JSAE Review 20 (1999) 343}348

Control and evaluation of active suspension forMDOF vehicle model

Keqiang Li!, Masao Nagai"!Chongqing University, 174, Sazheng Street, Sapingba, Chongqing, 400044 People's Republic of China

"Tokyo University of Agriculture and Technology, 24-16, Nakacho 2-cho, Koganeishi, Tokyo 184-8588, Japan

Received 15 October 1998; received in revised form 28 January 1999

Abstract

The current method for solving the problem of active suspension control for a vehicle often uses a quarter car or a half car model.This kind of model is not suitable for practical applications. In this paper, based on considering the in#uence of factors such as theengine, seat and passengers, a MDOF (multi-degree-of-freedom model) describing the vehicle motion has been set up, and a controllerfor this model is designed by using LQ control theory. Furthermore the appropriate control scheme is selected by testing variousperformance indexes. ( 1999 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.

1. Introduction

It is necessary that the demand for both safety and ridecomfort should be met when the performance of vehicle'slight weight and high speed are emphasized. Active con-trol technology has been researched and applied toa broad class of dynamic problems in ground trans-portation vehicles, such as vehicle structural vibrationsuppression. One of the important applications is thedevelopment of active suspensions [1].

The current method for dealing with the problem ofactive suspension control for a vehicle is often usinga quarter car or a half car model. As is well known, thedesign of any control system depends crucially on theaccuracy of the mathematical model used to describethe real system. There are some factors, such as theengine, seats and passengers in a vehicle model, whichperhaps have a great e!ect on the performance of thecontrol system. So it is not enough to use a simple modelof the vehicle for practical applications, without consid-ering these factors as well.

In this paper, based on considering the vibration in#u-ence of the engine, seats and passengers, a multi-degree-of-freedom (MDOF) model describing the vehiclemotion has been set up. After theoretical analysis andcomputer simulation, it was found that the engine strong-ly a!ects the vibrations of the car body and passengers insome cases. Therefore, it is more suitable to use a MDOFmodel instead of a 4DOF half car model in the design ofactive control for car vibration. Based on the derived

8DOF model including engine and other masses, a con-troller for this model is also designed. Furthermore theappropriate control scheme is selected by testing variousperformance indexes.

2. Vehicle model

Fig. 1(a) shows a half car model, which is a 4DOFvehicle model commonly used. Fig. 1(b) shows a 6DOFvehicle model, which adds seats and passengers to themodel of Fig. 1(a). Fig. 1(c) shows the 8DOF vehiclemodel used in this paper. This model adds an engine tothe model of Fig. 1(b).

2.1. Equation of motion [2]

The equation of motion for the model in Fig. 1(c) isgiven below:

(1) Vertical motion of each mass

m1&

xK1&"!C

1&*x5

10&!K

1&*x

10&#C

2&*x5

21&

#K2&

*x21&

!u&, (1)

m13

xK13"!C

13*x5

103!K

13*x

103#C

23*x5

213

#K23

*x213

!u3, (2)

m6&

xK6&"!C

6&*x5

65&!K

6&*x

65&, (3)

m63

xK63"!C

63*x5

653!K

63*x

653, (4)

0389-4304/99/$20.00 ( 1999 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.PII: S 0 3 8 9 - 4 3 0 4 ( 9 9 ) 0 0 0 1 9 - 3 JSAE9931297

Fig. 1. Vehicle models.

M2(bxK

2&#axK

23) /l

"!C2&

*x521&

!K2&

*x21&

!C23

*x5213

!K23

*x213

#u&#u

3#C

4&*x5

43&

#K4&

*x43&

#C43

*x5433

#K43

*x433

#C6&

*x565&

#K6&

*x65&

#C63

*x5653

#K63

*x653

, (5)

M4(b

%xK4&#a

%xK43

) /l%"!C

4&*x5

43&!K

4&*x

43&

!C43

*x5433

!K43

*x433

. (6)

(2) Pitch motion of car body and engine

J2uK2"!(!C

2&*x5

21&!K

2&*x

21&#u

&)a

#(!C23

*x5213

!K23

*x213

#u3)b

#(!C4&

x543&

!K4&

x43&

)e-

#(!C43

x5433

!K43

x433

)erl

!(!C6&

x565&

!K6&

x65&

)s-

!(!C63

x5653

!K63

x653

)srl, (7)

J4uK4"!(!C

4&*x5

43&!K

4&*x

43&)a

%

#(!C43

*x5433

!K43

*x433

)b%, (8)

where

u2"

x23!x

2&l

, u4"

x43!x

4&l%

,

a"¸2!¸

2&, b"¸

23!¸

2, l"¸

23!¸

2&,

a%"¸

4!¸

4&, b

%"¸

43!¸

4, l

%"¸

43!¸

4&,

e-"¸3&!¸

2, erl"¸

33!¸

2,

s-"¸6&!¸

2, srl"¸

63!¸

2.

2.2. State space equation

The state space equation can be written as

XQ (t)"AX(t)#Bu(t)#=XQ0(t), (9)

where the state variable vector X is de"ned as

X"M*x21&

, x52&

, *x10&

, x51&

, *x213

, x523

, *x103

, x513

, *x43&

,

x54&

, *x433

, x543

, *x65&

, x56&

, *x653

, x563

NT.

The equation of motion and state space equation fora 4DOF or 6DOF vehicle model can also be described asshown above.

2.3. Comparison of each model

To investigate the in#uence caused by the engine, seatsand passengers as additional masses to the vehicle model,the frequency response of each passive vehicle model arecalculated under the random road PSD disturbance. Therelationship between the front input and the rear input isx03(t)"x

0&(t!q) and q"l/v.

344 K. Li, M. Nagai / JSAE Review 20 (1999) 343}348

By comparing the results shown in Fig. 2 it can be seenthat:

(1) The engine strongly a!ects the vibration of the carbody and passengers, which is related to the ridecomfort.

(2) The in#uence of the seats and passengers on thevibration is more severe than the in#uence of otherparts of the car.

It is not enough to consider the factors of seats andpassengers in general, as their in#uence on vibration cannot be ignored. Therefore, it is indispensable to makea vehicle model close to the real one, when the preciseresult of vibration control is required [3]. In this paper,according to the 8DOF vehicle model considering theabove factors, the control schemes are studied as follows.

3. Design of active suspensions

When an active suspension controller is designed, theinput of the control system is assumed as

u(t)"Mu&, u

3NT"!KX(t),

where the feedback gain K will be determined by usingLQ optimal control theory.

3.1. Controller for the 8DOF model

This model consists of the 4DOF model and engineand seats (passengers) shown in Fig. 1(c). For this case thecontroller model is a full order one for the 8DOF vehiclemodel, the state space equation is given in Eq. (11)

X"M*x21&

, x52&

, *x10&

, x51&

, *x213

, x523

, *x103

, x513

, *x43&

,

x54&

, *x433

, x543

, *x65&

, x56&

, *x653

, x563

NT,

u(t)"!KX(t). (11)

The cost function is expressed as follows:

J"P=

0AA

*x21&

q1B

2#A

x52&

q2B

2#A

*x10&

q3B

2#A

x51&

q4B

2

#A*x

213q5B

2#A

x523

q6B

2#A

*x103

q7B

2#A

x513

q8B

2

#A*x

43&q9B

2#A

x54&

q10B

2#A

*x433

q11B

2#A

x543

q12B

2

#A*x

65&q13B

2#A

x56&

q14B

2#A

*x653

q15B

2

#Ax563

q16B

2#A

u

rB2

B dt,

where qi, r are the weighting coe$cients.

Fig. 2. Frequency response of vehicle vibration under, random road input.

3.2. Controller for conventional 4DOF model (A)

For the case shown in Fig. 1(a), the controller model isa reduced order one for the 8DOF vehicle model, the

K. Li, M. Nagai / JSAE Review 20 (1999) 343}348 345

state space equation is given in Eq. (12)

x"M*x21&

, x52&

, *x10&

, x51&

, *x213

, x523

, *x103

, x513

NT (12)

u(t)"!KX(t).

The cost function used in this paper is expressed asfollows:

J"P=

0AA

*x21&

q1B

2#A

x52&

q2B

2#A

*x10&

q3B

2#A

x51&

q4B

2

#A*x

213q5B

2#A

x523

q6B

2#A

*x103

q7B

2

#Ax513

q8B

2#A

u

rB2

B dt,

where qi, r are weighting coe$cients.

3.3. Controller for 4DOF model (B)

Both to avoid complexity and to improve precision inthe process of designing the controller, a controller usingthe simple model in Fig. 3 is designed. In this model, themotion of unsprung masses, the pitching of engine andthe vertical motion of front seat are neglected.

The state space equation is expressed as

X@"M*x21&

, x52&

, *x213

, x523

, *x43

, x54, *x

653, x5

63NT,

XQ @0"Mx5

1&, x5

13NT,

XQ @(t)"AX@(t)#Bu(t)#=XQ @0(t).

The cost function for this system is given in Eq. (13)with weighting coe$cients q

i, r:

J"P=

0AA

*x21&

q1B

2#A

x52&

q2B

2#A

*x213

q3B

2

#Ax523

q4B

2#A

*x43

q5B

2#A

x54

q6B

2

#A*x

653q7B

2#A

x563

q8B

2#A

u

rB2

B dt. (13)

As is well known, the feedback gain vector K of thecontrol system can be computed by solving a Riccatiequation. So the optimal control input u is obtained andis expressed as u"!KX@.

4. Results and discussions

Computer simulation is carried out for a passenger carrunning at 100 km/h on a road with: (1) bumpy distur-

Fig. 3. 4DOF controller model B.

bance; (2) random road disturbance expressed by powerspectral density (PSD). The vibrations corresponding toride comfort are derived under the road disturbancesmentioned above.

4.1. Response under bumpy disturbance

Fig. 4 illustrates the response of the vehicle at a velo-city of<"100 km/h under the bumpy road disturbance,whose bump height is 20 mm. Fig. 4(a), (b) and (c) are theresponses of the car body, seats (passengers) and engine,respectively, under this disturbance.

First, the result of the response of uncontrolled andcontrolled system is compared. The e!ect of the activesuspension is clearly shown because the response of ve-hicle vibration is greatly suppressed and quickly dampedin this case.

Next, to investigate the in#uence of di!erent controlschemes on the control e!ect, the result achieved witheach controller is compared. The best result is that usingthe controller designed for the 8DOF model, which con-sidered the engine, seats and passengers. As there aresome problems due to complexity and high cost whenusing the 8DOF model for practical applications, it isdesirable to use a lower order model for the controller.Comparing the controllers using the two di!erent 4 ordermodels, it can be seen that the use of the 4DOF model (B)resulted in a better control performance than that of the4DOF model (A).

4.2. Frequency response under random road disturbance

Fig. 5 illustrates the frequency response of the vehicleat a velocity of <"100 km/h under the random roadPSD disturbance. Fig. 5(a), (b) and (c) are the responses ofthe car body, seats (passengers) and engine, respectively,under this disturbance.

This is similar to the response under bumpy distur-bance, the vibrations are greatly suppressed by the activecontrol over most of the frequency range, in which thevibration strongly a!ects ride comfort. The control e!ectis not noticeable in the neighborhood of 10 Hz, due to theinvariant point of system vibration, caused by the un-spring mass.

346 K. Li, M. Nagai / JSAE Review 20 (1999) 343}348

Fig. 4. Time history of active and passive system, under impulse roadinput.

Fig. 5. Frequency response of active and passive system, under randomroad input.

K. Li, M. Nagai / JSAE Review 20 (1999) 343}348 347

From the above results, we can observe:

(1) By applying the controller designed for the 4DOF or8DOF model to the 8DOF vehicle model, it wasshown that the consideration of engine, seats andpassengers deeply in#uences the control e!ect.

(2) It can be seen that the controller designed using the4DOF model (B) suppressed vibrations in the areathat passengers can feel uncomfortable better thanthat of the controller designed using the 4DOFmodel (A).

(3) It can be seen that the controller designed using the4DOF model (B) reduced not only the vibrations ofthe engine and passengers but also of the car body.For both high precision and low cost, it would bebetter to choose to design the controller based on the4DOF model (B).

5. Conclusions

In this paper, the vibration and its control perfor-mance in#uenced by the existence of engine, seats and

passengers were analyzed and the control schemes andtheir performances were discussed in the process of de-signing a vehicle active suspension. The following con-clusions can be obtained:

(1) There is a remarkable di!erence of vibration responseamong 4DOF, 6DOF and 8DOF vehicle model onthe frequency range sensitive to ride comfort.

(2) As engine, seats and passengers have a great in#u-ence on the performance of the control system, thesefactors, especially the engine, should be considered inthe design of the controller.

References

[1] Masao Nagai, Recent researches on active suspensions for groundvehicles, JSME Int. J. Ser. C, Vol. 36, No.2, pp. 161}170. (1993).

[2] Noboru Miura, Mizuho Fukuda, Vehicle Design and SimulationAnalysis, pp. 194}205 Baifuukan Tokyo, (1990).

[3] Kayaba Industry Co., Ltd., Automotive Suspension, pp. 244}273.Sannkaidou, Tokyo, Japan (1991).

348 K. Li, M. Nagai / JSAE Review 20 (1999) 343}348