H2 Control for Active Suspension
to Improve Ride Comfort based on ISO 2631
M2015SC015 Kohei SUZUKI
Supervisor : Isao TAKAMI

Abstract2This paper presents a method of H2 control for an
active suspension to improve the ride comfort based on
ISO 2631. The half-car model is used to analyze thevertical motion and pitch motion. The time delay between front and rear wheels is approximated by using
the Pade approximation and high pass filter. Based on
the frequency weighting curves and the multiplying factors in ISO, the evaluated function is designed. The
eﬀectiveness of the proposed method is verified by simulations and experiments.

The characteristics of the half-car model are discussed.
The half-car model, which consists of two mass-springdamper systems, is shown in Fig. 1. To analyze thevertical motion and the pitch motion, the front side andrear side of the car body is connected by the negligible
bar of gravity. There is an actuator between the car
body and wheel. When the actuator is stopped, theactive suspension model works as the passive suspension
model.

1llr

INTRODUCTION

A suspension is a shock absorber between car bodyand wheels. There are a passive suspension and an active suspension. The passive suspension absorbs thevibration from the road surface by using springs anddumpers. The active suspension shows the higher performance than passive one by using an actuator in addition to the springs and the dumpers.
In this study, the H2 controller is proposed to improve
the ride comfort based on ISO 2631. The half-car model,
which can analyze the vertical motion and the pitch motion, is used. To improve the ride comfort based on ISO
2631, this study approaches by two methods. The firstone is Pade approximation for the time delay. The timedelay is important for the analysis of ride comfort because the pitch motion depends on the disturbance timings between front and rear wheels. The time delay forthe disturbance between front wheel and rear wheel isapproximated by using the first-order Pade approximation. However, when the first-order Pade approximation
is used, the high frequency bands of the vertical acceleration is underestimated for the original time delay. Then,the pitch angular acceleration is approximated based on
the resonance point of the original time delay. On theother hand, when the second-order Pade approximation
is used, the vertical acceleration is approximated basedon the resonance point of the original time delay. However, the pitch angular acceleration is underestimatedfor the original time delay. Based on the frequency
response of vertical acceleration with the second-order
Pade approximations, the high pass filter is designed
to shape the response of the first-order Pade approximation. The second method is loop-shaping based on
ISO 2631. The most uncomfortable frequency bands
are the ranges of 4-8[Hz] for the vertical accelerationand 0.6-0.8[Hz] for the pitch angular acceleration, respectively. By designing the frequency weights basedon these frequency bands, the loop-shaping is applied
to the vertical acceleration and the pitch angular acceleration. The weighted accelerations are evaluated by
the ratio of 10 : 4 called multiplying factors in ISO 2631.
The H2 controller, which can improve the response forthe impulse disturbance, is designed. The eﬀectivenessof proposed method is illustrated by simulations andexperiments.

MODELING

x2r

M2r (Rear Body)
Fr

x1r

x0r

K2r

x2f

M2f (Front Body)

K2f

M1r (Rear Tire)

C1r

Car Body
PositionFf

x2 (Center of Mass)

C2r

K1r

θ

lf

C2f

x1f

M1f (Front Tire)

K1f

Rear[r]

Wheel
Position

C1f

x0f

Front[f]Road
Surface

Figure 1: Half-Car Model
2.1

Physical Constants

The physical constants in the half-car model are shown
based on an experimental unit. The car body position
x2 and the wheel position x1 are considered as the displacements from the equilibrium points. The pitch angleof the car body is θ. The front and rear control inputs are Ff and Fr . The physical constants are shownin Tab. 1. The parameters with suﬃx f and suﬃx
r stand for the front side and rear side of the half-car
model, respectively. The front and rear lengths of thecar body are derived as lf = M2r l/(M2f + M2r ) and
lr = M2f l/(M2f + M2r ), respectively.
Table 1: Physical Constants
Unsprung mass
Sprung mass
Wheel stiﬀness
Suspension stiﬀness
Wheel damping
Suspension damping
Wheelbase (Length of car body)
2.2

M1f
M2f
K1f
K2f
C1f
C2f
l

M1r
M2r
K1r
K2r
C1r
C2r

[kg]
[kg]
[N/m]
[N/m]
[Ns/m][Ns/m][m]

Motion Equations

The motion equations of the half-car model are derived by using Newton’s motion equation. The nonlinear terms, sin θ and cos θ, are approximated as sin θ ≈ θand cos θ ≈ 1. The motion equations of x1f , x1r , x2 ,and θ are derived as Eqs. (1) to (4), respectively.
M1f ẍ1f = − K1f (x1f − x0f ) − C1f (ẋ1f − ẋ0f )+ K2f (x2f − x1f ) + C2f (ẋ2f − ẋ1f )− Ff(1)
M1r ẍ1r = − K1r (x1r − x0r ) − C1r (ẋ1r − ẋ0r )+ K2r (x2r − x1r ) + C2r (ẋ2r − ẋ1r )− Fr(2)
M2 ẍ2 = − K2f (x2f − x1f ) − C2f (ẋ2f − ẋ1f )− K2r (x2r − x1r ) − C2r (ẋ2r − ẋ1r )+ Ff + Fr(3)
M2f M2r lθ̈ = + M2f K2r (x2r − x1r )− M2r C2f (ẋ2f − ẋ1f ) + M2r Ff− M2r K2f (x2f − x1f )+ M2f C2r (ẋ2r − ẋ1r ) − M2f Fr(4)Here, M2 means the sum of the car body weights, M2f +M2r . The time delay τ [s] between front disturbance andrear disturbance is calculated as τ = l/(V · 1000/3600)
from the wheelbase l[m] and vehicle speed V [km/h].

3wr (s)
2 − τs=wf (s)
2 + τs12 − 6τ s + τ 2 s2
wr (s)
D2 (s) ==wf (s)
12 + 6τ s + τ 2 s2

D1 (s) =Suspension Dynamics P (s)

wf

ẋ1f

ẋ1r

x1r − x0r

x2f − x1f
T

ẋ2f

ẋ2r ]

y(t) = [ y1 (t)y2 (t) ]T = [ ẍ2w(t) = [ wf (t)Tu(t) = [ Ffwr (t) ] = [ ẋ0f

Fr ]T

ẋ0r ]

(6)
T

(7)
(8)

Time Delay e−sτ

The time delay e−sτ from the front disturbance wf (t)to the rear disturbance wr (t) is approximated by using
Pade approximation [2]. Because the pitch motion depends on the disturbance timings between front and rear
wheels, the time delay aﬀects the ride comfort. The firstorder Pade approximation D1 (s) and the second-order

Pw21(s) Pw22(s)

y1
y2

F(s)

yf

disturbance w(t) as Eq. (12).
[
] [
][
]
y1 (s)
Pw11 (s) Pw12 (s)wf (s)=(12)y2 (s)
Pw21 (s) Pw22 (s)wr (s)The high pass filter F (s) is derived based on the firstorder Pade approximation D1 (s) and the second-order
Pade approximation D2 (s) as Eq. (13).
F (s) = (Pw11 (s) + Pw12 (s)D2 (s))(Pw11 (s) + Pw12 (s)D1 (s))−1= (1 + D2 (s))(1 + D1 (s))−14(6 − τ s)
= 2 2
+ τs
τ s + 6τ s + 12
T1 s + 1≈
T2 s + 1

(5)
θ̈ ]T

(11)

Figure 2: Time Delay D1 (s) and High Pass Filter F (s)x2r − x1r

The state space representation P (s) is defined as Eq.
(9).
{ẋ(t) = Ax(t) + B1f wf (t) + B1r wr (t) + B2 u(t)P (s) :
y(t) = Cx(t) + Du(t)(9)
3.2

Pw11(s) Pw12(s)D1(s) wr

The state space representation P (s) including the suspension dynamics is derived from the motion equations.
The state variable vector x(t), the output vector y(t),the disturbance vector w(t), and the control input vector u(t) are defined as Eqs. (5) to (8).
x(t) = [ x1f − x0f(10)

When the first-order Pade approximation D1 (s) is used
for time delay e−sτ , the high frequency band of thevertical acceleration is underestimated, but the pitch
angular acceleration is accurately approximated for all
frequency band. On the other hand, when the secondorder Pade approximation D2 (s) is used for time delaye−sτ , all frequency band of the vertical acceleration isaccurately approximated, but the pitch angular acceleration isn’t underestimated for the high frequency band.
To shape the high frequency bands of the vertical acceleration for the first-order Pade approximation D1 (s),the high pass filer F (s) is designed as block diagram Fig.
2. Here, Pw (s) means the transfer function P (s) from

CONTROLLER SYNTHESISThe controller is designed to improve the ride comfort
based on ISO 2631 [1]. The suspension dynamics P (s) isderived from the motion equations. The time delay e−sτ
is approximated as the rational transfer function D1 (s)and the high pass filter F (s) by using Pade approximation. The frequency weight W (s) is designed basedon the frequency weighting curves in ISO. The designed
plants are integrated as the extended system G(s). The
evaluation outputs for the vertical acceleration and thepitch angler acceleration are evaluated based on the multiplying factors in ISO. The H2 controller is designed bysolving LMIs.
3.1

Pade approximation D2 (s) for time delay e−sτ are derived as Eqs. (10) and (11), respectively.

(13)
(14)

Here, the plant P (s) has the characteristic Pw11 (s) =Pw12 (s) because the front and rear weights of car bodyare considered as the same weights, M2f = M2r . The
high pass filter F (s) (Eq. (13)) is approximated as thefirst-order rational transfer function (14). The firstorder Pade approximation D1 (s) and high pass filter
F (s) are translated into the state space representations(15) and (16).
{ẋd (t) = Ad xd (t) + Bd wf (t)D1 (s) :(15)
wr (t) = Cd xd (t) + Dd wf (t){ẋf (t) = Af xf (t) + Bf y1 (t)F (s) :
(16)
yf (t) = Cf xf (t) + Df y1 (t)3.3Frequency Weight W (s)The frequency weight W (s) is designed based on thefrequency weighting curves in ISO 2631. According tothe frequency weighting curves in ISO, the most uncomfortable frequency bands for the vertical acceleration and the pitch angular acceleration are in the rangeof 4-8 [Hz] and 0.6-0.8 [Hz], respectively. The frequency
weights W1 (s) and W2 (s) for these accelerations are designed based on ISO. The frequency responses of the frequency weights, W1 (s) and W2 (s), are shown in Fig. 3.
The frequency weights for vertical acceleration and pitch

wf

D(s)

wr

y1

P(s)

u
u

F(s)

u

yf

W(s)

y2

yw

z

xw

x

xg

xd
xf

G(s)

xw

Gain [dB]10

Figure 4: Extended System G(s)53.60W1(s) VerticalW2(s) Pitch

0.2

0.8 2 4 8 16 32 80
Frequency [Hz]Figure 3: Frequency Weights W1 (s) and W2 (s)angular acceleration have the peak at 4-8[Hz] and 0.60.8[Hz], respectively. These frequency weights, W1 (s)and W2 (s), are defined as the state space representation
W (s) of MIMO.
{ẋw (t) = Aw xw (t) + Bw ỹ(t)W (s) :(17)yw (t) = Cw xw (t) + Dw ỹ(t)Here, the input vector ỹ(t) including the vertical acceleration and pitch angular acceleration is defined asTỹ(t) = [yf (t) y2 (t)] .
3.4

Extended System G(s)The extended system G(s) is designed by introducing
the new state variable vector xg (t) and the evaluation
output vector z(t) as Eqs. (18) and (19).
xg (t) = [ x(t)z(t) = [ u(t)xd (t)

xf (t)yw (t) ]T

xw (t) ]T

(18)
(19)

The new state space representation G(s) is derived as
Eq. (20).
{ẋg (t) = Ag xg (t) + B1g wf (t) + B2g u(t)G(s) :(20)z(t) = Cg xg (t) + Dg u(t)The block diagram of the extended system G(s) is shownin Fig. 4.
3.5

Weight Matrix W

The weight matrix W is designed based on the multiplying factors in ISO 2631. The weighted accelerations
for vertical acceleration and pitch angular acceleration
are evaluated as the ratio of 10 : 4. The weight matrix
W is designed as Eq. (21).
W = diag( W1

W1

W2

0.4W2 )

(21)

By introducing the weight matrix W , the new evaluation
output z̃(t) is defined as Eq. (22).
z̃(t) = W z(t) = W Cg xg (t) + W Dg u(t)= C̃xg (t) + D̃u(t)(22)

H2 Controller

The state feedback controller is designed to improve
the response for the disturbance. The situation that the
car passes on the road surface of the step wise is considered. However, the disturbance wf (t) = ẋ0f (t) is defined as the vertical velocity of the road surface. Whenthe road position is considered as the step wise disturbance, the vertical velocity of road surface is consideredas impulse disturbance. The H2 controller, which can
improve the response for the impulse disturbance, is designed.

4

SIMULATION & EXPERIMENTThe eﬀectiveness of proposed controller is evaluatedby simulations and experiments. The vehicle speed isconsidered as 50[km/h]. The weight matrix W is defined as W = diag( 0.6 0.6 0.7 0.28 ). The feedback gain K is obtained. The ride comfort is evaluatedby comparing the proposed controller, the conventional
controller, and the passive suspension. The conventional
controller means the controller without the frequency
weight W (s).
4.1Frequency Response (Simulation)

The ride comfort is evaluated by the frequency response. To evaluate the ride comfort exactly, the thefrequency weighting curves in ISO are used for the vertical acceleration and pitch angular acceleration. The
frequency weighting curves are defined as the transfer
functions H1 (s) and H2 (s). The transfer functions areT
integrated for the output equation y(t) = [y1 (t) y2 (t)]in Fig. 5. The ride comfort is exactly evaluated by the
x0f

s

wf

y1
e-sτ

P(s)
wr

y2

H1(s)
H2(s)

yh1
yh2

Figure 5: Analysis based on ISO(s)(s)
transfer functions xyh1
and xyh2. Figs. 6 to 8 indi0f (s)0f (s)
cate the frequency response for the vertical acceleration,
the pitch angular acceleration, and the total accelerationof these accelerations based on the multiplying factors,
respectively. The solid line, the dashed line, and thedotted line indicate the proposed controller, the conventional controller, the passive suspension, respectively.
As can be seen from the graph, the vertical accelerationand the pitch angular acceleration of the proposed controller are suppressed than the conventional controller
and the passive suspension. The total acceleration forGain [dB]

The RMS accelerations are shown as Tab. 2. As canbe seen from the table, the total RMS acceleration forproposed controller is suppress to about 43% than conventional controller.

Passive
Proposed
Conventional

40
20
0

Table 2: RMS Accelerations

-20
-40

0.6

2

4
8
16
Frequency [Hz]32

Passive Suspension
Conventional Controller
Proposed Controller

80

Figure 6: Vertical Acceleration
20

0
-10

Vertical Acceleration[m/s2]

-30
0.6

2

4
8
16
Frequency [Hz]32

80

Figure 7: Pitch Angular Acceleration

proposed controller is suppress to about 32% at 6[Hz]for conventional controller.
4.2

2

Passive
Proposed

1
0
-1
-2

0

1

RMS Analysis (Simulation)

The ride comfort is evaluated by RMS acceleration
based on ISO 2631. The overall RMS accelerations rw1
and rw2 for the vertical acceleration and the pitch angular acceleration are calculated as Eq. (23).
[
]1
[
]1
2 2
2 2
rw1 = Σj (r1j ), rw2 = Σj (r2j )(23)

Passive
Proposed
Conventional

40
Gain [dB]

(24)

20

4

5

and dotted line indicate the proposed controller and thepassive suspension, respectively. As can be seen fromthe graph, the vertical acceleration for the proposed controller is improved than the passive suspension.

CONCLUSION

The method of H2 control for the active suspension
is proposed to improve the ride comfort based on ISO
2631. The half-car model is used to analyze the verticalmotion and the pitch motion. The time delay is approximated by using the first-order Pade approximation. To
shape the vertical acceleration for the first-order Padeapproximation, the high pass filter is designed. Basedon the frequency weighting curves in ISO, the frequency
weights are designed for the vertical acceleration and thepitch angular acceleration. The weighted accelerations
are evaluated based on the multiplying factors in ISO.
The eﬀectiveness of the proposed controller is illustrated
by simulations and experiments.

References

0
-20
-40

2
3
Time[s]Figure 9: Experiment of Vertical Acceleration

5

Here, r1j and r2j indicate the RMS accelerations ofweighted vertical acceleration and weighted pitch angular acceleration based on the jth one-third octave bandin the range of 0.5-80[Hz]. The total RMS acceleration
rv is calculated as Eq. (24) based on the multiplying
factors (ratio of 10 : 4).
[ 2]1
rv = rw1+ (0.4rw2 )2 2

Total
2.00
0.98
0.43

Time Domain Analysis (Experiment)

-20

-40

Pitch
0.50
0.49
0.47

The eﬀectiveness of proposed controller is evaluatedby the experiment of time domain analysis. The situation that the car passes on the square wave is considered.
The period and amplitude of the square wave are 5[s]and 0.01[m], respectively. Fig. 9 indicates the experiment result of the vertical acceleration. The solid line

Passive
Proposed
Conventional

10
Gain [dB]

4.3

Vertical
1.99
0.96
0.39

0.6

2

4
8
16
Frequency [Hz]32

Figure 8: Total Acceleration

80[1] ISO2631-1, Mechanical vibration and shock evaluation of human exposure to whole-body vibration
Part 1 : General requirements, (1997)[2] T. Suzuki, M. Takahashi, Active Suspension Control Considering Lateral Vehicle Dynamics due
to Road Input at Diﬀerent Vehicle Speed (inJapanese), The Japan Society of Mechanical Engineers, Series C 78(786), pp.98-113, (2010)