H∞ Control of Active Suspension
with Ensuring Ride Comfort
M2015SC017 Tatsuo TODA
Supervisor: Isao TAKAMIAbstract
This paper presents a method of the H∞ control foran active suspension. The main purpose of this study
is to ensure the ride comfort. The comfort reactions
to vibration environment is defined in International Organization for Standardization (ISO) 2631-1. In thispaper, the comfort reaction is introduced as the constraint. The L2 norm which is regarded as the weighted
root mean square (r.m.s.) acceleration is suppressed tosatisfy the constraint. On the other hand, to improve
the ride comfort is also one of the purposes. The perturbation of the vertical force of the wheel which aﬀects
the lateral force is suppressed by considering the deflection of the wheel. The controller is derived by solving
a finite set of Linear Matrix Inequalities (LMIs). The
eﬀectiveness of the proposed controller is evaluated by
the simulation. The road surface profile based on ISO
8608 is introduced in the evaluation.

1controller is evaluated by simulations. The road surface
profile based on ISO 8608 is introduced in the evaluation
to conduct more realistic simulations [5].

2Modeling

In this section, a quarter vehicle model of an active
suspension shown in Fig. 1 is analyzed. The modelzsms

kscsFz us

m us

Introduction

A suspension is a component which connects a vehicle
body to a wheel.
It mainly consists of the springs and dampers. It canabsorb the vibration transmitted from the road surface.
As the types of suspension, there are passive suspension and active suspension. The active suspension showshigher performance than passive suspension by using anactuator in addition to the springs and dampers.
The first roll of the suspension is to improve the ride
comfort. According to ISO 2631-1, the ride comfort canbe evaluated by the accelerations of the human body [1].
The comfort reactions to vibration environment using
the weighted r.m.s. acceleration are defined in ISO 26311. The second roll of the suspension is to improve thedriving stability. It is improved by suppressing the perturbation of the vertical force of the wheel. If there
is a perturbation of the vertical force, the lateral force
required for turning or stopping is also changed [2].
Recently,there are many studies which focus on both
the ride comfort and the driving stability. In those studies, various control methods are proposed such as LQR
control [3], H∞ control [4], and so on. In some studies,
the loop-shaping for the specific frequency range basedon ISO 2631-1 is used to improve the ride comfort more
eﬀectively [4].
In this paper, The quarter vehicle model is analyzed.
The main purpose of this study is to ensure the ride comfort. The comfort reaction to vibration environment defined in ISO 2631-1 is introduced as the constraint. Thefrequency weight based on IO 2631-1 is used. The L2
norm which is regarded as the weighted r.m.s. acceleration is suppressed to satisfy the constraint. On theother hand, to improve the driving stability is also oneof the purposes. The perturbation of the vertical forceof the wheel is suppressed by considering the deflectionof the wheel. To satisfy those purposes, the H∞ controller is designed. The controller is derived by solving
a finite set of LMIs. The eﬀectiveness of the proposed

k us

c us
zr

Figure 1 Quarter vehicle model
consists of three layers, which are the vehicle body, thewheel, and the road surface from the top to the bottom.They are connected by springs and dampers. Thereis an actuator between the vehicle body and the wheel.
If there isn’t control input, the active suspension works
as a passive suspension.
Here zs [m], zus [m], and zr [m] are the displacementsof the vehicle body, the wheel, and the road surface
from their equilibrium points, respectively. F [N] is thecontrol input.
2.1

Physical Parameters

In this paper, an experimental unit which is a small
scale model of a vehicle is used. The physical constantsof the model are shown in Tab. 1.
Table 1 Physical constants
Description
SymbolValue
Sprung mass
ms
2.45[kg]
Unsprung mass
mus
1[kg]
Suspension stiﬀness
ks970[N/m]
Wheel stiﬀness
kus
2717[N/m]
Suspension dampingcs
7.5[Ns/m]
Wheel dampingcus
5[Ns/m]2.2

Motion equation

The motion equations of the model are derived basedon Newton’s motion equation. The motion equations ofzs (t) and zus (t) are derived as Eqs. (1) and (2).
ms z̈s = − ks (zs − zus ) − cs (żs − żus ) + Fmus z̈us =ks (zs − zus ) + cs (żs − żus ) − F− kus (zus − zr ) − cus (żus − żr )
2.3

(1)

The gain diagram of Eq. (4) is shown in Fig. 2. The
solid curve is the frequency weight G(s) used in thisstudy, and the dashed curve is the frequency weight usedin ISO 2631-1. It can be seen that the proposed fre-(2)10State space representation

The state space representation is derived from Eqs.
(1) and (2). The state variable vector x(t), the disturbance w(t), the control input u(t), and the output
vectors y1 (t) and y2 (t) are defined as follows. y1 (t)represents the vertical acceleration, and y2 (t) represents the perturbation of the vertical force of the wheel
and the control input. Here Nz means the perturbation of the vertical force which is calculated as Nz =kus (zus − zr ) + cus (żus − żr ).
x(t) = [ zus − zr zs − zusw(t) =żr , u(t) = Fżusy1 (t) =z̈s , y2 (t) = [ Nz

TF ]0000mus

ksmus
ks−ms


A =
 − kus
01−1

csms


0−1
0
 0 
Bw =  cus  , Bu =  − 1[[csmskus 0 cus
C2 =0 0 0
[ ]0Du2 =
1

3101

102

Gain[dB]

frequency[Hz]

cs
mus
cs−ms

quency weight G(s) covers the weight of ISO 2631, andit has the peak at 4-8[Hz].
To adopt this frequency weight to the model, thetransfer function G(s) is translated into the state space
representation as Eq. (5).
{ẋf (t) = Af xf (t) + Bf y1 (t)(5)G(s) :yf (t) = Cf xf (t) + Df y1 (t)
3.2





The new state variable vector x̃(t) for extended system
is defined as Eq. (6).

[Ã =[B̃u =3.3

O2×2Af]A
Bf C1Bu
Bf Du1][, B̃w =Bw
O2×1(7)]

Evaluation output for ride comfort

The evaluation output z1 (t) which represents theweighted vertical acceleration is defined as Eq. (8)z1 (t) = yf (t) = C̃1 x̃(t) + D̃u1 u(t)C̃1 = [ Df C1The frequency weight used in ISO 2631 has too highdimensions. The frequency weight for the vertical acceleration which has lower dimensions is designed. Thefrequency weight G(s) is designed as Eq. (4).

1
1
11, T2 =, T3 =, T4 =3.8
1670.5500(6)

˙x̃(t)=Ãx̃(t) + B̃w w(t) + B̃u u(t)Frequency weight

Here T1 to T4 are as follows.

Txf (t) ]Then, the state space representation of the extended
system is derived as Eq. (7).

]1cs−m, Du1 =sms][]
0−cus, Dw2 =00

1 T1 s + 1 T4 s + 1G(s) = ··3 T2 s + 1 T3 s + 1

Extended systemx̃(t) = [ x(t)Controller synthesis

T1 =100In this paper, the H∞ controller is proposed. The
ride comfort is ensured by constraining the weighted acceleration. On the other hand, the driving stability isimproved by suppressing the perturbation of the verticalforce.
3.1-4010-11ms0

ks0 −ms-30

Figure 2 Frequency weightmus mus

C1 =

-20

T01+cs− cusmus-10żs ]The state space representation P (s) is derived as Eq.
(3).
{ẋ(t) = Ax(t) + Bw w(t) + Bu u(t)y1 (t) = C1 x(t) + Du1 u(t)P (s) :(3)y2 (t) = C2 x(t) + Du2 u(t)proposedISO-2631

0(8)Cf ] , D̃u1 = Df Du1

In ISO 2631-1, the weighted r.m.s. acceleration aw isdefined as Eq. (9).
[aw =1T∫Ta2w (t)0] 12(9)(4)
Here aw (t)[m/s2 ] means the weighted acceleration as afunction of time. In this paper, z1 (t) is assumed asaw (t). T [s] is the duration of the measurement. According to ISO 2631, the ride comfort can be evaluatedwith the aw as Tab. 2.

Table 2 Comfort reactions
Less than 0.315[m/s2 ]not uncomfortable
0.315-0.63[m/s2 ]a little uncomfortable
0.5-1[m/s2 ]fairly uncomfortable
0.8-1.6[m/s2 ]uncomfortable1.25-2.5[m/s2 ]very uncomfortableGreater than 2[m/s2 ] extremely uncomfortableconditions are satisfied, the closed-loop system is stabilized by u(t) = Kx(t) = Y X −1 x(t) and the upperbound of H∞ norm is less than γ [4, 6].
minimize : γ 2subject to :X>0

He[ÃX + B̃u Y ] B̃w
T
B̃w
−µ2
C̃1 X + D̃u1 Y0He[ÃX + B̃u Y ] B̃w
T
B̃w
−γ 2
˜ 2Y
˜˜C2 X + DuD̃w2Eq. (10) shows the condition which enable to constraint the weighted r.m.s. acceleration less than theobjective value. Here awo means the objective value.
[[[1T1T∫T] 12∫T] 21z12 (t)0z12 (t)0∫
1 TT0w2 (t)< awo

4] 12 < [awo] 12 = µ∫
1 T
2T 0 w (t)||F (s)||∞ < µ
3.4(10)The weight matrix W is defined as Eq. (11).
w2 ])Simulation

In this section, the eﬀectiveness of the proposed controller is evaluated by the simulation.

Evaluation output for driving stability

W =diag([ w1(C̃1 X + D̃u1 Y )T
<00−1(C̃2 X + D̃u2 Y )TT<0D̃w2
2×2−I

4.1

Simulation setup

In ISO 8608, the road surface profile is defined. For thetime-domain analysis, the road surface profile of crass B
of is introduced in this study. Fig. 4 shows the roadsurface profile. The vehicle velocity is assumed as 36.8

(11)
0.03

z2 (t) =C̃2 x̃(t) + D̃w2 w(t) + D̃u2 u(t)C̃2 =[W C2O]
2×2(12), D̃w2 = W Dw2

The block diagram of the closed-loop system is shownin Fig. 3.

y1(t)
G(s)

z1(t)

w(t)
z2(t)

y2(t)
Wu(t)
x(t)
KFigure 3 Block diagram
3.5

0.01
0-0.01

D̃u2 = W Du2

P(s)

0.02
Displacement[m]Here w1 and w2 are the weights for the perturbationof the vertical force and the control input, respectively.
The evaluation output z2 (t) is derived as Eq. (12) byusing Eqs. (3) and (11).

LMI condition

The H∞ controller is designed to ensure the ride comfort and to improve the driving stability. These problems are formulated to solving following finite set ofLMIs. If there exist X and Y such that following LMI-0.02

0102030
Distance[m]405060Figure 4 Road surface profile[km/h].
In this paper, experimental unit which is the scalemodel of the vehicle is used. The resonance frequency issame, but the amplitude is diﬀerent. Tab. 3 shows thecomparison of the weighted r.m.s. acceleration of theexperimental unit value and the real vehicle value [4].
Both of them are the simulation results of the passive
suspension. It can be seen that the value of the experiTable 3 Experimental unit and real vehicle
Experimental unit Real vehicle
3.35[m/s2 ]0.779[m/s2 ]mental unit is 4.3 times larger than the value of the realvehicle. In this paper, ”not uncomfortable” of Tab. 2 isthe object. the value is modified 4.3 times larger for theexperimental unit. Therefore, the objective value awo is
1.35. By using these constants, the value of µ is 48.3.
On the other hand, the weight matrix W is chosen aseq. (13).
W =diag([ 0.08 0.04 ])(13)

The feedback gain K and the upper bound of γ are
obtained as follows.

4.2−343 6.20 −264

−94.8 −60.3 ]Simulation results

Figs. 5 to 7 show the results of the time-domain analysis. Here the solid curves and the dashed curves mean
the results of the active suspension and the passive suspension, respectively.
Fig. 5 show the vertical weighted acceleration of thevehicle body. It can be seen that the maximum amplitude of the weighted acceleration is suppressed by theactive suspension. Tab. 4 shows the comparison of the1050-5-10-15-20012345Time[s]Figure 6 Perturbation of vertical force
14Active

activepassive

30.5

2

Control input[V]Weighted acceleration[m/s2]

activepassive15
Vertical force[N]K = [ 3720γ =19.4

20

10-1

0-0.5-2-3-1-40

1

2

3

450

1

2

3

45Time[s]Time[s]Figure 5 Weighted acceleration1

Figure 7 Control input

weighted r.m.s. acceleration of each result. According
to Tab. 2, the results are evaluated as not uncomfortable and fairly uncomfortable, respectively. The results
indicate that the ride comfort is ensured by the proposed
controller.

evaluated by the simulation which uses the road surface
profile based on ISO 8608.

Table 4 Weighted r.m.s. acceleration
ActivePassive
1.21[m/s2 ] 3.35[m/s2 ]Fig. 6 shows the perturbation of the vertical forceof the wheel. The vertical force of the wheel without
disturbance is 34[N]. It can be seen that the maximum
amplitude of the perturbation of the vertical force is suppressed by the active suspension. The result indicatesthat the driving stability is improved.
Fig. 7 shows the control input. It can be seen thatthe control input is less than the limitation of the experimental unit (±10[v]). The result indicates that thecontroller can be used in the experiment.

5Conclusion

In this paper, the H∞ controller is designed for an active suspension. The main goal of the study is to ensure
the ride comfort. The comfort reactions to vibration
environment defined in ISO 2631-1 is introduced as theconstraint for the weighted r.m.s. acceleration. On theother hand, to improve the driving stability is one ofthe goal. The perturbation of the vertical force of thewheel is suppressed by considering the deflection of thewheel. The controller is derived by solving the finite setof LMIs. The eﬀectiveness of the proposed controller isReferences[1] ”ISO 2631-1 Mechanical vibration and shock - Evaluation of human exposure to whole-body vibration- Part 1: General requirements”, International Organization for Standardization, 1997.
[2] Y. Hanamura, K. Fujita, Y. Araki, M. Oya, H.
Harada, ”Control of Vehicle Maneuverability andStability of 4 Wheeled Vehicle by Active Suspension Control with Additional Vertical Load Control”, Transactions of the Japan Society of Mechanical Engineers, Series C 65 (629), pp. 236-243, 1999.
[3] M. P. Nagarkar, G. J. Vikhe Patil, ”Performance
Analysis of Quarter Car Active Suspension System:LQR and H∞ Control Strategies”, 2012 Third International Conference on Computing, Communication and Networking Technologies (ICCCNT’12),
1-6, 2012.
[4] H. Chen, and K.-H. Guo, ”Constrained H∞ Control of Active Suspensions: An LMI Approach.”
IEEE Transactions on Control Systems Technology
13.3, pp. 412-421, 2005.
[5] ”ISO 8608 Mechanical vibration - Road surface profiles - Reporting of measured data”, International
Organization for Standardization, 2016.
[6] Y. Kami, E. Nobuya, ”Iterative Methods for theMixed H2 /H∞ Control System Design Using H2Norm Decreasing Controller Sequences”, Transactions of the Society of Instrument and Control Engineers Vol. 40 No. 4, pp. 382-389, 2004.