A new nodeling of the macpherson suspension system and its optimal pole placement control

A New Modeling of the Macpherson Suspension System and its Optimal PolePlacement Control KeumShik Hong School of Mechanical Engineering Pusan National University Pusan, 609735 Korea DongSeop Jeon Graduate College Pusan National University Pusan, 609735 Korea HyunChul Sohn‡ Graduate College Pusan National University Pusan, 609735 Korea Abstract In this paper a new model and an optimal poleplacement control for the Macpherson suspension system are investigated. The focus in this new modeling is the rotational motion of the unsprung mass. The two generalized coordinates selected in this new model are the vertical displacement of the sprung mass and the angular displacement of the control arm. The vertical acceleration of the sprung mass is measured, while the angular displacement of the control arm is estimated. It is shown that the conventional model is a special case of this new model since the transfer function of this new model coincides with that of the conventional one if the lower support point of the damper is located at the mass center of the unsprung mass. It is also shown that the resonance frequencies of this new model agree better with the experimental results. Therefore, this new model is more general in the sense that it provides an extra degree of freedom in determining a plant model for control system design. An optimal poleplacement control which combines the LQ control and the poleplacement technique is investigated using this new model. The control law derived for an active suspension system is applied to the system with a semiactive damper, and the performance degradation with a semiactive actuator is evaluated. Simulations are provided. 1 Introduction In this paper, a new model of the suspension system of the Macpherson type for control system design and an optimal poleplacement control for the new model are investigated. The roles of a suspensionsystem are to support the weight of the vehicle, to isolate the vibrations from the road, and to maintain the traction between the tire and the road. The suspension systems are classified into passive and active systems according to the existence of a control input. The active suspension system is again subdivided into two types: a full active and a semiactive system based upon the generation method of the control force. The semiactive suspension system produces the control force by changing the size of an orifice, and therefore the control force is a damping force. The full active suspension system provides the control force with a separate hydraulic power source. In addition, the suspension systems can be divided, by their control methods, into a variety of types: In particular, an adaptive suspension system is the type of suspension system in which controller parameters are continuously adjusted by adapting the timevarying characteristics of the system. Adaptive methods include a gain scheduling scheme, a model reference adaptive control, a selftuning control, etc. The performance of a suspension system is characterized by the ride quality, the drive stability, the size of the rattle space, and the dynamic tire force. The prime purpose of adopting an active suspension system is to improve the ride quality and the drive stability. To improve the ride quality, it is important to isolate the vehicle body from road disturbances, and to decrease the resonance peak at or near 1 Hz which is known to be a sensitive frequency to the human body. Since the skyhook control strategy, in which the damper is assumed to be directly connected to a stationary ceiling, was introduced in the 1970s, a number of innovative control methodologies have been proposed to realize this strategy. Alleyne and Hedrick3 investigated a nonlinear control technique which combines the adaptive control and the variable structure control with an experimental electrohydraulic suspension system. In their research, the performance of the controlled system was evaluated by the ability of the actuator output to track the specified skyhook force. Kim and Yoon4 investigated a semiactive control law that reproduces the control force of an optimally controlled active suspension system while deemphasizing the damping coefficient variation. Truscott and Wellstead5 proposed a selftuning regulator that adapts the changed vehicle conditions at startup and road disturbances for active suspension systems based on the generalized minimum variance control. Teja and Srinivasa6 investigated a stochastically optimized PID controller for a linear quarter car model. Compared with various control algorithms in the literature, research on models of the Macpherson strut wheel suspension are rare. Stensson et al.8 proposed three nonlinear models for the Macpherson strut wheel suspension for the analysis of motion, force and deformation. Jonsson7 conducted a finite element analysis for evaluating the deformations of the suspension components. These models would be appropriate for the analysis of mechanics, but are not adequate for control system design. In this paper, a new controloriented model is investigated. Fig. 1 shows a sketch of the Macpherson strut wheel suspension. Fig. 2 depicts the conventional quarter car model of the Macpherson strut wheel suspension for control system design. In theconventional model, only the updown movements of the sprung and the unsprung masses are incorporated. As are shown in Fig. 1 and Fig. 3, however, the sprung mass, which includes the axle and the wheel, is also linked to the car body by a control arm. Therefore, the unsprung mass can rotate besides moving up and down. Considering that better control performance is being demanded by the automotive industry, investigation of a new model that includes the rotational motion of the unsprung mass and allows for the variance of suspension types is justified. The Macpherson type suspension system has many merits, such as an independent usage as a shock absorber and the capability of maintaining the wheel in the camber direction. The control arm plays several important roles: it supports the suspension system as an additional link to the body, completes the suspension structure, and allows the rotational motion of the unsprung mass. However, the function of the control arm is completely ignored in the conventional model. In this paper, a new model which includes a sprung mass, an unsprung mass, a coil spring, a damper, and a control arm is introduced for the first time. The mass of the control arm is neglected. For this model, the equations of motion are derived by the Lagrangian mechanics. The open loop characteristics of the new model is compared to that of the conventional one. The frequency responses and the natural frequencies of the two models are analyzed under the same conditions. Then, it is shown that the conventional 14 car model is a special case of the new model. An optimal poleplacement control, which combines the LQ control and the poleplacement technique, is applied to the new model. The closed loop performance is analyzed. Finally, the optimal poleplacement law, derived for the active suspension system, is applied to the semiactive suspension system which is equipped with a continuously variable damper for the purpose of investigating the degradation of the control performance. The results in this paper are summarized as follows. A new model for the Macpherson type suspension system that incorporates the rotational motion of the unsprung mass is suggested for the first time. If the lower support point of the shock absorber is located at the mass center of the unsprung mass, the transfer function, from road disturbance to the acceleration of the sprung mass, of the new model coincides with that of the conventional one. Therefore, the new model is more general, from the point of view that it can provide an extra degree of freedom in determining a plant model for control design purpose. In the frequency response analysis, the natural frequencies of the new model agree better with the experimental results. An optimal poleplacement control, which combines the LQ control and the poleplacement technique, is applied to the new model. The control law, derived for a full active suspension, is applied to the semiactive system with a continuously variable damper. It is shown that a small degradation of control performance occurs with a continuously variable damper.

A New Modeling of the Macpherson Suspension System

and its Optimal Pole-Placement Control

School of Mechanical Engineering

Pusan National University

Pusan, 609-735 Korea

Dong-Seop Jeon**

Graduate College Pusan National University Pusan, 609-735 Korea

Graduate College Pusan National University Pusan, 609-735 Korea

Abstract

In this paper a new model and an optimal pole-placement control for the Macpherson suspension system are investigated The focus in this new modeling is the rotational motion of the unsprung mass The two generalized coordinates selected in this new model are the vertical displacement of the sprung mass and the angular displacement of the control arm The vertical acceleration of the sprung mass is measured, while the angular displacement of the control arm is estimated It is shown that the conventional model is a special case of this new model since the transfer function of this new model coincides with that of the conventional one if the lower support point of the damper is located at the mass center of the unsprung mass It is also shown that the resonance frequencies of this new model agree better with the experimental results Therefore, this new model

is more general in the sense that it provides an extra degree of freedom in determining a plant model for control system design An optimal pole-placement control which combines the LQ control and the pole-placement technique is investigated using this new model The control law derived for an active suspension system is applied to the system with a semi-active damper, and the performance degradation with a semi-active actuator is evaluated Simulations are provided.

1 Introduction

In this paper, a new model of the suspension system of the Macpherson type for control system designand an optimal pole-placement control for the new model are investigated The roles of a suspension

* Email: kshong@hyowon.pusan.ac.kr.

** E-mail: dosjeon@hyowon.pusan.ac.kr.

‡ E-mail: hcson@hyowon.pusan.ac.kr.

system are to support the weight of the vehicle, to isolate the vibrations from the road, and to maintainthe traction between the tire and the road The suspension systems are classified into passive andactive systems according to the existence of a control input The active suspension system is againsubdivided into two types: a full active and a semi-active system based upon the generation method ofthe control force The semi-active suspension system produces the control force by changing the size

of an orifice, and therefore the control force is a damping force The full active suspension systemprovides the control force with a separate hydraulic power source In addition, the suspension systemscan be divided, by their control methods, into a variety of types: In particular, an adaptive suspensionsystem is the type of suspension system in which controller parameters are continuously adjusted byadapting the time-varying characteristics of the system Adaptive methods include a gain schedulingscheme, a model reference adaptive control, a self-tuning control, etc

The performance of a suspension system is characterized by the ride quality, the drive stability, thesize of the rattle space, and the dynamic tire force The prime purpose of adopting an activesuspension system is to improve the ride quality and the drive stability To improve the ride quality,

it is important to isolate the vehicle body from road disturbances, and to decrease the resonance peak

at or near 1 Hz which is known to be a sensitive frequency to the human body

Since the sky-hook control strategy, in which the damper is assumed to be directly connected to astationary ceiling, was introduced in the 1970's, a number of innovative control methodologies havebeen proposed to realize this strategy Alleyne and Hedrick[3] investigated a nonlinear controltechnique which combines the adaptive control and the variable structure control with an experimentalelectro-hydraulic suspension system In their research, the performance of the controlled system wasevaluated by the ability of the actuator output to track the specified skyhook force Kim and Yoon[4]investigated a semi-active control law that reproduces the control force of an optimally controlledactive suspension system while de-emphasizing the damping coefficient variation Truscott andWellstead[5] proposed a self-tuning regulator that adapts the changed vehicle conditions at start-upand road disturbances for active suspension systems based on the generalized minimum variancecontrol Teja and Srinivasa[6] investigated a stochastically optimized PID controller for a linearquarter car model

Compared with various control algorithms in the literature, research on models of the Macphersonstrut wheel suspension are rare Stensson et al.[8] proposed three nonlinear models for theMacpherson strut wheel suspension for the analysis of motion, force and deformation Jonsson[7]conducted a finite element analysis for evaluating the deformations of the suspension components.These models would be appropriate for the analysis of mechanics, but are not adequate for controlsystem design In this paper, a new control-oriented model is investigated

Fig 1 shows a sketch of the Macpherson strut wheel suspension Fig 2 depicts the conventionalquarter car model of the Macpherson strut wheel suspension for control system design In the

conventional model, only the up-down movements of the sprung and the unsprung masses areincorporated As are shown in Fig 1 and Fig 3, however, the sprung mass, which includes the axleand the wheel, is also linked to the car body by a control arm Therefore, the unsprung mass canrotate besides moving up and down Considering that better control performance is being demanded

by the automotive industry, investigation of a new model that includes the rotational motion of theunsprung mass and allows for the variance of suspension types is justified

The Macpherson type suspension system has many merits, such as an independent usage as ashock absorber and the capability of maintaining the wheel in the camber direction The control armplays several important roles: it supports the suspension system as an additional link to the body,completes the suspension structure, and allows the rotational motion of the unsprung mass However,the function of the control arm is completely ignored in the conventional model

In this paper, a new model which includes a sprung mass, an unsprung mass, a coil spring, adamper, and a control arm is introduced for the first time The mass of the control arm is neglected.For this model, the equations of motion are derived by the Lagrangian mechanics The open loopcharacteristics of the new model is compared to that of the conventional one The frequencyresponses and the natural frequencies of the two models are analyzed under the same conditions.Then, it is shown that the conventional 1/4 car model is a special case of the new model An optimalpole-placement control, which combines the LQ control and the pole-placement technique, is applied

to the new model The closed loop performance is analyzed Finally, the optimal pole-placementlaw, derived for the active suspension system, is applied to the semi-active suspension system which isequipped with a continuously variable damper for the purpose of investigating the degradation of thecontrol performance

The results in this paper are summarized as follows A new model for the Macpherson typesuspension system that incorporates the rotational motion of the unsprung mass is suggested for thefirst time If the lower support point of the shock absorber is located at the mass center of theunsprung mass, the transfer function, from road disturbance to the acceleration of the sprung mass, ofthe new model coincides with that of the conventional one Therefore, the new model is moregeneral, from the point of view that it can provide an extra degree of freedom in determining a plantmodel for control design purpose In the frequency response analysis, the natural frequencies of thenew model agree better with the experimental results An optimal pole-placement control, whichcombines the LQ control and the pole-placement technique, is applied to the new model The controllaw, derived for a full active suspension, is applied to the semi-active system with a continuouslyvariable damper It is shown that a small degradation of control performance occurs with acontinuously variable damper

2 Conventional Model

Fig 2 shows the conventional model that depicts the vertical motions of the sprung and the unsprungmasses All coefficients in Fig 2 are assumed to be linear The equations of motion are

a r u t u s p u s s u u

d a u s p u s s s s

f z z k z z c z z k z m

f f z z c z z k z m

−

− +

− +

−

=

− +

) ( ) (

d r

a B z B f f

B Ax

u t

s p

s

s

m

c m

k m

0

0

1 0

) (

) (

) ( ) (

s d

k s c s k s z

s z s

r

s a

t s p t

t s s u s p

u s u

s k k s c k

s k m k m m s c m m s m m s d

+ +

+ +

+ +

+

} )

{(

) (

) (

3 A New Model

The schematic diagram of the Macpherson type suspension system is shown in Fig 3 It is composed

of a quarter car body, an axle and a tire, a coil spring, a damper, an axle, a load disturbance and acontrol arm The car body is assumed to have only a vertical motion If the joint between thecontrol arm and the car body is assumed to be a bushing and the mass of the control arm is notneglected, the degrees of freedom of the whole system is four The generalized coordinates in thiscase are z s, d, θ1 and θ2 However, if the mass of the control arm is ignored and the bushing isassumed to be a pin joint, then the degrees of freedom becomes two

As the mass of the control arm is much smaller than those of the sprung mass and the unsprungmass, it can be neglected Under the above assumption, a new model of the Macpherson typesuspension system is introduced in Fig 4 The vertical displacement z s of the sprung mass and the

rotation angle θ of the control arm are chosen as the generalized coordinates The assumptionsadopted in Fig 4 are summarized as follows.

1 The horizontal movement of the sprung mass is neglected, i.e the sprung mass has only thevertical displacement z s

2 The unsprung mass is linked to the car body in two ways One is via the damper and the other isvia the control arm θ denotes the angular displacement of the control arm

3 The values of z s and θ will be measured from their static equilibrium points

4 The sprung and the unsprung masses are assumed to be particles

5 The mass and the stiffness of the control arm are ignored

6 The coil spring deflection, the tire deflection and the damping forces are in the linear regions oftheir operating ranges

Let (y , A z A), (y , B z B) and (y , C z C) denote the coordinates of point A, B and C, respectively, whenthe suspension system is at an equilibrium point Let the sprung mass be translated by z s upward,and the unsprung mass be rotated by θ in the counter-clockwise direction Then, the followingequations hold

) (cos(θ−θ0 − −θ0

B l

)) sin(

) (sin(θ−θ0 − −θ0

) (cos(θ−θ0 − −θ0

C l

)) sin(

) (sin(θ−θ0 − −θ0

2 2 cos ' ) (l A l B l A l B α

l= + −

2 1 2

2 2 cos( ' )) (

' = l A+l B− l A l B α−θ

l

where l is the initial distance from A to B at an equilibrium state, and l' is the changed distancefrom A to B with the rotation of the control arm by θ Therefore, the deflection of the spring, therelative velocity of the damper and the deflection of the tire are, respectively

2

1 2

2

2 2

)}

' cos(

' cos )

' cos(

' (cos

{ )) ' cos(

' (cos 2

) ' ( ) (

θ α α θ

α α

θ α α

− +

− +

⋅

−

−

− +

l

b

b a a b

a

l l l

(5a)

2 )) ' cos(

( 2

) ' sin(

'

θ α

θ θ α

l

b a

b l

l l

&

&

r C

s r

where, a l =l2A+l B2, b l = 2l A l B

3.1 Equations of Motion

The equations of motion of the new model are now derived by the Lagrangian mechanics Let T,

V and D denote the kinetic energy, the potential energy and the damping energy of the system,respectively Then, these are

) (

2

1 2

C C u s

m

2 2

) ( 2

1 ) ( 2

1

r C t

k

2 ) ( 2

u C u s u

2

+ +

+

2 0 0

2 2

2

] ) sin(

) (sin(

[ 2 1

] )) ' cos(

' cos ))

' cos(

' cos

( 2 )) ' cos(

' (cos 2

[ 2 1

r C

s t

l

l l l l

l s

z l

z k

b

b a a b

a k V

−

−

−

− +

+

− +

− +

−

−

− +

−

=

θ θ

θ

θ α α θ

α α

θ α α

(7b)

)) ' cos(

( 8

) ' ( sin22

θ α

θ θ α

l p b a

b c D

&

(7c)Finally, for the two generalized coordinates q1=z s and q2 =θ , the equations of motion are obtained

as follows

t

C u C

u s u s

f z l

z k

l m l

m z m m

+

−

−

− +

+

) sin(

) (sin(

) sin(

) cos(

) (

0 0

2 0 0

θ θ

θ

θ θ θ θ

l

l l

s

r C

s C

t

l l

l p s C

u C u

f l d

c

d b

k

z l

z l

k

b a

b c z l

m l m

−

=

−

− +

− +

−

−

− +

− +

] ) ) ' cos(

( )[

' sin(

2

1

) )) sin(

) (sin(

)(

cos(

)) ' cos(

( 4

) ' sin(

) cos(

2

0 0

0

0 2

0 2

θ α θ

α

θ θ

θ θ

θ

θ α

θ θ α θ

2− α+θ

= l l l

l a a b

Now, introduce the state variables as [ ] [ ]T

s s T

z z x x x

written in the state equation as follows

) , , , , , , (

) , , , , , , (

4 3 2 1 2 4

4 3

4 3 2 1 1 2

2 1

d r a

d r a

f z f x x x x f x

x x

f z f x x x x f x

x x

) ) ( sin )

(

) ( ) cos(

) ' sin(

2

1 ) sin(

{ 1

0 3 0

3 2 3

3 0 3 3 2

0 3 2

1 1

d C a

B C

t p

s C

u

f l x

f l z x l k x h c

x g x x k

x x l m D f

−

− +

⋅

−

− +

θ

θ α

θ

&

} ) cos(

) (

) ) cos(

) ( ) ' sin(

) (

2 1

) ( ) (

) cos(

) sin(

{ 1

0 3

0 3 3

3

4 3 2

4 0 3 0 3 2 2

2 2

d C

u a B u s

C t s s

u s

p u s C

u

f x l m f l m m

z x l k m x g x k

m m

x x h c m m x x x

l m D f

θ

θ α

θ θ

−

− +

+

⋅

− +

− +

−

+ +

1=m l +m l x −θ

D s C u C

)(sin2 3 02

2 2

2=m m l +m l x −θ

D s u C u C

2 3

3

)) ' cos(

( ) (

x d

c

d b

x g

l l

l l

−

− +

=

α

)) ' cos(

( 4

) ' ( sin )

(

3 3 2 2

3

x b

a

x b

x

h

l l

r C

z x x z

2

1 () () (), (0))

24 23 21

4 2

3 2

2 2

1 2

4 1

3 1

2 1

1 1

0

1 0 0 0 0

0 0 1 0

1 0 0 0

0 0 1 0

a a a

a a a

x

f x

f x

f x f

x

f x

f x

f x f A

e

x

− +

) (

0

) ( sin

) cos(

0

0 0

0 2 2 2 2

0 2 0

0

2 1

1

θ

θ θ

C u C u s

B u s

C u C s

B T

f a a

l m l m m

l m m

l m l m l

f

f f

f B

−

− +

) cos(

0

) ( sin

) ( sin 0

0 0

0 2 2 2 2

0

0 2 0 2

0

2 1

2

θ θ θ θ

C u C u s

C t s

C u C s

C t T

z r r

l m l m m

l k m

l m l m

l k z

f z

f B

−

− +

) cos(

0

) ( sin 0

0 0

0 2 2 2 2

0

0 2

0

2 1

3

θ θ θ

C u C u s

C u

C u C s

C T

f d d

l m l m m

l m

l m l m

l

f

f f

f B

d

and

1 0 2 21

) ( sin

D l k

)}

) ' cos (

)(

( cos ) sin(

' sin

)]

( sin [

)]

cos(

) ( sin

) ) ' cos (

2

' sin )(

cos(

' sin ( 2 1

) ' )(cos(

)) ' cos(

(

( 2

1 {[

1

2 0

2 0

0 2 0

0 2 2

2

2 0

0 2

2 23

α θ

θ α

θ θ

θ

α

α θ

α

θ α α

l l

l l

C s u

C u C s C

t

l l

l s

l l

l l

s

d c

d b

l k m

l m l m l

k

d c

d k

d c

d b

k D a

− +

−

− +

− +

=

) ' cos (

4

' sin

l p

b a

b c D a

) cos(

D l k m

)}

) ' cos (

)(

sin(

' sin )

( 2 1

)]

( sin [

)]

cos(

) ) ' cos (

2

' sin (

' sin ) ( 2 1

) ) ' cos (

( ' cos ) (

2

1 {[

1

2 0

2 2

0 2 2 2 2 0

2

2 2

2 2

43

α θ

α

θ θ

α

α α

α α

l l

l l

C s u u s

C u C u s C

t s

l l

l s

u s

l l

l l

s u s

d c

d b

l k m m m

l m l m m l

k m

d c

d k

m m

d c

d b

k m m D a

− +

− +

+

− +

⋅

− +

− +

−

− + +

−

=

) ' cos (

4

' sin ) (

l p u s

b a

b c m m D a

y( ) = & θ Then the output equation is

) ( ) ( ) ( ) ( ) (t Cx t D1f t D2z t D3f t

−

=

0

) ( sin

) cos(

0 2 0

θ C u C s

B l m l m

−

=

0

) ( sin

) ( sin

0 2 0 2

θ C u C s

C t l m l m

l k

−

=

0

) ( sin2 0

C l m l m

l

4 Comparison of Two Models

In the conventional model, where the road input is z& r, the output variables were assumed to be theaccelerations of the sprung mass z&&s and the unsprung mass z&&u In (12), however, while the roadinput is the displacement z r, the outputs are the acceleration of the sprung mass z&&s and the angulardisplacement of the control arm θ Thus, the output variable that can be compared between the twomodels is the acceleration of the sprung mass z&&s To be able to compare the two models, the roadinput in the new model is modified to the velocity z& r

First, it is shown that the conventional model and the new model coincide if the lower supportpoint of the shock absorber in the new model is located at the mass center of the unsprung mass Let

C

B l

l = , l B =l Acosα and θ0=0o Then, equation (11) has the form

0 3

2

1 () () (), (0))

−

−

=

u s

p u s

u t

s u

s u s

C u t

s

C p

s

C s

m m

c m m m

k m m

k m m l

m k

m

l c m

l k A

) (

) (

0

1 0

0 0

0 0

0 0

1 0

,

T

C u s

u s

m m m

t l m

0 0

0 0

s

C p

s

C s

m

l c m

l k

For comparing the two models, the following parameter values of a typical Macpherson typesuspension system are used:

Kg

ms = 453 , mu = 71 Kg, cp = 1950 N ⋅ sec/ m,

m N

Table 1 Comparison of the two models for a typical suspension system

New model Conventional model

C

B l

m l

m l

C

B

37 0

34 0

-1.50 ± 7.70i -10.92 ± 48.30i

Resonances

(Damping ratio)

0.97 Hz (0.30) 8.33 Hz (0.27)

0.97 Hz (0.30) 8.33 Hz (0.27)

1.25 Hz (0.20) 7.88 Hz (0.23)

The frequency responses of the two models, with the same road input, are compared in Fig 5.There are substantial differences in the resonance frequencies and peaks between the two models Atendency of the new model is that the smaller the l / C l B is, the lower the resonance frequency is Allthe above observations are summarized as follows:

(1) The conventional model is considered as a special case of the new model where lB = lC