlinear nature of the MR damper to generate a force, a

model like that for its inverse dynamics is difficult to obtain

mathematically. Because of this reason, a neural network with

fuzzy logic controller is constructed to copy the inverse dy-

namics of the MR damper.

Neuro-fuzzy controller is an artificial neural network, which

is used to aggregate rules and provides control result for the

This paper was recommended for publication in revised form by Associate Editor

Hyoun Jin Kim*Corresponding author. Tel.: +989378550656, Fax.: +987224223895

E-mail address: [email protected]

KSME & Springer 2012

Fig. 1. A schematic model of MR damper.


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324 S. H. Zareh et al. / Journal of Mechanical Science and Technology 26 (2) (2012) 323~334

designed fuzzy logic controller. Application of fuzzy inference

systems as a fuzzy logic controller (FLC) has gradually been

recognized as the most significant and fruitful application for

fuzzy logic and fuzzy set theory.

Finally, the results of applied control strategies are com-

pared. There are three models for modeling passenger cars

suspension systems: quarter-car, half-car and full-car model.In quarter-car model, it is assumed that each of four-wheel has

independent suspension to simulate the actions of an active

vehicle suspension system, and therefore, quarter-car models

are using for many simulations of suspension system. The

model of quarter-car for vehicle suspension system has been

used by Wang et al. [1]. Narayanan et al. [2] applied half-car

model for simulating semi-active suspension system. They

modeled MR damper parameters by the modified BoucWen

model and determined them to fit the hysteretic behavior.

Vibration control of passenger car utilizing half-car was dis-

cussed by Yahaya et al. [3]. To improve the accuracy of car

model Zabihollah et al. [4] modeled the vehicle suspension

system by a full car model; in which the accuracy of model

improved compared to quarter-car and half-car models.

Semi-active and active control methods have been devel-

oped using different actuators such as electrorhrological (ER)

and MR dampers. Salem et al. [5] controlled an active quarter-

car suspension system by fuzzy logic controller. Hyun et al.

[6] utilized an adaptive LQG control for semi-active suspen-

sion system for quarter-car model. Semi-active control is used

by Chen et al. [7] where the MR damper utilized as actuator of

suspension system.

Jialin et al. [8] presented and designed a full-state LQR con-

troller for a half-car suspension system composed of actuators

in parallel with conventional spring-damper passive suspen-sion. Yang et al. [9] modeled and controlled an intelligent

active suspension system using fuzzy controller by adaptive

filter method. Zhou and Sun [10] have done a semi-active

vibration control on five degrees of freedom suspension sys-

tem using adaptive fuzzy PID control. They combined the

advantages of PID and fuzzy controllers and applied them to

the vibration control of engineering vehicle. The parameters of

PID tuned on line by fuzzy controller.

Golnaraghi et al. [11] controlled a semi-active quarter car

suspension system intelligently. They utilized an inverse map-

ping model to estimate the current based on an artificial Neu-

ral Network and incorporated into the fuzzy logic controller.

Sadati et al. [12] designed a neuro-fuzzy controller for a vehi-

cle suspension system. They controlled a half car model with

four degrees of freedom using feedback error learning.

The previous studies made full use of the advantages of the

neural-network and the fuzzy logic controller and solved the

different problems in suspension systems. Few researches

involved combination of the two techniques to solve the time-

delay and the inherent nonlinear nature of the MR dampers in

semi-active strategy for full car model with high degrees of

freedom. In this paper, four MR dampers are added in a sus-

pension system between body and wheels parallel with pas-

sive dampers. For the intelligent system, fuzzy controller

which inputs are relative velocities across MR dampers that

are excited by road profile for predicting the force of MRdamper to receive a desired passengers displacement and

velocity is applied. When predicting the displacement and

velocity of MR dampers, a four-layer feed forward neural

network, trained on-line under the LevenbergMarquardt

(LM) algorithm, is adopted. In order to verify the effective-

ness of the proposed neuro-fuzzy control strategy, the uncon-

trolled system and the clipped optimal controlled suspension

system are compared with the neuro-fuzzy controlled system.

Through a numerical example under actual road profile excita-

tion, it can be concluded that the control strategy is very im-

portant for semi-active control, the neuro-fuzzy control strat-

egy can determine currents of the MR damper quickly andaccurately, and the control effect of the neuro-fuzzy control

strategy is better than that of the other control strategy.

2. Full car model

In the full-car model, 11-DOFs are assumed; all wheels and

passengers are dependent on each other and on the cars body.

It is assumed that each wheel has an effect on the spring and

damper of other wheels, and two axes of vehicle are relevant.

MR actuator is utilized to damp the effect of road profile on

the passengers. Note that MR shock absorber is added to the

axel and car body. In a full-car model, the effect of body rota-

tions around roll and yaw axis is simulated. The suspension

system using a full-car model has 11-DOFs, four of them for

the four wheels, three for body displacement and its rotations

and the last four for passengers. A schematic of a full-car

model with 11-DOFs and addition MR damper is shown in

Fig. 2.

The dynamic equations of each DOF are given as in Eq. (1)-

(11). As a result, the state space form of the equation is shown

in Eq. (12). The state space form and the corresponding matri-

ces are observed in Eq. (13)-(19). E is a location matrix of

actuators. Matrices K, S and T are defined as stiffness, damp-

Fig. 2. A full-car model with 11-DOFs.


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S. H. Zareh et al. / Journal of Mechanical Science and Technology 26 (2) (2012) 323~334 325

ing coefficient and input matrix due to wheels stiffness, re-

spectively.

1 1 1 1 1 1 5 1 11 1 12

1 1 1 5 1 11 1 12 1 1

( )

0

t

t i

m x k k x k x k r k r

b x b x b r b r k x

+ +

+ =

&&

&&& &(1)

2 2 2 2 2 2 5 2 21 2 22

2 2 2 5 2 21 2 22 2 2

( )

0

t

t i


b x b x b r b r k x

+ + +

+ + =

&&

&&& &(2)

3 3 3 3 3 3 5 3 31 3 32

3 3 3 5 3 31 3 32 3 3

( )

0

t

t i


b x b x b r b r k x

+ + +

+ + =

&&

&&& &(3)

4 4 4 4 4 4 5 4 41 4 42

4 4 4 5 4 41 4 42 4 3

( )

0

t

t i


b x b x b r b r k x

+ + + +

+ + + =

&&

&&& &

(4)

5 1 1 2 2 3 3 4 4 1 2

3 4 5 6 7 8 5 5 6 6 7

7 8 8 9 1 11 2 21 3 31 4 41 5 51

6 61 7 71 8 81 1 12 2 22 3 32

4 42 5 52 6 62 7 72 8 82 1 1

2 2 3 3 4 4

(

)

(

) (

)

bM x k x k x k x k x k k

k k k k k k x k x k x

k x k x k r k r k r k r k r

k r k r k r k r k r k r

k r k r k r k r k r b x

b x b x b x

+ +

+ + + + + +

+ + +

+ + +

+ +

&&

&

& & & 1 2 3 4 5

6 7 8 5 5 6 6 7 7 8 8 9

1 11 2 21 3 31 4 41 5 51 6 61

7 71 8 81 1 12 2 22 3 32 4 42

5 52 6 62 8 82 8 82

(

)

(

) (

) 0

b b b b b

b b b x b x b x b x b x

b r b r b r b r b r b r


b r b r b r b r

+ + + + +

+ + +

+ + + + +

+ +

+ =

& & & & &

&

&

(5)

1 1 11 1 2 21 2 3 31 3 4 41 4

1 11 2 21 3 31 4 41 5 51 6 61

7 71 8 81 5 5 51 6 6 61 7 7 71 8

2 2 2 2 28 81 9 1 11 2 21 3 31 4 41 5 51

2 2 26 61 7 71 8 81 1 11 12 2 21 22

(

)

(

) (

I k r x k r x k r x k r x


k r k r x k r x k r x k r x

k r x k r k r k r k r k r

k r k r k r k r r k r r

+ +

+ + + +

+

+ + + + + +

+ + + +

&&

3 31 32 4 41 42 5 51 52 6 61 62 7 71 72

8 81 82 1 11 1 2 21 2 3 31 3 4 41 4

1 11 2 21 3 31 4 41 5 51 6 61 7 71

8 81 5 5 51 6 6 61 7 7 71 8 8 81 9

2 21 11 2 21 3 3

)

(

)

(

k r r k r r k r r k r r k r r

k r r b r x b r x b r x b r x

b r b r b r b r b r b r b r

b r x b r x b r x b r x b r x

b r b r b r

+ + +

+ + +

+ + + +

+ +

+ + +

& & & &

& & & & &

2 2 2 21 4 41 5 51 6 61

2 27 71 8 81 1 11 12 2 21 22 3 31 32

4 41 42 5 51 52 6 61 62 7 71 72 8 81 82

) (

) 0

b r b r b r

b r b r b r r b r r b r r

b r r b r r b r r b r r b r r

+ + +

+ + +

+ + + + =

&

&

(6)

2 1 12 1 2 22 2 3 32 3 4 42 4 1 12

2 22 3 32 4 42 5 52 6 62 7 72

8 82 5 5 52 6 6 62 7 7 72 8 8 82 9

1 12 11 2 22 21 3 32 31 4 42 41 5 52 51

6 62 61 7 72 71 8 82 81 1

(

)

(

) (

I k r x k r x k r x k r x k r


k r x k r x k r x k r x k r x

k r r k r r k r r k r r k r r

k r r k r r k r r k

+ + +

+ + +

+ +

+ + +

+ +

&&

2 212 2 22

2 2 2 2 2 23 32 4 42 5 52 6 62 7 72 8 82

1 12 1 2 22 2 3 32 3 4 42 4 1 12

2 22 3 32 4 42 5 52 6 62 7 72

)

(

r k r


b r x b r x b r x b r x b r


+

+ + + + + +

+ + +

+ + +

& & & &

(7)

8 82 5 5 52 6 6 62 7 7 72 8 8 82 9

1 12 11 2 22 21 3 32 31 4 42 41 5 52 51

2 26 62 61 7 72 71 8 82 81 1 12 2 22

2 2 2 2 2 23 32 4 42 5 52 6 62 7 72 8 82

)

(

) (

) 0

b r x b r x b r x b r x b r x

b r r b r r b r r b r r b r r

b r r b r r b r r b r b r


+ +

+ + +

+ + +

+ + + + + + =

& & & & &

&

&

5 6 5 5 5 6 5 51 5 52

5 5 5 6 5 51 5 52 0m x k x k x k r k r

b x b x b r b r

+

=

&&

&&& &(8)

6 7 6 5 6 7 6 61 6 62

6 5 6 7 6 61 6 62 0

m x k x k x k r k r

b x b x b r b r

+ +

+ + =

&&

&&& &(9)

7 8 7 5 7 8 7 71 7 72

7 5 7 8 7 71 7 72 0

m x k x k x k r k r

b x b x b r b r

+ +

+ + =

&&

&&& &(10)

8 9 8 5 8 9 8 81 8 82

8 5 8 9 8 81 8 82 0

m x k x k x k r k r

b x b x b r b r

+ + +

+ + + =

&&

&&& &

(11)

1 1 2 2 3 3 4 4 5 5

6 6 7 7 8 8 9 9 10

11 1 12 2 13 3 14 4 15

5 16 6 17 7 18 8 19 9 20

21 22

, , , , ,

, , , , ,

, , , ,

, , , , ,

,

x x x x x x x x x x

x x x x x x x x x

x x x x x x x x x

x x x x x x x x x x

x x

= = = = =

= = = = =

= = = = =

= = = = =

= =

& & & &

& & & & &

&&

(12)

x Ax Bu Gw

y Cx

= + +

=

&

(13)

1 1

0 I

A K M S

= (14)

1

0B

T

=

(15)

1

0G

E

=

(16)

1 2 3 4

5 6 7 8 1 2 11 11

bm m m m M M diag

m m m m I I

=

L(17)

1 23 4

[ , 0, 0,0; 0, , 0, 0;0, 0,

,0;0,0,0, ; (7,4)]t t

t t

T k k

k k zeros

=(18)

11 21 31 41 12 22 32 42

[ (4,4);1,1,1,1; (4,4);

, , , ; , , , ]

E eye zeros

r r r r r r r r

=

(19)

whereMb, m1, m2, m3, m4, m5, m6, m7and m8 stand for the mass

of the car body, masses of four wheels and mass of passengers,

respectively. I1 and I2 are the moments of inertia of the car

body around two axes, respectively. The terms k1, k2, k3, k4, k5,

k6, k7 and k8 are stiffnesses of the springs of the suspension


4/12


system and stiffnesses of the springs of passengers seat, re-

spectively. The terms kt1, kt2, kt3 and kt4 are stiffnesses of the

tires. The terms b1, b2, b3, b4, b5, b6, b7 and b8 are coefficients

of car and passengers seat dampers, respectively. Then, br1, br2,

br3 and br4 are coefficients of the MR dampers, respectively.x1,

x2, x3, x4, x5, x6, x7, x8, x9, and indicate the DOFs of the

suspension system model, respectively. The terms xi1, xi2, xi3

andxi4 indicate load profile disturbance, respectively.

The numerical values of models dimensions for obtaining

the responses are shown in Table 1 [4].

3. LQR controller strategy

LQR controller responds to changes in the location of the

poles of the system to the optimal place. Time response, over-

shoot and steady state errors depend on the location of the

systems poles. LQR controller controls the system by a ma-

trix gain Eq. (20). This gain is achieved from Eq. (21). To

solve this energy equation of the system, Riccati equation isused, and this relation is given in Eq. (22), in which Q is a

symmetric positive semi-definite matrix and Ris a symmetric

positive definite matrix. The result of solving Riccati equation

is matrix S. Gain of pole placement is achieved from Eq. (23)

by using S matrix.

( )x A BK x Gw

y Cx

= +

=

&

(20)

0

1( ( ) ( ) ( ) ( ))

2

T Tx t Qx t u t Ru t dt

= + (21)

1 0T TA S SA SBR B S Q+ + = (22)1 TK R B S= (23)

Gain Kby B matrix made a square matrix, subtracted from

A matrix, and changes dynamic properties and the pole of the

control system. The A matrix presents the dynamic properties

of the system. New A matrix shows the system with new posi-

tion pole. Therefore by finding optimal gain, pole of the sys-

tem is shifted to the optimal position. Here, Q22*22 is an iden-

tity matrix (I22*22) because all states are important and the best

responses are obtained by this value. R4*4 is 45*I4*4, which is

obtained by trial and error to receive the desired response.

4. LQG controller strategy

In order to control a system with disturbance and noise by

modern control method a controller with noise filter must be

used. Optimal control, linear quadratic Gaussian, is the most

appropriate to control the full car model. The LQG controller

is simply the combination of a Kalman filter (i.e., a linear-

quadratic-estimator (LQE) with a linear-quadratic-regulator).

To eliminate the effect of disturbance on the suspension

system, the LQE part of the LQG controller is utilized. Per-

haps, the main step to design the LQE is to identify the ampli-

tude and position of disturbance. As a result, disturbance vec-tor and position of disturbance entrance should be identified.

The disturbance of the system is entered to the system as the

input part. In the present work, it is assumed that some states

of the system can be detected by sensors attached to the parts

of the suspension system (seven states of twenty two states

that are denoted as Clqg). Consequently, the controller of the

suspension system is not a full state control. The state space of

the suspension system is controllable and observable.

To design the LQG controller, first, the disturbance vector

needs to be identified. Then, the matrices W and V are pro-

duced; the process noise (disturbance) is simulated by road

Table 1. Numerical values of model.

Symbol Quantity Value

Mb Mass of car body (kg) 670

I1 Inertia around yaw (kg/m^2) 800

I2 Inertia around roll (kg/m^2) 1100

m1-4 Masses of Wheels (kg) 30

m5-8 Masses of passengers (kg) 120

k1-4 Stiffnesses of car springs (N/m) 17500

k5-8 Stiffnesses of seats (N/m) 1750

b1-4 Viscosity of car dampers (Ns/m) 1460

b5-8 Viscosity of seats damper (Ns/m) 700

kt1-4 Stiffnesses of wheels (N/m) 175500

r11Length distance between the front left

wheel and center of mass (m)1.9975

r12Width distance between the front left


r21Length distance between the front right

wheel and center of mass (m) 1.9975

r22Width distance between the front right


r31Length distance between the back right


r32Width distance between the back right


r41Length distance between the back left


r42Width distance between the back left


r51Length distance between the driver seat

and center of mass (m)1.9975

r52 Width distance between the driver seatand center of mass (m)

0.8025

r61Length distance between the front right

seat and center of mass (m)1.9975

r62Width distance between the front right


r71Length distance between the back left


r72Width distance between the back left


r81Length distance between the back right


r82Width distance between the back right



5/12


profile that is explained in next subsection; the measurement

noise is simulated by fraction of road profile. Kalman filter is

a state estimator given a state-space model of the plant and the

process and measurement noise covariance data. The Kalman

estimator provides the optimal solution to the following con-

tinuous or discrete estimation problems.

x Ax Bu Gw= + +& (24)

lqgy C x v= + (25)

The observer structure with known inputs u, white process

noise w, and white measurement noise v, is in the form of:

( )lqgx Ax Bu L y C x= + + & (26)

where x is an LQG optimal estimate ofx.

In the next step, the energy equation of the LQE must be

solved, as expressed in Eq. (27). In order to solve the energy

equation, Riccati equation is produced. Riccati equation isgiven in Eq. (28). Solving Riccati equation, the P matrix will

be achieved. The P matrix is utilized to find the observer gain,

L, in Eq. (29).

0

[ ]T TJ x Vx u Wu dt

= + (27)

1 0T Tlqg lqg AP PA PC V C P W+ + = (28)

1TlqgL PC V

= (29)

where [ ]TW E ww= , [ ]TV E vv= are the plant disturbance

and measurement noise covariances. The input u itself is gen-erated by a state feedback law as:

.u Kx= (30)

Finally, using LQG controller in close-loop system, it leads

the original system convert to following equations:

0

0

lqg lqg

x A BK x

LC A BK LC xx

G w

L v

=

+

&

&

(31)

11

(11 11) (11 11)

(11 11)lqg

zeros zerosC

zeros C

=

(32)

11

(5 5) (5 4) (5 2)

(4 5) (4 4) (4 2) .

(2 5) (2 4) (2 2)

I zeros zeros

C zeros zeros zeros

zeros zeros I

=

(33)

Here,Iis a identity matrix. The optimal force vectorfc regulated

only by the state vectorx that is calculated by Eq. (34) [13].

c LQRf K x= (34)

5. Clipped optimal algorithm

The clipped optimal control strategy for an MR damper

usually involves two steps. The first step is to assume an ideal

activelycontrolled device and construct an optimal controller

for this active device. In the second step, a secondary control-

ler finally determines the input voltage of the MR damper.That is, the secondary controller clips the optimal force in a

manner consistent with the dissipative nature of the device.

The block diagram of clipped optimal algorithm is shown in

Fig. 3.

The clipped optimal control approach is to append a force

feedback loop to induce the MR damper to produce approxi-

mately a desired control forcefc. The linear quadratic regulator

algorithm has been employed both for active control and for

semi-active control. Using this algorithm, the optimal control

forcefc for f, which is force generated by a MR damper, may

be obtained by minimizing the following scalar performance

index

0

( ) .

ft

T T

t

x Qx F RF dt= + (35)

Q and R are weighting matrices and their values are se-

lected depending on the relative importance given to different

terms in their contributions to the performance index J.

Solving the optimal control problem with J defined by Eq.

(35), results in a optimal force vectorfc regulated only by the

state vectorx, such that

1( )Tcf R B P x Gx= = (36)

where matrix G represents the gain matrix; and the matrix P is

the solution of the classical Riccati equation given by Eq. (37)

1 0 .T TPA A P PBR B P Q+ + = (37)

The force generated by the MR damper cannot be com-

manded. When the MR damper is providing the desired opti-

mal force (i.e., f = fc ), the voltage applied to the damper

Fig. 3. Clipped optimal algorithm block diagram.


6/12


should remain at the present level. If the magnitude of the

force produced by the damper is smaller than the magnitude of

the desired optimal force and the two forces have the same

sign, the voltage applied to the current driver [14] varies con-

tinuously in the range of [0-Vmax]. The secondary controller for

continuously varying the command voltage can be stated as

{ }( )i ci ci i iv V H f f f = (38)

max

max max

,

,

i ci cici

ci

f for f fV

V for f f

=

>(39)

where Vmax is 12v and fmax is maximum force produced by

the damper (=3,000 N); is coefficient relating the voltage to

the force (= Vmax/fmax);H (.) is the heaviside step function ex-

pressed as 0 or 1 [14]; fi is the force produced by ith

MR

dampers which is applied to the structure.

In this paper, a simple mechanical model consisting of a

Bouc-Wen element in parallel with a viscous damper is used,

as shown in Fig. 4. This model has been verified to accuratelypredict the behavior of a prototype shear-mode MR damper

over a wide range of inputs in a set of experiments, and is also

expected to be appropriate for modeling a full-scale MR

damper.

The equations governing the force fi exerted by this model

are as follows:

0if c x z= +& (40)

1n nmz x z z x z A x

= +& & && & (41)

wherex is the displacement of the device andzis the evolu-

tionary variable that accounts for the history dependence of

the response. The parameters , , n andAm adjusted to deter-

mine the linearity in the unloading and the smoothness of the

transition from the pre-yield region to the post-yield region.

Device model parameters and c0 are determined by the de-

pendency on the control voltage u, as follows:

( ) a bu u = = + (42)

0 0 0 0( ) .a bc c u c c u= = + (43)

Moreover, to account for a time-lag in the response of the

device to the changes in the command input, the first-order

filter dynamics are introduced to the system as follows:

( )u u = & (44)

where v is a command voltage applied to the control circuitand is the time constant of the first-order filter.

The numerical values of parameters are shown in Tables 2

and 3 [23].

The hysteretic behavior of the MR damper model according

to the input voltage is shown in Fig. 5.

6. Road profile simulation

The random road excitation is generated using white noise

as a road irregularity as a disturbance. The power spectral

density (PSD) function of road irregularity is assumed to be in

the form of Eq. (45).

2

2 2( ( ) )

rh

r

VS

V

=

+(45)

where 2

is the variance of the road profile, is the excitation

frequency of the road input Vis the vehicle forward constant

velocity and r is a coefficient depending on the type of road

surface. The typical properties of unpaved road profile are

shown in Table 4 [2].

The PSD of the road surface is obtained by using MATLAB.

Table 2. Numerical values of MR damper model.

a (N cm-1) b (N cm

-1 V-1) (cm-2) (cm-2)

140 695 363 363

Table 3. Numerical values of MR damper model.

c0a (Nscm-1) c0b (Nscm

-1 V-1) n (s-1) Am

21 3.5 1 190 301

Fig. 5. Hysteretic behavior of an MR damper.

Fig. 4. Mechanical model of a shear mode type MR damper.


7/12


7. Neuro fuzzy strategy

Unfortunately, due to the inherent nonlinear nature of the

MR damper to generate a force, a model like that for its in-

verse dynamics is difficult to obtain mathematically. Because

of this reason, a neural network with fuzzy logic controller is

constructed to copy the inverse dynamics of the MR damper.

Unlike conventional controllers, such controllers do not re-

quire mathematical model and they can easily deal with the

nonlinearities and uncertainties of the controlled systems. Also,

a Levenberg-Marquardt neural controller has been designed

for variable geometry suspension systems with MR actuators.

In the present research, an optimal controller (LQR) is de-

signed for the control of a semi-active suspension system for a

full-model vehicle, using a neuro-fuzzy along with Leven-

berg-Marquardt learning. The purpose in a vehicle suspension

system is reduction of transmittance of vibrational effects

from the road to the vehicles passengers, hence providing ride

comfort. To accomplish this, one can first design a LQR con-

troller for the suspension system, using an optimal control

method and use it to train a neuro-fuzzy controller. This con-

troller can be trained using the LQR controller output error on

an online manner.

Once trained, the LQR controller is automatically removed

from the control loop and the neuro-fuzzy controller takes on.

In case of a change in the parameters of the system under con-

trol, the LQR controller enters the control loop again and theneural network gets trained again for the new condition [14].

An important characteristic of the proposed controller is that

no mathematical model is needed for the system components,

such as the non-linear actuator, spring, or shock absorbers.

The basic idea of the proposed neuro-fuzzy control strategy

is that the force of the MR dampers is determined by a fuzzy

controller, whose inputs are the measured displacement and

velocity response provided by a neural network. The architec-

ture of this strategy is shown in Fig. 6, which consists of four

parts to perform different tasks. The first part is the neural

network to be trained on-line. The numbers of the sample data

pairs are 3500, the training data pairs increase step by stepduring the entrance disturbance from road profile, predicted

voltage by clipped method.

The neural network is trained to generate the one step ahead

prediction of the displacement xk+1 and the velocity xk+1. In-

puts to this network are the delayed outputs (xk+3,xk+2,xk+1,xk,

xk+3, xk+2, xk+1, xk), the delayed force which is predicted by

fuzzy controller (fk+1), and the disturbance input (dk). At the

initial time, the inputs of the network will be taken to have the

value of zero in accordance with the actual initial circum-

stance. Before online training, the network trained off-line to

achieve update weights near to desired.

The second part is the fuzzy controller, whose inputs are the

measured displacement and velocity across MR dampers. The

disturbance can be calculated by road profile model. The out-

put of the fuzzy controller is control force of the MR dampers.

The main aim of this part is to determine control force of the

MR dampers quickly in accordance with the input excitation.

How to design the fuzzy controller will be explained in the

following section. In order to reach this aim, it is required to

predict the responses of passengers in accordance with the

optimal responses. At the same time, the actual responses will

feed back to the neural network and the weights and bias will

be revised real time. In this research work, the calculated re-

sults by optimal control history analysis method are used to

simulate the actual measured responses. The errors between

the predicted responses and the actual responses are used to

update the weights of the neural network on-line.

7.1 The neural network based on Levenberg-Marquardt

(LM) algorithm

The MR damper model discussed earlier in this research es-timates damper forces based on the inputs of the reactive ve-

locity. In such case, it is essential to develop an inverse dy-

namic model that predicts the corresponding control force to

be sent from dampers so that an appropriate damper force can

be generated [16].

Neural network is a simplified model of the biological

structure found in human brains. This model consists of ele-

mentary processing units (also called neurons). It is the large

amount of interconnections between these neurons and their

capabilities to learn from data to enable neural network as a

strong predicting and classification tool. In this study, Three-

layer feed forward neural network, which consists of an inputlayer, one hidden layer, and an output layer is selected to pre-

dict the responses with MR dampers. The net input value netk

of the neuron kin some layer and the output value Ok of the

same neuron can be calculated by the following Eqs. (46) and

(47):

k jk jnet w O= (46)( )k k kO f net = + (47)

where wjk is the weight between thejth

neuron in the previous

Fig. 6. Architecture of the Neuro-Fuzzy control strategy.

Table 4. Typical properties of unpaved road profile.

2 (m2 ) (rad/s ) V (m/s) r(rad/m)

3e-4 200 16.66 0.45


8/12


layer and the kth

neuron in the current layer, Ojis the output of

thejth

neuron in the previous layer,f(.) is the neurons activa-

tion function which can be a linear function, a radial basis

function, and a sigmoid function, and yk is the bias of the kth

neuron. Feed forward neural network often has one or more

hidden layers of sigmoid neurons followed by an output layer

of linear neurons. Multiple layers of neurons with nonlineartransfer functions allow the network to learn nonlinear and

linear relationships between input and output vectors. In the

neural network architecture as shown in Fig. 6, the logarithmic

sigmoid transfer function is chosen as the activation function

of the hidden layer, Eq. (48):

( )

1( ) .

1 k kk k k net

O f net e

+= + =

+(48)

The linear transfer function is chosen as the activation func-

tion of the output layer, Eq. (49):

( ) .k k k k k O f net net = + = + (49)

We note that neural network needs to be trained before pre-

dicting responses. As the inputs are applied to the neural net-

work, the network outputs (.) are compared with the targets

(.). The difference or error between both is processed back

through the network to update the weights and biases of the

neural network so that the network outputs match closer with

the targets. The input and output data are usually represented

by vectors called training pairs. The process as mentioned

above is repeated for all the training pairs in the data set, until

the network error converged to a threshold minimum defined

by a corresponding performance function. In this research, themean square error (MSE) function is adopted (desired MSE is

1e-5).

LM algorithm is adapted to train the neural network, which

can be written as Eq. (50):

2

12

1i i

ii

E Ew w I

ww

+

= +

(50)

where i is the iteration index, / iE w is the gradient descent of

the performance functionEwith respect to the parameter ma-

trix

i

w , 0 is the learning factor, andIis the unity matrix.During the vibration process, the neural network updates

the weights and bias of neurons real time in accordance with

sampling pairs till the objective error is satisfied, i.e. the prop-

erty of the system is acquired. As we know, the main aim of

the neural network is to predict the dynamic responses of the

system and to provide inputs of fuzzy controller and data of

calculating control force of MR dampers. Thus outputs of the

neural network are predicted values of displacement xk+1 and

velocityxk+1. In order to predict the dynamic responses of the

system accurately, the most direct and important factors which

affect the predicted dynamic responses are considered, i.e. the

delayed outputs (xk+3,xk+2,xk+1,xk ,xk+3,xk+2,xk+1, xk), the pre-

dicted force (fk+1), and the disturbance input (dk). LM algo-

rithm is encoded in neural networks toolbox in MATLAB

software.

7.2 Design of fuzzy controller

The first step in designing a fuzzy controller is to determinethe basic domains of inputs and outputs. The desired dis-

placement and velocity responses are chosen as inputs of the

fuzzy controller. The output of fuzzy controller is the control

force of the MR damper, which basic domain is -700N

300N same as the working force of the MR damper calculated

using LQR.

The membership functions are usually chosen in accordance

with the characteristics of the membership functions and de-

signing experience. For simplifying the calculation, triangle or

trapezoid form functions are usually adopted as the member-

ship functions. The triangle membership function is more

Fig. 7. Membership function of front-left damper velocity (m/s).

Fig. 8. Membership function of front-right damper velocity (m/s).

Fig. 9. Membership function of back-left damper velocity (m/s).

Fig. 10. Membership function of back-right damper velocity (m/s).


9/12


sensitive to inputs than the trapezoid form function, in expec-

tation that the control forces of the MR dampers are sensitive

to excitations and responses [13], but in this case are used

Gaussian and triangle form because considered form had bet-

ter response by trial and error. In this research, Gaussian and

triangle functions are adopted as the membership functions of

velocity. The membership function curves of velocity are

shown in Figs. 7-10.

8. Results

The full-car model with MR damper and disturbance is

modeled by the dynamic equations and state space matrices.

One of the desired points of this study is to decrease the am-

plitude of passengers displacements, when the suspension

system excited from the road profile. Therefore the effect of

LQR and LQG controllers and neuro-fuzzy strategy are simu-

lated for road excitation with calculated their amplitude, and

then compared with each other. The random road surface are

generated compatible with the power spectral density given in

Eq. (45) using a sum of bumper with 7 cm height and 4 cm

width. The displacement and acceleration trajectories for

front-left passengers seat that is excited by bumper with 7 cm

height and 4 cm width with 60 and 30 km/h constant velocity

in unpaved road under front left wheel are shown in Figs. 11-

14, respectively. Notice that, in all graphs, time duration is

selected for the best resolution and critical responses are hap-

pened when car strikes with bumper.

The graphs which are presented show that LQR controller

designed cannot eliminate the effect of the distributed road,

but the LQG controller can eliminate the effect of the distrib-

uted road. However, this modeling and controller which have

been theoretically designed may be experimental model can-

not work such as the theoretical model. Also, the trajectories

of neuro-fuzzy strategy show that this strategy reduces the

amplitude of vibration lower than the passive system and also

to some extent as well as optimal controllers; because dis-

placements and acceleration are predicted by feed forward

neural networks. The primary oscillations are due to the less

number of network input to train, on the other hand, there are

not strong history in transient, therefore the transient part of

response not as well as steady state part. To investigate, are

there any primary unwanted oscillations due to use of neuro-

fuzzy strategy at the other bumpiness after the first one, is

Fig. 11. Displacement of front-left seat from front left wheel excited

with 60 km/h.

Fig. 12. Acceleration of front-left seat from front left wheel excited

with 60 km/h.


with 30 km/h.

Fig. 14. Acceleration of front-left seat from front left wheel excited

with 30 km/h.


10/12


added another after five seconds. The displacement of front-

left seat due to two bumpers is shown in Fig. 15.

The graph which is presented is shown that the unwanted

oscillations due to use of neuro-fuzzy strategy at the beginning

of first excitation to some extent removed at the beginning of

second bumper. The road holding for front-left damper that is

excited by bumper with 7 cm height and 4 cm width with 60

km/h and 30 km/h constant velocity under front left wheel are

shown in Figs. 16 and17, respectively.

The stability of automobile due to use of neuro-fuzzy strat-

egy is better than other two strategies, because the oscillation

of front left wheel due to road excitation in neuro-fuzzy algo-rithm is less than other strategies. The trajectories of require-

ment forces to obtain the desire displacements and accelera-

tion is shown in Fig. 18.

The force is calculated by use of optimal controllers to some

extent have the same trend. But the forces of neuro-fuzzy can-

not follow them; because, optimal forces depend on twenty

two state variables and the forces obtained by the fuzzy part of

neuro-fuzzy strategy depend on one state variable (relative

velocity across MR damper). One of the main advantages of

using neuro-fuzzy, the control effort of dampers is less than

LQR and LQG responses. The clipped optimal method re-

sponses to generate a requirement voltage for two different

velocities are shown in Figs. 19 and 20, respectively. These

are obtained by another neuro-fuzzy strategy (the output of

fuzzy part is voltage).

The voltages are calculated by use of optimal controllers to

some extent have the same trend. The voltages are calculated

using neuro-fuzzy with less oscillations, therefore saving en-

ergy and cost.

9. Conclusions

Usual suspension systems are utilized in the vehicle, and

damped the vibration from road profile. However, passive

suspension systems have long settling time. When cars are

moved along bumpy road, the passive suspension system

driver cannot react in effective time. As a result, the usual

suspension system cannot damp the excitation with small time

interval. In order to remove this problem the properties of the

suspension system should be variable. This task is done by

adding MR dampers as an actuator to the suspension system.

In order to send commands to the actuator, LQR controller

is utilized. It can decrease the amplitude of the vibration of

passenger seats, but it cannot eliminate the effect of bumpy

roads as a disturbance. Therefore, LQG controller was de-

signed. This controller by LQE estimator can estimate the


with 60km/h due to two bumpers.

Fig. 16. Road holding for front-left damper excited with 60 km/h.

Fig. 17. Road holding for front-left damper excited with 30 km/h.

Fig. 18. Generated force by front-left MR damper from front left wheel

excited with 60 km/h.


11/12


state of the system without disturbance effect. Then the esti-

mated system is controlled by LQR part of LQG controller.

Consequently, the disturbance effect is eliminated, and the

response of this controller in the theoretical model is much

performed. It is seen that the control forces with LQG algo-

rithm and LQR algorithm completely suitable for road excita-

tion, meaning that the Kalman filter gives accurate estimations

of structural states. As can be seen, LQG can eliminate the

effect of disturbances better than LQR.

Unfortunately, due to the inherent nonlinear nature of the

MR damper to generate force, a model like that for its inverse

dynamics is difficult to obtain mathematically. Because of this

reason, a neural network with fuzzy logic controller is con-

structed to copy the inverse dynamics of the MR damper.

According to the graphs that show above, the trajectories of

neuro-fuzzy strategy can reduce the amplitude of vibration to

some extent as well as optimal controllers with less control

effort and oscillation.

Acknowledgment

The authors wish to express their gratitude to the Interna-

tional campus of Sharif University of Technology for the sup-

port provided for this research.

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Seiyed Hamid Zareh is a MSc. Student

in Mechatronics at Sharif University of

Technology, School of Science and

Engineering, Iran. He was born in 1983

and also graduated in mechanical engi-neering in 2004 at Imam Hossein Uni-

versity of Tehran, Iran.

Documents

Intelligent Semi-Active Vibration Control Suspension System